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AMATYC 39th Annual Conference Friday night Ignite Event: Twenty slides are automatically advanced every 15 seconds while the speakers have exactly five minutes to share their passion!
Citation preview
Lecture Is Dead
Long Live Lecture
How and why to make class time more exciting and rewarding for your students
Rob EbyBlinn College ndash Bryan TX Campus
Lecture is Booooring So add commercial breaks
First 10 vs last 40 recall is laughable
Lecture is Booooring So add commercial breaks
First 10 vs last 40 recall is laughable
If All You Do Is Lecture
15
35
Ten and Two Hike 10 minute lecture 2 minutes to chew on it
Similar to commercial breaks
BUT you engage the students
Think of a TV program
What type of commercials
Minute papers
Clickers ndash poll anywhere and such
Turn to classmate
Example in book
ldquoWhat is wrong hererdquo
Group quizzes
end of or after class ideas Minute papers
What do you think was the goal today
Clearest or muddiest point
Write your own question
Exit quizzes
Solve and Classmates Grade
Make Them Read
Readings or videos out of class
GUIDE THE READING
Follow up with quizzes
Help students learn how to learn
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Ten and Two Hike 10 minute lecture 2 minutes to chew on it
Similar to commercial breaks
BUT you engage the students
Think of a TV program
What type of commercials
Minute papers
Clickers ndash poll anywhere and such
Turn to classmate
Example in book
ldquoWhat is wrong hererdquo
Group quizzes
end of or after class ideas Minute papers
What do you think was the goal today
Clearest or muddiest point
Write your own question
Exit quizzes
Solve and Classmates Grade
Make Them Read
Readings or videos out of class
GUIDE THE READING
Follow up with quizzes
Help students learn how to learn
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
What type of commercials
Minute papers
Clickers ndash poll anywhere and such
Turn to classmate
Example in book
ldquoWhat is wrong hererdquo
Group quizzes
end of or after class ideas Minute papers
What do you think was the goal today
Clearest or muddiest point
Write your own question
Exit quizzes
Solve and Classmates Grade
Make Them Read
Readings or videos out of class
GUIDE THE READING
Follow up with quizzes
Help students learn how to learn
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
end of or after class ideas Minute papers
What do you think was the goal today
Clearest or muddiest point
Write your own question
Exit quizzes
Solve and Classmates Grade
Make Them Read
Readings or videos out of class
GUIDE THE READING
Follow up with quizzes
Help students learn how to learn
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Make Them Read
Readings or videos out of class
GUIDE THE READING
Follow up with quizzes
Help students learn how to learn
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Public Speaking 101
1 Multiply by three
2 Subtract five
( ) 3 5f x x
1 UNDO Multiply by three
2 UNDO Subtract five
1( ) 5f x x 1 1( ) ( 5)
3f x x
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Pictures Not just any pictures Good pictures
Why is the sum of the first n odd numbers always a square
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Pictorial Superiority Effect Our brains are hard wired for pictures
Things written in text are not considered a picture
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
INFORMATION
72 Hours after exposure Recall from hearing only
Recall from hearing and picture
INFORMATION
TION
FORMATIONIN
INFORMA
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Pictorial Superiority Effect
( 3)( 5)x x 2 2( 3 13)( 5 42)x x x x
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
2 2( 3 13)( 5 42)x x x x
2
2
2 2
(13)( 5 42)
(3 )( 5 42)
( )( 5 42)
x x
x x x
x x x
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Brain Rules The brain seems to rely partly
on past experience in deciding how to learn new things
Make sure they understand what is new each time
Our senses evolved to work together
We learn best if we stimulate several senses at once
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Patterns and Connections We are better at seeing
patterns and abstracting the meaning of an event than we are at recording detail
Emotional arousal helps the brain learn
So make it emotional
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Memory and Brain Rules Most memories disappear
within minutes
How do we make sure it gets into long-term memory
Incorporate new information gradually
Repeat it in timed intervals
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Brain RulesBabies are the model of how we learn
observation
hypothesis
experiment
conclusion
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Darn Kids these days
This is not just about ldquokids these daysrdquo this research is decades old
Brains more wired for linear bursts than deep thinking (always on etc)
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Most Desired Skills - Forbes
No 1 Critical Thinking
No 2 Complex Problem Solving
No 3 Judgment and Decision-Making
No 4 Active Listening
No 5 Computers and Electronics
No 6 Mathematics Knowledge of arithmetic algebra geometry calculus statistics and their application
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Find out more httptinyurlcomk3sbgh5
RobEbymathdude
jeby blinnedu
Blinn College ndash Bryan Campus
(next door to Texas AampM)
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
The Dos and Donrsquots of Personal Branding OnlineJon OaksMacomb Community Collegewwwjonoakscom
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
12 Good Wordshellip hellip and 7 Bad Ones
Dave SobeckiMiami University Hamilton
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Exercise
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Hard
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Cancel
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Input and Output
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Explain
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Mimic
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Context
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Backwards
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Real-World
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Story
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Variable
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ALEKS
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Assume
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Discover
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Interpret
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Rate-of-change
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Formal
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Roadblocks
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
And two bonus words that are always appropriatehellip
Thank you
FacebookcomTeachingbackwards
teachbackwards
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Problem SolvingDONrsquoT Just Tell Them What to Do
Joe BrowneOnondaga Community CollegeSyracuse NY 13215brownejsunyoccedu
Exercises vs Problems
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ExercisePractice a rule or procedure
Factor this polynomial
Solve this equation
Find the distance between these two points
Differentiate these functions
The student is told exactly what to do
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ProblemA problem usually requires more thinking and analysis In most cases there is no formula that immediately gives the answer
Find a point on a given line which is equidistant from two given points
x
y
A
B
P
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Steps in problem solving process
1 Understand the problem
2 Devise a plan (strategy)
3 Carry out the plan
4 Look back
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
When given a problemhellip Where do students want to start
1 Understand the problem2 Devise a plan3 Carry out the plan4 Look back
Where should they start
72
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Ever Heard This
ldquoOnce I know what to do itrsquos easy My trouble is knowing where to startrdquo
ldquoDonrsquot make me try to understand this just tell me what to dordquo
73
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
In the ldquoreal worldrdquohellip
1 Whose job is it to ldquounderstand the problemrdquo(Usually company officers and executives)
2 Whose job is it to ldquodevise a plan or strategyrdquo(Usually the managers and engineers)
74
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
In the ldquoreal worldrdquohellip
3 Whose job is it to ldquocarry out the planrdquo(Usually the assistants clerks and laborers)
4 Whose job is it to ldquolook backrdquo and see if a satisfactory outcome has been achieved(Usually the officers and executives again)
75
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Who are the lowest paid
1 Company officers and executives
2 Managers and engineers
3 Assistants clerks and laborers
4 Company officers and executives
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Who are most likely to be replaced by automation outsourcing or a computer1 Company officers or executives
2 Managers or engineers
3 Assistants clerks or laborers
4 Company officers or executives
77
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
So if we focus almost entirely on step 3 ldquocarry out the planrdquo what are we preparing our students for
78
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
A thorough knowledge of what skills techniques and procedures are possible is necessary especially to those who ldquodevise a planrdquo
But this is not sufficient
79
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
To fully prepare our students for a productive (and well paying) career we must emphasize all facets of problem solving
80
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Math + Facebook = SuccessNancy Che MahanSanta Ana College
NancyCheMahangmailcom
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Ack ProblemshellipNo office HoursNo websiteKeeping track of different campusrsquo systems passwords websites
Keeping track of Studentsrsquo requests names faces
Needing a place to upload documents
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
My Experimentatio
n
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHAT can you use this Facebook class group for1
Repository
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHAT can you use this Facebook class group for1 Repository
2 Announcements
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHAT can you use this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHAT can you this Facebook class group for1 Repository
2 Announcements
3 Class Discussions
4 Etc
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHY Facebook1 Math is always in front of them
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHY Facebook2 Efficient
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
WHY Facebook3 Social Environment
Face it - Facebook is FUN
And Math can sure use some fun
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
HOW to get started
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
HOW to get started
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
HOW to get startedhellip
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
OTHER Important Tidbits Policy I have a very clear written policy
on the use of this forum stated on the About section of the class FB page I warn them that any inappropriate use of the forum results in immediate removal from the group (although this has never been an issue)
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
OTHER Important Tidbits Optional The use of FB is optional but I
always strongly recommend it For those who are FB-haters I tell them that they can create an account with their school email and only use it for this purpose Once in a blue moon I have someone who is adamantly against FB so I just tell them that its their responsibility to get info from a classmate or myself
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
OTHER Important Tidbits Privacy
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
What STUDENTS think
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
What STUDENTS think
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
What STUDENTS think
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
httpnancychemahanblogspotcom201307math-facebookhtml
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
How Math is Like Golf
Laurie K McManus PhDProfessor of MathematicsSt Louis CC ndash Meramec
lmcmanusstlcceduwwwmcmathprofcom
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
There are rules that must be followed
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
You must use the proper equipment
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Your skills improve with regular practice
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Sometimes itrsquos necessary to consult an expert
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Your skills do not improve by watching someone else play the
game
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Patience hellip
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Persistence may be rewarded
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Sometimes you have to calculate
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
You must be prepared to interpret the result of your
calculations
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
A good score can be very satisfying
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
It can be a social activity
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Sometimes itrsquos fun
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
And now for something completely different hellip
Mary Beth OrrangeorrangeecceduAMATYC Board Secretary
Dear Math
I am sick and tired of finding
your ldquoxrdquo Just accept the fact
she is gone Move on Dude
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
314 of sailors are Pi Rates
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Q Why cant you say 288 in publicA Its two gross
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Very Bad Math JokesQ What did the zero say to the
eightA Nice belt
Q How does a cow add
A It adds one udder to an udder
Q Why is the snake so good at math
A Because hersquos an adder
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Even Worse Math JokesQ Why is the math book so upsetA Because it has lots of problems
Q What did the math plan grow
A Square roots
Q How do you make seven evenA Take away the ldquosrdquo
Q What did the dollar say to the 4
quartersA Yoursquove changed
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
There are 10 kinds of mathematicians Those who can think binarily and those who cant
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Whats your favorite thing about
mathematics Knot theory
Yeah me neither
In Alaska where it gets very cold pi is only 300 As you know everything shrinks in the cold They call it Eskimo
pi
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Q What does the little
mermaid wear A An Alge-bra
Q Whats a polar bear A A rectangular bear after a coordinate transformation
Q What is a dilemma A A lemma that proves two
results
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Motto of the society Mathematicians Against Drunk
Deriving
Math and Alcohol dont mix so PLEASE DONT DRINK AND
DERIVEQ Do you already know the latest stats jokeA Probably
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoThe trouble with Mobius is that he thinks there is only one side
to every questionrdquo
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoUh yeah Homework Help Line I need you to explain the quadratic equation in
roughly the amount of time it takes to get a cup of
coffeerdquo
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Cat Theorem A cat has nine tails
Proof No cat has eight tails A cat has one tail more than no cat Therefore a
cat has nine tails
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Life is complex It has real and imaginary components
The shortest math joke let
epsilon be lt 0
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Q What is the first derivative of a cow A Prime Rib
Q What is purple and commutative A An abelian grape
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Math problems
Call 1-800-[(10x)(13i)2]
sin(xy)2362x]
The number you have dialed is imaginary Please rotate your phone by 90 degrees and try
again
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Thatrsquos all
folks
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Math Is Hard
Luke WalshCatawba Valley Community CollegeHickory North Carolinalwalshcvccedulukeselfwalker
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoYoung man in mathematics you donrsquot understand thingshellip
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
hellipYou just get used to themrdquo
~John Von Neuman
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
I am a math instructor
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
gtNORlt
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
I TEACH MATHWHATrsquoS YOUR SUPERPOWER
OPERATORS STANDING BY
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
i lt 3 MATH
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
A MorganB RussellC Gauss
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
D HilbertE BellF Viete
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
G H Hardy
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
R i
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
1 2 3hellipn
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoI have never been good at mathrdquo
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoLetters are not mathrdquo
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoWhen I was in schoolhelliprdquo
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquohellipwe didnrsquot do this type of mathrdquo
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoIn all of my classes I have Arsquoshelliprdquo
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquohellipexcept for mathrdquo
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquoI have been here for two yearshelliprdquo
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ldquohelliptrying to avoid mathrdquo
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Math is hard
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Yesmath is hard
2
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
E(math)
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Fair PriceWhy play
lt0
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
T T TT F FF T TF F T
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Soap Box Net
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Yes math is hard
And so is life
ldquoYou just get used to themrdquo
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
LOGIC
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Creating an Aha Moment with Function Models
Dennis C EbersoleNorthampton Community College
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Linear ModelsSlope-Intercept Form
8
6
4
2
-2
-4
-6
-5 5
y = 2x + 3
Form if the y-intercept is (00)
8
6
4
2
-2
-4
-6
-5 5
y = 2x
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric Representations
y = 2x
x yndash2 ndash4ndash1 ndash20 01 22 43 6
y = ndash3x
x yndash2 6ndash1 30 01 ndash32 ndash63 ndash9
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric Representations
Table with Initial Value Not 0
x y5 ndash16 27 58 89 11
Table After Translation of Axes and the Associated Equation
x ndash 5 y + 10 01 32 63 94 12y + 1 = 3(x ndash
5)
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Graphic Representations
Convert Graph to a Table
12
10
8
6
4
2
-2
5 10 15
Now Translate the Table and Find the Equation of the Line
x y x - 4 y - 84 8 0 06 7 2 ndash18 6 4 ndash2
10 5 6 ndash312 4 8 ndash4
y ndash 8 = ndash12(x ndash 4)
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Verbal RepresentationsThe Problem Statement
Jan has had a plumber do work twice in the last month The first time she was charged $140 for a 1-hour job The second time she was charged $320 for a 3-hour job Find a linear model showing the charge as a function of the number of hours on the job
Associated Table Translated Table and Symbolic Representation
x y x - 1 y - 140
1 140 0 0
3 320 2 180
y ndash 140 = 90(x ndash 1)
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Quadratic Models Vertex Not at Origin
General Form
6
4
2
-2
-4
-6
-5 5
- 4x + 32
y = 2x
Vertex of Parabola at Origin One Parameter
6
4
2
-2
-4
-6
-5 5
2y = -2x
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric RepresentationsVertex at Origin
x y
ndash2 12
ndash1 3
0 0
1 3
2 12
x y
ndash2 ndash8
ndash1 ndash2
0 0
1 ndash2
2 ndash8
22xy
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x y
0 0
1 25
2 40
3 45
4 40
5 25
6 0
Translated Table Yields Equation in Vertex Form
x ndash 3 y ndash 45ndash3 ndash45ndash2 ndash20ndash1 ndash50 01 ndash52 ndash203 ndash45
y - 45 = ndash5(x ndash 3)2
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Converting Graphic Representations
Convert Graph to Table
6
4
2
-2
-4
-6
-5 5
Translate Table and Find the Equation in Vertex Form
x y x ndash 2 y + 40 0 ndash2 41 ndash3 ndash1 12 ndash4 0 03 ndash3 1 14 0 2 4
y + 4 = 1(x ndash 2)2
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Absolute Value Models
6
4
2
-2
-5
One Parameter
6
4
2
-2
-4
-5 5
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Absolute Value Models
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-5 5
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric RepresentationsVertex at (0 0)
x yndash3 6ndash2 4ndash1 20 01 22 43 6
x yndash3 ndash9ndash2 ndash6ndash1 ndash30 01 ndash32 ndash63 ndash9
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Numeric RepresentationsVertex at (h k)Vertex is max or min function value
x yndash1 90 71 52 33 54 75 9
Translate vertex to (0 0) and find equation in y = a|x| form
x ndash 2 y ndash 3ndash3 6ndash2 4ndash1 20 01 22 43 6
y ndash 3 = 2|x ndash 2|
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Graphic Representationsto NumericVertex is max or min
Find points 1 away
6
4
2
-2
-4
-5 5
Convert Table to One with Vertex at (0 0) Find Equation in y = a|x| Form
x y x + 2 y + 3
ndash3 ndash1 ndash1 2
ndash2 ndash3 0 0
ndash1 ndash1 1 2
0 1 2 4
y + 3 = 2|x + 2|
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Symbolic to NumericRepresentationsFind a and Vertex (h k) in y ndash k = a|x ndash k|
Create Table Using Previous Patterns
x y
ndash4 1 ndash 3 = ndash2
ndash3 4 ndash 3 = 1
ndash2 4
ndash1 1
0 ndash2
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Symbolic to Numeric Representations II
Convert Equation to Form Use Patterns to Create Table
x y
ndash1 1
0 ndash1
1 ndash3
2 ndash1
3 1
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Other Functions ModelsExponential Model
8
6
4
2
-2
-4
-6
-5 5
Same Model After Translation
8
6
4
2
-2
-4
-6
-5 5
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Other Function ModelsSine Function Models
8
6
4
2
-2
-4
-6
-5 5
After Translation of Axes
8
6
4
2
-2
-4
-6
-5 5
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Questions Comments Suggestions
Email medebersolenorthamptonedu
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Discovering the Art of Mathematics
Mathematical Inquiry in the Liberal Arts -- Innovative Materials and Pedagogical Support
Julian Fleron Phil Hotchkiss Volker Ecke and Christine von Renesse Westfield State University
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Discovering the Art of Mathematics Project
Julian Fleron Phil Hotchkiss Volker Ecke Christine von Renesse Westfield State University Massachusetts
This project is based upon work currently supported by the National Science Foundation under NSF1225915 (TUES) and previously supported by NSF0836943 (CCLI) and a gift
from Mr Harry Lucas
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Our Vision
Mathematics for Liberal Arts students will be actively involved in authentic mathematical experiences that
are both challenging and intellectually stimulating
provide meaningful cognitive and metacognitive gains and
nurture healthy and informed perceptions of mathematics mathematical ways of thinking and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Rubikrsquos Cube
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Perspective Drawings
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Board Work
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Projecting Cubes
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
String Art
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Maypole Dancing
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Anamorphic Art
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Slice Forms
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Student Artwork
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Dancing Tessellations
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Math and Music Palindromes and Tuning
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Spirographs
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Tangles and Knots
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Prime number sieves
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
Gabrielrsquos Wedding Cake
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
More Student Art
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
ResourcesOpportunities
11 free books for MLA Art amp Sculpture Calculus Dance Games amp Puzzles Geometry Knot Theory Music Number Theory Patterns The Infinite Reasoning amp Proof
Topic Index (other math classes)
Individual mentoring or collaboration (stipend)
Beta-testreview (stipend)
Traveling workshops
wwwartofmathematicsorg
