Making Mathematics Meaningful Marty Romero UCLA Graduate School of Education [email protected] 2011 UCLA Curtis Center Mathematics and Teaching Conference

Making Mathematics Meaningful

Marty Romero

UCLA Graduate School of Education

[email protected]

2011 UCLA Curtis Center

Mathematics and Teaching Conference

March 5, 2011

What is Mathematics?“It is only within the last thirty

years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns. What a mathematician does is examine abstract patterns – numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, purely utilitarian or of little more than recreational interest, They can arise from the world around us, from the depths of space or time, or from our inner workings of the human mind.” (Delvin – The Language of Mathematics Pg. 3 )

What is Mathematics?“employers want young people who can

use ‘statistics’ and three-dimensional geometry, systems thinking, and estimation skills. Even more important, they need the disposition to think through problems that blend quantitative work with verbal, visual and mechanical information; and the ability to deal with situations when something goes wrong.”(Boaler – What’s Math Got to Do with It? Pg 6 )

“State business leaders consistently report that they need a pipeline of prospective entry-level employees who can read, write, solve problems, communicate with others, think critically, and be responsible for their work (Tulchin & Muehlenkamp, 2007; de Cos, Chan, & Salling, 2009). Other organizations echo these concerns. The Partnership for 21st Century Skills has highlighted a wide range of high-level skills that are important in the fastest growing job sectors. These skills include critical thinking and problem solving, excellent communication, collaboration, and creativity and innovation skills (http://www.21stcenturyskills.org).”

x S2 23 24 45 26 47 68 89 210 411 612 813 1014 1215 1416 1617 218 419 620 821 1022 1223 1424 1625 1826 2027 2228 2429 2630 2831 3032 32

)2(2 )]1(int[log2 xxy

Algebra 2 POW – The Birthday Cake

Determining the maximum /minimum number of pieces in which it is possible to divide a circle (birthday cake) for a given number of cuts is called the circle cutting or pancake cutting problem. Use the table below to help organize your work to come up with your answers.

1 Slice 2 Slices 3 Slices 4 Slices 5 Slices

Diagram(s) Diagram(s) Diagram(s) Diagram(s) Diagram(s)

Minimum Regions: ___





Maximum Regions:___

Maximum Regions:___

Maximum Regions:___

Maximum Regions:___

Maximum Regions:___

Algebra 2 POW – The Birthday Cake

Slices

Min Max

1

2

3

4

5

n

Extra Serving: The problem of dividing a circle by lines can also be generalized to dividing a plane by circles. As can be seen above, the maximal numbers of regions obtained from n = 1, 2, 3, circles are given by 2, 4, 8. What is the maximum number of regions a plane can be divided by 4 circles. Show the drawing and explain the numerical pattern.

Common Core

College/Career ReadinessExpect students to:

•make inferences

•interpret results

•analyze conflicting explanations of phenomena

•support arguments with evidence

•solve complex problems that have no obvious answer

•reach conclusions

•offer explanations

•conduct research

•engage in the give-and-take of ideas

•think deeply about what they are being taught (National Research Council, 2002).

http://www.corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/

Project: Follow Me Consecutive NumbersWhich natural numbers can be expressed as the sum of two or more consecutive natural

numbers?Use your calculator and your equation solving skills to find the solutions.

n Sum n Sum

1 21 10+11, 6+7+8,1+2+3+4+5+6

2 22

3 1+2 23

4 24

5 25

6 26

7 27

8 28

9 29

10 30

11 31

12 32

13 33

14 34

15 35

16 36

17 37

18 38

19 39

20 40

Describe in a complete sentences three patterns that you found after finishing the table.1.

2.

3.

Which numbers cannot be written as a consecutive sum? What is special about these numbers?

Write 95 as the sum of 5 consecutive natural numbers. Write 64 as the sum of 4 consecutive numbers.

In algebra, the sum of any two consecutive numbers is x + (x+1)= 2x +1. Complete the table below to express the sum of different lengths of consecutive numbers.

Sum of Consecutive Numbers

Expression Result

2 Numbers x + (x+1) 2x+1

3 Numbers x + (x+1) + (x+2)

4 Numbers

5 Numbers

6 Numbers

10 Numbers(Try to use a pattern

to find the result)

Project: Follow Me Consecutive Numbers – Page 2

Solve each problem below by a guess and check method and then by writing and solving an equation. Show your work in the table

Word Problem

Equation Simplified Equation Solution

Find two consecutive

integers whose sum is 45.

x + (x+1) = 45 2x+1 = 45 The numbers are 22 and 23



Find three consecutive




Find two consecutive odd integers whose

sum is 92.

x + (x+2) = 92

Find two consecutive even integers whose

sum is 54.

Find three consecutive odd integers whose

sum is 369.

Follow Me- Homework Consecutive Integers Word Problems


Word Problem




x + (x+1) = 45 2x+1 = 452x + 1 – 1 = 45 – 1

2x = 44X = 22

The numbers are 22 and 23







Find two consecutive odd integers whose

sum is 92.

x + (x+2) = 92 2x + 2 = 922x + 2 – 2 = 92 – 2

2x = 90X = 45

The numbers are 45 and 47


sum is 154.


sum is 306.

Follow Me- Practice Quiz


Word Problem









sum is 514.


sum is 156.

Follow Me - Quiz

Tom and Jerry together have $111. If Tom has $17 more than Jerry, how much does each person have?

TOM ($) JERRY ($) MONEY TOGETHER ANSWER

1st # 2nd # SumAnswer

7533rd #

The sum of three consecutive integers is 753. What are the integers?

Algebra 2 Problem Solving - Word Problems

Use a guess and check table to write an equation that could be used to solve the following word problems.


Use a guess and check table to write an equation that could be used to solve the following word problems.1.A 150-centimeter board is cut into two pieces. One piece is 24 centimeters longer than the other. How long is each piece?2.Admission to the school dance was $3 in advance and $4 at the door. There were 30 more tickets sold at the door than in advance, and ticket sales totalled $1590. How many of each kind of ticket were sold?3.Find three consecutive odd integers such that the sum of the smallest and seven times the largest is 68.4.Maya has three times as many dimes as nickels. She has four more quarters than nickels. If the value of the coins is $9.40, how many nickels does Maya have?5.Gloria is twice as old as her brother Jaime. Jaime is two years older than Susanna. The sum of the ages of the three children is 26. Find the age of each child.6.Ms. Speedi has 40 pets, all canaries and cats. The number of birds is twenty-two more than twice the number of cats. How many of each animal does she have?7.The longest side of a triangle twice the length of the shortest side. The third side is two centimeters longer than the shortest side and the perimeter of the triangle is 38 centimeters. Find the lengths of the three sides.8.A rectangle has perimeter 48 feet. The length is three times the width. Find the length and width of the rectangle. (Remember, the perimeter of a rectangle is the sum of all four sides.)9.Francie and Angie are twins. Their little sister, Denise is half as old as they are. The sum of all three sisters' ages is 45. Find the age of each girl.10.When Alex banged on the Coke machine, all the coins came out, but no Coke. There were nickels, dimes, and quarters. There were three times as many dimes as nickels, and 40 more quarters than nickels. If the total amount of money was $17.20, how many coins fell out of the machine?


1. Jessica and Jazylynn are twins. Their little sister, Jezebel is half as old as they are. The sum of all three sisters' ages is 45. Find the age of each girl.

2. Eddie and Alejandro together have $125. If Eddie has $15 more than Alejandro, how much does each person have?

3. A 102-centimeter board is cut into two pieces so that one piece is five times as long as the other. How long is each piece?

4. Admission to the school dance was $3 in advance and $4 at the door. There were 30 more tickets sold at the door than in advance, and ticket sales totaled $1590. How many of each kind of ticket were sold?

5. Find three consecutive odd integers such that the sum of the smallest and seven times the largest is 68.

6. Carla has three times as many dimes as nickels. She has four more quarters than nickels. If the value of the coins is $9.40, how many nickels does Carla have?

7. A rectangle has perimeter 48 feet. The length is three times the width. Find the length and width of the rectangle. (Remember, the perimeter of a rectangle is the sum of all four sides.)

Use a guess and check table to write an equation that could be used to solve the following word problems.

.05( ) .10(3 ) .25( 4) 9.40x x x 7( 4) 68x x

3 3 48x x x x

452

xx x

( 15) 125x x

5 102x x 3 4( 30) 1590x x

Match the word problem to the algebraic expression. Put the number of the problem in the box below the equation.

How many squares are there on an 8x8 checkerboard? And the answer is not 64.

GEOMETRY POW #1 – The Checkerboard

Oh yeah, the answers is not 65 either.

Wilson High School Wilson High School Activity CardActivity Card

Wilson High School wants to boost interest in sports and school activities. It has decided to sell an activity card that will allow the holder to enter all sporting events free as well as get discounts on other school activities. The entire student body was surveyed and asked the question, “What is the most you would pay for an activity card?” The results of the survey are given below. Use the data to determine the optimal ticket price.

Maximum Price Total Number Willing to Pay for Activity Card

507585105115135150170

765620565460405285210115

Algebra Quadratic Recap – Putting It All Together

Modeling with Quadratics Investigation

x yUse the table to record points from the parabola in the picture.

General Form Vertex Form Factored/Root Form

Wall Ball Investigation:

Find an Equation Algebraically

x (horizontal)y (vertical)

Polynomials in the real world – Lagrange Polynomials Investigation

___)(___,0x 0y

1x 1y

2x 2y

3x 3y

The equation to model the top part of the roller coaster is:

)()()()()( 33221100 xyxyxyxyxF

30

3

20

2

10

10 )(

xx

xx

xx

xx

xx

xxx

31

3

21

2

01

01 )(

xx

xx

xx

xx

xx

xxx

32

3

12

1

02

02 )(

xx

xx

xx

xx

xx

xxx

23

2

13

1

03

03 )(

xx

xx

xx

xx

xx

xxx

___)(___,

___)(___,

___)(___,

0 0 1 1 2 2 3 3( ) ( ) ( ) ( ) ( )

( )

F x y x y x y x y x

F x

Polynomial Function:

A polynomial function is a function of the form

f(x)=anxn + an-1xn-1 + … + a1x + a0

where an ≠ 0, the exponents are all whole numbers, and the coefficients are all real numbers.

an is called the leading coefficient, n is the degree, a0 is the constant term.

71260

1321

630

1

1260

23)( 23 xxxxF

Marty Romero, Wallis Annenberg High School, Adapted from Math Horizons Magazine, February 2007

The cast of “Mathematical Survivor” is a collection of n not necessarily distinct real numbers {x1,x2,x3,…,xn}, where n=2. From this collection, we select any two numbers, say xa and xb, delete these from the collection, and insert the number xaxb+xa+xb into the collection. The process to find the mathematical survivor includes continuing the selection of two random numbers from the new collection and performing the deletion and insertion process. Proceed until the collection has a single number left. This number is the survivor.

Mathematical Survivor Project: Last One Standing

Part 1Find the survivor for the set of numbers {2, 4, 6, 8}. You can randomly choose the two numbers to start with. Once you find the survivor, repeat the process two more times to find the surviving number. Make sure to alter your choice of numbers to delete.

EXTRA: How many different ways can a survivor be produced when starting with 4 numbers in your collection. You know there has to be at least 3 different ways since you completed part 1.Part 2

Using you inductive reasoning skills, make a conclusion about the survivor from the set of numbers {2, 4, 6, 8}. Choose your own set of 4 numbers and find the survivor. Find the survivor two more times for your set. What conclusions can you make now?

Part 3

The next obvious question is whether or not a mathematical survivor of a collection of numbers is predictable at the outset and is it totally independent of the selection process made throughout the process. The answer to the question is YES. The theorem below explains the results.

If S = {x1, x2,…, xn} is a collection of n not necessarily distinct real numbers, where nù2,then the mathematical survivor of S is guaranteed to be

(x1+1)(x2+1)…(xn+1)-1

Use the theorem to verify that you found the correct survivor for the sets in Part1 and Part 2.

Part 4

Use the theorem above to find the survivors for the following sets.

{2,2}

{2,2,2}

{2,2,2,2}

{2,2,2,2,2}

{5,5}

{5,5,5}

{5,5,5,5}

{5,5,5,5,5}

Use the results from above to find the survivor for {c,c,…,c} (n copies), where c is some number .EXTRA, EXTRA

Find the mathematical survivor for the following set: 1 1 1 1

{1, , , ,..., }2 3 4 n

What Goes Around Comes Around Project: Hailstone Numbers

A particularly famous problem in number theory, the hailstone problem, has fascinated mathematicians for several decades. It has been studied primarily because it is so simple to state yet apparently intractably hard to solve. This problem is also known as the 3n+1 problem, the Collatz algorithm, and the Syracuse problem.

1 2 3 4 5 6 7 8 9 10

1 7 2

11 12 13 14 15 16 17 18 19 20

Number

Steps to 1

Number

Steps to 1

If the number is even, divide by 2, if it is odd, multiply by 3 and add 1.

Answer the following questions

1. What is the pattern for the number of steps for 2, 4, 8, and 16? Predict how many steps the number 32 will have? How many steps for 128?

2. What is the pattern for any of the numbers that are doubles of each other? For example, 5 and 10, 7 and 14?

3. Predict how many steps are needed for the numbers 40, 80 and 100. How many steps does it take 76 to get to 1?

4. Fill in the missing numbers from the number chain. There are two answers.

_____, 15, 46, 23, …

_____, 15, 46, 23, …

Of the first 1,000 integers more than 350 have their maximum at 9,232. Find one of the integers that has a maximum of 9232.

5.

What Goes Around Comes Around Project: Hailstone Numbers

Hailstone Numbers By Ivars Peterson Muse, February 2003, p. 17.

Nothing could be grayer, more predictable, or less surprising than the endless sequence of whole numbers. Right? That's why people count to calm down and count to put themselves to sleep. Whole numbers define booooooooring. Not so fast. Many mathematicians like playing with numbers, and sometimes they discover weird patterns that are hard to explain. Here's a mysterious one you can try on your calculator. Pick any whole number. If it's odd, multiply the number by 3, then add 1. If it's even, divide it by 2. Now, apply the same rules to the answer that you just obtained. Do this over and over again, applying the rules to each new answer. For example, suppose you start with 5. The number 5 is odd, so you multiply it by 3 to get 15, and add 1 to get 16. Because 16 is even, you divide it by 2 to get 8. Then you get 4, then 2, then 1, and so on. The final three numbers keep repeating. Try it with another number. If you start with 11, you would get 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, and so on. You eventually end up at the same set of repeating numbers: 4, 2, 1. Amazing! The numbers generated by these rules are sometimes called "hailstone numbers" because their values go up and down wildly—as if, like growing hailstones, they were being tossed around in stormy air—before crashing to the ground as the repeating string 4, 2, 1.Mathematicians have tried every whole number up to at least a billion times a billion, and it works every time. Sometimes it takes only a few steps to reach 4, 2, 1; sometimes it takes a huge number of steps to get there. But you get there every time. Does that mean it would work for any whole number you can think of—no matter how big? No one knows for sure. Just because it works for every number we've tried doesn't guarantee that it would work for all numbers. In fact, mathematicians have spent weeks and weeks trying to prove that there are no exceptions, but they haven't succeeded yet. Why this number pattern keeps popping up remains a mystery.

Objectives:

Algebra 2 Lecture Page 1

KNOWLEDGE:Students will know…The different representations for logarithms of different bases.How to correctly verbalize logarithmic notation using proper math terminologyWhat the common logarithm isThe definition of logarithmHow to represent a logarithmic equation in exponential form

Objectives:

Answer the question and evaluate each numerical expression.Do Now:


What number multiplied with itself is equal to 81?

144 2

1

49

Silent Board Game: Without talking to anyone, determine the numbers that complete the table.

X 16 1 4 2 32

Y 4 -3 1 -4 -5

4

1

2

1

Definition of __________________:

Record the relationship between the two variables . Then plot the points from the table to see the graphical relationship.

110


Evaluating LogarithmsExample 1:

Logarithmic Expression

How do you say it? What question is being asked?

Answer

Log base 2 of 16 is … 2 raised to what number is 16?

16log2162? 416log162 2

4

3log3

100log10

4

1log4

Quick Note (Common Logarithm):

Guided Practice: Evaluate each logarithmic expression81log3 7log49 1.0log

Rewrite each logarithmic equation in exponential form.Example 2:

Logarithmic Form How do you say it? Exponential Form

Log base 4 of 16 is 2?216log4

31000log

15.0log2

1642

Make up your own examples and have a partner rewrite the logarithmic equation in exponential form

When I was in high school, my Algebra 2 teacher Mr. Gunner used to say that a logarithm is just an exponent. What did he mean by that? Write complete sentences using the appropriate vocabulary to explain what he meant.

Think and Reflect:

Documents

Making Mathematics Meaningful Marty Romero UCLA Graduate School of Education [email protected] 2011 UCLA Curtis Center Mathematics and Teaching Conference