Mathematics 20-2 - Activating Students in High School Math Web viewMathematics . 20-2. ... In many ways this reflects what we observe in the small details and total pattern of life

MATHEMATICS 20-2

INDUCTIVE AND DEDUCTIVE REASONING

High School collaborative venture withEdmonton Christian, Institutional Services, Jasper Place,

Millwoods Christian, Queen Elizabeth and Victoria Schools

Edmonton Christian: Jenn JohnsonInstitutional Services: Eric HansonJasper Place: Jessica NoselskiMillwoods Christian: Ken ScharfQueen Elizabeth: David Hernandez-RiveraVictoria: Gina MacKechnie

Facilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)

2010 - 2011TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea 4

Mathematics 20-2 Inductive and Deductive Reasoning Page 2 of 47

Enduring Understandings

Essential Questions

4

4

Knowledge

Skills

5

6

STAGE 2 ASSESSMENT EVIDENCE

Transfer TaskDetective Challenge

Teacher Notes for Transfer TaskTransfer TaskGlossary and RubricPossible Solution

78

1416

STAGE 3 LEARNING PLANS

Lesson #1 Identifying Patterns and Writing Conjectures 18

Lesson #2 Inductive Reasoning & Counterexamples 25

Lesson #3 Deductive Reasoning 30

Lesson #4 Proofs by Induction 37

Lesson #5 Proofs by Deduction 42


Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?

Mathematics 20-2 Inductive and Deductive Reasoning

STAGE 1 Desired Results

Big Idea:

In the world around us it is important to see all types of problems from different perspectives. We can use inductive and deductive reasoning to make general assumptions and to validate an argument.

Enduring Understandings:

Students will understand …

Patterns exist in many forms and can be identified and described. Inductive reasoning is based on patterns and assumptions that can leave an

argument open to flaws. Deductive reasoning attempts to show that a conclusion follows from a set of

premises. Inductive and deductive reasoning can be applied to contextual problems. Number sense and logic can be used to validate/disprove arguments.

Essential Questions:

When is it appropriate to use deductive and/or inductive reasoning? What are some real-life applications of deductive and inductive reasoning? What is the role of a counterexample? Is there a unique proof for each argument?


Knowledge:

EnduringUnderstanding

SpecificOutcomes

Description ofKnowledge

Students will understand…

Inductive reasoning is based on patterns and assumptions that can leave your argument open to flaws.

Deductive reasoning attempts to show that a conclusion follows from a set of premises.

Inductive and deductive reasoning can be applied to contextual problems.

Number sense and logic can be used to validate/disprove arguments.

*N1, N2Students will know …

how to identify a pattern what deductive reasoning is what inductive reasoning is when to use inductive reasoning when to use deductive reasoning what makes a counter example valid if an argument is valid how to identify an error that if an error occurs, two outcomes can arise:

o the entire case is disprovedo an exception is revealed

8888I*N = Number


Implementation note:Teachers need to continually askthemselves; if their students are acquiring the knowledge and skills needed for the unit.

Skills:

EnduringUnderstanding

SpecificOutcomes

Description ofSkills

Students will understand…





*N1, N2Students will be able to…

use appropriate tools to demonstrate their arguments

demonstrate their arguments use appropriate tools to prove/disprove an

argument prove using algebraic methods recognize false conjectures state a conjecture by observing patterns use

o divisibility ruleso number propertieso algebra and number relations in order to

justify an argument

*N = Number


Implementation note:Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.

Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

STAGE 2 Assessment Evidence

1 Desired Results Desired Results

Detective Challenge

Teacher Notes

There is one transfer task to evaluate student understanding of the concepts relating to inductive and deductive reasoning. A photocopy-ready version of the transfer task is included in this section.

Notes are included in the sample solution following the transfer task.


Detective Challenge - Student Assessment Task

Stanley Skeezeball has recently been a victim of a theft crime. He is accusing Johnny B. Good, his former business partner, of stealing company property and money from their office on the 11th floor of The Big Tall Towers Office Building.

The lead detective has obtained information. Inspect the Crime Report and use your knowledge of logic and reasoning to help the Edmonton Police Department. Answer the following questions and present your findings in the Intern Report.

o List the information given.

o Summarize how the crime was committed?

o Use inductive reasoning to prove who committed the crime

o Use inductive reasoning to prove if Johnny B. Good was involved.

o Is he guilty?

o Provide a counterexample to support your conclusions.

o Explain how your inductive reasoning may lead to a false conclusion.

Crime/Incident ReportCR IME

Type of Incident:Break and Enter, Theft, Vandalism

Year2010

Day of WeekWednesday

Time8:30 pm

Location of Incident (or address) CityPrivate Office of Stan Skeezeball and Johnny B. Good EdmontonBig Tall Towers Office Building – 11th floor

W ITNESS

Witness Name:Beefy "Bulldog" Jones (Security Guard)Witness Statement:The following witness statement is an account of the events according to Beefy ‘Bulldog’ Jones (Security Guard) from his kiosk in the front lobby.

At 8:30 pm a key card was used to gain entry to the front lobby of The Big Tall Towers Office Building Male 1 approximately 6’3” with short brown hair entered the mail lobby. As Mr. Jones returned from a nightly walk he

witnessed the male enter the stairwell on the North side of the building at 8:31 pm. He only saw the male from behind At 8:35 pm a different key card was used to gain entry to the front lobby of The Big Tall Towers Office Building. Mr. Jones witnessed a second male enter the main lobby and proceed to the elevator at 8:36 pm. Male 2 was approximately

5’10” and wore a ball cap. At 8:37 pm the security camera caught male 1 walking the corridors on the 11th floor. At 8:38 pm male 2 exited the elevator on the 10th floor (security camera). At 8:42 pm male 2 was seen walking down the corridor on the 11th floor (security camera). At 8:45 pm Mr. Jones departed the kiosk to complete a nightly walk around the perimeter of the building. Male 2 entered the elevator on the 10th floor at 8:52 pm. Male 2 entered lobby and departed from the building at 8:54 pm. Mr. Jones witnessed male 1 departing from the back entry of the building while on his nightly walk around the building.

Male 1 departed at 8:58 pm.Witness Name:Mr. Sid Clean (Custodian)

Witness Statement:

Mr. Sid Clean (Custodian) provided the following witness statement. Mr. Clean began his shift at 6:00 pm; he was assigned to clean floors 10, 11 and 12. Mr. Clean witnessed a male (6’3” with short brown hair) on the 12th floor just before his break. Mr. Clean stated he normally takes his break at 8:45 pm. The break room is on the 11th floor. During his break Mr. Clean heard banging from down the hall. Mr. Clean received a text message from his girlfriend at 8:51 pm right before he went to the bathroom on the 10th floor. On his way to the bathroom Mr. Clean stopped right outside the bathroom door and saw a male with a ball cap on walking

towards the elevator.

MO

INFORMAT ION

Total # of witnesses at crime: 2Placeof attack: Business Vehicle Street/Alley Lot/Park/Yard Vessels Other Description of Surrounding area:

Residential Business Industrial/Mftg. Recreational Institutional Open Space School Marine/Water Other Force Tool Weapon

Specify:Hammer

How Used:Broke Lock on door

Type of Structure N/AHigh Rise Office Building

Security UsedSecurity guard on duty.

Suspect Actions

Point of EntryKey card used to enter lobby of office building

PROPERTY

ItemNo. Article Name

Stolen Miscellaneous Description ValueReceived

1 Laptop stolen MacBook Pro - neon pink, sticker on cover "I love Butterflies"Includes personal information from clients and banking records and codes

$1200

2 Money stolen Taken from safe, $50 bills bundled in groups of 25, 2 bundles $2500

SUSPECTS

Arrested Y N

Suspect #1Johnny B. Good

SexMale

Age38

DOBFebruary 29, 1972

Height5'10''

Weight195 lbs

BuildAverage

Hair colorbrown

Eye colorbrown

Additional information/Further suspect description (i.e., glasses, tattoos, teeth, birthmarks, jewellery, scars, etc.NoneSuspect’s clothingRed New York Yankees Baseball Cap, Dark Grey Jacket, Black PantsHair Length/TypeShort, brown hair, slightly balding

Hair StyleCrew Cut

Facial HairNone

ComplexionNormal

General AppearanceN/A

NARRAT IVE

Evidence:No fingerprints foundHammer found near office doorOfficer’s statement/investigationUpon further investigation it was discovered that the security tapes were damaged beyond repair.

Intern Report

List the information given:

Summarize the crime that was committed…

Write a proof statement about who committed the crime (male 1 or male 2?):

Statement Reason

Write a proof statement about who Johnny B. Good is (male 1 or male 2?):

Statement Reason

Johnny B. Good is _________________. (guilty/not guilty)

Counterexample (How could you prove that the other person is guilty/not guilty?):

False Conclusion (how could your inductive reasoning be incorrect? Where are the gaps in your logic?)

Glossary

conjecture - A generalization made through inductive reasoning

deductive reasoning – A type of reasoning by which generalizations are drawn from specific patterns in observed data

inductive reasoning – A type of reasoning by which generalizations are drawn from general patterns in observed data

pattern – A repeated sequence or arrangement about which predictions can be made

Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

http://www.learnalberta.ca/content/memg/index.html

http://www.learnalberta.ca/content/memg/Division02/Pattern/index.html

Assessment

Mathematics 20-2

Inductive and Deductive Reasoning

Rubric

Level

Criteria

4Excellent

3Proficient

2Adequate

1Limited *

Explanations of proofShows thoroughunderstanding;explanations areeffective and thorough.

Shows understanding;explanations are appropriate.

Shows partialunderstanding;explanations areoften incomplete orsomewhat confusing.

Shows very limitedunderstanding;explanationsare omitted orinappropriate.

Explains Choice Shows a solution for the problem; provides a logical and insightful explanation.

Shows a solution for the problem; provides an adequate explanation.

Shows a solution for the problem; provides explanations that are complete but vague.

Shows a solution for the problem; provides explanations that are incomplete or confusing.

Solves and explains the solution using appropriate terminology conjecture, deductive inductive reasoning).

The student(s) has/have a good understanding of terminology and are able to correctly solve and explain the solution.

For the most part student(s) has/have an understanding of terminology and can solve and explain the solution.

Student(s) has/have a partial understanding of terminology and have errors in explaining the solution.

Student(s) has/have limited understanding of terminology and cannot correctly explain or solve the question.

Communicates findings in a thoughtful and convincing manner. Student(s) show(s) clear understanding of inductive and deductive reasoning

Communicates findings in a thoughtful and convincing manner, using specific mathematical vocabulary.

Communicates findings in an interesting and coherent manner, using appropriate mathematical vocabulary.

Communicates findings in a predictable and simplistic manner using inconsistent mathematical vocabulary.

Communicates findings in a confusing and vague manner using inappropriate mathematical vocabulary.

When work is judged to be limited or insufficient, the teacher makes decisions about appropriate intervention to help the student improve.


Possible Solution to Detective Challenge




Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

STAGE 3 Learning Plans

Lesson 1

Identifying Patterns and Writing Conjectures

STAGE 1

BIG IDEA:


ENDURING UNDERSTANDINGS:


Patterns exist in many forms and can be identified and described.

ESSENTIAL QUESTIONS:

What are some real-life applications of deductive and inductive reasoning?

KNOWLEDGE:

Students will know …

how to identify a pattern

SKILLS:

Students will be able to …

state a conjecture by observing patterns

Lesson Summary

Students will be able to identify patterns pictorially and numerically and write a conjecture regarding the observed properties.


Lesson Plan

Initial Activity:

From the information provided, find the names and positions of the first eight to finish the marathon. Sean finishes the marathon in fourth place; he finishes after John, but before Sandra. Sandra finishes before Robert but after Liam. John finishes after Rick but before Alex. Anne finishes two places after Alex. Liam is sixth to finish the race.

Answer:

1st – Rick

2nd – John

3rd – Alex

4th – Sean

5th – Anne

6th – Liam

7th – Sandra

8th – Robert

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&ID2=AB.MATH.JR.PATT.PATT&lesson=html/object_interactives/patterns/use_it.html






Ask Students:

What is a pattern and how do we know something has a pattern? Give examples of patterns in real life?

Ask Students:

What is one looking for when they ask someone to write a conjecture?

Show examples of patterns and have students write conjectures of what they expect to happen if the patterns continue. Here are some examples you could use:

Fractal Patterns in Nature

Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern. In many ways this reflects what we observe in the small details and total pattern of life in all its physical and mental varieties.

Ferns Mountains in Tibet

Source: http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

Sunflower and Fibonacci

Source: http://brittgow.globalteacher.org.au/files/2010/03/fibonacci-sunflower.jpg


http://brittgow.globalteacher.org.au/files/2010/03/fibonacci-sunflower.jpg

http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

Draw Your Own Fractal

Source: http://math.youngzones.org/Fractal webpages/fractal_intro.html

Other Patterns

1. 2, 8, 27, 64, …

2. 34, 7, 29, 11, 23, 16, 16, 22, 8 …

3. 1, 2, 3, 7, 22, …

4. F, S, T, F, F, S, . . .

5. 7, -5, 2, 1, -3, 7, -8, 13, ?, 19


http://math.youngzones.org/Fractal%20webpages/fractal_intro.html

Going Beyond

The teacher can bring in various music pieces and talk about the relationship of music with patterns.

Show TED talk regarding fractals in nature:

http://www.ted.com/talks/ron_eglash_on_african_fractals.html




Another way students can extend their learning is through a game called SET. Select "Daily Puzzle" from: http://www.setgame.com/November 23, 2010 sample:

Resources

Principle of Mathematics - NelsonSection 1.2-1.3 (pages 16-23)

Provide examples of number patterns and/or patterns that can be seen in nature.


http://www.setgame.com/

Supporting


Video of fractals in nature: http://www.youtube.com/watch?v=kkGeOWYOFoA

Assessment

Students need to demonstrate an ability to recognize patterns and write a conjecture describing the pattern and predicating what will occur if the pattern continues.

Glossary

conjecture - A generalization made through inductive reasoning

pattern – A repeated sequence or arrangement about which predictions can be made

Other


Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

http://www.learnalberta.ca/content/memg/index.html

http://www.learnalberta.ca/content/memg/Division02/Pattern/index.html

https://portal-owa.epsb.ca/exchweb/bin/redir.asp?URL=http://www.youtube.com/watch?v=kkGeOWYOFoA




http://www.youtube.com/watch?v=kkGeOWYOFoA

Lesson 2

Inductive Reasoning & Counterexamples

STAGE 1

BIG IDEA:






When is it appropriate to use deductive and/or inductive reasoning?

What are some real-life applications of deductive and/or inductive reasoning?

What is the role of a counterexample?

KNOWLEDGE:


what inductive reasoning is when to use inductive reasoning what makes a counter example valid

SKILLS:


demonstrate their arguments recognize false conjectures use appropriate tools to prove/disprove an

argument

Lesson Summary

1. Students will be able to recognize examples of inductive reasoning, by describing if the reasoning provided is weak or strong.

2. Through counterexamples students will disprove a given conjecture.


Lesson Plan

Initial Activity:

Bank A Bank B

The mortgage rates for March decreased from February

Ask Students: What conclusions can you make from this graph about mortgages?

What is Inductive Reasoning?

Inductive reasoning is reasoning based on detailed facts and general principles, which are eventually used to reach a specific conclusion. There are two types of conjectures, strong and weak. The strong inductive reasoning is specific and does not lead to a false conjecture. A weak inductive reasoning leaves it self open to false conjecture.

Example of Strong Inductive Reasoning:

All the tigers observed in a particular region have yellow stripes, therefore all the tigers native to this region have yellow stripes.

Examples of Weak Inductive Reasoning:

I always run the red light; therefore everybody runs the red light.


0.00%1.00%2.00%3.00%4.00%5.00%6.00%7.00%

Feb Mar

15 Year30 Year5 Year

0.00%1.00%2.00%3.00%4.00%5.00%6.00%7.00%

Feb Mar


More Examples of Inductive Reasoning:

Students should be able to:

Classify each statement of the following as strong or weak inductive reasoning examples.

Explain why each example is strong or weak.

1. "Every time you eat shrimp, you get cramps. Therefore I get cramps because I ate shrimp."

2. "Mikhail is from Russia and Russians are tall. Therefore Mikhail is tall."

3. "When chimpanzees are exposed to rage, they tend to become violent. Humans are similar to chimpanzees. Therefore they tend to get violent when exposed to rage."

4. "All men are mortal. Socrates is a man. Therefore Socrates is mortal."

5. "The woman in the neighboring apartment has a shrill voice. I can hear a shrill voice from outside. Therefore the woman in the neighboring apartment is shouting."

Counter Examples:

Introduce the idea that not all inductive/deductive reasoning is true. Only ONE counter example is needed to disprove a conjecture.

For each inductive reasoning example, have the students:

Write two examples that confirm the conjecture.

Write one counter example.

1. In Edmonton we had a white Christmas in 2008, 2009, and 2010.

2. A number that is not positive is negative.

3. The square of a number is always greater than that number.

4. A larger number divided by a smaller number is always greater than 1.

5. A number raised to a whole number exponent will always produce a number greater than or equal to the base.


Conclusion:Bank A Bank B

Ask Students:

Is this graph a strong or weak inductive reasoning example?What can be done to the graph to make it a strong inductive reasoning example?

Going Beyond

Student Activity: Make your own strong and weak inductive reasoning example with solutions.http://www.learnalberta.ca/content/me20tle/TLE11/patterns/patterns_desc.html

Another way students can extend their learning is through a game called Set. Select "Daily Puzzle" from: http://www.setgame.com/November 23, 2010 sample


0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

Feb Mar


0.00%

1.00%2.00%3.00%

4.00%5.00%6.00%

7.00%

Feb Mar



http://www.learnalberta.ca/content/me20tle/TLE11/patterns/patterns_desc.html

Resources

Principle of Mathematics - NelsonSection 1.3 (pages 18-23)

http://www.buzzle.com/articles/inductive-reasoning-examples.html

Supporting


Assessment

Written formative assessments

Glossary

inductive reasoning – A type of reasoning by which generalizations are drawn from patterns in observed data

Other



http://www.buzzle.com/articles/inductive-reasoning-examples.html

Lesson 3

Deductive Reasoning

STAGE 1

BIG IDEA:









KNOWLEDGE:


what deductive reasoning is when to use deductive reasoning

SKILLS:



demonstrate their arguments. use appropriate tools to prove/disprove an

argument

Lesson Summary

Students will be able to describe and recognize examples of deductive reasoning.

Through counterexamples students will disprove a given conjecture.


Lesson Plan

Define Deductive Reasoning and provide a few examples of deductive reasoning:

Deductive reasoning – A type of reasoning by which generalizations are drawn from patterns in observed data

or- Making a conclusion based on statements we believe to be

true.

Deductive Reasoning Examples

1. All oranges are fruits. All fruits grow on trees. Therefore, all oranges grow on trees.

2. All bachelors are single. Johnny is single. Hence, Johnny is a bachelor.

Source: http://www.buzzle.com/articles/deductive-reasoning-examples.html

Given: Triangle ABC and line CD are parallelProve: ∠1 + ∠2 + ∠3 = 180o

Proof: ∠1 = ∠4 ∠2 = ∠5∠4 + ∠3 + ∠5 = 180o

∠1 + ∠3 + ∠2 = 180o


http://www.buzzle.com/articles/deductive-reasoning-examples.html

Deductive Reasoning Puzzles

Have students work through a deductive puzzle

1. The Hospital Staff

The staff at the hospital consists of 16 doctors and nurses, including me. The following fact applies to the staff members; whether you include me or not does not make a difference. The staff consists of:

more nurses than doctors more male doctors than male nurses. more male nurses than female nurses. at least one female doctor.

What is the sex and occupation of the speaker? (Answer: woman nurse)

Source: Test Your Logic (by George J Summers, Dover Publications, 1972) (Pan American and International Copyright Conventions)

2. The Woman Freeman will Marry:

Freeman knows five women: Ada, Bea, Cyd, Deb, and Eve.

The women are in two age brackets: three women are under 30 and two

women are over 30.

Two women are teachers and the other three women are secretaries.

Ada and Cyd are in the same age bracket.

Deb and Eve are in different age brackets.

Bea and Eve have the same occupation.

Cyd and Deb have different occupations.

Of the five women, Freeman will marry the teacher over 30.

Whom will Freeman marry? (Answer: Deb)

Source: Test Your Logic (by George J Summers, Dover Publications, 1972)

(Pan American and International Copyright Conventions

3. Given 34, 7, 29, 11, 23, 16, 16, 22, 8, _, _, _, what are the next three numbers?

4. Given 7, -5, 2, 1, -3, 7, -8, 13, ?, 19 What is the value of the missing term?


5. A snail is at the bottom of a well and wants to get out. He manages to crawl up the wall 3 feet each day, but at night slips back down 2 feet when he rests. If the well is 30 feet deep, how long will it take him to get out?

6. An Old friend of the family left $666,666 to 2 fathers and 2 sons to be split equally. After careful consideration they each happily received $222,222. Why and how was this possible?

7. If a snail crawls halfway around a circle then turns around and crawls halfway back, is it now back where it started?

8. What day of the year has 25 hours in it?

Source for 3-8: http://www.justriddlesandmore.com/logicquestions.html

9. Triangles Puzzle 1: Look at the two top triangles, and then work out what number should replace the question mark in the bottom triangle.

Source:

http://www.justriddlesandmore.com/logicquestions.html






Inductive Reasoning vs. Deductive Reasoning

Inductive reasoning makes conclusions based on several examples showing that same result.

Deductive reasoning makes conclusions based on statements we believe to be true.

Deductive reasoning is one of the two basic forms of valid reasoning, the other one being inductive reasoning. The main difference between these two types of reasoning is that, inductive reasoning argues from a specific to a general base whereas deductive reasoning goes from a general to a specific instance. Thus, deductive reasoning is the method by which conclusions are drawn using proofs.

Going Beyond

Activity:

Students make their own strong and weak inductive reasoning with solutions

Another way students can extend their learning is through a game called SET. Select "Daily Puzzle" from: http://www.setgame.com/November 23, 2010 sample:



See "Reasoning and Analyzing Conjectures" on the following site for additional information (you may have to install the MathReader):


Resources



Supporting

Set game: Select "Daily Puzzle" from: http://www.setgame.com/



Assessment

Students need to demonstrate an ability to recognize examples of inductive reasoning and provide true or false conjectures.







http://www.learnalberta.ca/CheckMySystem.aspx?lang=en


Glossary

deductive reasoning – A type of reasoning by which generalizations are drawn from patterns in observed data

inductive reasoning – A type of reasoning by which generalizations are drawn from patterns in observed data

Other


Lesson 4

Proofs by Induction

STAGE 1

BIG IDEA:







Is there a unique proof for each argument? When is it appropriate to use deductive and/or

inductive reasoning?

KNOWLEDGE:


what inductive reasoning is when to use inductive reasoning if an argument is valid

SKILLS:


use appropriate tools to prove/disprove an argument


Lesson Summary

Students will be able to support inductive and deductive reasoning based on patterns and assumptions when give a contextual problem.

Lesson Plan

Review what inductive reasoning is. Students can provide examples for inductive reasoning.

1. What is the relationship between the diameter and the circumference of a circle?

Solution:

Cd = π

2. What is the pattern when adding two odd numbers together?

a. Add 3 + 5 =b. Add 7 +13 = c. Add 9 + 21 =

Solution:The sum of two odd numbers is an even number. (Answers will vary for the following tasks).


Try other examples to prove or disprove adding two odd numbers is an even number.

Show the proof for this example.

Does a pattern exist when 3 odd numbers are added together?

Does a pattern exist when 4 or 5 odd numbers are added together?

Are there any patterns that become apparent when adding odd numbers together?

3. What are the sums of the angles of a quadrilateral?

Solution: 360o

4. Compare the diagonal length to side length ratio of a square.

Solution: diagonal = √s2+s2=√2 s2=√2 s

ratio:

√2 ss

=√2


5. Does this diagonal length to side length ratio exist for other quadrilaterals?

Solution: No. The diagonal = √ l2+w2.

Going Beyond

Resources


Supporting

Assessment



Glossary



Other


Lesson 5

Proofs by Deduction

STAGE 1

BIG IDEA:










KNOWLEDGE:


what inductive reasoning is when to use inductive reasoning if an argument is valid

SKILLS:



prove using algebraic methods state a conjecture by observing patterns use

divisibility rules number properties

Lesson Summary

Students are to support their answers to contextual problems using deductive reasoning.


Lesson Plan

Introduction Activity A:

Example:

1. Pick any number

2. Multiply that number by 3

3. Add 45

4. Double your number

5. Divide your number by 6

6. Subtract by the number you initially picked

7. The number you have left is 15

Task: Prove 15 is the final answer.

Solution:

1. Pick any number n

2. Multiply that number by 3 3n

3. Add 45 3n + 45

4. Double your number 6n + 90

5. Divide your number by 6 n + 15

6. Subtract by the number you initially picked n + 15 – n

7. The number you have left is 15 15

Introduction Activity B:

Example:

1. Pick any number

2. Subtract by 1

3. Multiply by 3

4. Add 12

5. Divide by 3

6. Add 5


7. Subtract by the number you initially picked

8. The answer is 8

Task: Prove 15 is the final answer.

Solution:

1. Pick any number n

2. Subtract by 1 n – 1

3. Multiply by 3 3n – 3

4. Add 12 3n + 9

5. Divide by 3 n + 3

6. Add 5 n + 8

7. Subtract by the number you initially picked n + 8 – n

8. The answer is 8 9

Task: Have students write their own number magic trick.

Solution: Will vary.

Prerequisite:

Deduction:

Reasoning based on statements, which are known to be true. Like theorems

Lesson:

1. Prove deductively that when two odd numbers are added, the result is an even number.

Solution:

2n + 1 = first odd number

2m + 1 = second odd number

(2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1)

Since 2 is a factor, the result must be an even number.


2. Prove deductively that the difference between an odd number and an even number is always odd.

Solution:

(2n + 1) – (2m) = 2n – 2m + 1 = 2(n – m) + 1

Whenever 1 is added to an even number, the result is odd.

3. ABCD is a square and ΔABE is an equilateral triangle. Explain why ΔAED is isosceles.

Solution:

AB = AD Properities of a square

AB = AE Properties of an equalteral triangle

AD = AE Transitive property

Therefore ΔADE is Isosceles (Triangle with 2 equal sides)

4. [extension problem]:

The difference of the squares of two odd numbers is divisible by 4.

Solution:

First odd number = 2n – 1

Second odd number = (2n – 1) + 2 = 2n + 1 (2n + 1)2 - ( 2n - 1) 2 = [(2n + 1) + ( 2n - 1)] [(2n + 1) - ( 2n - 1)]

= 4n (2)

= 8n

Going Beyond

Prove by Deduction

1. Pick a 3-digit number with all three digits being the same. Ex. 333, 999, etc.

2. Add the three digits together

3. Divide your three digit number by the sum you found

4. Your answer is 37


Proof:

Let x be the repeating digit

100x + 10x + x = your number

x + x + x = 3x

(100x + 10x + x) / 3x

= 100x/ 3x + 10x/ 3x + x/ 3x

= 33 1/3 + 3 1/3 + 1/3

= 37

Resources


Supporting

Assessment


Glossary



Other


Documents

Mathematics 20-2 - Activating Students in High School Math Web viewMathematics . 20-2. ... In many ways this reflects what we observe in the small details and total pattern of life