FOUNDATIONS OF MATHEMATICS

UNIT 1

Inductive and Deductive Reasoning

(Chapter 1)

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Email: jmckenzie@sd79. bc.ca Website: https://msmckenzie. weebly.com/f6m-ll. html

Table of ContentsUnit 1: Inductive and Deductive Reasoning..................................................................................................3

CURRICULAR COMPETENCIES COVERED IN THIS UNIT .............................................................................3

CONTENT LEARNING OUTCOMES COVERED IN THIS UNIT .......................................................................3

UNIT REFLECTION.........................................................................................................................................^

J. McKENZIE FOUNDATIONS OF MATHEMATICS 11

Unit 1: Inductive and Deductive Reasoning

CURRICULAR COMPETENCIES COVERED IN THIS UNITReasoning and Modeling IRM)

. Develop thinking strategies to solve problems and play games

. Explore, analyze and apply mathematical ideas using reason, technology and other tools

. Think creatively and with curiosity and wonder when exploring problemsUnderstanding and Solving (US)

. Develop, demonstrate and apply mathematical understanding through play, story, inquiryand problem solving

. Apply flexible and strategic approaches to solve problems

. Solve problems with persistence and a positive disposition

. Engage in problem-solving experiences connected with place, story, cultural practices andperspective relevant to local First Peoples communities, the local community and othercultures.

Communicating and Representing (CmRp). Explain and justify mathematical ideas and decisions in many ways

. Represent mathematical ideas in concrete, pictorial, and symbolic forms

. Use mathematical vocabulary and language to contribute to discussions in the classroom

. Take risks when offering ideas in classroom discourse

Connecting and reflecting (CnRfl

. Reflect on mathematical thinking

. Connect mathematical concepts with each other, other areas, and personal interests

. Use mistakes as opportunities to advance learning

. Incorporate First Peoples worldviews, perspectives, knowledge, and practices to makeconnections with mathematical concepts

CONTENT LEARNING OUTCOMES COVERED IN THIS UNIT

1A:1 can make conjectures by observing patterns and identifying properties and justify the reasoning. (1. 1)IB: Explain why inductive reasoning may lead to a false conjecture. (1. 1)1C: Determine if a given argument is valid and justify the reasoning (1. 2)ID: Provide and explain a counterexample to disprove a given conjecture. (1. 3)IE: Prove algebraic and number relationships such as divisibility rules, number properties, mental math

strategies or algebraic number tricks. (1. 4)IF: Prove a conjecture using deductive reasoning. (1. 4)1G: Identify errors in a given proof (eg. A proof that ends with 2=1) (1. 5)1H: Solve a problem that involves inductive or deductive reasoning. (1. 6)11: Determine, explain and verify strategies to solve a puzzle or to win a game. (1. 7)U: Create a variation of a puzzle or a game and describe a strategy for solving the puzzle or winning the

game. (1. 7)

Email: jmckenzle@sd79. bc.ca Webslte:https://msmckenzie. weebly. com/f6m-ll. html

UNIT REFLECTION

<-

<-

RED I still have difficulty understanding the learning outcome.

YELLOW I am somewhat ok in my understanding of the work,

GREEN I am strongly confident in my understanding of work.

Formative: DURING

1A: I can make conjectures by observingpatterns and identifying properties andjustify the reasoning. (1. 1)

1B: Explain why inductive reasoning maylead to a false conjecture.

1C: Determine if a given argument is validand justify the reasoning (1. 2)

ID: Provide and explain a counterexampleto disprove a given conjecture. (1. 3)

IE: Prove algebraic and numberrelationships such as divisibility rules,number properties, mental mathstrategies or algebraic number tricks. (1.4IF: Prove a conjecture using deductivereasoning. (1. 4)

1G: Identify errors in a given proof (eg. Aproof that ends with 2=1) (1. 5)

1H: Solve a problem that involvesinductive or deductive reasoning. (1. 6)

11: Determine, explain and verifystrategies to solve a puzzle or to win agame. (1. 7)

U: Create a variation of a puzzle or agame and describe a strategy for solvingthe puzzle or winning the game. (1. 7)

Formative: REVIEW

(If red/yellow, explain what needs to be done)

FOM11 1.1 Making Conjechires: Inductive Reasoning

If the same result occurs over and over again, we may conclude that it will always occur. Thiskind of reasoning is called inducttve reasoning.

Inductive reasoning can lead to a conjecture, which is a testable expression that is based onavaUable evidence but is not yet proved.

Example 1: Use inductive reasoning to make a conjecture about the product of an odd integerand an even integer.

Example 2: Make a conjecture about mtersecting lines and the angles formed.

Example 3: Make a conjecture about the sum of two odd numbers.

Assignment: pg. 12 #3, 5, 6, 9, 10-12, 14, 16, 20

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FOM11 1.2 Exploring the VaUdity Of Conjectures

Some conjectures initially seem to be valid, but are shown not to be valid after more evidence isgathered.

Example 1: Make a conjecture about the lines below:

Am lh«*e horirintal Itaw lacatd ardo they stop*?

Example 2: Make a conjecture about the grey rectangles:

The best we can say about a conjecture reached through inductive reasoning is that there isevidence either to support or deny It.

Assignment: pg. 17 #1-3

^

..... An Introduction to

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digits 1 to 6. Place them in such away that the numbers 1 to 6 onlyappear once in each row and column

In the more difficult puzzles there are many occasionswhen two or more possible numbers can go in each celtand you »vill usually have to pencil these possibilities inusing small writing before deciding which goes where

Before this stage there are always some cells that canhave only one solution. It is better to try and find allthese solutions first, before moving on to findsolutions for cells that have multiple possible solutions.

In this 2x3 box there is only oneplace to put the 5. Look along The

rows and columns to see why.

5 1

5 34 2

25 3

4 2

Use the same method to place a 5 in every2 by 3 box. (Remember you can only have

one 5 in each row and column

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Use the same method to place all the numbers that only have onesolution. Think logically, start from 1 and try to place it in every

2x3 box. Then move through all The numbers pladng only the onesvou know for cerhiin will fit.

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As in the smaller puzzles there may beoccasions when two or more possible numberscan go in each cell. Always try to fill in all thecells that only have one possible solution firs+1

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rows and columns to see why.

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»'use little numbers in the top of the box like this to show which numbers could

go there. (This will come in handy in more difficult puzzles.)

Homework Name:Block:

Corn lete the Follow! 2x3 Sudoku Puz2les(Remember to go about it in a logical manner)

EASY

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Hint:

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FOM11 1.3 Using Reasoning to Find A Counterexample to a Conjecture

We know that inductive reasoning can lead to a conjecture, which may or may not be true. Oneway a conjecture may be proven false is by a counterexample,

Example 1: If possible, find a counterexample for each conjecture. If not, write "true"

a. Conjecture: Every mammal has fur.

b. Conjecture: The acute angles m a right triangle are equal.

c. Conjecture: A polygon has more sides than diagonals.

d. Conjecture: The square of every even number is even.

e. Conjecture: An even number is any number which is not odd.

Example 2: Three conjectures are given.For which conjectures is this diagram a counterexample?

A. The opposite sides of a parallelogram are equal.

B. A quadrilateral cannot have both a 90° angle and an obtuse angle.

C. Every trapezoid has 2 pairs of equal angles.

Assigmnent: pg. 22 #1, 3-6, 10, 12, 14, 17

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^

FOM11 1.4 Proving Conjectures: Deductive Reasoning

When we make a conclusion based on statements that we accept as tone,we are using deductive reasoning.

Example 1: Use deductive reasoning to prove that the product of an odd integer and an eveninteger is evea.

Example 2: Use deductive reasoning to prove that opposite angles of intersecting lines areequal.

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Example 3: Use deductive reasoning to prove that the difference between consecutive perfectsquares is always an odd number.

Example 4: Weight-lifting bulds muscle. Muscle makes you strong. Strength improvesbalance. Inez lifts weights. What can be deduced about Inez?

Assigmnent: pg. 31 #1, 2, 4-7, 10, 11, 15, 19

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FOM11 1.4.2 Deductive Reasoning Part II

When we make a conclusion based on statements that we accept as true,we are using deductive reasoning. The rules we follow when perfonning algebraic

manipulations are things that we accept (and know) as tme.So we are using deductive reasoning to prove a statement is always true.

Statements that we know are true:

Any Integer multiplied by 2 is an even number.- This means that 2x or 2(any combination of variables and coefficients) will always be even.

If you add 1 to any even integer you will get an odd number.- This means that 2x + 1 or 2(any combination of variables and coefficients) + 1 will aliaaxs be odd.

Consecutive Numbers follow each other in numerical order

- This means that x, x+l, x-*-2, x+3are4 numbers that come one after the other numerically.

- 2x, 2x+ 2, 2x+ 4, 2x+ 6are4 consecutive even numbers

- 2x+l, 2x+3, 2x+5, 2x+7are4 consecutive odd numbers

Example 1: Use deductive reasoning to prove that the sum of an odd number and an even numberis always odd.

Finishing a Proof:

- If proving an answer Is even it should look like this -^ 2(any combination of variable terms)- If proving an answer is odd it should look like this -> 2(any combination of variable terms) + 1- If proving an answer is divisible by 3 it should look like this -» 3(any combination of variable terms)- If proving an answer is divisible by 4 it should look like this -> 4(any combination of variable terms)- If proving an answer is dwislble by 5 It should look like this -> 5(any combination of variable terms)

etc......

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Example 2: Prove that the square of an even integer is always even

Example 3: Prove that the result of the number trick below is always the number you start with.

- Choose a number

-Add2

-Multiply by 3

- Subti'act 6

- Subtract your original number

-Divide by 2

Example 4: The sum of a two digit number and its reversal is a multiple of 11.

Assignment: Deductive Reasoning Worksheetl4>'

Name:

Statements that we know are true:

Any integer multiplied by 2 is an even number.

- This means that 2x or 2(any combination of variables and coefficients) will always be even.

If you add 1 to any even Integer you will get an odd number.

- This means that 2x + 1 or 2(any combination of variables and coefficients) + 1 will always be odd.

Consecutive Numbers follow each other in numerical order

- This means that x, x+ l, x+ 2, ic+ 3 are 4 numbers that come one after the other numerically.- 2jc, 2x + 2, 2x+ 4, 2x+ 6are4 consecutive even numbers

- 2x+l, 2x+3, 2x+5, 2x+7are4 consecutive odd numbers

Finishing a Proof:

- If proving an answer is even it should look like this -> 2(any combination of variable terms)- If proving an answer is odd it should look like this -> 2(any combination of variable terms) + 1- If proving an answer is divisible by 3 it should look like this -^ 3(any combination of variable terms)- If proving an answer is divisible by 4 it should look like this -> 4(any combination of variable terms)- If proving an answer is divisible by 5 it should look like this -> 5(any combination of variable terms)

etc......

Important Tips to remember:

- Proving Mathematical Concepts using Deductive Reasoning uses ALGEBRA- Remember to square something means to multiply by itself! If there is more than 1 term being

squared you MUST use the box method (or arrows) to multiplySum = Add

Difference = Subtract

- Product = Multiply- Quotient = Divide

- If the question involves more than one "number" you must use a different letter(variable) for each term! (use. same variable for any consecutive numbers)

For example, to prove something about the sum of 2 odd numbers and an even number It would looklike this:

(2x+l)+ (2y + 1) + 2zOdd + Odd + Even

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Prove the following deductively:

1. Conjecture: Tlie sum of five consecudve integers is always divisible by five.

(Prove the conjecture by manipulating the expression to look like this 5(........)

-c+(x+ l)+ (x+ 2) + (x+ 3) + (x+4)

2. Prove that the sum of two even numbers and an odd number is always odd,(Prove the conjecture by manipulating the expression tD look like this 2(........ ) + 1

2x+ 2y + (2z+ 1)

3. Prove that the sum of any two odd integers is even.

4. Prove that the negative of any even Integer is even.

.

5. Prove that the difference between an even integer and an odd integer is odd.

6. Prove that the product of an odd integer and an even integer is always even.

7. Prove that the result of the number trick below is always the number you start with,

- Choose a number- Double it-Add 6- Double it again- Subtract 4

- Divide by 4- Subtract 2

8. Prove that whenever you square an odd integer, the result is odd.

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9. Prove that the sum of three consecutive integers is always a multiple of 3

10. Prove that the difference between the square of any odd integer and the integerItself is always an even integer.

11, Write any two digit number. Reverse the order of the digits and subtract it from theflrst number, (example: 81 - 18)

a) Investigate this for different starting numbers and make a conjecture

b) Prove that the difference is always a mulUple of 9.

Hint: A two digit number can be written as the sum of the digit in the tens place value, and the digit inthe unit place value -> a number xy Is (lOx + y) the reverse number yx would be (IQy + x)

^

FOM11 1.5 Proofs That Are Not VaUd

A single error in a deductive proof will make it invalid. Some common errors are:

. Dividing by zero.

. Circular reasoning.

. Confusing reasoning.

Example 1: i .

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Example 2:

Why is this proof invalid?

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Example 3: Isaac claims that -3=3.

Proof: Assume -3=3.

(-3)2=329=9

Therefore: -3 = 3.

Where did Isaac go wrong?

Assigmnent: pg. 42 #1, 3, 5, 6, 7, 10

^

FOM11 1.6 Reasoning to Solve Problems

Reminder: A conjecture is a conclusion based on examples.

We know that inductive reasoning can lead to a conjecture that may be proven by deductivereasoning. However, conjectures may be false, and can be disproven by a counterexample.

Example 1: Decide whether the process used is inductive or deductive reasoning:

a. Show the sum of two even numbers is even by using several examples.

b. No mathematician is boring. Ann is a mathematician. Therefore, Ann is notboring.

c. One counterexample proves that a conjecture is false.

d. You show why your statement makes sense.

e. You give evidence that your statement is true.

f. Six other examples to show fhat your conjecture is true.

g. What three corns have a value of $0.60?

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Example 2; Al, Bob, Cal, and Dave are on four sports teams.

. Each play on just one team.

. They play football, basketball, baseball, and hockey.

. Bob is a goalie.

. The tallest player pkiys basketball, and the shortest baseball.

. Cal is taller than Dave, but shorter than Al and Bob.

What sports does each play?

Example 3: Art, BU1, Cecil, and Don live in the same apartment. They are a manager, teacher,artist and musician. Art and Cecil watch TV with the teacher. Bill and Don go tothe hockey game with the manager. CecU jogs with the manager and teacher.Who is the manager?

Assigmnent: pg. 48 #1, 3, 5, 6, 8, 9, 10, 13, 16

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FOM11 Notes

A Logic Puzzle is a word problem which requires the use of Mathematical Deductive Reasoning to solve.Deductive Reasoning, is the process of working from one or more general statements to reach a logically

certain conclusion.

Example 1; The Boxes

There are three boxes. One is labeled "APPLES" another Is labeled "ORANGES". The last one is labeled

"APPLES AND ORANGES". You know that each is labeled Incorrectly. You may ask me to pick one fruit

from one box which you choose. How can you label the boxes correctly?

Example 2: Mary's mum has four children.

The first child is called April.

The second May.

The third June.

What is the name of the fourth child?

Often Logic Puales Include clues to help you find the solution, j

If a logic puzile seems too difficult, It Is often helpful to use a table to keep track of the clues.

Example 3: Danny is having a birthday party with 6 of his

family members. They are his grandmother, mother,

aunt, brother, father, and uncle. Their names in random

order are Ben, Lily, Jeff, Betty, Jane, and Luke. Look at

the clues to discover the names of Danny's family

members.

CLUES:1. Ben is not Dann/s uncle.

2. Danny's grandmother's name starts with B.

3. Luke is not Danny's brother.

4. Lily is not his aunt.

5. Danny's father's name is Jeff.

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Example 4: Three little pigs, who each lived In a different type of house, handed out treats forHalloween. Use the clues to figure out which pig lived In each house, and what type of treat each pighanded out.

CLUES;

1. Petey Pig did not hand out popcorn.2. Pippin Pig does not live in the wood house.

3. The pig that lives In the straw house, handed out popcorn.4. Petunia Pig handed out apples.

5. The pig who handed out chocolate, does not live in thebrick house.

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Drawing Pictures often helps to sort out clues.

Example S: Alex, Bret, Chris, Derek, Eddie, Fred, Greg, Harold, and John are nine students who live in athree storey building, with three rooms on each floor. A room in the West wing, one in the centre, andone In the East wing. If you look directly at the building, the left side Is West and the right side Is East.Each student is assigned exactly one room. Can you find where each of their rooms is:

CLUES:

1. Harold does not live on the bottom floor.

2. Fred lives directly above John and directly next to Bret (who lives in the West wing).3. Eddie lives In the East wing and one floor higher than Fred.4. Derek lives directly above Fred.

5. Greg lives directly above Chris.West Win Centre East Wing

3rt floor

2nd floor

1" floor

-^

Example 7: (The Big One)

During a recent music festival, four DJs entered the mixing contest. Each wore a number, either 1, 2, 3 or4 and their decks were different colours. Can you determine who came where, which number they woreand the colour of their deck?

CLUES:

1. DJ Skinf Lint came first, and only one DJ wore the same number as the position he finished in.

2. DJ Slam Dunk wore number 1.

3. The DJ who wore number 2 had a red deck and DJ Jam Jar didn't have a yellow deck.

4. The DJ who came last had a blue deck.

5. DJ Park'n Ride beat DJ Slam Dunk.

6. The DJ who wore number 1 had a green deck and the DJ who came second wore number 3.

DJ Skinf Lint

DJ Slam Dunk

DJ Jam Jar

DJ Park'n Ride

1

2

3

4

Red Deck

Yellow Deck

Blue Deck

Green Deck

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FOM 11 Name:

1. Four students named Al, Bjom, Carl, and Don each have a favourite sport. The sports areswimming, running, bowling, and golf. Use the clues to match each student with his sport.

CLUES:

1. The runner met Bjorn and the golfer for lunch.2. Neither Bjorn's sport nor Carl's sport requires a ball.3. Don Is In the same math class as the golfer's cousin.

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Bjorn

Carl

Don

2. You are running In a local race and you overtake the person in 2nd place. What place areyou now?

3. A farmer needs to take a fbx, a chicken, and a bag of grain across the river.The boat is tiny and can only carry one passenger at a time (the farmer is not a passenger).If he leaves the fox and the chicken alone together, the fox will eat the chicken.If he leaves the chicken and the grain alone together, the chicken will eat the grain.How can he bring all three safely across the river?

4. Susan, Irina, Traci, and Debbie are a pilot, a dentist, a doctor, and a writer. Use the cluesto match each person with her profession.

CLUES:

1. Debble Is a friend of the doctor.2. Susan and the writer sail with Trad.3. Neither Susan nor Irina has patients. s

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Susan

Irlna

Traci

Debbie

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5. At GWG school, I was chatting to some students and noticed a number of things

CLUES:

1. Jessica has auburn coloured hair2. The girl with black hair was wearing a green dress.3. Lucy Is not blonde and Lauren does not have brown hair.4. Chloe was wearing a blue dress.5. The blonde girl was not wearing red and Lauren was notwearing green.6. I can't remember which girl was wearing a yellow dress.

Can you determine the colours of the girl'sdresses and their hair?

Jessica

Lauren

LucyChloe

Auburn

Blonde

BlackBrown

chilli!u:^mt3<Bain

6. During a crazy weekend of palntball, four friends were having great fun. The paint camein blue, green, yellow and red. Colncldentally, the four friends had T-shirts In those samecolours.

CLUES:

1. Brenda used blue paint balls.2. The person In the green T-shirt used yellow paint balls.3. James was not wearing a red T-shirt.4. Dlane used green paint balls and wore a blue T-shirt.5. Simon was the only person who used paint which wasthe same colour as his T-shirt.

Can you tell which colour paint they each used and thecolour of their respective T-shirts?

Brenda

Dlane

James

Simon

Blue Shirt

Green Shirt

Yellow Shirt

Red Shirt

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7. Four people are running a marathon. Manuel is 30 m behind Margaret. Careen is 20 m behind Tom.Margaret is 75 m ahead of Careen. How far ahead of Tom is Manuel.

^

FOM11 1.7 Analyzing Puzzles And Games

Both inductive and deductive reasoning are useful for determining a strategy to solve a puzzle orwm a game.

Example 1: Use four 9's in a math equadon that equals 100.

Example 2: The following figure is made up of 12 sticks. Can you move just two sticks andcreate seven squares?

Example 3: Put the numbers 1 to 8 in each square so that each side adds to (he middle term.

12 13 14 15

^

Kakuro is an arithmetic puzzle in a grid. You must place the digits 1 to 9 into a grid of squaresso that each horizontal or vertical run of white squares adds up to the due printed either tothe left of or above the run.

No digit can be repeated within any single run. Runs end when you reach a non-white square.Every puzzle has a single uiiique solution and can be solved purely by logic - no guessing lgrequired.

Example 4: Conylete the following Kakuio puriles by filling in the grey squares.

14 7 33 9

121013 9 4 ,o12 5 7

15

21

/ 2 61479

4 23 1

16

10

7 92 1

14

15

21

33

13 9 4 12

10 5 7

531428 7B 2 6

716

3

9

7

2

10

9

1

Assigmnent: pg. 55 #4, 5, 6, 7, 9, 10, 1 1

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FOM11 Name;

Chapter 7 TaskHow many Brothers and Sisters?

/20

Rob, Yu, and Wynn challenged each other to create a number trick that endedwith the number of siblings they have. Their number tricks are given below.

Rob's Number Trick

Choose a number

Add 3Multiply by 2Subtract 2Multiply by 5Divide by 10Addssubtract the startingNumber

Wynn's Number Trick

Choose a number

Multiply by 4AddsDivide by 4subtract the startingnumber

Yu's Number Trick

. Choose a number

. Subtract 2

. square your answer

. Multiply by 0

. Divide by 5

A. Choose a number trick from the above list. Which student's number trick did you choose?

Test their number trick with three (3) different starting numbers. (3 marks)

B, Make a Conjecture about the number of siblings that student has (1 mark)

^

U} hat would your Number Trick be?C. How many siblings do you have? (number of brothers and/or sisters you have) (1 mark)

D. Create a number trick that always ends with the number of siblings you have.- Use at least three different operations (+, -, x, -, etc)- and at least four steps (not including "Choose a number^

(7 marks)

E. Use Deductive Reasoning to develop a proof that your number b-ick will work for ANY chosennumber. (Prove that your number trick will always work using Algebra) (8 marks)

Let» be the chosen number

°)^

FOM11 Ch. 1 Practice Test Name:Inductive and Deductive Reasoning

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

1. Justin gathered the following evidence.

17(22)-374 14(22) =308 36(22) =792 18(22) =396

Which conjecture, if any, is Justin most likely to make fi-om this evidence?

a. When you multiply a two-digit number by 22, the last and first digits of the product are (hedigits of the original number.

b. When you multiply a two-digit number by 22, the first and last digits of the product are thedigits of the original number.

c. When you multiply a two-digit number by 22, the first and last digits of the product form anumber that is twice the original number.

d. None of the above conjectures can be made from this evidence.

2. Which conjecture, if any, could you make about the sum of two odd integersand one even integer?

a. The sum will be an even integer.b. The sum will be an odd integer.c. The sum will be negative.d. It is not possible to make a conjecture.

3. Kerry created (he following tables to show patterns.

Multi lea of 3 12 15 18 21Sum of the Di "ts 3693In each case, the sum of fhe digits of a multiple of 3 is also a multiple of 3.

Multi les of 3-3= 9 18 27 36 45Sum of the M ts 9999fa each case, the sum of the digits of a multiple of 3 . 3, or 9, is also a multiple of 9.

Based on this evidence, which conjecture might Kerry make? Is (he conjecture valid?

a. The sum of the digits of a multiple of 2 . 3, or 6, is also a multiple of 6;yes, this conjecture is valid.

b. The sum of the digits of a multiple of 2 . 3, or 6, is also a multiple of6;no, this conjecture is not valid.

c. The sum of the digits of a multiple of 3 '3 . 3, or 27, is also a multiple of 27;no, fhis conjecture is not valid.

d. The sum of the digits of a multiple of 3 *3 . 3, or 27, is also a multiple of 27;yes, this conjecture is valid.

^

4. Sasha made the following conjecture:

All polygons with six equal sides are regular hexagons.

Which figure, ifeifter, is a counterexample to this conjecture? Explain.

B

a. Figure A is a counterexample, because all six. sides are equal and it is a regular hexagon.b. Figure B is a counterexaiq)le, because all su sides are equal and it is a regular hexagon.c. Figure B is a counterexample, because all six sides are equal and it is not a regular hexagon.d. Figure A is a counterexample, because all six sides an equal and it is not a regular hexagon.

5. Afhena made the following conjecture.

The sum of a multiple of 4 and a multiple of 8 mast be a multiple of 8.

Is the following equation a counterexample to this conjecture? Explain.

12+24=36

a. Yes, it is a counterexample, because 36 is a multiple of 8b. No, it is not a connterexanyle, because 36 is a multiple of 8.c. No, it is not a counterexaiq)le, because 36 is not a multiple of 8.d. Yes, it is a counterexanq)le, because 36 is not a multiple of 8.

6. All birds have backbones, Birds are the only animals that have feathers.Rosie is not a bird. What can be deduced about Rosie?

1. Rosie has a badcbone.2. Rosie does not have featfaers.

a. Neither Choice 1 nor Choice 2b. Choice 1 onlyc. Choice 1 and Choice 2d. Choice 2 only

^.

7. Which offlie following choices, if any, uses deductive reasoning to showthat tfae sum of two odd integers is even?

a. 3+5-8 and 7+5= 12b. (2x+l)+(2^+l)=2(x+y+l)c. 2x+2y+l=2(x+y)+ld. None of the above choices

8. What type of error, if any, occurs in the following deduction?

Saturday is not a school day for most students.Therefore, students should not wear red clothing on Saturdays.

a. a fiilse assumption or generalizationb. an error in reasoningc. an error in calculationd. There is no error in fhe dedaction.

9. Alison created a number trick in which she always ended with the original number. When Alisontried to prove her trick, however, it did not work. What type of error occurs in the proof?

n

2n2nn

n

n

+

+

+

+

4

4

8

4

1

Use n to

Add 4.Multi 1Add 4.DivideSubtract

entan number

b 2.

2.5.

a. a false assumption or generalizationb. an error in reasoningc. an error in calculation

d. There is no error in the proof.

10. Which type of reasoning does the following statement demonstrate?

Over the past 11 years, a tree has produced peaches each year.Therefore, the tree will produce peaches this year.

a. inductive reasoningb. deductive reasoningc. neither inductive nor deductive reasoning

11. Determine the unknown term in this pattern.

8, 17, 14, 23, _, 29, 26, 35

a. 21b. 22c. 20d. 25

^\

12. Which number should q>pear in the centre of Figure 4?

12463522

24 192 450

43426523Figure 1 Figure 2 Figure 3 Figure 4

a. 41b. 24c. 36d. 11

13. Which number should go in the grey square in this Sudoku puzzle?

1

753263 84

9

75 689

a,

b.c.

d.

Short Anawer

4 262 5

5

7

1

3

7

4

1

9

14. What conjecture could you make about the product of two odd integersand one even integer?

^

15. Make a conjecture about tfae relative size of the three figures.Check the validity of your conjecture.

^MII1-:1(^

16. Cheyenne told her litde brother, Joseph, that horses, cats, and dogs are all mammals.As a result, Joseph made (he following conjecture;

All animals wifh four legs are mammals.

Use a counterexample to show Joseph that his conjecture is not valid.

17. Kendra made the following conjecture:

The sum of any three integers is greater than each integer.

Do you agree or disagree? Briefly justify your decision with a counterexample if possible.

Ml

18. Try the following number trick with different numbers. Make a conjecture about the trick.

. Choose a number.

. Multiply by 3.

. Add 5.

. Multiply by 2.

. Subteact 10.

. Divide by 6.

19. In a Kaknro puzzle, you fill m the empty squares with the numbers fi-om 1 to 9.

. Each row of squares must add up to the circled number to the left of it.

. Each colmnn of squares must add up the circled number above it,

. A number cannot appear more than once m the same smn.

Conq>lete this Kakuro puzzle by filling in the grey squares.

14

15

21

33

13 9 4 12

10 5 7

531428 78 2 6

7 916

10

7 93 2 1

^

Problem

20. Are d and e equal? Prove your answer.

e

8

21. Akibh, Barbara, Cathy, and Donna all go to the same high school. One likes history the best, one likes math thebest, one likes computer science (he best, and one likes English the best. Use the statements below to determinewho likes conq)uter science tfae best.

. Aldlah and Cafhy eat lunch with the student who likes conqmter science.

. Donna likes history the best.

45

Ch. 1 Practice Test - Inductive and Deductive ReasoningAnswer Section

MULTIPLE CHOICE

I.

2.3.4.5.6.7.8.9.

10.11.12.13.

D

A

c

D

D

D

B

B

c

A

c

B

B

PROBLEM

SHORT ANSWER

14.For exanyle, (he product will be an even integer.

15.For example, I conjectured that the figures were different sizes,but when I measured them with a ruler, it turned out that flieywere the same size.

16-For example, lizards have four legs, and they are not mammals.

17.For example, disagree; -3 + (-4) + 2 = -5, and -5 is less thaneach integer.

IS.For example, the answer is always the original number.

19.

14

20. No, they are not equal. Angle d and the right angle are supplementary,BO d must also be a right angle.

If angle e is a right angle, then the side opposite to it will be ahypotenuse.

Using the Pytfaagorean theorem:6a+8a=3d+64

6'+8'-100

da+8a-10aBut the length of the side opposite e is 9 units, not 10, so e is not a right angle.

Therefore, angle a and angle b are not equal.

13 915 5

8

21 16 44 3

4

38

7

4

2

1

10

1

2

716

3

5

4

6

9

7

2

7

2

10

9

1

21.Barbara likes conqiuter sdence the best.

4^