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Foundations 11 Inductive and Deductive Reasoning

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

____ 1. Which conjecture, if any, could you make about the sum of two even integers and one odd integer?

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

____ 2. Jessica noticed a pattern when dividing these numbers by 4: 53, 93, 133.

Determine the pattern and make a conjecture.

a. When the cube of an odd number that is 1 more than a multiple of 4 is divided by 4, the decimal part of the result will be .75.

b. When the cube of an odd number that is 1 less than a multiple of 4 is divided by 4, the decimal part of the result will be .75.

c. When the cube of an odd number that is 1 more than a multiple of 4 is divided by 4, the decimal part of the result will be .25.

d. When the cube of an odd number that is 1 less than a multiple of 4 is divided by 4, the decimal part of the result will be .25.

____ 3. Jackie made the following conjecture.

The square of a number is always greater than the number.

Which choice, if either, is a counterexample to this conjecture?

1. 0.52 = 0.252. (–5)2 = 25

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

____ 4. Randolph made the following conjecture.

The sum of a multiple of 4 and a multiple of 8 must be a multiple of 2.

Which choice, if either, is a counterexample to this conjecture?

1. 4 + 8 = 122. 8 + 8 = 16

a. Choice 2 onlyb. Choice 1 and Choice 2

c. Choice 1 onlyd. Neither Choice 1 nor Choice 2

____ 5. All cats are mammals. All mammals are warm-blooded. Tashi is a cat. What can be deduced about Tashi?

1. Tashi is warm-blooded.2. Tashi is a mammal.

a. Choice 1 and Choice 2b. Neither Choice nor Choice 2c. Choice 1 onlyd. Choice 2 only

____ 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 backbone.2. Rosie does not have feathers.

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

____ 7. Which of the following choices, if any, uses inductive reasoning to show that the sum of three even integers is even?

a. 2x + 2y + 2z = 2(x + y + z)b. 2 + 4 + 6 = 12 and 4 + 6 + 8 = 18c. x + y + z = 2(x + y + z)d. None of the above choices

____ 8. Which of the following choices, if any, uses inductive reasoning to show that an odd number and an even number sum to an odd number?

a. 3 + 6 = 9 and 4 + 5 = 9b. 2x + 2y + 1 = 2(x + y + 1)c. (2x + 1) + 2y = 2(x + y) + 1d. None of the above choices

____ 9. Which of the following choices, if any, uses inductive reasoning to show that an odd number and an odd number sum to an even number?

a. 2x + 2y = 2(x + y)b. 2x + 2y + 1 = 2(x + y) + 1c. 6 + 6 = 12 and 4 + 6 = 10d. None of the above choices

____ 10. What type of error, if any, occurs in the following proof?

3 = 3 – 1

2(3) = 2(3 – 1)2(3) + 1 = 2(3 –1) + 1

6 + 1 = 4 + 17 = 5

a. a false assumption or generalizationb. an error in reasoningc. an error in calculationd. There is no error in the proof.

____ 11. What type of error, if any, occurs in the following deduction?

Diamond jewellery is expensive.Beyondé has expensive jewellery.Therefore, Beyondé has diamond jewellery.

a. a false assumption or generalizationb. an error in reasoningc. an error in calculationd. There is no error in the deduction.

____ 12. Determine the unknown term in this pattern.

2, 6, 18, 54, ____, 486, 1458

a. 108b. 162c. 216d. 196

____ 13. Determine the unknown term in this pattern.

17, 14, ____, 8, 5, 2, –1

a. 14b. 11c. 13d. 12

____ 14. Choose the next figure in this sequence.

Figure 1 Figure 2 Figure 3

a.

b.

c.

d.

____ 15. Which number should appear in the centre of Figure 4?

Figure 1 Figure 2 Figure 3 Figure 4

a. 15b. 240c. 120d. 6

____ 16. Which number should go in the grey square in this Sudoku puzzle?

a. 5b. 7

c. 1d. 3


a. 2b. 3c. 1d. 4


a. 6b. 8c. 2d. 4

____ 19. In a Kakuro puzzle, you fill in the empty squares with the numbers from 1 to 9.

• Each row of squares must add up to the circled number to the left of it.• Each column of squares must add up the circled number above it.• A number cannot appear more than once in the same sum.

Complete this Kakuro puzzle by filling in the grey squares.

a. 9, 7, 4, 1b. 1, 4, 8, 8c. 2, 3, 7, 9d. 1, 4, 7, 9

____ 20. Colin and Erynn are playing darts. Colin has a score of 75.To win, he must reduce his score to zero and have his last counting dart be a double.Which of the following scores on the dart board, in order, would give him the win?

a. 15, 20, double 20b. 0, 15, triple 20c. double 20, 20, 15d. 10, 20, double 15

Short Answer

21. Xavier claims that whenever you add an even integer to the square of an even integer, the result is an even number. Is his conjecture reasonable? Briefly justify your decision.

22. Perry works at a bakery shop in Regina. He goes to the farmer’s market and buys enough fruit to bake 20 saskatoon berry pies and 10 rhubarb pies. Which conjecture has Perry most likely made?

23. Cheyenne made the following conjecture:

As you travel farther south, toward the equator, the climate gets hotter.

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

24. Ryan made the following conjecture:

Every even number can be written as the sum of two consecutive integers.

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

25. Try the following calculator trick with different numbers. Make a conjecture about the trick.

• Start with your age.• Multiply it by 3.• Multiply it by 7.• Multiply it by 37.• Multiply it by 13.

26. What type of error occurs in the following deduction?Briefly justify your answer.

Given: Cheryl made blueberry muffins. Blueberry muffin recipes call for flour and milk. Anton is baking carrot muffins.Deduction: Anton’s recipe does not call for flour and milk.

27. What type of error occurs in the following deduction?Briefly justify your answer.

Jay-Qs likes to watch videos by Émail Zola. Therefore, the next video Jay-Qs watches will be by Émail Zola.

28. Does the following statement demonstrate inductive reasoning or deductive reasoning?

Every high school student in western Canada has to take math. You are a high school student in western Canada, therefore you have to take math.

29. Determine the unknown term in this pattern.

12, 7, 14, ____, 16, 11, 18

30. Franz and Gregor are playing darts. Gregor has a score of 52.To win, he must reduce his score to zero and have his last counting dart be a double. Give a strategy that Gregor might use to win.

Problem

31. Make a conjecture about the meaning of the symbols and colours in the flag of the territory of Nunavut, which is shown here. (You may need to obtain a colour version of this flag from a book or the Internet.)

32. The squares of two odd integers are added together. Develop a conjecture about whether the sum is odd or even. Provide evidence to support your conjecture.

33. Tyler made the following conjecture:

A polygon with four right angles must be a rectangle. Matthew disagreed with Tyler’s conjecture, however, because the following figure has four right angles, and it is not a rectangle.

How could Tyler’s conjecture be improved? Explain the changes you would make.

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

35. The divisibility rule for 9 is:

If the sum of the digits of a number is divisible by 9, then the original number is divisible by 9.

Prove this rule is true for numbers with four digits.

36. Terry wrote the following equations. The results led him to make the conjecture that the sum of three consecutive perfect cubes is not divisible by 3.

Let n be the first integer.n3 + (n + 1)3 + (n + 2)3 = n3 + (n3 + 3n2 + 3n + 1) + (n3 + 6n2 + 12n + 8)n2 + (n + 1)2 + (n + 2)2 = 3n3 + 9n2 + 15n + 10n2 + (n + 1)2 + (n + 2)2 = (3n3 + 9n2 + 15n + 9) + 1n2 + (n + 1)2 + (n + 2)2 = 3(n3 + 3n2 + 5n + 3) + 1

But in checking his work, Terry found the following counterexample, so he knew he had made an error.

1 + 8 + 27 = 36

Determine what Terry’s error is. What does the proof show?

37. Six people enjoyed a meal at an East Indian restaurant. The waiter brought a bill for $60. Each person at the table paid $10. Later the manager realized that the bill should have been for only $50, so she sent the waiter back to the table with $10. The waiter could not figure out how to divide $10 six ways, so he gave each person $1 and kept $4 for himself. Each of the six people paid $9 for the meal.9 • 6 = 54The waiter kept $4.54 + 4 = 58

What happened to the other two dollars?

38. Michale says she can prove that 4 = 0. Here is her proof.

Let both x and y be equal to 1.Step 1: x = yStep 2: x2 = y2

Step 3: x2 – y2 = 0Step 4: (x – y)(x + y) = 0

Step 5: = Step 6: 1(x + y) = 0Step 7: x + y = 0Step 8: 2 + 2 = 0Step 9: 4 = 0

Which step in Michale’s proof is invalid? Why is it invalid?

39. Solve this Sudoku puzzle using the numbers 1 to 9. Fill the grid so that each column, row, and block contains all the numbers. No number can be repeated within any column, row, or block.

40. Solve this Sudoku puzzle using the numbers 1 to 9. Fill the grid so that each column, row, and block contains all the numbers. No number can be repeated within any column, row, or block.

Foundations 11 Inductive and Deductive ReasoningAnswer Section

MULTIPLE CHOICE

1. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.1OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.TOP: conjectures and Inductive Reasoning KEY: conjecture| inductive reasoning

2. ANS: C PTS: 1 DIF: Grade 11 REF: Lesson 1.1OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.TOP: conjectures and Inductive Reasoning KEY: conjecture| inductive reasoning

3. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.3OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.TOP: Disproving conjectures: CounterexamplesKEY: conjecture| disproving conjectures| counterexamples

4. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.3OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.TOP: Disproving conjectures: CounterexamplesKEY: conjecture| disproving conjectures| counterexamples

5. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

6. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

7. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

8. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

9. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

10. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.5OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a given proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs| deductive reasoningKEY: valid proofs| invalid proofs| deductive reasoning

11. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.5OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a given proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs| deductive reasoningKEY: valid proofs| invalid proofs| deductive reasoning

12. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

13. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

14. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

15. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

16. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

17. ANS: C PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

18. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

19. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

20. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

SHORT ANSWER

21. ANS:

Yes, it is reasonable, because, for example, 22 + 4 = 8, 42 + 2 = 18, and 62 + 4 = 40.In each case, the result is an even number.

PTS: 1 DIF: Grade 11 REF: Lesson 1.1OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.TOP: conjectures and Inductive Reasoning KEY: conjecture| inductive reasoning

22. ANS:For example, people are more likely to buy saskatoon berry pies.


23. ANS:For example, agree: the closer you get to the equator, the warmer the climate gets.

PTS: 1 DIF: Grade 11 REF: Lesson 1.3OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.TOP: Disproving conjectures: CounterexamplesKEY: conjecture| disproving conjectures| counterexamples

24. ANS:For example, disagree: it is not possible to write “4” as the sum of two consecutive integers.

PTS: 1 DIF: Grade 11 REF: Lesson 1.3OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.TOP: Disproving conjectures: CounterexamplesKEY: conjecture| disproving conjectures| counterexamples

25. ANS:For example, the answer is always your age repeated three times.

PTS: 1 DIF: Grade 11 REF: Lesson 1.4OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).TOP: Proving conjectures| deductive reasoningKEY: conjecture| proving conjectures| reasoning| deductive reasoning

26. ANS:There is an error in reasoning: we are not told what Anton’s recipe requires, so we cannot say whether it calls for flour and milk or not.

PTS: 1 DIF: Grade 11 REF: Lesson 1.5OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a given proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs| deductive reasoningKEY: valid proofs| invalid proofs| deductive reasoning

27. ANS:There is an error in reasoning: Jay-Qs may well like to watch videos by other performers.


28. ANS:deductive reasoning

PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning. TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

29. ANS:9

PTS: 1 DIF: Grade 11 REF: Lesson 1.6OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning. TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

30. ANS:For example, score: triple 10, 2, double 10, in order.

PTS: 1 DIF: Grade 11 REF: Lesson 1.7OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

PROBLEM

31. ANS:For example, gold might represent mineral wealth, red might represent the country of Canada, white might represent snow, and blue might represent the sky. The inukshuk might represent the stone monuments that are built to guide travellers and to mark sacred places. The star might represent the North Star, Niqirtsuituq, a traditional guide for navigation.


32. ANS:The sum of the squares of two odd integers is even. For example, I began with different possibilities to see if there was a pattern.

52 + 52 = 25 + 25 or 50, which is even.32 + 32 = 9 + 9 or 18, which is even.52 + 72 = 25 + 49 or 74, which is even.

In each case, the sum is even, so I developed the conjecture that the sum will always be even. I used reasoning to look for support for this conjecture.

I know that the product of two odd numbers is always odd, so the square of an odd number will always be odd.

odd number(odd number) = odd number

So, both squares will be odd. I also know that the sum of two odd numbers is an even number.

odd number + odd number = even number

This evidence supports my conjecture.


33. ANS:For example: A polygon with more than four right angles cannot be a rectangle.

A polygon with fewer than four right triangles may not be a rectangle.

For example, the conjecture could be changed as follows:A polygon with exactly four right angles must be a rectangle.

PTS: 1 DIF: Grade 11 REF: Lesson 1.3OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture. TOP: Disproving conjectures: CounterexamplesKEY: conjecture| disproving conjectures| counterexamples

34. ANS:No, they are not equal. Angle d and the right angle are supplementary, so d must also be a right angle.

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

Using the Pythagorean theorem:

But 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.


35. ANS:

For example:Let abcd represent a four-digit number.Assume that the sum of a + b + c + d is divisible by 9.

abcd = 1000a + 100b + 10c + dabcd = 999a + 1a + 99b + 1b + 9c + 1c + dabcd = 999a + 99b + 9c + (a + b + c + d)abcd = 9(111a + 11b + c) + (a + b + c + d)

The term 9(111a + 11b + c) is a multiple of 9, so it is divisible by 9. Also, the sum of a + b + c + d is divisible by 9.The number abcd must be divisible by 9.


36. ANS:For example, in the second line, Terry collected like terms incorrectly. The ones value should be 9, not 10.

The proof should be:n3 + (n + 1)3 + (n + 2)3 = n3 + (n3 + 3n2 + 3n + 1) + (n3 + 6n2 + 12n + 8)n2 + (n + 1)2 + (n + 2)2 = 3n3 + 9n2 + 15n + 9n2 + (n + 1)2 + (n + 2)2 = 3(n3 + 3n2 + 5n + 3)

This proof shows that the sum of three consecutive perfect cubes is divisible by 3.


37. ANS:The question is misleading. Each person initially paid $10 for the meal, but got $1 back. So, each person paid $9 for the meal. The meal cost $50. The waiter kept $4.

6(9) – 4 = 50


38. ANS:Step 5 is invalid, because in this step Michale is dividing by 0 (since x = y), which makes the equation meaningless.

PTS: 1 DIF: Grade 11 REF: Lesson 1.5OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a given

proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs| deductive reasoningKEY: valid proofs| invalid proofs| deductive reasoning

39. ANS:


40. ANS:


Documents

mathcaddy.weebly.commathcaddy.weebly.com/uploads/2/1/4/4/21446970/... · Web viewOBJ:1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number