1 exploration: writing a conjecturedeductive reasoning uses facts, definitions, accepted properties,...

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5.2 For use with Exploration 5.2

Name _________________________________________________________ Date __________

Essential Question How can you use reasoning to solve problems?

Recall that conjecture is an unproven statement about a general mathematical concept that is based on observations.

Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern.

a.

b.

c.

Work with a partner. Use the Venn diagram to determine whether the statement is true or false. Justify your answer. Assume that no region of the Venn diagram is empty.

a. If an item has Property B, then it has Property A.

b. If an item has Property A, then it has Property B.

1 EXPLORATION: Writing a Conjecture

2 EXPLORATION: Using a Venn Diagram

1 2 3 4 5 6 7

Property C

Property A

Property B

151

Inductive and Deductive Reasoning


5.2

Name _________________________________________________________ Date _________

c. If an item has Property A, then it has Property C.

d. Some items that have Property A do not have Property B.

e. If an item has Property C, then it does not have Property B.

f. Some items have both Properties A and C.

g. Some items have both Properties B and C.

Work with a partner. Draw a Venn diagram that shows the relationship between different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, rhombuses, and kites. Then write several conditional statements that are shown in your diagram, such as “If a quadrilateral is a square, then it is a rectangle.”

Communicate Your Answer 4. How can you use reasoning to solve problems?

5. Give an example of how you used reasoning to solve a real-life problem.

3 EXPLORATION: Reasoning and Venn Diagrams

2 EXPLORATION: Using a Venn Diagram (continued)

Property C

Property A

Property B

152

Inductive and Deductive Reasoning (continued)


5.2 For use after Lesson 5.2

Name _________________________________________________________ Date __________

In your own words, write the meaning of each vocabulary term.

conjecture

inductive reasoning

counterexample

deductive reasoning

Core Concepts Inductive Reasoning Recall that conjecture is an unproven statement about a general mathematical concept that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.

Notes:

Counterexample To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding just one counterexample. A counterexample is a specific case for which the conjecture is false.

Notes:

153Copyright © Big Ideas Learning, LLC

All rights reserved.

5.2 Notetaking with Vocabulary For use after Lesson 5.2

Name _________________________________________________________ Date __________


conjecture

inductive reasoning

counterexample

deductive reasoning


Notes:


Notes:


5.2 For use after Lesson 5.2

Name _________________________________________________________ Date __________


conjecture

inductive reasoning

counterexample

deductive reasoning


Notes:


Notes:

153Copyright © Big Ideas Learning, LLC

All rights reserved.

5.2 Notetaking with Vocabulary For use after Lesson 5.2

Name _________________________________________________________ Date __________


conjecture

inductive reasoning

counterexample

deductive reasoning


Notes:


Notes:

154Copyright © Big Ideas Learning, LLCAll rights reserved.

5 .2 Notetaking with Vocabulary (continued)

Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture.

Laws of Logic Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion is also true.

Law of Syllogism

If hypothesis p, then conclusion q.

If hypothesis q, then conclusion r.

If hypothesis p, then conclusion r. then this statement is true.

Notes:

Extra Practice In Exercises 1–4, describe the pattern. Then write or draw the next two numbers, letters, or figures.

1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.

If these statements are true,

Practice A



Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture.

Laws of Logic Law of Detachment


Law of Syllogism



If hypothesis p, then conclusion r. then this statement is true.

Notes:


1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.


Practice A153

Practice


5 .2

Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoninga logical argument. This is different from inductive reasoningand patterns to form a conjecture.

Law of Syllogism



If hypothesis p, then conclusion r.

Notes:


1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.


Practice A


5 .2

Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoninga logical argument. This is different from inductive reasoningand patterns to form a conjecture.

Law of Syllogism




Notes:


1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.


Practice A



Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoninga logical argument. This is different from inductive reasoning


Law of Syllogism




Notes:


1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.

Practice A

Worked-Out Examples

Example #1

Make and test a conjecture about the given quantity.

Example #2

Use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true.

Copyright © Big Ideas Learning, LLC Integrated Mathematics I 433All rights reserved. Worked-Out Solutions

Chapter 9

23. If a fi gure is a rhombus, then the fi gure has two pairs of opposite sides that are parallel.

27. The law of logic used was the Law of Detachment.

28. The law of logic used was Law of Syllogism.

29. 1 + 3 = 4, 3 + 5 = 8, 7 + 9 = 16

Conjecture: The sum of two odd integers is an even integer. Let m and n be integers, then (2m + 1) and (2n + 1) are odd integers.

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

= 2(m + n + 1)

Any number multiplied by 2 is an even number. Therefore, the sum of two odd integers is an even integer. 30.

Let m and n be integers. Then (2m + 1) and (2n + 1) are odd integers.

(2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 Any number multiplied by 2 is an even number, and adding 1

will yield an odd number. Therefore, the product of two odd integers is an odd integer.

31. inductive reasoning; The conjecture is based on the assumption that a pattern, observed in specifi c cases, will continue.

32. deductive reasoning; The conclusion is based on mathematical defi nitions and properties.

33. deductive reasoning; Laws of nature and the Law of Syllogism were used to draw the conclusion.

34. inductive reasoning; The conjecture is based on the assumption that a pattern, observed in specifi c cases, will continue.

35. The Law of Detachment cannot be used because the hypothesis is not true; Sample answer: Using the Law of Detachment, because a square is a rectangle, you can conclude that a square has four sides.

36. The conjecture was based on a pattern in specifi c cases, not rules or laws about the general case; Using inductive reasoning, you can make a conjecture that you will arrive at school before your friend tomorrow.

37. Using inductive reasoning, you can make a conjecture that male tigers weigh more than female tigers because this was true in all of the specifi c cases listed in the table.

39. 1: 2 = 1(2) 2: 2 + 4 = 6 = 2(3) 3: 2 + 4 + 6 = 12 = 3(4) 4: 2 + 4 + 6 + 8 = 20 = 4(5) + 8 + 10 = 30 = 5(6)

rst n positive even integers is n(n + 1).

rst two) is the sum

= 76 47 + 76 = 123 76 + 123 = 199 The next three numbers in the pattern are 76, 123, and

199.

c. The Fibonacci sequence follows the same pattern:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

41. Argument 2: This argument uses the Law of Detachment to say that when the hypothesis is met, the conclusion is true.

42. Pattern 1: Multiply each term by 2.

1 — 4 ⋅ 2 = 1 — 2 , 1 — 2 ⋅ 2 = 2 — 2 = 1, 1 ⋅ 2 = 2, 2 ⋅ 2 = 4, 2 ⋅ 4 = 8 Pattern 2: Add 1 — 4 to the previous term.

1 — 4 + 1 — 4 =

2 — 4 =

1 — 2

1 — 2 + 1 — 4 =

2 — 4 +

1 — 4 =

3 — 4

3 — 4 + 1 — 4 =

4 — 4 = 1

1 + 1 — 4 = 4 — 4 +

1 — 4 =

5 — 4

5 — 4 + 1 — 4 =

6 — 4 =

3 — 2

Pattern 3: Multiply each term by half the reciprocal of the previous term.

1 — 4 ⋅ ( 1 — 2 ⋅ 4 ) = 1 — 4 ⋅ 2 = 1 — 2 1 — 2 ⋅ ( 1 — 2 ⋅ 2 ) = 1 — 2 ⋅ 2 = 1 — 2 1 — 2 ⋅ ( 1 — 2 ⋅ 2 ) = 1 — 2 ⋅ 2 = 1 — 2 1 — 2 ⋅ ( 1 — 2 ⋅ 2 ) = 1 — 2 ⋅ 2 = 1 — 2 43. The value of y is 2 more than three times the value of x;

y = 3x + 2; Sample answer: If x = 10, then y = 3(10) + 2 = 32; If x = 72, then y = 3(72) + 2 = 218.



Name _________________________________________________________ Date _________

Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoningand patterns to form a conjecture.

Law of Syllogism




Notes:


1. 20, 19, 17, 14, 10, 2. 2, 3, 5, 7, 11, − −

3. C, E, G, I, K, 4.


Practice A

432 Integrated Mathematics I Copyright © Big Ideas Learning, LLC Worked-Out Solutions All rights reserved.

Chapter 9

6. Sample answer: If x = 1 — 2 , then x2 = ( 1 — 2 )

2 = 1 — 4 .

1 — 4 is less than

1 — 2 ,

not greater.

7. Sample answer: The sum of −1 and −3 is −1 + (−3) = −4. The difference of −1 and −3 is −1 − (−3) = −1 + 3 = 2. Because −4 < 2, the sum is not greater than the difference.

8. ∠R is obtuse.

9. If you get an A on your math test, then you can watch your favorite actor.

10. Conjecture: The sum of a number and itself is 2 times the number. If n is the number, then n + n = 2n.

Inductively: 4 + 4 = 8, 10 + 10 = 20,45 + 45 = 90, n + n = 2n

Deductively: Let n be any number. By the Refl exive Property, n = n. If n is added to each side by the Addition Property, then n + n = n + n. Combining like terms yields 2n = 2n. Therefore, n + n = 2n, which means the sum of any number and itself is 2 times the number.

11. Deductive reasoning is used because the Law of Detachment is used to reach the conclusion.

9.2 Exercises (pp. 456–458)

Vocabulary and Core Concept Check

1. Because the prefi x counter- means “opposing,” a counterexample opposes the truth of the statement.

2. Inductive reasoning uses patterns to write a conjecture. Deductive reasoning uses facts, defi nitions, accepted properties, and the laws of logic to form a logical argument.

Monitoring Progress and Modeling with Mathematics

3. The absolute value of each number in the list is 1 greater than the absolute value of the previous number in the list, and the signs alternate from positive to negative. The next two numbers are: −6, 7.

4. The numbers are increasing by successive multiples of 2. The sequence is: 0 + 2 = 2, 2 + 4 = 6, 6 + 6 = 12,12 + 8 = 20, 20 + 10 = 30, 30 + 12 = 42, etc. So, the next two numbers are: 30, 42.

5. The pattern is the alphabet written backward. The next two letters are: U, T.

6. The letters represent the fi rst letter of each month of the year, and they are in the order of the months. The next two letters are: J, J.

7. The pattern is regular polygons having one more side than the previous polygon.

8. The pattern is the addition of 5 blocks to the previous fi gure. One block is added to each of the four ends of the base and one block is added on top. So, the next two fi gures will have 16 blocks and then 21 blocks.

integer.

Tests: 3 + 4 = 7, 6 + 13 = 19

11. The quotient of a number and its reciprocal is the square of

that number.

Tests: 10

— ( 1 — 10 )

= 10 — 1 ⋅ 10 — 1 = 100 = 102

( 2 — 3 )

— ( 3 — 2 )

= 2 — 3 ⋅ 2 — 3 =

4 — 9

= ( 2 — 3 ) 2

12. The quotient of two negative numbers is a positive rational

13. Sample answer: Let the two positive numbers be 1 — 2 and 1 — 6 .

The product is 1 — 2 ⋅ 1 — 6 = 1 — 12 . Because 1 — 12 < 1 — 2 and 1 — 12 < 1 — 6 , the product of two positive numbers is not always greater than

either number.

14. Sample answer: Let n = −1.

−1 + 1 — −1

= 0

0 > 1

15. Each angle could be 90°. Then neither are acute.

16. If line s intersects — MN at any point other than the midpoint, it is not a segment bisector.

17. You passed the class.

18. not possible; You may get to the movies by other means.

19. not possible; QRST could be a rectangle.

20. P is the midpoint of — LH .

21. not possible

22. If 1 — 2 a = 1 1 — 2 , then 5a = 15.

154

Practice (continued)

The product of any two even integers is an even integer.

Tests: 2 ⋅ 8 = 16, 22 ⋅ 20 = 440

The product of any two even integers.

The sum of two odd integers.


5.2

Name _________________________________________________________ Date __________

In Exercises 5 and 6, make and test a conjecture about the given quantity.

5. the sum of two negative integers

6. the product of three consecutive nonzero integers

In Exercises 7 and 8, find a counterexample to show that the conjecture is false.

7. If n is a rational number, then 2n is always less than n.

8. Line k intersects plane P at point Q on the plane. Plane P is perpendicular to line k.

In Exercises 9 and 10, use the Law of Detachment to determine what you can conclude from the given information, if possible.

9. If a triangle has equal side lengths, then each interior angle measure is 60 . ABC° has equal side lengths.

10. If a quadrilateral is a rhombus, then it has two pairs of opposite sides that are parallel. Quadrilateral PQRS has two pairs of opposite sides that are parallel.

In Exercises 11 and 12, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible.

11. If it does not rain, then I will walk to school. If I walk to school, then I will wear my walking shoes.

12. If 1, then 3 3.x x> >

If ( )23 3, then 3 9.x x> >

155

(continued)Practice


9.2 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, describe the pattern. Then write or draw the next two numbers, letters, or figures.

1. A, 26, B, 25, C, 24, … 2.

In Exercises 3 and 4, make and test a conjecture about the given quantity.

3. the sum of two absolute values 4. the product of a number and its square

5. Vertical angles are always complementary. Find a counterexample to show that the conjecture is false.

In Exercises 6 and 7, use the Law of Detachment to determine what you can conclude from the given information, if possible.

6. If you eat a healthy breakfast, then you will not be hungry until lunchtime. You are not hungry until lunchtime.

7. Adjacent angles share one common ray. and AOB DOB∠ ∠ are adjacent angles.

In Exercises 8 and 9, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible.

8. If a polygon has three sides, then it is a triangle. If triangle has two congruent sides, then it is an isosceles triangle.

9. If it is Tuesday, then you mow the grass. If you mow the grass, then you water the flowers.

In Exercises 10 and 11, decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning.

10. All mammals have hair. Cats are mammals. So, all cats have hair.

11. Each time you go to school you walk. You went to school today, so you walked.

12. Is it possible to have a series of true conditional statements that lead to a false conclusion? Explain.

13. The table shows the cost per pound of several varieties of organic and nonorganic produce at your local grocery store. What conjecture can you make about the relation between the cost of organic produce and the cost of nonorganic produce? Explain your reasoning.

Organic Nonorganic

Bananas $0.49 $0.29

Carrots $1.19 $0.89

Strawberries $3.99 $2.99

Practice B

156

1 exploration: writing a conjecturedeductive reasoning uses facts, definitions, accepted properties,...

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