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Pre-AP Geometry Worksheet 1.7

Pre-AP Geometry Name: ___________________ Worksheet 1.7: Inductive Reasoning Date: _______ Period: ____ 1) The pattern of dots shown below continues infinitely, with more dots being added at each step.

First Step Second Step Third Step Which expression can be used to determine the number of dots in the nth step? A 2n B n(n + 2) C n(n + 1) D 2( n + 1)

2) The figure at the right shows a partial view of Pascal’s triangle. Which row of numbers best represents the seventh row in Pascal’s triangle? A 1 5 10 10 5 1

B 1 6 15 20 15 6 1

C 1 7 21 35 35 21 7 1

D 1 8 28 56 70 56 28 9 1

3) A pattern exists as a result of raising i, an imaginary number, to n, an integer greater than or equal to 1. Based on the table, which of the following best represents i raised to the 16th power? A 1− B -1 C -i D 1

Factor and solve. Show all work. 4) 2x2 + 5x + 3 = 0 5) 3x2 – 7x + 2 = 0

Pascal’s Triangle 1

1 1 1 2 1

1 3 3 1 1 4 6 4 1

Row 1: Row 2: Row 3: Row 4: Row 5:

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Pre-AP Geometry Worksheet 1.7

6) 2x2 - 7x + 5 = 0 7) 5x2 – 3x = 2 Solve. Show all algebra work. 8) The base of a triangle is 3 more than twice the height of the triangle. The area of the triangle is 10 square meters. Find the length of the base and the height. 9) The width of a rectangle is given as 3x – 5 and the length of the same rectangle is given as 2x – 3. If the area of the rectangle is 35 square meters, find x, the length, and the width. 10) Given: m�1 = 3x2; m�2 = 10x + 48

Find: x, m�1, and m�2

Problem 5

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

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Verifying a Conjecture Is False Using a Counterexample

What is a counterexample for each conjecture?

A If the name of a month starts with the letter J, it is a summer month.

Counterexample: January starts with J, and it is a winter month.

B You can connect any three points to form a triangle.

Counterexample: If the three points lie on a line, you cannot form a triangle.

C When you multiply a number by 2, the product is greater than the original number.

The conjecture is true for positive numbers, but it is false for negative numbers and zero.

Counterexample: -4 # 2 = -8 and -8 w -4.

These three pointssupport the conjecture . . .

. . . but these three points are a counterexample to the conjecture.

Find a pattern for each sequence. Use the pattern to show the next two terms.

1. 5, 10, 20, 40, c 2. 1, 4, 9, 16, 25, c 3. 1, -1, 2, -2, 3, c 4. 1, 12, 14, 18, c 5. 1, 12, 13, 14, c 6. 15, 12, 9, 6, c 7. O, T, T, F, F, S, S, E, c 8. J, F, M, A, M, c 9. 1, 2, 6, 24, 120, c 10. Washington, Adams, Jefferson, c 11. dollar coin, half dollar, quarter, c 12. AL, AK, AZ, AR, CA, c 13. Aquarius, Pisces, Aries, Taurus, c 14. 15.

16. Draw the next figure in the sequence. Make sure you think about color and shape.

17. Find the perimeter when 100 triangles are put together in the pattern shown. Assume that all triangle sides are 1 cm long.

18. Analyze Mathematical Relationships (1)(F) Below are 15 points. Most of the points fit a pattern. Which does not? Explain.

A(6, -2) B(6, 5) C(8, 0) D(8, 7) E(10, 2) F(10, 6) G(11, 4) H(12, 3)

I(4, 0) J(7, 6) K(5, 6) L(4, 7) M(2, 2) N(1, 4) O(2, 6)

Scan page for a Virtual Nerd™ tutorial video.

What numbers should you guess and check?Try positive numbers, negative numbers, fractions, and special cases like zero.

46 Lesson 2-1 Patterns and Conjectures

Use the sequence and inductive reasoning to make a conjecture.

19. What is the color of the thirtieth figure? 20. What is the shape of the fortieth figure?

Apply Mathematics (1)(A) Use inductive reasoning to make a prediction about the weather.

21. The speed at which a cricket chirps 22. Lightning travels much faster than is affected by the temperature. If you thunder, so you see lightning before

hear 20 cricket chirps in 14 s, what you hear thunder. If you count 5 s is the temperature? between the lightning and the thunder, how far away is the storm?

Find one counterexample to show that each conjecture is false.

23. ∠1 and ∠2 are supplementary, so one of the angles is acute.

24. △ABC is a right triangle, so ∠A measures 90.

25. The sum of two numbers is greater than either number.

26. The difference of two integers is less than either integer.

27. Apply Mathematics (1)(A) Look for a pattern in the Chinese number system.

a. What is the Chinese name for the numbers 43, 67, and 84?

b. Explain Mathematical Ideas (1)(G) Do you think that the Chinese number system is base 10? Explain.

28. Display Mathematical Ideas (1)(G) Write two different number sequences that begin with the same two numbers.

STEM

Temperature (!F)Number of Chirpsper 14 Seconds

5

10

15

45

55

65

Dis

tanc

e of

Sto

rm (m

i)

Seconds Between Lightning and Thunder

6

4

2

010 20 30 400

Chinese Number System

ChineseWord

yı

èr

san

wu

lìu

qı

1

2

3

4

5

6

7

NumberChinese

Word

shí

shí-yı

shí-èr

èr-shí

èr-shí-yı

san-shí

10

11

12

20

21

Number

8 30san-shí-yı31

ba9 jıu

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sì

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¯

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