(For help, go the Skills Handbook, page 715.) GEOMETRY LESSON 1-1 1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers.

(For help, go the Skills Handbook, page 715.)

GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1

1. Make a list of the positive even numbers.

2. Make a list of the positive odd numbers.

3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . .

4. Which do you think describes the square of any odd number? It is odd. It is even.

Patterns and Inductive ReasoningPatterns and Inductive Reasoning

1-1

Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . .Some are even and some are odd.

1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . .

2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . .

3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64; 92 = (9)(9) = 81; 102 = (10)(10) = 100

4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd.

Solutions



1-1

-Reasoning based on patterns you observe

-Creating logical generalizations

-Reasoning from detailed facts to general principles

Inductive Reasoning



1-1

Definitions

Find a pattern for the sequence. Use the pattern to

show the next two terms in the sequence.

384, 192, 96, 48, …



1-1

Each term is half the preceding term. So the next two terms are

48 ÷ 2 = 24 and 24 ÷ 2 = 12.

Find a pattern for the sequence. Use the pattern to

show the next two terms in the sequence.

384, 192, 96, 48, …



1-1

A conclusion you reach using inductive reasoning

Conjecture



1-1

Definitions

Make a conjecture about the sum of the cubes of the first 25

counting numbers.

Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern.

13 = 1 = 12 = 12

13 + 23 = 9 = 32 = (1 + 2)2

13 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2

13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2

13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2

The sum of the first two cubes equals the square of the sum of the first two counting numbers.



1-1

This pattern continues for the fourth and fifth rows of the table.13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2

13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2

So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 + 2 + 3 + … + 25)2.

The sum of the first three cubes equals the square of the sum of the first three counting numbers.

(continued)



1-1

A single example that proves a conjecture to be false.

Counter Example



1-1

Definitions

The first three odd prime numbers are 3, 5, and 7. Make and

test a conjecture about the fourth odd prime number.

The fourth prime number is 11.

One pattern of the sequence is that each term equals the preceding term plus 2.

So a possible conjecture is that the fourth prime number is 7 + 2 = 9.

However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false.



1-1

The price of overnight shipping was $8.00 in 2000, $9.50 in

2001, and $11.00 in 2002. Make a conjecture about the price in 2003.

Write the data in a table. Find a pattern.

2000

$8.00

2001 2002

$9.50 $11.00

Each year the price increased by $1.50.

A possible conjecture is that the price in 2003 will increase by $1.50.

If so, the price in 2003 would be $11.00 + $1.50 = $12.50.



1-1



Pages 6–9 Exercises

1. 80, 160

2. 33,333; 333,333

3. –3, 4

4. ,

5. 3, 0

6. 1,

7. N, T

8. J, J

9. 720, 5040

10. 64, 128

11. ,

1 16

1 32

1 36

1 49

12. ,

13. James, John

14. Elizabeth, Louisa

15. Andrew, Ulysses

16. Gemini, Cancer

17.

18.

15

16

19. The sum of the first 6 pos. even numbers is 6 • 7, or 42.

20. The sum of the first 30 pos. even numbers is 30 • 31, or 930.

21. The sum of the first 100 pos. even numbers is 100 • 101, or 10,100.

13

1-1

28. ÷ = and is

improper.

29. 75°F

30. 40 push-ups;

answers may vary.

Sample: Not very

confident, Dino may

reach a limit to the

number of push-ups

he can do in his

allotted time for

exercises.

31. 31, 43

32. 10, 13

33. 0.0001, 0.00001

34. 201, 202

35. 63, 127

36. ,

37. J, S

38. CA, CO

39. B, C

13

12

13

13

12

12/ /

/

12

13

32

32

3132

6364



1-1

22. The sum of the first 100 odd numbers is 1002, or 10,000.

23. 555,555,555

24. 123,454,321

25–28. Answers may vary. Samples are given.

25. 8 + (–5 = 3) and 3 > 8

26. • > and • >

27. –6 – (–4) < –6 and

–6 – (–4) < –4

40. Answers may vary. Sample: In Exercise 31, each number increases by increasingmultiples of 2. In Exercise 33, to get the next term, divide by 10.

41.

You would get a third line between and parallel to the first two lines.

42.

43.

44.

45.

46. 102 cm

47. Answers may vary. Samples are given.a. Women may soon outrun

men in running competitions.b. The conclusion was based

on continuing the trend shown in past records.

c. The conclusions are based on fairly recent records for women, and those rates of improvement may not continue. The conclusion about the marathon is most suspect because records date only from 1955.



1-1

50. His conjecture is probably false because most people’s growth slows by 18 untilthey stop growing somewhere between 18 and 22 years.

51. a.

b. H and I

c. a circle

48. a.

b. about 12,000 radio stations in 2010

c. Answers may vary. Sample: Confident; the pattern has held for several decades.

49. Answers may vary. Sample: 1, 3, 9, 27, 81, . . .1, 3, 5, 7, 9, . . .

52. 21, 34, 55

53. a. Leap years are years that are divisible by 4.

b. 2020, 2100, and 2400

c. Leap years are years divisible by 4, except the final year of a century which must be divisible by 400. So, 2100 will not be a leap year, but 2400 will be.



1-1

54. Answers may vary.Sample:

100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101

The sum of the first 100 numbers is

, or 5050.

The sum of the first n numbers is .

55. a. 1, 3, 6, 10, 15, 21b. They are the same.c. The diagram shows the product of n

and n + 1 divided by 2 when n = 3. The result is 6.

100 • 1012

n(n+1)2

55. (continued)d.

56. B

57. I

58. [2] a. 25, 36, 49

b. n2

[1] one part correct



1-1

59. [4] a. The product of 11 and a three-digit number that begins

and ends in 1 is a four-digit number

that begins and ends in 1 and has middle digits that are each one greater than the middle digit of the three-digit number.

(151)(11) = 1661(161)(11) = 1771

b. 1991

c. No; (191)(11) = 2101

59. (continued)[3] minor error in

explanation

[2] incorrect description in part (a)

[1] correct products for (151)(11), (161)(11), and (181)(11)

60-67.

68. B

69. N

70. G



1-1

(For help, go the Skills Handbook, page 715.) GEOMETRY LESSON 1-1 1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers.

Documents