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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
(For help, go the Skills Handbook, page 715.)
Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . .
Some are even and some are odd.
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.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Solutions
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.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Definitions
Inductive Reasoning
-Reasoning based on patterns you observe
-Creating logical generalizations
-Reasoning from detailed facts to general principles
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Find a pattern for the sequence. Use the pattern to
show the next two terms in the sequence.
384, 192, 96, 48, …
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Find a pattern for the sequence. Use the pattern to
show the next two terms in the sequence.
384, 192, 96, 48, …
Each term is half the preceding term. So the next two terms are
48 ÷ 2 = 24 and 24 ÷ 2 = 12.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Definitions
Conjecture
A conclusion you reach using inductive reasoning
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
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
13 + 23
13 + 23 + 33
13 + 23 + 33 + 43
13 + 23 + 33 + 43 + 53
=1
=9
= 36
= 100
= 225
= 12
= 32
= 62
= 102
= 152
= 12
= (1 + 2)2
= (1 + 2 + 3)2
= (1 + 2 + 3 + 4)2
= (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.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
(continued)
The sum of the first three cubes equals the square of the sum of the
first three counting numbers.
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.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Definitions
Counter Example
A single example that proves a conjecture to be false.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
The first three odd prime numbers are 3, 5, and 7. Make and
test a conjecture about the fourth odd prime number.
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.
The fourth prime number is 11.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 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
2001
2002
$8.00 $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.
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
Pages 6–9 Exercises
1. 80, 160
12. 1 , 1
2. 33,333; 333,333
13. James, John
3. –3, 4
14. Elizabeth, Louisa
4.
1, 1
16 32
5 6
15. Andrew, Ulysses
5. 3, 0
16. Gemini, Cancer
6. 1, 1
17.
3
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.
7. N, T
8. J, J
19. The sum of the first 6 pos.
even numbers is
6 • 7, or 42.
18.
9. 720, 5040
10. 64, 128
11. 1 , 1
36 49
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Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
22. The sum of the first
100 odd numbers is
1002, or 10,000.
28. 1 ÷ 1 = 3 and 3 is
2
3
2
2
improper.
29. 75°F
25–28. Answers may vary.
Samples are given.
25. 8 + (–5 = 3) and 3 >/ 8
26.
1 • 1 > 1 and 1 • 1 > 1
/
/ 2
3 2
3 2 3
27. –6 – (–4) < –6 and
–6 – (–4) < –4
32. 10, 13
33. 0.0001, 0.00001
23. 555,555,555
24. 123,454,321
31. 31, 43
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.
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34. 201, 202
35. 63, 127
36. 31 , 63
32 64
37. J, S
38. CA, CO
39. B, C
Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
40. Answers may vary.
Sample: In Exercise
31, each number
increases by increasing
multiples of 2. In Exercise
33, to get the next term,
divide by 10.
42.
43.
44.
41.
45.
You would get a third line
between and parallel to
the first two lines.
46. 102 cm
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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.
Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
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, . . .
50. His conjecture is
52.
probably false
because most
53.
people’s growth
slows by 18 until
they stop growing
somewhere between
18 and 22 years.
51. a.
b. H and I
c. a circle
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21, 34, 55
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.
Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
54. Answers may vary.
Sample:
55. (continued)
d.
100 + 99 + 98 + … + 3 + 2 + 1
1 + 2 + 3 + … + 98 + 99 + 100
101 + 101 + 101 + … + 101 + 101 + 101
56. B
The sum of the first 100 numbers is
57. I
100 • 101 , or 5050.
2
The sum of the first n numbers is n(n+1) .
2
55. a. 1, 3, 6, 10, 15, 21
b. 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.
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58. [2] a. 25, 36, 49
b. n2
[1] one part correct
Patterns and Inductive Reasoning
GEOMETRY LESSON 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
59. (continued)
[3] minor error in
explanation
60-67.
[2] incorrect description
in part (a)
[1] correct products for
(151)(11), (161)(11),
and (181)(11)
68. B
b. 1991
69. N
c. No; (191)(11) = 2101
70. G
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