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Chapter 1
Tools of Geometry
Sec 1 – 1
Patterns and Inductive Reasoning
Objectives:
1) To use inductive reasonings to make
conjectures.
Inductive Reasoning – Is reasoning that is
based on patterns you observe.
– If you observe a pattern in a sequence you can
use inductive reasoning to find the next term.
Ex. 1: Find the next term in the sequence:
48 ___
96
Rule: x2
A) 3, 6, 12, 24, ___,
Rule: +1, +2, +3, +4, …
B) 1, 2, 4, 7, 11, 16, 22, ___,
29 ___
37
C)
Inductive Reasoning assumes that an
observed pattern will continue. This may
or may not be true.
– Ex: x = x • x
– This is true only for x = 0 and x = 1
Conjecture – A conclusion you reach
using inductive reasoning.
Ex. 2: Make a conjecture about the sum
of the first 30 odd numbers.
1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
1 + 3 + 5 +...+ 30
=1
= 12
=4
= 22
=9
= 32
= 16
= 42
= 25
= 52
= 900
= 302
To show that a conjecture is always true, you must
prove it.
To show that a conjecture is false, you have to find
only one example in which the conjecture is not true.
This case is called a counterexample.
A counterexample can be a drawing, a statement, or a
number.
Counter Example – To a conjecture is an
example for which the conjecture is incorrect.
Ex.1-3: The first 3 odd prime numbers are 3,
5, 7. Make a conjecture about the 4th.
11
– 3, 5, 7, ___
– One would think that the rule is add 2, but that
gives us 9 for the fourth prime number.
Is that true?
– What is the next odd prime number?
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.
1-1
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.
Homework
Page 6
1-53 odd and 2,26,40,44,48
GEOMETRY LESSON 1-1
Pages 6–9 Exercises
1. 80, 160
12. 1 , 1
5 6
2. 33,333; 333,333 13. James, John
3. –3, 4
14. Elizabeth, Louisa
4.
15. Andrew, Ulysses
1 1
16 , 32
5. 3, 0
16. Gemini, Cancer
6.
17.
1
1, 3
7. N, T
8. J, J
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.
18.
9. 720, 5040
10. 64, 128
1 1
11. 36 , 49
1-
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.
1-1
34. 201, 202
35. 63, 127
36. 31 , 63
32 64
37. J, S
38. CA, CO
39. B, C
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
1-1
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.
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
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.
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.
1-1
58. [2] a. 25, 36, 49
b. n2
[1] one part correct
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