Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1
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.
Make sure your name is in your book!
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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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What is Geometry
•Geometry is more than the study of shapes. It is the study of
truths.
•These truths are constant, no matter what the situation.
•Geometry uses reason (logic) to prove truths and to build
upon them to prove even more truths.
•The study of geometry is a study of how to think logically.
There are two types of logical strategies:
1. Inductive Reasoning
2. Deductive Reasoning
Chapter 1 Section 1
• Goals:
• Use inductive reasoning to
make a conjecture.
Vocabulary 1.1
•Inductive Reasoning •investigating using the observation of
patterns
•Conjecture
•A conclusion reached based upon
inductive observation
•Counterexample
•An example that shows the conjecture is
not correct
•Prime Number
•A Positive number with no factors other
than itself and 1.
(The smallest prime number is 2.)
Use 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, …
#
384
192
96
48
24
÷2
÷2
1. Write the sequence
2. What value is +,-,x, or ÷ each
time?
÷2
÷2
÷2
12
Each term is half the preceding term. So the next two terms are
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48 ÷ 2 = 24 and 24 ÷ 2 = 12.
Use 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.
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
= (1)2
= (1+2)2
= (1+2+3)2
= (1+2+3+4)2
= (1+2+3+4+5)2
The sum of the cubes equals the square of the sum of the
counting numbers.
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Use 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.
By applying the assumed pattern and then testing the result against the initial
directions, we have found a counterexample.
A counterexample applies the presumed pattern and gives a false result.
Only ONE counterexample is needed to prove a conjecture is false.
Conjecture: odd prime numbers are found by adding 2 to each odd prime.
Counterexample: 7 is odd prime, 7+2 = 9, 9 is not prime.
Result: Conjecture is false.
The fourth prime number is 11.
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When points on a circle are joined, they produce unique
regions within the circle:
Points
2
3
4
5
6
Regions
2
4
8
16
??
Will the # of regions always be twice as many as the
previous number?
6 points yields 30 regions.
30 is NOT 2x16!
Conjecture is false.
Use 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.
year
2000
2001
2002
2003
$$
8.00
9.50
11.00
12.50
Write the data in a table. Find a pattern.
+ 1.50
+ 1.50
+ 1.50
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.
Re-Cap
•Inductive Reasoning •Based upon observation of patterns
•Conjecture
•A conclusion reached based upon
inductive observation
•Counterexample
•An example that shows the conjecture is
not correct
•Prime Number
•Number with no factors other than itself
and 1.
Tips for Inductive Reasoning:
•Make a list
•Make a table when comparing two sets of numbers
•Look for simple numbers patterns
Additional Practice
GEOMETRY LESSON 1-1
Find a pattern for each sequence.
Use the pattern to show the next
two terms or figures.
Use the table and inductive reasoning.
1. 3, –6, 18, –72, 360
–2160; 15,120
2.
3. Find the sum of the first 10 counting numbers.
55
4. Find the sum of the first 1000
counting numbers.
500,500
Show that the conjecture is false by finding one
counterexample.
5. The sum of two prime numbers is an
even number.
Sample: 2+3=5, and 5 is not even
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