1. 2.2 Patterns & Inductive Reasoning
Geometry
2.2
Patterns & Inductive Reasoning
Essential Question: How can you prove a conjecture false?

2. Geometry 2.2 Patterns and Inductive Reasoning
Topic/Objectives:
Find and describe patterns.
Use inductive reasoning to make conjectures.
To understand “What is a conjecture”
Know what a counterexample is.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

3. Geometry 2.2 Patterns and Inductive Reasoning
Find a Pattern
What’s the next number?
1) 17, 15, 12, 8, … 3
2) 1, -2, 4, -8, … 16
3) 48, 12, 3, …
3/4
Now check your answers.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

4. Geometry 2.2 Patterns and Inductive Reasoning
You have just done
Inductive
Reasoning
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

5. Geometry 2.2 Patterns and Inductive Reasoning
Look for a pattern
Make a conjecture
Conjecture: An unproven statement based on observations.
Verify the conjecture
Inductive Reasoning is making conjectures based on patterns
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

6. Geometry 2.2 Patterns and Inductive Reasoning
Making a Conjecture
The sum of two odd numbers is _____.
Think of some odd numbers:
Now think of the sums of some of them:
even
See any Pattern?
= 26
3 + 5 = 8
even
7 + 9 = 16
even
even
= 24
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

7. Geometry 2.2 Patterns and Inductive Reasoning
Our Conjecture:
The sum of
any two odd
numbers is
EVEN.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

8. Your turn. Study the examples and make a conjecture.
1 = 1
1 + 3 = 4
= 9
= 16
The sum of the first n odd integers is____. 12 22 32 42 n2
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

9. Geometry 2.2 Patterns and Inductive Reasoning
To prove a conjecture is TRUE, you must be able to prove it for all cases.
Proof
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

10. To prove a conjecture is false, you need only one
counterexample.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

11. Geometry 2.2 Patterns and Inductive Reasoning
Counterexample
An example that proves a statement to be false.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

12. Geometry 2.2 Patterns and Inductive Reasoning
Example
Conjecture:
The sum of any two integers is positive.
BUT…
1 + 5 = 6
= 7
0 + 1 = 1
= 88
= -3 which is not positive.
So, we found a counterexample, and the conjecture is false.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

13. Your turn. Find a counterexample to the statement.
For all real numbers, n2 > n.
Counterexamples:
12 is not > 1
02 is not > 0
0.52 is not > 0.5
9 > 3
25 > 5
100 > 10
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

14. Geometry 2.2 Patterns and Inductive Reasoning
Unproven Conjectures
Conjectures that have not been proven true, but no counterexample can be found.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

15. Goldbach’s Conjecture
Every even number greater than 2 can be written as the sum of two prime numbers.
Prime number: only divisible by 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19…
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

16. Goldbach’s Conjecture
Every even number greater than 2 can be written as the sum of two prime numbers.
6 = 3 + 3
8 = 3 + 5
18 =
20 =
3 + 17
24 =
5 + 19
5 + 13
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

17. Geometry 2.2 Patterns and Inductive Reasoning
Christian Goldbach (1690 – 1764) made his conjecture almost 300 years ago, and it still hasn’t been proven, despite efforts by some of the greatest mathematicians of all time. Many people today suspect that it is true, but that it cannot, and never will, be proven true.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

18. Geometry 2.2 Patterns and Inductive Reasoning
Important!
Just because something is true for several specific cases does not prove that it is true in general.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning

19. Geometry 2.2 Patterns and Inductive Reasoning
What have you learned?
Take a moment, and quietly write down one thing you have learned today that you didn’t know before.
April 6, 2019
Geometry 2.2 Patterns and Inductive Reasoning