1. 1.1 Patterns and Inductive Reasoning
Geometry
Mrs. Blanco

2. Objectives: Find and describe patterns.
Use inductive reasoning to make real-life conjectures.

3. Ex. 1: Describing a Visual Pattern
1)Sketch the next figure in the pattern. 1 2 3 4 5

4. Ex. 1: Describing a Visual Pattern - Solution
The sixth figure in the pattern has 6 squares in the bottom row. 5 6

5. Ex. 1: Cont…
2) Find the distance around each figure. Organize your results in a table. 3) Describe the patterns in the distances. 4) Predict the distance around the twentieth figure in this pattern.

6. Ex. 2: Describe a pattern in the sequence of numbers
Ex. 2: Describe a pattern in the sequence of numbers. Predict the next number.
1, 4, 16, 64, …
b. 10, 5, 2.5, 1.25, …
c , 16, 4, 2, …
d. 48, 16, , , …
256 (multiply previous number by 4)
0.625 (divide previous number by 2)
(take the square root of previous number)
16/27 (divide previous number by 3)

7. Using Inductive Reasoning
Much of the reasoning you need in geometry consists of 3 stages:
Look for a Pattern: Use diagrams and tables to help
Make a Conjecture. Unproven statement that is based on observations
Verify the conjecture—make sure that conjecture is true in all cases.

8. Ex. 3: Complete the Conjecture
Conjecture: The sum of the first n odd positive integers is ______.
First odd positive integer:
1 = 12
1 + 3 = 4 = 22
= 9 = 32
= 16 = 42
The sum of the first n odd positive integers is n2.

9. Note:
To prove that a conjecture is true, you need to prove it is true in all cases.
To prove that a conjecture is false, you need to provide a single counterexample.

10. Ex. 4: Finding a counterexample
Show the conjecture is false by finding a counterexample.
For all real numbers x, the expressions x2 is greater than or equal to x.
The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to In fact, any number between 0 and 1 is a counterexample.

11. Another: Ex. 4: Finding a counterexample
Show the conjecture is false by finding a counterexample. 2) The difference of two positive numbers is always positive.
The conjecture is false. Here is a counterexample: =-1

12. Page 6 #1-11 and Page 8 #34-37