1. Honors Geometry Section 1.0 Patterns and Inductive Reasoning

2. Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. Much of the reasoning in geometry consists of three steps.

3. 1. 2. 3. Recognize a pattern. Make a conjecture about the pattern.
Recognize a pattern.
Make a conjecture about the pattern.
A conjecture is an educated guess based on past observations.
Prove the conjecture.

4. Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words. 2, 6, 18, 54…

5. Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words. 1, 3, 5, 7, 9… 1, 1.1, 21.1, 21.12, , …

6. A sequence can be specified by an equation or “rule”
A sequence can be specified by an equation or “rule”. For the first example (2,6,18,54,…), the sequence can be specified by the rule where n = 1,2,3, etc. corresponding to the 1st term, 2nd term, 3rd term , etc.

7. Example 2: Write a rule for the nth term for the 2nd and 3rd sequences in example 1. 1, 3, 5, 7, 9

8. Reasoning based on past observations is called inductive reasoning.
Keep in mind that inductive reasoning does not guarantee a correct conclusion.

9. Later in the course, we will prove a conjecture is true using deductive reasoning. To prove a conjecture is false, you need to show a single example where the conjecture is false. This single example is called a
counterexample.

10. Example 2: Show the conjecture is false
Example 2: Show the conjecture is false The product of two positive numbers is always greater than the larger number.     If m is an integer*, then m2 > 0.  

11. You should know the following sets of numbers: whole numbers: integers: rational numbers: irrational numbers: