1. 1 1-1 Patterns and Inductive Reasoning Objectives: Define: –Conjectures –Inductive reasoning –Counterexamples Make conjectures based on inductive reasoning Find counterexamples to prove that a conjecture is false

2. 2 Definition of Conjecture, Inductive Reasoning Conjecture – An unproven guess based on observations Inductive Reasoning – Reasoning that is based on patterns you observe Example 0: Sketch the next figure in the pattern ?

3. 3 Solution to Example 0 Problem solving methodology: Look for a pattern. Make a conjecture based on the pattern. Verify that the conjecture is true in all cases.

4. 4 Quick Check 1a Write the next two terms in the sequence: 1 2 4 7 11 16 22 ? ?? 1 + 1 = 2 1 x 2 = 2 2 + 2 = 4 2 x 2 = 4 4 + 3 = 7 4 x 2 = 8 (NO) 7 + 4 = 11 11 + 5 = 16 16 + 6 = 22 22 + 7 = 29 29 + 8 = 37 Step 1. Look for a pattern. Step 2. Make a conjecture. Step 3. Verify that the conjecture is true for all cases

5. 5 Example 2. Using Inductive Reasoning Make a conjecture about the sum of the first 30 odd numbers. 1 = 1 2 1 + 3 = 4 = 2 2 1 + 3 + 5 = 9 = 3 2 1 + 3 + 5 + 7 = 16 = 4 2 Step 1. Look for a pattern. Step 2. Make a conjecture. 1 + 3 + 5 + 7 +……+ 59 = 30 2 = 30 x 30 = 900

6. 6 Example 3a. Finding a Counterexample to Disprove a Conjecture Not all conjectures are true. You can prove that a conjecture is false by finding one counterexample, which is an example that contradicts the conjecture. Conjecture a.: The square of any number is greater than the original number. 2 2 = 2 x 2 = 4, 3 2 = 3 x 3 = 9, 1 2 = 1 x 1 = 1 (Counterexample; proves the conjecture is false)

7. 7 Example 3b. Finding a Counterexample Conjecture b. You can connect any three points to form a triangle. Counterexample

8. 8 Unproven Conjectures Not every conjecture in mathematics has been proven. http://en.wikipedia.org/wiki/Goldbach’s_conjecture

9. 9 Learning Check and Summary What type of reasoning is based on patterns you observe? An unproven guess you reach using inductive reasoning is called a ? To prove that a conjecture is false, what do you try to find?

10. 10 Assignment Class Work –Workbook: Daily Notetaking Guide 1-1 p. 2; Omit Ex. 2, 3 Practice 1-1 p. 249, Omit 7-12, 19, 20 Homework –Workbook: Daily Notetaking Guide 1-3 p. 10 Daily Notetaking Guide 1-4 p. 13