1. Lesson 1-1: Patterns & Inductive Reasoning 8/27/2009

2. Vocabulary TermDefinitionOwn Words Inductive Reasoning Conjecture Counterexample Reasoning based on observed patterns. A conclusion based on inductive reasoning. An example that shows a conjecture is not true (false). (Copy completed table in notes.)

3. – Sunday was sunny and hot, Monday was sunny and hot, Tuesday was sunny and hot, what will today be? Conjecture: every day will be sunny and hot – 2, 3, 5 What will the next number be? Conjecture: the next number is 8 CONGRATULATIONS!!! You just used inductive reasoning based on what you observed (saw).

4. We use inductive reasoning to find patterns Find patterns from 2 main areas: – Numbers See what is going on from one number to the next (add, subtract, multiply, divide?) – Shapes Ask how the shape changed: turned? Added parts? removed parts?

5. 1)5, 10, 20, 40,… 2) What is going on from one number to the next? NOT increasing by 5 but doubling (or multiplying by 2) Next two numbers: 40 * 2 = 80, and then 80 * 2 = 160 What is going on from one number to the next? Realizing the first number is really 1 over 1 Pattern: denominator is doubling (times by 2), numerator stays 1 Next two numbers: Think: 80160 Think: In-Class Examples Find the next 2 items in each pattern.

6. 3) How does the figure change? It adds another square turned on its points then another turned back on its side. Worksheet: Skip #17 If you get a story problem see if you can draw a picture to help you find the pattern.

7. Review our two examples from yesterday about the sunny days and 2, 3, 5. Were those conjectures correct? Conjectures are not always true (correct) – They are still conjectures though! We prove a conjecture is not true by showing a counterexample – Sunny days counterexample: it will rain someday – 2, 3, 5 counterexample: the next number could be 7 or 8 7 if your conjecture was prime numbers 8 if your conjecture was 2 + 3 = 5, so 3 + 5 = 8

8. Finding counterexamples: – Find numbers that fit the criteria but do not reach the correct result (conclusion) – Hint: throw the word “not” after the is Example on page 7: find a counterexample. 25)The sum of two numbers is greater than either number. Need to find two numbers whose sum is not greater than either number. 2 + 3 = 5; 5 > 2 and 5 > 3 … not 2 and 3 No positive numbers work! Let one be negative: -2 + 5 = 3; the sum, 3, is not greater than either number Counterexample: -2 and 5

9. Assignment: Practice 1-1 (handout)