Lesson 1-1: Patterns & Inductive Reasoning

Vocabulary Term Definition Own 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.)

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).

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 How does the shape changed: turned? Added parts? removed parts?

Find the next 2 items in each pattern. In-Class Examples Find the next 2 items in each pattern. 5, 10, 20, 40,… 2) 80 160

3) Assignment Practice 1-1 If you get a story problem see if you can draw a picture to help you find the pattern.

Were those conjectures correct? 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

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