1.2 Patterns and Inductive Reasoning. Ex. 1: Describing a Visual Pattern Sketch the next figure in the pattern. 123 45.

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1.2 Patterns and Inductive Reasoning

Ex. 1: Describing a Visual Pattern Sketch the next figure in the pattern

Ex. 1: Describing a Visual Pattern - Solution The sixth figure in the pattern has 6 squares in the bottom row. 56

Ex. 2: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. a. 1, 4, 16, 64

Ex. 2: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. b How do you get to the next number? That’s right. You add 3 to get to the next number, then 6, then 9. To find the fifth number, you add another multiple of 3 which is +12 or 25, That’s right!!!

Goal 2: Using Inductive Reasoning Much of the reasoning you need in geometry consists of 3 stages: 1. Look for a Pattern: Look at several examples. Use diagrams and tables to help discover a pattern. 2. Make a Conjecture. Use the example to make a general conjecture.

Goal 2: Using Inductive Reasoning A conjecture is an unproven statement that is based on observations. 3. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES.

Ex. 3: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ?. How to proceed: List some specific examples and look for a pattern.

Ex. 3: Making a Conjecture First odd positive integer: 1 = = 4 = = 9 = = 16 = 4 2 The sum of the first n odd positive integers is n 2.

Note: To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

Ex. 4: Finding a counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expressions x 2 is greater than or equal to x.

Ex. 4: Finding a counterexample- Solution Conjecture: For all real numbers x, the expressions x 2 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 0.5. In fact, any number between 0 and 1 is a counterexample.

Note: Not every conjecture is known to be true or false. Conjectures that are not known to be true or false are called unproven or undecided.

Complete in Notebooks Page , 13-15, 17-19