1.1 Patterns and Inductive Reasoning

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Presentation transcript:

1.1 Patterns and Inductive Reasoning Geometry Mrs. Blanco

Objectives: Find and describe patterns. Use inductive reasoning to make real-life conjectures.

Ex. 1: Describing a Visual Pattern 1)Sketch the next figure in the pattern. 1 2 3 4 5

Ex. 1: Describing a Visual Pattern - Solution The sixth figure in the pattern has 6 squares in the bottom row. 5 6

Ex. 1: Cont… 2) Find the distance around each figure. Organize your results in a table. 3) Describe the patterns in the distances. 4) Predict the distance around the twentieth figure in this pattern.

Ex. 2: Describe a pattern in the sequence of numbers Ex. 2: Describe a pattern in the sequence of numbers. Predict the next number. 1, 4, 16, 64, … b. 10, 5, 2.5, 1.25, … c. 256, 16, 4, 2, … d. 48, 16, , , … 256 (multiply previous number by 4) 0.625 (divide previous number by 2) (take the square root of previous number) 16/27 (divide previous number by 3)

Using Inductive Reasoning Much of the reasoning you need in geometry consists of 3 stages: Look for a Pattern: Use diagrams and tables to help Make a Conjecture. Unproven statement that is based on observations Verify the conjecture—make sure that conjecture is true in all cases.

Ex. 3: Complete the Conjecture Conjecture: The sum of the first n odd positive integers is ______. First odd positive integer: 1 = 12 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 The sum of the first n odd positive integers is n2.

Note: To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counterexample.

Ex. 4: Finding a counterexample Show the conjecture is false by finding a counterexample. For all real numbers x, the expressions x2 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.

Another: Ex. 4: Finding a counterexample Show the conjecture is false by finding a counterexample. 2) The difference of two positive numbers is always positive. The conjecture is false. Here is a counterexample: 2-3 =-1

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