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1.1 Patterns and Inductive Reasoning
Geometry Mrs. Blanco
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Objectives: Find and describe patterns.
Use inductive reasoning to make real-life conjectures.
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Ex. 1: Describing a Visual Pattern
1)Sketch the next figure in the pattern. 1 2 3 4 5
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Ex. 1: Describing a Visual Pattern - Solution
The sixth figure in the pattern has 6 squares in the bottom row. 5 6
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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.
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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 , 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)
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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.
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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 = 9 = 32 = 16 = 42 The sum of the first n odd positive integers is n2.
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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.
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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 In fact, any number between 0 and 1 is a counterexample.
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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: =-1
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Page 6 #1-11 and Page 8 #34-37
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