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2.1 Use Inductive Reasoning

2.1 Use Inductive Reasoning. Counting Cubes Task. Describe a pattern you see in the cube buildings. Use your pattern to write an expression for the number of cubes in the nth building. Use your expression to find the number of cubes in the 5 th building.

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2.1 Use Inductive Reasoning

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  1. 2.1 Use Inductive Reasoning

  2. Counting Cubes Task • Describe a pattern you see in the cube buildings. • Use your pattern to write an expression for the number of cubes in the nth building. • Use your expression to find the number of cubes in the 5th building. • Check your results by constructing the 5th building and counting the cubes. • Look for a different pattern in the buildings. Describe the pattern and use it to write a different expression for the number of cubes in the nth building. Building 2 Building 3 Building 1

  3. Vocabulary • A conjecture is an unproven statement that is based on observations • You are using inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.

  4. A student from one of my classes last year made the following conjecture: To find the number of blocks B, you multiply the building number N by 4 since there are 4 arms. 4N = B Is this conjecture always true, sometimes true, or never true? How do you know?

  5. Vocabulary • A counterexample is a specific case for which a conjecture is false.

  6. Find a counterexamplefor the following: • Everything in the room with hands is a clock. • The perimeter of a rectangle is larger than its area. • The sum of two numbers is always greater than the larger number.

  7. Christian Goldbach (1690-1764) A Germanmathematician who also studied law. He is remembered today for Goldbach's conjecture.

  8. Goldbach’s Conjecture All odd numbers are either prime, or can be expressed as the sum of a prime and twice a square. Is Goldbach’s conjecture always true, sometimes true, or never true?

  9. Goldbach’s Conjecture All odd numbers are either prime, or can be expressed as the sum of a prime and twice a square. Examples: prime

  10. Always true or sometimes true? How many examples do we need to know Goldbach’s conjecture is always true or sometimes true? Does Goldbach’s conjecture work for 25? 27? 31? 105?

  11. Counterexample to Goldbach’s The first counterexample is 5,777. Goldbachproposed this conjecture in 1752. It was found a century later in 1856 by Moritz Abraham Stern

  12. Moral of the story: Examples are not enough to PROVE that a conjecture is always true.

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