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Man Vs. Machine: A Prime Example of Number Sense

Man Vs. Machine: A Prime Example of Number Sense. AMTNYS October 27-29, 2011. Presented By: Adam Sprague and Stephanie Wisniewski SUNY Fredonia. Round 1: Repeat or Terminate. Terminate. Repeat. Repeat. The Number Sense Involved.

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Man Vs. Machine: A Prime Example of Number Sense

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  1. Man Vs. Machine: A Prime Example of Number Sense AMTNYS October 27-29, 2011 Presented By: Adam Sprague and Stephanie Wisniewski SUNY Fredonia

  2. Round 1: Repeat or Terminate Terminate Repeat Repeat

  3. The Number Sense Involved We recognize a terminating decimal as a fraction with in the denominator. Since the prime factorization of is , the denominator can also be written as where is any integer. For example: Since the denominator could be deduced to a power of ten, , we were able to determine this was a terminating decimal.

  4. The Number Sense Involved A fraction repeats when the denominator cannot be rewritten as a power of ten. For example: Due to the uniqueness of prime factorization it will repeatsince is not a prime factor of .

  5. Round 2: Is This Number a Perfect Square Yes, Perfect Square No, Not a Perfect Square Yes, Perfect Square No, Not a Perfect Square

  6. The Number Sense Involved Every perfect square can be broken down to its prime factors each raised to an even power. We have a perfect square when all the powers of the prime factors are even, such as in , , and . This is not the case for or because their prime factors are not all of even powers.

  7. The Number Sense Involved

  8. Round 3:Sums of Consecutive Integersand Perfect Squares In 2002, the 12th-grade American Mathematics Competition (AMC 12) asked the following problem: The sum of 18 consecutive positive integers is a perfect square. What is the smallest possible value for this sum? [http://www.unl.edu/amc/]

  9. Guess & Check Method 18 Consecutive Integers

  10. What If We Asked… Can the sum of 16 consecutive positive integers be a perfect square? Let us try our guess and check approach;

  11. The Number Sense Involved Let represent the consecutive positive integers. The sum of consecutive positive integers is Now, what can we determine from this product?

  12. The Number Sense Involved Since and we know that is an odd number, the prime factorization of our sum will always have a factor of . Since we also know that a perfect square has all prime factors with even exponents we know that it is not possible for consecutive integers to have a sum which is a perfect square.

  13. In Conclusion… Thank you. Adam Sprague Stephanie Wisniewski asprague@fredonia.eduswisniewski@fredonia.edu SUNY Fredonia

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