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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. - PowerPoint PPT Presentation
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Man Vs. Machine: A Prime Example of Number
Sense
Presented By: Adam Sprague and Stephanie Wisniewski
SUNY Fredonia
AMTNYSOctober 27-29, 2011
Round 1: Repeat or Terminate
𝟏𝟕𝟓
𝟏𝟒𝟎
𝟏𝟐𝟒𝟖𝟓𝟓𝟔𝟗
Terminate
Repeat
Repeat
The Number Sense InvolvedWe 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.
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 repeat since is not a prime factor of .
Round 2: Is This Number a Perfect
Square𝟖𝟏
𝟖𝟏𝟎
𝟖𝟏𝟎𝟎
𝟖𝟏𝟎𝟎𝟎
No, Not a Perfect Square
Yes, Perfect Square
Yes, Perfect Square
No, Not a Perfect Square
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.
The Number Sense Involved
Round 3:Sums of Consecutive Integers
and 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/]
Guess & Check Method
18 Consecutive Integers
What If We Asked…
Can the sum of 16 consecutive positive integers be a perfect square?Let us try our guess and check approach;
The Number Sense InvolvedLet represent the consecutive positive integers. The sum of consecutive positive integers is
Now, what can we determine from this product?
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.