# Coordinate Geometry, Parity and Invariance: Leaping Frogs!

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“Parity” is the mathematical concept of “evenness” and “oddness.” For example, we know that all whole numbers are either odd or even, and that if you add or subtract evens and odds, certain forms of parity will emerge. For example, if you add two odd or even numbers, you will end up with an even number; if you add an even and an odd, you’ll end up with an odd number. To make things more interesting, if you add an odd number of even numbers (say, 2 + 4 + 6), you’ll end up with an even number. However, if you add an odd number of odd numbers (like 1 + 3 + 5), you’ll end up with an odd number. What happens if you add an even number of odd numbers or an odd number of even numbers? You get the point.

“Invariance” is the concept that if you change certain parts of a mathematical equation, the result will not be changed. For example, if you are adding two numbers and you switch the ones place around (say, change 52 + 38 to 58 + 32) the answer will be the same. You can also use this principle in subtraction: if you move both the minuend and subtrahend up or down the same amount, the answer will be the same: thus, 500 - 287 = 499 - 286, which eliminates the need for re-grouping across zeroes! Thus, we can say that the answer to a subtraction problem is “invariant” on whether the two numbers have been moved up or down the number line the same amoun

This is a very excellent and interesting activity that was described by Mark Saul in his book, "Camp Logic." The problem involves placing three frogs on a coordinate grid and then recording their path as they leapfrog over one another to get to a target. It turns out that certain arrangements of frogs or certain locations of targets can or cannot be reached, depending on the location of frogs and target.

By using the concept of "parity" (the combination of odd and even coordinates), students can predict which problems can and cannot be solved, and even set up situations which they know will work and which won't.

These activities are really fun and very, very motivating. Your students will enjoy learning about coordinates and using the coordinate plane in order to investigate how parity and invariance work.

GREAT FOR ENRICHMENT WORK, GIFTED STUDENTS AND SUMMER MATH CAMPS!
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30 pages
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4 days
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