Many of you may be aware of this "trick," but I thought I'd pass this along:
I like to "amaze" my students with my mad mathemagician skills. One way that I do this is with Pythagorean Triples. I have the students give me any odd number, and I can tell them two other numbers that satisfy the Pythagorean Theorem with the original odd number (the original odd number is the shortest leg of the right triangle). I did this "trick" with my advanced geometry class and of course, they wanted to know how I was able to do it. I don't tell them for quite some time. I let them try to figure out how I am able to quickly find two other numbers, with few calculations. After a while, I let the class in on my little secret. They asked me if I could do the trick with even numbers, and I told them that I did not know a trick for even numbers. Well, today, one of my advanced geometry students said that they had found a way to do a similar trick using an even number. We tried his trick and it worked! Every time! I was so impressed with him and his thinking process to figure out the trick for even numbers. Here's the trick:
Take any odd number. Square the odd number. Divide by two. Add 0.5 to the quotient. Subtract 0.5 from the quotient. The sum and difference of 0.5 and the quotient are the two other integers that satisfy the Pythagorean Theorem. Example: Take 11. Square it: 121. Divide by 2: 60.5. The other two integers that satisfy the Pythagorean Theorem with 11 are 60 and 61. So, 11, 60, and 61 is a Pythagorean Triple.
Take any even number. Square the even number. Divide by four. Add 1 to the quotient. Subtract 1 from the quotient. The sum and difference of 1 and the quotient are the two other integers that satisfy the Pythagorean Theorem. Example: Take 8. Square it: 64. Divide by four: 16. Add 1: 17. Subtract 1: 15. So, 8, 15, and 17 is a Pythagorean Triple.
Pretty cool, huh? It also lead to an interesting conversation in class as to why the square of every even number is divisible by 4. I just love it when a student finds math interesting and wants to do math outside the classroom!!