After introducing my geometry students to the Pythagorean Theorem, I introduced them to the idea of Pythagorean Triples. I showed them that the numbers 3, 4, and 5 will work in the equation: 3^{2}+4^{2}=5^{2}. Then I had students pick a couple of numbers, and we generated a brand new Pythagorean Triple with some fun algebra. At least, I thought it was fun, and students were curious where I was going with my algebraic manipulations. There are several methods, but the one I used went like this.

Pick two interesting numbers. Make them integers, not too big, but not too small. Did I hear 7 and 19? Perfect.

First, multiply those two numbers, and double your answer:

2 ∙ 7 ∙ 19 = 266

Next, find the (absolute value of the) difference of the squares of those two numbers:

19^{2} – 7^{2} = 312

Now, find the sum of the squares of those two numbers:

19^{2} + 7^{2} = 410

And just like that, we have a brand new Pythagorean triple:

266^{2} + 312^{2} = 410^{2}

Go ahead – check it on your calculator if you’d like. Pretty nice, huh? I’ll leave it to you to do the algebra to see that it will always work. The most amazing thing about this little task is that students were immediately convinced (and rightly so) that even though it may be hard to find Pythagorean triples, there are an infinite number of them.

And then the game started. I gave each group of 3-4 students a stack of 24 numbers that can be turned into 8 sets of Pythagorean triples. (Of course, there’s only one way to get all 8 sets, but since there are some doubles, it’s possible to match up some that will throw you off).

Matching and making Pythagorean triples. Surprisingly engaging, given its a bunch of calculations. #MTBoS pic.twitter.com/FuOblPPiFS

— Ethan Weker (@Ethan_MidPen) November 17, 2016

For both of my geometry classes, all students (no exaggeration!) were engaged bell to bell, furiously testing and playing with the numbers, discussing strategies, and trying to find the eight correct sets. Not only that, although students were in separate groups, they soon realized that they’d be more successful working together. I heard so many great ideas about how to attack the problem that I wasn’t surprised when, even though it came down to the wire, one of the classes found all 8 sets!

This, to me, was a great low floor, high ceiling activity. Every student was able to work on calculations – calculating squares, calculating sums, etc. And some students really worked through some great strategies – what units digits can sum to the biggest number’s units digit? Can we work backwards from that? And of course, the little bit of algebra review in the beginning was helpful as I found later, when I had students repeat the process to develop their own new Pythagorean triples. It was easy for them to check their answers at the end and then look for any mistakes they may have made, and there’s something about big numbers that makes students truly proud (and rightfully so, even when they are using a calculator to do the calculating work).

I’ve had some requests for the Pythagorean triples that I used, so here they are:

3, 4, 5

6, 8, 10

5, 12, 13

7, 24, 25

8, 15, 17

11, 60, 61

13, 84, 85

20, 99, 101

You’ll see some repeated numbers which can throw kids off a bit. Also, the challenges often came from numbers that were close together more than numbers that were large. Of course, larger numbers are fun to work with (probably with a calculator). None of my students put the numbers into a spreadsheet, so I’m thinking I may show them how to use a spreadsheet to play with the numbers a little more easily. Spreadsheet skills are always a good thing for people to know well (in my opinion anyway), and we often expect students to know them, but they don’t show up in any obvious place in our curriculum. So, note for next year I guess.

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