The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a

^{2}+b

^{2}= c

^{2}. Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.

Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)

[This was my intro to the online circle, whose recording was included here. Sadly, zoom recordings expire, and it is no longer available.]

Way back in 2007, I read Bob and Ellen Kaplan's book,

*, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.*

**Out of the Labyrinth: Setting Mathematics Free**I was also exploring online, and ended up putting together a book that collected some of the best resources I had found:

*.*

**Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers**Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).

We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.