Monday, March 23, 2020

Online Math Circle: Pythagorean Triples

The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a2+b2 = c2. 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, Out of the Labyrinth: Setting Mathematics Free, 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.

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.


  1. I just was reading your 'Online Math Circle: Pythagorean triples . the recording it says does not exist.I too am craving for discussions with passionate teachers . I do math enrichment for curious students in the elementary grades who enjoy doing math and was in particular interested in Pythagorean triples . So I wondered how do you introduce them and present them as an accessible mystery . How do we know what we know and how can students in the early grades like 1-4 make the first steps driven by their own natural creativity to get a first inkling about the problem of the Pythagorean theorem and its whole number solutions. Can we make the transition from the triples to find a general proof of the theorem by Pythagoras for any number or the length of the sides of the 3 squares of numbers children know in the grades 1-4 and not just the whole numbers. Thank you for your answer . Peter Koehler

  2. This is Sue. Google seems to be letting its blog tools break. I cannot seem to sign in to post as myself.

    The whole internet is a bit unstable. Zoom recordings don't seem to be permanent. So I deleted that link. (My blog is full of old links that no longer work.)

    The algebra needed for Pythagorean triples is beyond most elementary students.

    The proofs of the Pythagorean theorem are not related to the triples. My favorites are mentioned in another post. ( Google has removed the pictures I included, but you might be able to make sense of it.


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