Homeschool Math: How to Model the Pythagorian Theorem with the Decanomial Square

Neither kid has yet studied the Pythagorean theorem in math, but they DID both study Pythagoras a couple of weeks ago, and what better way to bring him to life than to model his most famous theorem?

And fortunately, if you focus on the Pythagorean triples, the Pythagorean theorem is also actually really easy to model, and quite accessible to even a younger learner. To two kids who've studied area, square numbers, and triangles, it's a snap!

First, of course, you need to build the decanomial square. Notice that the kid starts with the square, then adds the matching pieces in descending order:

When those are all placed, she finds the next largest square and carries on:

It's also good to model your work on gridded centimeter paper. This makes the translation between model and equation much clearer:

Find the square whose side matches side a. That square is a squared. The square whose side matches side b is side b squared. The hypotenuse is c. Once you've got the squares in place, you can start to do the calculation:

You can either work the calculation first, or find the square whose side matches the hypotenuse first. Either way, your work should match:

Although that's the only Pythagorean triple that's modeled in whole pieces in the decanomial square, you can use the decanomial square and/or Base 10 blocks and Cuisenaire rods to piece together the squares of the larger Pythagorean triples:

And to prove that a squared plus b squared really does equal c squared, break down the blocks that make up a squared and b squared--

--and put those pieces on top of c squared. You'll have to puzzle them together a bit, but in the end, they should fit perfectly!

There are lots of other fun ways to model the Pythagorean theorem, although since they also mostly rely on the 3, 4, 5 Pythagorean triple, it can get tedious if you do too many of them with the kids. It's more fun to make larger square models of the other triples to test, or to use the Pythagorean theorem in real-life situations.

Here are some other resources that we've enjoyed:

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