Today I decided to revisit that project. We started by looking at the same idea from algebra:

Does ?

At first we talked about the two equations using ideas from algebra and arithmetic.

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Now I asked the boys for their geometric intuition and then showed them the 3d printed graphs of the two functions.

This part ran a little long while my younger son was stuck on a small but important point about the graph – I didn’t want to tell him the answer and it took a couple of minutes for him to work through the idea in his mind.

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Next I showed them 3d prints of and and asked them to tell me which one was which. It is really neat to hear the reasoning that kids use to go from shapes to equations.

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For the last part of the project I asked the boys to come up with their own algebra “mistakes” for us to explore. My older son chose to compare the graphs of and .

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My younger son chose the two equations and . Changing the + to a – in our first set of equations turns out to have some pretty interesting geometric consequences – “it looks sort of like a saddle” was a fun comment.

One especially interesting idea here was exploring where . We used Mathematica’s ContourPlot[] function to explore those two lines because those lines weren’t immediately obvious on the saddle.

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I’m happy to have had the opportunity to revisit this old project. I think exploring simple algebraic expressions is a fun and sort of unexpected application of 3d printing.