Using 3d printing to help explore a few ideas from introductory algebra

Last spring I was playing around with some different 3d printing ideas and found a fun way to explore a common algebra mistake:

Does (x + y)^2 = x^2 + y^2

comparing x^2 + y^2 and (x + y)^2 with 3d printing

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

Does x^2 + y^2 = (x + y)^2 ?

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 z = (x + y)^2 – 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 x^3 + y^3 and (x + y)^3 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 \sqrt{x^2 + y^2} and x + y.

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My younger son chose the two equations x^2 - y^2 and (x - y)^2. 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 x^2 - y^2 = 0. 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.

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