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

Yesterday we did a project exploring a common algebra mistake -> assuming that $(x + y)^2 = x^2 + y^2$. That project is here:

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

Today I thought it would be fun to explore the same idea using 3d printing. During the day I made prints of the two surfaces

(i) $z = x^2 + y^2$, and

(ii) $z = (x + y)^2$.

Here they are:

For the project I asked the boys to try to figure out what the two graphs looked like over the domain -2 < x < 2, and -2 < y < 2, and they showed them the shapes. Though not really by design, the choice of a square for the domain turned out to lead to an interesting discussion at the end.

Here's how the project went:

(1) What does $z = x^2 + y^2$ look like?

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(2) What does $z = (x + y)^2$ look like?

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(3) What about the actual shapes surprises you?

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I really enjoyed the combination of these two projects. Hopefully seeing the shapes of the surfaces becomes one little extra reminder that the two commonly confused expressions $x^2 + y^2$ and $(x + y)^2$ are not the same.