Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare (x + y)^2 and x^2 + y^2:

3d prints

That project is here:

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

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> \sqrt{x^2 + y^2} and \sqrt{x^2} + \sqrt{y^2}

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that \sqrt{x^2} = |x|. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.

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With the difficult part of the algebra behind us we moved on to talking about the surface z = x^2 + y^2. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.

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Next we tried to figure out what the surface z = \sqrt{x^2} + \sqrt{y^2} would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him

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Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂

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I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

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