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Introducing Solids – Origami Style

My geometry classes are starting a brand new unit – Solids. Rather than starting out with vocabulary and definitions, I decided to start with origami and good golly am I glad I did so.

TL;DR We made origami cubes, square pyramids, and triangular prisms. Students with less mathematical-status were able to show off & students with high mathematical-status struggled. Beautiful. Students got curious about the relationship between edges, vertices and faces in solids.

The first lesson I will offer is that teaching the class en masse is a terrible idea. Some kids get it quickly, some kids are super lost and need lots of help. I led students through making a cube yesterday under the doc cam and it was a mixed bag. Last night I made videos of me making the shapes and students watched those on iPads today. It went much more smoothly. I was able to bounce around and help who needed help and students who didn’t need the assistance were able to move at their own pace.

Here is a video of folding the cube:

(Just kidding, I want to get this post up for MCTM, but the internet is too slow to upload the video. I’ll get it up when I can)

Here is a video of folding the square pyramid:

After students had made each unit, they counted faces, edges and vertices – my goal was to get them to Euler’s formula, which we did, but it required a lot of zometool units and a fair amount of prodding on my part. In hindsight, a valuable part of this activity was discussing what edges, faces, and vertices were.

Some students hadn’t made very clear creases on their cubes and struggled to find the edges, although they knew the cube should have 6…no wait…8…12? Really? We made some predictions about relationships between edges and faces for other polyhedra. I gave them a table to record the results.

The following day,  I brought in a bunch of Zometool polyhedra and had students add to their tables. I was thoroughly unable to get students to find Euler’s formula (for those unfamiliar, for all convex polyhedra, V + F – E = 2) on their own, but I did ask them to compute V+F-E for each entry in their table, and there was some satisfying oohs and ahh’s and “What?!”s when they kept getting 2.

I would love to tell you that I’d thought of this at the time, but I didn’t. In hindsight, I could have had them plot their coordinates into geogebra as (x, y, z) coordinates- it makes a nice plane that would have been good to wonder about. 

What DID happen during all of this was that status in the class got turned on its head. Which was absolutely great. Students who have generally been super successful in my class really struggled to make origami. Students who have really struggled seemed to do a little bit better. It felt welcoming to them. I had one student who had mastered the blow up cube, and he became an absolute superstar helping other kids. He doesn’t normally have a good time in math class. The whole thing was worth it even for that one kid. It was great.

I had a ton of kids say something along the lines of “THIS ISN’T EVEN MATH,” which of course was like candy to me to talk about what the heck math is and how we define it. We talked about spatial reasoning. We talked about order and rules-when they’re important, and when they can be broken.

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