Weird Geometry

Warning, this post contains a large amount of links that are intended to open student’s minds and then blow them away. Use at your own discretion.

I started my geometry class this year with some weird geometry activities. The first was this task from the Harvard project, Balanced Assessments in Mathematics. It asks them to imagine life on a cubical planet. Then after looking more at scale, measurement, and the distance and midpoint formulas, we dove into some taxicab geometry with these problems. We then spent time learning about proofs of angles and segments until we could start talking about Euclid’s Parallel Postulate. I started off talking about proofs by looking at Elon Musk arguing that we are living in a game. We analyzed the argument, wrote two column “proofs” to synthesize it. It has been a wild ride and we are wrapping it all up with the lesson we did the other day, and the project we are finishing now. The lesson comes from this one I wrote a while ago where they blow up balloons and do some geometry on them. Here are the questions they needed to answer for this.

When we got to the final question, “What is the maximum sum of the angles in a triangle in spherical geometry,” some of my students said, “The triangles should still add up to 180 degrees!” But just kidding, their protractors said otherwise. Then they started asking how to figure out the maximum and I pointed to one of the poles on their balloons and asked what happens when they make triangles from that point. One of my students figured out that their could be an angle inside the triangle that measures 180 degrees. He said, “But that wasn’t really a triangle,” I asked, “Why not? It’s got three angles right?” Then he went from there and figured out that all three points of the triangle could lie somewhere on the equator, creating a triangle whose angle measures add to 540 degrees and that had to be the maximum. I was pretty happy with this but then he said, “But then when you try to find the minimum sum, it turns out that you can’t get as low as 180 degrees but you can get close to it! Actually, you can get as close to 180 as you want! And if you get to a point that you think is the closest, there is another point closer so that you can get infinitely close to 180 degrees but never get there! That’s crazy!” And this is a kid who doesn’t always do to well on quizzes, but he figured out the hardest question I’ve asked and went further with it on his own.

There you have it, give a kid a balloon and he reinvents calculus. Not too bad for six weeks of work!

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