I had a really good visit with one of my independent study students today. He started off by telling me about putting points in space so that they’re mutually equidistant—point, two points, triangle, tetrahedron, etc. Then he asked about what would happen in other shaped spaces—like on a torus or sphere. These are 2D spaces where it seems like maybe we could arrange more than three points equidistantly.
Somewhere along the way as we discussed this, he said something about some infinities being bigger than others. So I trotted out the counting numbers versus the integers, and we had a cool chat about interleaving and zig-zagging matching-up. Then we were back to spaces a bit. Then we started thinking about listing decimals and fractions (you can just barely see some fractions through the paper in the photo above). And then we abstracted our quadrant of fractions to a quadrant of dots, and he tried to find a way to string them all up. After trying a couple of things, he found a path different from the one I’m used to. It was a nice feeling to give him some space to figure out a path for himself, rather than just show it to him as usually happens (including with me in charge).
Bobbing and weaving through lots of good stuff. We both felt pepped up coming out of it and ready to do some more on it next week.