In previous years, I’ve had time to really go in depth into the polar version of conic sections. It may seem abstruse, but I love that one equation gives rise to all the conic sections. From there, and a deep analysis of the polar world, we then moved to the rectangular coordinate versions of these creatures. This year, I have no time. So I reduced a two day investigation into a 20 minute “notice and wonder” exercise. And although less rich, I saw such engagement from a good number of kids. They were exclaiming “WEIRD!” and “WHAT!?!”
Here is, in essence, an animation each group made on Desmos (which I turned into a gif here), which they exclaimed some wonderment at — why were all these weird things happening?!
What they were plotting on Desmos was and , and they modified the values of and .
They brought up great noticings and wonderings: circles and parabolas, graphs with two “pieces,” oblique asymptotes, distance from the origin was affected by the variable, squashed circles getting elongated… Good stuff.
Although it was painful to cut some of this out, because I think it ties together so many of the previous ideas we’ve encountered, I at least got them to appreciate that these graphs are weird, beautiful, and oh yeah, all related to a single equation and hence interconnected. And that was certainly a huge (and fun-to-watch) good thing!