# Polar Conics

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 $r=\frac{a}{1-k\cos(\theta)}$ and $r=\frac{a}{1-k\sin(\theta)}$, and they modified the values of $a$ and $k$.

They brought up great noticings and wonderings: circles and parabolas, graphs with two “pieces,” oblique asymptotes, distance from the origin was affected by the $a$ 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!