Teach 180: Powers of i (Day 143)

Today I was visiting another teacher's classroom for an informal classroom observation.  They were talking about complex numbers and started to work with powers of i.  When I have taught complex numbers in the past, we calculate powers of i by noticing the pattern.  We recognize that there is a cycle of 4 and that i to any power can be determined based on where you are in the cycle.  In this class, the teacher was showing the complex plane for graphing complex numbers as she was talking about powers of i.  Then it dawned on me.  Why not visually represent powers of i and consider the cycle of 4 by seeing the complex number move around the complex plane?

Was there a Desmos Activity Builder lesson that already did this?  I did a quick search and came up empty.  That's when I created this in Desmos - Powers of i. If you hit the play button in line 6, you can see the power increasing by 1 and the number being plotted in the complex plane as it cycles, 1, i, -1, -i...  Here is an short screencast to show the animation.

Two things that would make this better are for the powers to display in a little box on the complex plane, as in i¹², i¹³, etc., and for the coordinates of the point to be displayed as i, -1, -i, 1, etc. as the point moves around the screen.  Looks like I'll need to get into the computational layer of an Activity Builder for that.  Then, I would need to think about what questions to ask students.  Perhaps it is just as easy as "What do you notice?"  "What do you wonder?" and "Can you use this to find i¹⁰³?"