In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here are a couple of similar problems that arise in Precalculus:

- Convert the point from Cartesian coordinates into polar coordinates.
- Convert the complex number into trigonometric form.

For both problems, a point is identified that is 5 steps to the right of the origin and then 5 steps below the axis (or real axis). To make this more kinesthetic, I’ll actually walk 5 paces in front of the classroom, turn right face, and then walk 5 more paces to end up at the point.

I then ask my class, “Is there a faster way to get to this point?” Naturally, they answer: Just walk straight to the point. After some work with the trigonometry, we’ll establish that

- in Cartesian coordinates is equivalent to in polar coordinates, or
- $5-5i$ can be rewritten as in trigonometric form.

Once this is obtained, I’ll walk it out: I’ll start at the origin, turn clockwise by 45 degrees, and then take steps to end up at the same point as before.

Continuing the lesson, I’ll ask if the numbers and , or if some other angle and/or distance could have been chosen. Someone will usually suggest a different angle, like or . I’ll demonstrate these by turning 315 degrees counterclockwise and walking 7 steps and then turning 675 degrees and walking 7 steps (getting myself somewhat dizzy in the process).

Finally, I’ll suggest turning only 135 degrees clockwise and then taking 7 steps *backwards*. Naturally, when I do this, I’ll do a poor man’s version of the moonwalk:

For more information, please see my series on complex numbers.