My polar graphing unit in Precalculus has always started in the same lackluster way: With me telling students how to graph polar coordinates. We then launch into some point-by-point graphing, followed by various explorations and challenges involving graphing polar equations, and we’re off to the races.
This year I wanted to try something different. Instead of telling students how to plot polar coordinates, I wanted them to discover the mechanics by using technology to plot a handful of points.
It wasn’t exactly profound, but this brief introductory lesson felt like an improvement. I started by displaying these images:
We then fired up Desmos, with students working in pairs. Once everyone successfully plotted the first point, I turned them loose on this:
That’s my “can you plot points in the polar coordinate plane” assessment from last year. I don’t allow students to use a calculator on it, at least not when it’s a real assessment. As a learning tool, especially without the usual direct instruction intro, this page paired nicely with a bit of technology.
My favorite part from this brief lesson came at the end when we discussed what to do with negative radii and/or negative angles. In the past, it was a lot of “do this” and “do that” and “don’t forget this.” Here, I invited students to share their observations and make conjectures about points involving negative values.
And the payoff was in what happened next: Instead of “yes, that’s right” or “nope, try again” from me as the expert, we turned back to Desmos to test (and in most cases refine) our conjectures. While there’s still some learning to be done here, I think we’re got off to a decent start.
Next up, in reality: A Desmos-driven, noticing-and-wondering exploration with six types of polar equations. If all goes according to plan, I’ll blog about it soon.
Next up, in my ideal world: In the future, I’d prefer to squeeze an extra lesson in prior to the aforementioned/upcoming exploration. This in-between lesson would involve each student receiving an equation, finding its value every 10 degrees (from 0 to 360), and plotting those points by hand on a polar grid. I think this would serve as a nice link between the “hey, now I can graph polar points!” lesson described above, and the “oh, sweet! Desmos can graph these equations in milliseconds” exploration that follows. Maybe next time…