Teaching polar flowers
(math, not arctic botany, sorry)
My precalculus course is mine to design (with collaboration from two other teachers). At the end of an extended trig unit (chunks of it were review for some kids, all of it was useful. But did I say “long”? I meant that) it was time to move on. But I suggested, and we agreed, for a fun three days with polar coordinates.
The best day? The last. That was the “messing around” day. Try things. Use the TI. Find something neat. Try to figure out why it was neat.
The next best day? Day 2. Roses.
Steve Lazar says this is called “guided inquiry” – anyone know? It sounds ok to me. What happened? I told them we would graph r = cosθ, and we did. I used a rough plot on the board, and I asked for assistance, and I slowly slowly plotted point after point, let the curve double on itself, and finished the circle (without proving we had a circle – need to think about that). Some of them plotted along, more just watched carefully. I dwelled extra-long on the negative values of r. Did we plot r = sinθ as well? I don’t recall. But we moved to r = cos3θ, and again, I actively engaged as much of the class as possible in determining the next point, and the next….
I turned to the calculators. We carefully set windows (this is the calculator skill I think my students are best at – almost none of them even think of using Zoom-Fit anymore), with all sorts of reminders to let θ run from 0 to 2π, and graphed r = cosθ, animated. And they saw the circle double over itself, and they oohed. And they graphed r = cos3θ and they saw the petals doubled over themselves, and they aahed.
And this is the clever part. We next graphed r = 1/4 + cos3θ. Ladies and gentlemen, check a graph if you need to. This was a sweet detour. Do you see the hidden petals emerging? Now, there is no question. Watching the animation is nice, but unblocking is cooler: r = cos3θ is clearly a rose with 6 petals. After that, r = cos4θ was anticlimactic.
- Do I need to show that r = cosθ is a circle? Next time.
- Better r = sin3θ than r = cos3θ, because, believe it or not, graphed on the TI, the cosine graph generated ridiculous snickering. And on the next day, for free polar play, a couple of boys tried to adjust the graphs to look a bit more – hm – anatomical? I guess it is motivating, but, um, you know.
- The prize graph that I share is the sum of an Archimedes spiral and a five petal rose. I usually choose 10 windings. It looks COOL to me, and to them. And then they play with the parameters. r = C + kθ + AcosBθ. You might try r = 1 + θ/4 + cos5θ. I let θ go from 0 to 20π, and experiment with the window, trying both equal x- and y- limits, and squared off coordinates.
I am sure other people have done this, but I made it up independently, when my first school gave me a TI and tried to get me to teach myself how to teach with it. Instead I taught kids to make goofy figures, and my AP took it away.
- Recently Dan Meyer wrote on this topic. As a student teacher he punted on this lesson (procedure and pattern matching only). His suggested improvements ARE improvements, but much more was possible. Do I care about the number of petals on a rose, or do I care about understanding periodic behavior? (No false dichotomy here, an answer of “both” is perfectly reasonable.) My answer: I don’t care about the petals. The pretty graphs are motivation for something else.
On the other hand, my answer for “teacher in front vs kids on their own” is: Both. My answer for “hand graphing or calculator graphing” is: Hand first, then both. My answer for “working in groups vs working individually” is: Individually, with constant interaction/checking/questions with neighbors (desks are adjacent, touching) + I’m not stopping anyone from establishing small groups as they go. All of this is nuanced. All takes time to develop, refine.
Students gaining control over an unusual graph/equation relationship is good. Controlling the technology they use, also good. The gain made is not floral, it is the ability to master and control a bit of mathematics at a time.