(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.

Some afterthoughts:

• 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.

1. March 27, 2011 pm31 3:33 pm 3:33 pm

Very nice… I wonder if it is not true that being able to set the window for a function may be the best illustration of an understanding of the function that is possible to assess.. certainly it is one of the big steps to understanding..

pat

2. March 29, 2011 pm31 4:40 pm 4:40 pm

I’ve just finished teaching the polar coordinates section of the (UK, Edexcel) further maths course for the first time (audience: ten highly-motivated, -talented and -skilled 17-18-year-olds who spend fully half their curriculum time in maths), and so have been thinking about this stuff to some extent myself.

What I should have done more of: motivation. Their mechanics section doesn’t get to orbital mechanics, but we could have sketched it and I should have. I still may, once I finish the syllabus (four teaching weeks until they finish school, so time is at a premium – and I still have to cover the sizable chunk of Mobius maps and related stuff they need).

What I’m glad I didn’t do better: the book stated but didn’t prove that r=a cos θ is the circle it is. I proved it on the fly, algebraically, by substituting into the cartesian circle equation and hitting it with the trig identity stick repeatedly (they are all highly competent with said stick). The next day I had a “doh!” moment when the near-trivial proof-by-circle-theorem hit me, and showed it to them. I’m glad because it reinforces “there’s more than one way”, and “try something see if it works”, and “obvious != best”, and “your teacher is human”.

Our syllabus seems to be one of the rare ones that adheres to the “negative r values are spurious” convention, which makes a certain amount of sense imo (and rather simplifies the pattern-spotting for the flowers while making more complicated – and so interesting and demanding – tasks like finding the area of a complete flower). Also makes them think about sketches as well as using the calculator.

Nice post as ever, JD :).

• March 31, 2011 am31 6:46 am 6:46 am

Proof by circle theorem?? Please tell.

I tried (for myself):
$r = c0s \theta$
$x^2 + y^2 = cos (Tan^{-1}\frac{y}{x}$
$x^2 + y^2 = x$
complete the square
$x^2 - x + y^2 = 0$
$(x- \frac{1}{2})^2 + y^2 = \frac{1}{4}$
I must be missing something obvious, right?

• April 2, 2011 pm30 6:41 pm 6:41 pm

My trig proof went:
$x=r \cos\theta$, $y=r\sin\theta$, so for $r=2\cos theta$ we have:
$(x-1)^2+y^2 = r^2 – 2x + 1 = 1$ (effectively).

The circle theorem proof: draw the circle, draw in an angle in the semicircle. That gives you a general point on the circle, and since the angle in the semicircle is right we have $r = d\cos\theta$…

• April 2, 2011 pm30 6:42 pm 6:42 pm

I should say: that trig proof’s probably slightly tidier than the version that fell out of my head on the moment, though the strategy’s identical.