More lazy email(ish) excerpts. This one from a conversation about polar coordinates, and my Calculus II students’ struggles with them last semester.
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I don’t fully understand the usefulness of polar coordinates as traditionally presented. I feel like a great deal of the focus is on the pretty pictures we get when trig functions interact with polar coordinates. Cardioids and all that.
What do we gain from graphing this in PreCalc?
When I taught Calculus 2 this past semester, it was clear that my students were struggling to sort out differences between cartesian coordinates and polar coordinates. They knew how to convert between polar and cartesian coordinates, but they didn’t seem to know why one would do that, nor did they seem to see polar coordinates as a self-contained system. Polar coordinates were always (in their minds) in relation to cartesian coordinates.
My students struggled to think about an angle as an independent variable that could change (and correspondingly a radius as a dependent variable that could change).
They couldn’t view a function defined in polar coordinates as a dynamic relationship. They could identify points one at a time. They could make their graphing calculators display polar graphs. But they couldn’t think about the process of tracing out a polar graph. This seemed to be true even for students who could talk dynamically about cartesian graphs (increasing, decreasing, approaching an asymptote-this was terminology my students could apply to cartesian graphs, but not to polar ones).
The precise conceptual nature of the relationship between polar coordinates and cartesian parametric equations is unclear to me. My students saw some relationships that they couldn’t quite articulate. I’m interested in exploring this territory a bit.
For the record, I was flying blind through this material last semester. It was the first time I had taught polar coordinates in any serious way. I have never taught precalculus. I had never taught Calculus 2 before. So I kept bumping into obstacles that I hadn’t imagined would be there.
Overthinking my teaching 
As many times as I’ve taught Calc II, I’ve never addressed these issues. This is the last thing we do, so it always gets scrunched for time, and I just try to make sure they’ve at least encountered it. I guess I’ve saved all my pedagogical energy for earlier topics.
When learning Cartesian coordinates, a lot of the emphasis is also on how certain pretty pictures are represented by simple equations in polar coordinates (lines, circles, boxes, hyperbolas, etc.). A dynamic understanding of these is more difficult in both settings, but considering you spend so much time on it in Cartesian coordinates, it is assumed, probably unreasonably, that students will be able to adapt their dynamic understanding from one setting to the other.
My favorite thing to do in polar coordinates is probably the computation of the Gaussian integral:
http://math.etsu.edu/multicalc/prealpha/Chap4/Chap4-5/part4.htm
but unfortunately that’s Calc III stuff.
I’d imagine the most important mathematical application is the representation of complex numbers using complex exponentials.
As for physical applications, I tend to think of 3-D coordinates as being much more useful. Spherical coordinates are obtained from polar coordinates by introducing the azimuthal angle. They are useful, for instance, in describing fundamental physical laws (like gravity, electro-magnetism, etc., where these forces act in a spherically symmetric manner). They are also useful, according to textbooks, it dealing with airplanes, robot arms, etc. (but I’m no engineer so don’t take my word for it).
I don’t know how much of this could be made accessible to a typical Calc II student.
Seems like a clear case of “trust me, you’ll need this later.” Polar coordinates are wonderful tools for doing things that it’s hard to imagine describing to a precalculus student. The best motivation I can manage when I teach polar coordinates at that level is to say that they’re used to describe shapes whose rectangular descriptions would be inconvenient in certain ways.
I don’t have students do very much with graphing tools, because I don’t see much of an educational benefit to them in this topic — really, they’re more of a destructive shortcut. And I strongly discourage students from memorizing the pointless textbook table of “types” of polar equations (e.g. cardioids, limaçons, lemniscates, spirals, etc.). Instead, I help them to understand, as deeply as they can, why the graph of a given polar equation must look the way it does. Toward that end, I’ve found it helpful to warn them early on that the exam will definitely have them working with one or more polar equations that aren’t like anything in the textbook, or anything “covered” in class.
Why not start with a natural application, like the sensitivity of a unidirectional microphone, antenna, or rangefinder.
http://www.maxbotix.com/pictures/XL/XL%20Sensor%20Beam%20Patterns.gif has rangefinder sensitivity in Cartesian coordinates (with discretization errors).
http://en.wikipedia.org/wiki/Microphone#Microphone_polar_patterns
has (idealized) polar plots for different microphone designs. You can see measured microphone responses on spec sheets, like http://www.earthworksaudio.com/microphones/flexmic-series-2/fmr500hc/
http://www.astronwireless.com/topic-archives-antenna-radiation-patterns.asp
shows an antenna plot using the same data in both a usual x-y plot and as a polar plot, showing the advantage in visualization for the polar plot.
At our school we have minimized the amount of time we spend with our Pre-Calculus students on this topic for some of the reasons identified above. Granted, we have some time to play since we choose to teach BC Calculus as a second year course after AB Calculus. This way, when our BC kids encounter the topic they have plenty of time to digest it. I think that the repeating nature of the graphs and the fact that a different path direction yields the same final picture is the basis of much of the confusion. With cartesian coordinates (which have been part of their thinking for years) this idea of the direction of the graph or the idea that within a certain period the graph is simply repeated is not part of the concept at all. My guess is that this is where the concerns lie.
I want to check out the links above when time allows – they may cause me to entirely rethink our precalc decisions about polars.
I know this is a “trust me, you’ll use this later” scenario, but students will definitely need a good grasp of polar coordinates if they take integral calculus later on, because there are a lot of integrals that are just near impossible –if not impossible– to solve analytically using cartesian coordinates.
As something more approachable for your students, most pre-cal students are bright and curious so you can talk to them about using polar coordinates as a means of visualizing the universe as an infinitely big two-dimensional circle instead of an infinitely big rectangle. You could go a step further and talk about how in 3 dimensions, with Cartesian coordinates, we look at the world as infinitely big cube, but instead we could use spherical coordinates to look at the universe as an infinitely big ball. Then you could go a step further and talk about how none of these models actually model the way the universe really is. (And this is where I would personally begin to falter and would need to study some more if I ever am introducing polar coordinates.)
I wrote a blg post on the use of polar coordinates (actually complex numbers, but related ) as used in 2d image registration, where a “practical use” for these types of numbers comes into play.
post is at http://www.dominionsw.com. Link on the right