After my last unit, which didn’t go as well as I’d have liked, I have been teaching a unit on polar. Where one space (rectangular) transmogrifies into another space (polar).
I’ve been content with what the kids have done this unit. Again, they did all the discovery.
- How to change a point in rectangular coordinates to polar
- How to change a point in polar to rectangular
- How to graph a polar point — and find a bunch of different ways to represent that point in polar coordinates (showing that polar coordinates are not unique, unlike rectangular coordinates)
- How to convert from a rectangular equation to the polar equation
- How to convert from a polar equation to a rectangular equation
- How to graph simple polar equations by hand
- How to conceptually understand what the polar graphs will look like, and why
- How to graph complex numbers on the complex plane
- How to represent complex numbers in polar form (“cis”)
- How to multiply and divide complex numbers using polar form
- How to take find roots of unity (haven’t taught yet… throwing it in tomorrow…)
This unit, I’ve been more conscientious about collecting their work almost each night, so I could see their problems. One big thing was that I identified that rectangular points in the 2nd and 3rd quadrants provided difficulty when converting to polar (because of the inverse tangent function, which only yields angles in the 1st and 4th quadrant). Because of that, I was able to target that and bring that up in class, and incorporate more points there.
I think I made some solid — but very basic — materials for polar. I’m going to share them below, and explain each one of them… [In Word format, in case you want to use/modify them, they are here: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
1. Introduction to a new space: This is my introduction to polar. I introduce the idea of isotropy (without using the word) and say that we need things to anchor us so we can refer to them. One way — the way we’ve always done it — is using how for north/south we’ve gone from the origin, and how far east/west we’ve gone from the origin. Our regular rectangular coordinates. But… BUT there is another way. I have them play a few rounds of polar battleship on Geogebra to get a sense of it, without telling them anything. Then, without any direct instruction, students move on to actually convert from rectangular coordinates to polar. I do a little direct instruction about how to plot polar coordinates, when is negative and when is negative. And then they do some plotting, and then convert from polar coordinates to rectangular coordinates. Most importantly, they figure out how to do these conversions on their own.
2. Graphing in Polar: The warm up refreshes kids on the equation for a circle, but it also importantly has them graph a horizontal and a vertical line. When we talk about the warm up, I really emphasize that for the equation , what does that mean about the coordinate? (“It can be ANYTHING! There is no restriction placed on !”) The reason this is so important is that when they graph , I hope they say “well is stuck at , but the angle can be anything” (and similarly, for , “I suppose is stuck at , but can be anything.”). The rest is pretty much saying: okay, we’ve got our conversions for individual points… can we use them to convert whole equations? (Yes.)
3. Understanding Polar Graphs: Now, after all of this, we’re finally going to graph our first not-so-simple polar equation by hand. We do the very basic first one together, and then they’re off to the races! I wasn’t sure how much I truly cared about them graphing by hand, but it’s clear to me now that doing a few of these by hand really brings to life what graphing in polar truly means. Especially when you get negative r values.
4. Polar Graphing Exploration: This was a day and a half of conceptual lifting, done by the students. There is the most amazing polar applet created by David Little. More than anything else, this one applet has truly let me understand how polar graphs are created. It’s simple, and amazing. I didn’t want to teach my kids about the different names for the different graphs, and to identify what the graph will look like based on the equation. Instead, I wanted them to be able to understand why some graphs have a dent, why some look like a loop with a loop-de-loop inside of it, why some are spirals, and why some are flowers.
So I had them use the applet — and the packet — to explore. And I have to say: they were really making solid connections, and having good conversations, as they were going through it. If you use it, the one thing you want to tell the kids is to “not touch a” and “after you finish analyzing one graph, change b back to b=0.1$”. Honestly, this was the most “risky” of my classes because instead of staying traditional, I said “here, learn it.”
5. Shape of a Polar Graph: After they finished their exploration, I wanted to see what they took away from it, so I gave this out. I collected it and read through them. Some kids fared better on it than others. I didn’t have time to mark them up and give lots of individual feedback, so instead I created a solution packet and I gave it to them — letting them compare, ask questions, etc.
6. Polar Practice: After the last assessment, because I wasn’t totally confident on my kids’s ability to do the basic types of problems, I whipped this set of problems up, which I gave to them, collected, and harshly marked up. This was the best feedback for me, because I got to see what kids were confident about, and what they struggled with.
7. Complex Numbers, the Complex Plane, and Polar Coordinates: I start bringing complex numbers into the mix. I first have them spend a few minutes trying to multiply and divide complex numbers, but it starts getting tedious and annoying (it’s supposed to). We talk about what a complex plane looks like, how I can graph complex numbers on them, and how complex numbers not only have a rectangular (a+bi) form, but also a polar form (involving and ). They don’t seem to have any trouble getting that.
Then: BAM! I immediately have them do a problem, look for a pattern, and make a conjecture. The theorem we’re going for says that if you have two complex numbers written in polar form, if you multiply these numbers, you simply have to multiply the values and add the values. BAM. Awesome. So they see this. We talk about whether it’s a proof or not (it’s not). We talk about another example to evidence it, and then I let them loose on complex multiplication and division problems.
8. Basic Practice Problems for Complex Numbers: These are the practice problems I gave for students to work on. I also taught them how to use their calculators to input, store, and multiply/divide complex numbers.
9. Advanced Practice with Complex Numbers: This is a short sheet that has kids prove DeMoivre’s theorem.
10. Complex Roots: I wanted to teach roots of unity, but I don’t have a lot of time, so I made this sheet up — and we’re going to walk through it together. (Tomorrow.) We almost never do that. But it’s our last class before the test, and I think it’s just so cool that I have to show it to them.
Reflecting back, I feel kinda bad that I didn’t design this sheet backwards. Start by having students draw a perfect, regular pentagon on the coordinate plane (letting one point be (1,0) and the center at (0.0)). Then have them find the coordinate of the vertices of the pentagon. Then talking through the vertices to conclude they all are roots of unity (if we consider the plane a complex plane). But eh, I didn’t.