Last week I mentioned dull workflows. Here's a great example. We spent a lot of time on limacons, cardioids, and roses, your usual suspects. Where polar gets really interesting is conic sections and the idea of eccentricity. Manipulating these kind of equations is tough with pencil and paper. Normally I introduce the idea by having students build a solar system first. It's a fun introduction but there's a major problem. Students understand much about what they're doing if a bunch of planetary research is their introduction. They produce nice work and everything, but I should have let them play around with general ellipses first.
This year I almost didn't learn my lesson but some weirdly timed field trips changed my mind. Basically, there wasn't enough time to get the solar system thing in before Spring Break. I had a day to play with so I figured, why don't we play around.
- access a pre-built desmos array
- reproduce some shapes
- make connections between the physical size of the shapes and the parameters
- attempt to verbalize the possible effects
- discuss eccentricity with little idea what that word might mean
The Slider Issue
Previously I've had a desmos set up on the board and had them copy it. This time I was like, duh, make it in advance and so I had them access this:
Normally, I wouldn't really go the slider route. There's a lot of room for them to just kind of fiddle at random until something worked. The problem with polar conics is that you have to make adjustments to two variables and the relationship between those variables in the equation is not trivial. I want success to require some effort, but to come quickly. The primary point of this is to have a discussion about what they did and how to classify these shapes.
I gave very few instructions. I pointed them to the ellipse generator and told them the first idea was to reproduce the graphs and record how it happened. Nearly everyone got them all within 10 minutes. After that, we were done with the iPads. Told you, dull.
I put the 8 ellipses up on the screen. I had them tell me what a and k would reproduce them. I asked them if they could tell me what they thought the two parameters did. Main theory was a controlled size and k controls how "squishy" it is. We talked about how wide or tall they thought each shape was. I stood back for a sec and let them look at those numbers versus the a values (a is half the value of the longest dimension).
Then we talked about k. What is this weird number? Why is it a decimal? Why did the shape disappear with values of 1 and 0?
And now the big question. Look at the set of shapes. Which would classify as the most eccentric? Least eccentric? At no point did I ever hint at what "eccentric" might even mean in this case.
That's the part of the workflow I care about. The iPads were the most efficient way to play with the idea. We barely used them. But it let me have a discussion with them and hammer out theories about eccentricity without ever having to tell them what that word means.
At the end of discussion I teased the solar system idea. I think we are better prepared for the concepts involved there. The thought that their elementary school teacher might have mislead them was extremely concerning.