Wednesday featured an object lesson on why I should plan more but talk less.
In precal, we started parametric equations. My typical introduction for parametric equations is to throw a line on the board labeled with time, like this:
I tell the kids to imagine an ant crawling across a picnic blanket. At t = 0, he’s at (0,0). At t = 1, he’s at (2,3). At t = 2, he’s at (4, 6). And so on. What are equations that describe his x and y movement?
Over the last couple of days, though, I’ve been thinking about how we might get students moving or touching things or doing something different with this. I thought maybe the tile floors in the atrium might be a grid, and we could go out and turn it into a coordinate plane and move around on it. But the tiles didn’t align. Or if I’d planned ahead, we could have laid out tape on the floor and made a grid. But yesterday was busy and I didn’t get it done.
So, while the students are doing their warmup, I walking around the room trying to figure out where I can find a coordinate plane. I stare out the window into the parking lot and see the grid in the parking lot from the parking lines. It wasn’t perfect, but it worked.
First period didn’t have time, but in third period, I told them to leave everything and follow me for a field trip. I stood on a point and declared myself the origin. I gave them a set of parametric equations and told them to go figure out where they’d be when t was 0. Then we did t = 1, 2, 3, 4. Then I gave them another set of equations that would produce the same path, but a different orientation and a different starting point. Again, t = 1, 2, 3, 4. I started asking them, “Have you been here before?” They eventually realized that it was the same path, but they were going the opposite direction. Then we trooped back inside and had the normal lesson.
Over all, it went well. When we were talking about orientation and parametrizing equations, I could talk about how those two sets of equations produced the same path, but different orientation and different starting points. I got a lot of “Oh, that makes sense,” looks – a lot more than I did last year when I just drew colored arrows on the board. Orientation is where they usually got lost, so I’m hoping it’ll pay dividends.
On the other hand, planning more would have been good. For one, standing on a random point for the origin was confusion. I should have declared a light pole the origin because it was an easier reference. Also, they kept forgetting the equations. I should have thought them out ahead of time and had them write them down. Little things, and they were willing to bear with me, but it could have been smoother.
As for talking less…we worked on eliminating the parameter today. I split that up into two methods: substitution for algebraic equations and matching them with trig identities for equations with trig functions in them. In the past, I’ve done a lot of prompting, or assumed that they wouldn’t get it on their own.
This year, I kept quiet. I threw a problem up on the board and asked for ideas.
They came up with it.
In each class, a student said, “Well, can we use ?” I said, “Sure, what do you want to do with it?” “Well, we could solve for the trig part and plug in…” I was so proud of them! And it wasn’t from the students I would expect it from, either. I need to have more faith in them, and give them more opportunities to show me that they can think.