## Wednesday, September 11, 2013

### The First Three Days of Trigonometry

Day One:

We watch this video.

We actually watch it twice. It's cool. We talk about some of the interesting things. I ask them what they were noticing. Some mentioned the shape of the ride, others that there's a bit of shaking in the seats, and others were thinking about what exactly happens when that guy swings off the top of the ride. (No, seriously, go watch the video if you haven't yet. It's fun.)

I ask them my question: How does the height of a person change as they go around this ride?

They sketch a graph in their notebooks. I forget to take pictures of it for the blog. Instead, I remember to take this blurry picture of the board where we recorded the camps:

So that's probably unhelpful. But on the left side we've got the three ideas:

1. The graph of height vs. time will be curvy, but sort of steep in the middle.
2. The graph will not just be curvy; it will literally be composed of half circles, due to the circular motion.
3. The graph will be nice and spiky.
I urged the class to temporary elide the differences between the two curvy graphs, and join brothers in arms to form a single Kurvy Kamp, allied against those who thought the graph was spiky.

Here's something interesting: we actually only had one customer for the spiky graph on that initial go around, when kids were sketching graphs in their notebooks. But as we talked about it some more, a few more voices for spikiness piped up.

We talked about what sorts of predictions spiky vs. curvy graphs make. A few kids voiced the idea that the spiky graph predicted that there was very little time spent at the top of the ride. We watched the video again.

Settled: the graph is curvy, not spiky.

(PS We also had a nice conversation about where people chose to put the zero height. People were trying to avoid having to use negative numbers in their choices of baselines. No one, it should be pointed out, chose the center of the wheel as the zero height. That's a non-obvious choice that needs to be motivated, not just thrust into the situation.)

(Here's a picture:)

Day Two:

A weekend passes, and I'm thinking that we're going to spend a few minutes in class reminding ourselves of what we did last week, and then we're going to try to make things more precise.

I put on that video again. My plan is to ask them to imagine a bigger wheel, and to ask them to graph that, just to get back in the swing of things. But as we're watching that video, somebody just up and says,
"I think we need to be careful about whether we're talking about the peoples' heads or the carts they're in."
Well, see, now that's interesting. Pretty soon we've found ourselves confused about how the motion of the people and the carts are related, and I ask them to whip out their notebooks and draw graphs that keep track of three heights: the height of the heads, the height of the seats, and the height of a notch on that little inner wheel near the center of the ride.

The class was pretty much split between these two possibilities:

This is a round that the folks who thought that everything peaks synchronously definitely won. Their arguments were clear and convincing, and mostly hinged on the fact that you could connect the heads of the people, the seats of the carts and the edge of the wheel with a straight line.

And then somebody said that they thought the edge of the circle was going faster than the inner wheel. That's interesting too, and the class was split on whether they were buying it or not.

At the heart of all this? A deep confusion about how to square the fact that there is constant rotational speed with the idea that there could be non-uniform speeds at different points around the ride. I think that the difficulty of pairing easy, clean constant circular motion with messy, more subtle notions of speed was at the heart of the spiky/curvy debate the day before.

Things are basically settled by the time I ask students whether they'd give a Gold Medal to the guy who runs the outer ring of a racetrack in 10 seconds, or the guy who ran the inner ring in as much time.

My initial plan was to just urge them to figure out ways of being more precise with our measurements of heights, but I was glad for the way that class went. In the last few minutes we start talking about getting exact values for the height, but we run short on time.

They say that they know about special right triangles, so I give them a homework assignment on special right triangles.

Day 3:

The thing is, yeah, they don't really remember special right triangles. I figured they wouldn't, so I give them this Warm Up problem set at the beginning of class:

And then I ask how we can get more precise with our graphs. Enough is enough with this messing around. Let's settle this.

We decide to set the initial height at 0. And then it repeats every 6 seconds or so. So at least we've got those two points. And what about half-way, at 3 seconds? Well, yeah, that would be the maximum height. What about half-way to that, at 1.5 seconds?

Another unexpected surprise from a student. He gets up and says, look, I think it would be best if we split up the Ferris Wheel into eighths. And it takes an equal amount of time for a seat to go from one of those sections to the next. And then we should just number the heights around those eighths.

I ask him, for clarity: "Is a Height of 2 half of the Height of 4?" He says that it is. And I ask him if a Height of 1 is half way to the Height of 2.

So I ask him to graph it.

Me: You understand, this is spiky, right?
Kid: Yeah. But maybe spiky is right!

I love it. Brilliant.

And now we've got a real argument going on. What's so fascinating to me is how fervent the spiky folks are. They're not just excited by that whole "split the wheel into eighths" argument. They're also tossing out things like "Well the wheel is moving constantly and so the height is changing at a single rate!"

But I've been patiently waiting for this moment for three days. I tell the kids, correct me if I'm wrong, but I think that our disagreement basically comes down to this:

[Except, imagine this without all triangles and angles and stuff.]

When you do half a turn here, does that mean that you've gone up half the height?

And we start busting out triangles, and the bell rings.

1. Cool. Good discussions. Nice job guiding the spiky kids to figure out on their own the problem with their model... I know for me sometimes it's tempting to just jump in and tell them the answer and not give them enough time to get there on their own. So way to go. Sounds like a great start to trig!

2. Love this! Many of my college precalc students have already had trig, so I don't think I could get the spiky or not argument going. But maybe if I show them this during another unit...

We're starting triangle trig now, and doing the circle trig toward the end of the semester. I think they hold onto it better split up that way.

3. Nice development of trig functions. Great conversations. I have used similar strategies to have kids argue over what the graph might look like, but never with such a great video. Cool find!

4. I find it interesting that every time I read your blog, I find myself feeling like I am reading thoughts that are in my own head or on a lesson plan file I keep at school. I use a pretty similar approach to introducing Trig. functions and I too use the human powered Ferris Wheel video because the kids think it is awesome (mine might be a different one, not sure). I also use the video of a knex Ferris Wheel (http://www.youtube.com/watch?v=AB7mGHPV7E8). This video is awesome for the spiky debate. At one point in the video, the makers of the wheel put a camera on the Ferris Wheel so the observer gets to "ride.' From that ride, you can definitely tell that the rate at which one would ride is not constant and you definitely get a feel for the curve, not spiky nature, of the wheel.

1. Oh, I really like that POV idea.

Also, do you have a blog, Chris?

2. It is an awesome video and could easily be turned into a Meyer style "3-ACT," particularly because the creators of the Ferris Wheel give statistics about the wheel at the end.

As for a blog, not yet. I have contemplated it for a few years now. I might join Sam Shah's new initiative, so we shall see.

Thanks for the great blog posts about real teaching!