After eighteen months focusing on pre-calc, I was assigned three trigonometry classes for this year (again, over two semester cycles). In both cases, I got a single class at first, giving me a chance to get my feet wet, and then a bigger dose later, so I could really start to experiment.
I didn’t know much more than the basics of trigonometry when I began this class, and I’m not much of a planner. So I was often learning the math a few steps ahead of my students. For example, I had absolutely no idea what linear or angular velocity was until the late afternoon the day before I introduced the concept. But hey, I’m a quick study.
What the book said:
The math symbols just when whoosh over my head (figures often do) but the ice skaters, that made sense. Thanks, William McClure!
I instantly thought of John Ford’s classic Fort Apache, which may cause you to wonder if I actually understand the concept after all. Unless you’re a really big fan of the movie and also know some math, in which case you’re thinking “Oh, yeah, the Grand March”. Long ago, I’d observed that Shirley Temple had to hustle to keep up with her screen dad Henry Fonda in the Grand March (around the 57 second mark) and then saw the same catch up effect in all the subsequent quartets making the turn. At least a decade before I ever conceived of becoming a teacher, I thought “that makes sense. She has further to go around, so she has to go faster.”
I often kick off a section with a scenario that asks a question. Sometimes the question is a short, intuitively easy problem or activity that the students can do with little analysis. Other times it’s a long, extended dive into multiple concepts, drawing on a lot of previous knowledge. But the scenario is always designed to introduce the new concept. (I read about this idea in ed school, but as it was a good year or so before I began to incorporate the practice into my teaching, I can’t remember the reading or the author. Since I save everything, I hereby vow to go back into my readers and see if I can dig up the info.)
So I originally intended to do a short, intuitively easy demonstration on the different velocities, but I couldn’t find I couldn’t find any decent videos other than the Grand March itself, which was a little to subtle. Merry-go-rounds would be great, if I could just find a video of kids on this, with some standing easily in the middle and some holding on at the ends…but no luck. I tried skating videos of crack the whip, and much as I’d love to use a Winslow Homer painting, I knew I’d have to do all the explaining. I abandoned my initial idea of presenting the phenomenon and asking the kids to explain it.
Plan B: longer, more complex problem. After perusing the book and googling, at some point I found a Ferris wheel problem asking about velocity. I wish I could remember where, because I am certain I invented this problem almost entirely, and I’ve love to include the genesis. The question that sparked mine provided the total time to complete one revolution (15 minutes, I think), and the dimensions of the Ferris wheel. It then said that someone had traveled 6 minutes, or maybe 4, and asked what his linear velocity was. It gave too much away.
But from that question (or something close), I went WHOA and morphed my plan entirely. No more small illustration, but a long extended dive into–or onto–a Ferris wheel. Because, as any trig teacher can tell you, the Ferris wheel is the mother lode for application problems, a rich source of ideas that can be turned to a number of uses. My kids had already been through Ferris wheel problems calculating heights. So once I was pointed in the direction of Ferris wheels, a multi-faceted problem was an easy next step: one that combined right triangle trig, arc length, and linear velocity, the last in an intuitive way. Booyah.
And figure it out they did.
My students had just learned how to find the length of an arc, whereas I, who figured it out intuitively, just took the needed percentage of the circumference. I had spent no small amount of time over the past few days explaining that the algorithm for radians, which is the product of the “angle over 180” and the radius, is exactly the same thing as taking the corresponding fraction of the circumference. I was still a bit taken aback to see them multiplying 125 by three quarters pi. Oh, wait. Yeah. Okay.
I wasn’t taking pictures through the class, unfortunately, but grabbed these shots the next day. The one on the left is most complete, but for some reason they flipped the heights. The work on the right is also done well, but they did more of it on the calculator. I remember making them talk me through their thinking.
Meanwhile, a struggling group sketched aimlessly, hoping I wouldn’t notice that they weren’t working. I reminded them of the right triangle trig, helped them to find the angle measure, then asked them to think about what it meant. This group used the circumference instead of the algorithm and made progress although they didn’t finish the problem completely by the time I called everyone back together.
At forty minutes or so, all but one group had finished the entire problem. I had to help two of the eight groups significantly; the rest just needed mild reassurance. Outstanding work, a math teacher’s propaganda day.
At that point, I defined linear velocity, which they had intuitively understood as they worked the question. Once you associate arc length with the time to travel, it’s only natural to think about the speed.
The kids all began their calculations, using Sammy’s speed.
“Hold on,” I said, calling everyone’s attention. “Didn’t you use the radius to calculate Sammy’s velocity?”
“Sure, but they’re going the same speed, right?”
“Did you use the radius to calculate Sammy’s speed?”
And I had a bunch of kids looking at me like this:
“The bird and Sammy are going the same speed!”
“But did you use the….”
“Come on! They’re going the same speed! How can the bird be going a different speed? They’re both on the same Ferris wheel!!”
“What if I’d not mentioned Sammy and we started with this problem? What would you have done?”
With much skepticism, they worked the same method and realized that the bird was, indeed, going slower. (You can see some of the work on first picture of boardwork, above. That group had finished first and I gave them the problem verbally. Everyone else started it after I called everyone back together and explained linear velocity, so their work was on paper.
And so, I introduced angular velocity. Sammy and the bird were traveling a different distance in the same time, so their speeds were clearly different. Howevever, they were both completing one complete cycle, or circumference, in 16 minutes, so their angular velocity is the same. As we watch Sammy and the bird, we see them covering a circle in the same amount of time and this fools us into thinking they’re going the same speed.
I could tell they weren’t convinced.
“So how fast is the bird going?”
“The math says the bird is going 7.85 feet a minute, which is about .09 miles per hour.”
“Well, let’s be more precise: .0892 miles per hour, right? How fast is Sammy going?”
“Half a mile…”
“.5569 miles per hour.”
“How much faster is that?”
I won some breathing room from all that doubt when the kids determined that the speeds had the same ratio as the radii. But I could see doubt.
The math proved Sammy and the bird had different velocity. But how could I get them to accept the math?
I came up with an idea for the next day. Which I’ll cover in the next post.