The Grand March in the Quad: Linear vs. Angular Velocity, Part II

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So my students had done a magnificent job the previous day. And I hadn’t done too badly my own self. The original Sammy problem is, I submit, a masterpiece that integrates three different concepts without tipping its hand.

But the students’ skepticism came through loud and clear. They understood the math. They grasped the significance of the radius to the speed. But ain’t no way they bought the notion that Sammy was going faster than the bird.

I mulled this over the evening, and went back to youtube looking for videos. The Grand March? Too much time to wait for a very small demo.

But wait. What if I did my own Grand March? And in the words of the great Oracle Jones of the noted Western mockumentary The Hallelujah Trail:


THERE, now I see it!”

Alas, no whiskey to be found.

But I had the idea. And it all depended on George.
George, seen here in my algebra 2 class last year, is a top student and a fantastic young man. He also possesses a battery-operated wheelchair.

The next morning, before class, I went looking for him.George1

“George. I have a really cool idea, I think, but I need your help. Can you set your wheelchair to a particular speed?”

“Sure.” George doesn’t even ask why. He’s used to me.

“Okay, and this is a weird question–but can you, like, tip over? Do I have to worry if you go round and round on a 3-foot radius circle that it will….tilt?”

He’s kind, and doesn’t mock me.

“Ms. K, it weighs a lot. It can’t tilt over.”

“Phew. I was having nightmares.”

So before class started, I got some chalk from an old-school colleague and using a tape measure, marked out a circle around the courtyard drain, with a 3′ radius.

When the bell rang, I was ready.

“So. Yesterday I noticed skepticism about the bird’s speed. You understand the evidence, but you trust your lyin’ eyes more. I came up with a way to illustrate the proof so you won’t have to take math’s word for it.”

Then for the first time in my over five years as a teacher, I took my kids outside. Very unnerving. (Yes. I’m a big weenie.)

So the basic idea: George sets a speed and follows my traced circle at a very slow pace. The rest of the kids line up on opposite sides of the quad, and one by one they join in with George. Two kids go in on each of George’s rotation, one from each side.

The class was skeptical, but game.(In fact, this trigonometry class could not have been a better guinea pig for my first time teaching the subject. Every day, they jumped right in.)

On the first day, I just did proof of concept. I wasn’t sure how to get everyone to link together, so everyone held the edge of a tape measure.

The kids did a great job and the activity just exceeded my wildest expectations—and best of all, took less than 20 minutes from start to finish. I took pictures, and showed them to anyone walking by, including the entire admin team. This is my favorite shot from the first day:

FWDay113stdnts

The end of the tail really captures the movement. Austin the Action Figure!

We went back in and worked basic problems on angular and linear velocity for the rest of the day.

Then I realized that I really missed an opportunity. I was so worried my idea wouldn’t work that I didn’t take advantage of the obvious real-life problem at hand. What was our Grand March angular and linear velocity? How fast was George going? What about the speed of those at the end of the chain?

So the next day, we went out and did it again. But the kids had some changes. Nuff of those idiotic tape measures, Ms. Kerr. We are all comfortable with our sexuality, and will link arms. Football players and all.

Note–some of these pictures are taken by me with my tablet, which has a pretty low quality camera for a Samsung; other stills were taken from video that two of my students filmed.

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I knew we’d added as many kids as we could when I saw Alexis nearly getting creamed by the quad wall.

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So I yelled at them all to go full speed for the last half for the grand finale, the picture at the top. Here they are a couple seconds later.

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You can see the “whip” effect in many of the pictures. It would work even better if we weren’t running into the quad wall towards the end.

The kids had a blast. Between the two days, almost all the kids participated in a “grand march”.

Then everyone went in and learned how fast they’d been going. I measured a bunch of them shoulder to shoulder and took an average of 36″, or 3 feet for every two students.

FWLinearVelocity

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This lesson was a stunning success, and not just because of the fun and games. I had created memories and math that students would remember—and they did, all the way through to the final. I couldn’t wait to try it again with my two trig classes in the spring semester.

But while part 1, the Ferris wheel problem, went just as well both times, the outside activity was just a bit flat. Our German exchange student, Simon, was my TA this semester after having taken my trig class last fall (he’s the first one next to George, above). He played the anchor position, since I didn’t have George, and did it very well.

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If you notice, two students in the last picture, above, have dropped out later:

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This despite the fact that Simon was holding the same pace that George was (we had timers to confirm).

It wasn’t a disaster, and we had plenty of time to do it again. In both my spring classes, I had kids drop out, which simply hadn’t occurred the first time last fall. They seemed to have fun, but there wasn’t the same joy I saw in the fall. Simon agreed that the spring students didn’t seem to be as absolutely thrilled.

However, I had one of the best “told you so” comments in my third block class. I was explaining that the first student had to move very, very slowly—around 18 seconds per cycle–so that everyone could keep up.

“Um, keep up? Eighteen seconds around?” Braxton said. “That’s not going to be a problem.”

“Okay, everyone remember he said that!” I ordered. Which made a nice little teaching point when we got back to the room after the grand march.

Still, I wish I could have made the spring classes as absolutely perfect as the fall one was.

When I finally got around to writing this up, I suddenly had a revelation. Look at the top picture again. At least 10 of the 17 students in that picture are athletes—4 of them in two sports. I suddenly remembered all the students towards the end of the whip bouncing on their toes, warming up, waiting for the line to come around.

I mentally riffed through all my students in the two classes this time round, and yep–far fewer athletes. And here their trig teacher is demanding physical activity.

While I was always calling kids randomly, I had a much higher shot at getting an athlete in my fall class.

So same activity, same lack of warning–but not the same level of absolute ready-to-go spirit I had in the fall. I’m going to have to think about how to get them prepared to enjoy themselves, get some guidelines, maybe warn them ahead of time.

But even with less absolute magic, the kids understood and enjoyed the lesson.

I just need more space! Maybe I’ll try the football field in the fall.

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The Ferris Wheel, Sammy and the Bird: Linear vs. Angular Velocity, Part I

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:
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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.

So the next day, right off the bat, I projected part one of the problem:
FerrisWheelPt1

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.

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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.

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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.

Time for part 2:
FerrisWheelPt2

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:

FWShawn

“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.