"Oh, no!" you'll say. "What about all that astronomical stuff? What about the percentage of the moon that's visible on a given night?"

Well, two things. First, why would you want to model the percentage of the moon that's visible with a sinusoidal function? If I really want to know what the moon's going to look like on January 17, 2015, then I'm just going to subtract a bunch of 29.5 day intervals from 1/17/2015 until I land back on my data.[1]

The second problem is this:

The moon's visibility isn't sinusoidal. Then again, of course it isn't. If it were really sinusoidal, then its orbit would be circular.[2]

What is a Trigonometry teacher to do? Practically nothing interesting in the world shows truly circular motion. (Oh, pendulums?) And even the things that do show circular motion are rarely worth modeling.

We're stuck with Ferris Wheels. So, find cooler Ferris Wheels.

We watched the video, and I asked them to graph height vs. time on a Post-it.

I tossed these under the doc camera, and we narrowed down our options. I chose two at a time and asked the kids to compare them, which usually resulted in us throwing out one of the graphs. When we got stuck, I suggested that we separate the two wheels and figure out an equation that determines the motion of each. After some thinking, a kid suggested adding the equations together. I asked her to show us what she meant, and she produced this:

This lesson was fun, tough, and a genuine context for some Daily Desmos-style sinusoidal modeling.

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What's next? There are a lot of rides out there, but many of them seem to be versions of this Double Ferris Wheel. Maybe the next step is to get weirder. Like, what sort of ride would have this height graph?

I'm running out of rides. Ideas?

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[1] In other words, all you need to model is the periodicity of the moon's cycle. There's nothing that pushes people or students to pay attention to its sinusoidal nature.

[2] In class we model the visibility of the moon, and I tried to escape these problems by asking "Is the curve of the moon's visibility sinusoidal or not? Is it like our Ferris Wheel's motion, or is it a different pattern?"

My absolute favorite carnival ride is the tilt-a-whirl. The visualization of displacement for that looks like something made by a spirograph (which I suppose it is). What I would be VERY interested in seeing would be the velocity and acceleration visualizations.

ReplyDeleteMy favorite carnival ride is the Zipper, an oblong frame that rotates as the cars are simultaneously pulled around the perimeter on a chain. My dad and I rode a ride like this when I was in middle school. First time I heard him swear... and he did so continuously for three minutes.

ReplyDeleteHere's a video: http://www.youtube.com/watch?v=Y3OiAlvpw2w

Also, here's a cool article on Fourier transforms and combinations of sinusoids, like in your double-ferris wheel. Might might at least have some cool visualizations to share w/ students?

http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face

First of all, I get the need to find more applications that fit perfectly, but I also would argue that tossing out applications based on planetary motion just because the models aren't perfect isn't an ideal move. Periodic functions fit the motion pretty well and as mathematicians we seek models that do a good job of predicting things, not always a perfect one. Heck, at the very least, there is an awesome history of science lesson hidden in the discrepancies. With that said, modeling temperatures, tides, etc. are still in my grab bag for trigonometric functions.

ReplyDeleteNext, with so many machines in our lives based on rotating circles, there has to be a plethora of problems hidden in machinery, computers, etc. (think of a piston for instance).

I love your visualizations by the way, but I wonder about the last graph you posted. From a physics perspective, I would call the ride "instant death." After all, if you look at the x-intercepts, the slope is changing pretty rapidly, if not instantly. That is some fierce acceleration, thus a wicked amount of force. I love it when this idea comes up with my Pre-Calculus kids when we first start looking at graphs, their slopes, and the meaning of them. Graphs like that make the idea of differentiability in Calculus so much easier to discuss and understand (when they get there that is).

Excellent points, Chris.

DeleteI agree with you that it's unwise to toss out applications just because the models aren't perfect. And I agree that there is an awesome lesson in the discrepancies. (I even tried to teach that lesson. It's basically how I structured my sequence of moon-modeling classes.)

I think my bigger claim here is that it's just not so much fun to make predictions about periodic things. Like, they're periodic, they repeat, OK we get it. It's a cycle, and it repeats every month.

(Compare that to modeling exponential things, where, if not for functions, we really don't have any solid way to organize our predictions about the future.)

Also: your point about "Instant Death" is awesome. Isn't that all the more reason to ask that question in class, then?

I get the whole predicting the predictable (periodic) is not as much fun, but that is if we are looking more for the general idea. If all the sudden we start requiring specifics, we can motivate the need to model periodic functions (I am thinking a ton about machinery here). I wrote up a completely made up problem about a rotating disc at a pop bottling company that had three different capping mechanisms on it. The idea that such things could (and probably do) exist and that they are controlled by a computer makes the need to mathematically model and predict accurately even more necessary.

DeleteAs for the 'instant death" graph, I agree completely with the fact that students should be introduced to it, which is why I use graphs like it year after year. I steal my graph directly from the old Core Plus text. The sad thing is that the text never prompts students to think about whether or not the graph is realistic (a problem I remedy). It is a great argument to have with students (or make them have with each other).

I loved this post but am squished for time (now, at the end of the semester). You did inspire me to at least introduce my lesson on where the sine graph comes from with a ferris wheel video. I think the students got it better than in the past because they were able to connect with something concrete. I used this video: http://www.youtube.com/watch?v=0FjddxnLMmY, which I think we all loved. A student did something to make it automatically repeat, so we could think carefully about how the height changes with time.

ReplyDeleteCool video, and (funny enough) it's one that I've used in class also. I actually started off the year in Trig with that video, and we had a huge fight about whether the height graph would be curvy or pointy. Fun times.

DeleteI give them too much. I should be encouraging fights like that!

ReplyDelete