Part of my mandate as a teacher of Principles of Mathematics 12 (ie. roughly Precalc, for you Americans) includes teaching mathematical modeling. My textbook is filled with little subsections that boldly proclaim, “MODELING!”. It’s one of the mathematical processes that are supposed to be woven together across all of the curriculum content I’m covering.
I am wrapping up a unit on trig functions and it’s time to hook into the “MODELING REAL-WORLD SITUATIONS” content. Now, I personally have a serious love-hate relationship with trig functions. Being a graduate of a computer engineering program means I’ve seen them a LOT in my formal education. Trig integrals have nearly killed me on multiple occasions. On the other hand, playing with trig functions in an electronics lab is awesome, and being able to visually comprehend trig graphs is probably the only reason I managed to pass a course on Communication Systems. So part of me really, really wants them to get this.
But there’s one big problem: when I look through the textbook for real-world, all I see is textbook perfection. The opener they use is tide-level data from Nova Scotia – except that they’ve stripped the real data down to this:
This is a complete and utter fabrication. That sine wave is friggin’ perfect.
Messy peaks that don’t always line up. Some kind of weird alternating pattern hiding in them as well that totally makes sense if you stop for a second and think about how far the earth has turned in 12h.
You know what? It’s not perfect, it’s reality. And our model, based on a single sinusoid, is never ever going to match that reality perfectly. And that’s just fine, but for some reason the textbook seems deathly afraid of letting students realize this. The really ironic bit is that the real data is already incredibly close to the model, and yet they still couldn’t bring themselves to let students deal with even a tiny bit of messy reality.
So that’s what led me to this. I’ll just start off by saying this image probably only scores a C on the WCYDWT rubric but somehow this kind of worked anyway.
Opening question was an obvious one: “Which part of the graph do you notice first?”
After students pointed out the weird downward spike in the middle, I moused over that part and talked a bit about the numbers and what this was graphing. (The site this image comes from has that graph in a Flash applet that gives exact values when you mouseover the graph.) The story went something like this:
This graph shows the number of users connected to the online PC gaming service Steam over the past 48 hours. That spike in users online probably represents about 500,000 really ticked-off customers who can’t get at their online game.
We can see they got people back online pretty quickly. Which is good, because you don’t want to give 500,000 ticked-off time to start posting on forums on a Sunday afternoon. YOU DON’T WANT TO ANGER THE INTERWEBS.
So imagine you’re working at Steam. It’d be really nice to have some kind of system that alerts you automatically when something like this happens, because you don’t want to be at the office all weekend watching this.
So, ignoring the programming for now and just thinking about the math … how can we come up with a system that catches this?
Brainstorming session ensued! Ideas – great ideas! – came from the room and hit the whiteboard. We started off with four big ideas that were just point-form statements:
- goes down too quickly
- drops below a threshold
- below the avg for that time of day
- doesn’t fit the pattern
This made me so happy. From there we turned some of these into something we could calculate; we talked about turning these ideas into “math”. One thing I loved about this brainstorm is that the third item on the list was outside the scope of the class, but at least as good of a solution as what I was guiding them to.
Then we unpacked the big one: “doesn’t fit the pattern”. What pattern? Can we model it? Cue discussion / lecture on trig graphing where I showed them how to construct a sinusoidal model, and afterward we checked how accurate it was. (Not very, but probably enough to fit our task of catching a large drop in user connectivity.)
My own evaluation? This lesson isn’t a great WCYDWT – it required a lot of me talking, and I had to do some storytelling. The question wasn’t short. It got students participating who weren’t normally confident of their skills, but it didn’t get everyone involved.
But I had students asking, nearly begging me at the start of class to wrap up before 7pm to catch the Canucks game. This lesson went straight through to 7:15 and they didn’t even notice until they were a couple minutes into individual work afterward. Something must have gone right.