Friday, November 9, 2012

Ant on a Wheel

The hook for circular functions I had envisioned turned out to be pretty great, even better than I had imagined. The problem about the ant on the wheel was a hit, and really brought out some nice misconceptions from the kids. When I passed out the worksheet, I told them that they needed to estimate intelligently (not randomly guess) for the values in the table in #1, and that it should take a bit of time to complete if they were doing it correctly. They took their guesses, and most of them wrote things like

t = 0 --> h = 60
t = 0.5 --> h = 40
t = 1 --> h = 20
t = 1.5 --> h = 0
t = 2 --> h = 20
t = 2.5 --> h = 40
t = 3 --> h = 60
t = 3.5 --> h = 40
t = 4 --> h = 20
t = 4.5 --> h = 0

Then, when we discussed as a class, I drew a big wheel on the board and a vertical scale next to it to show height up to 60. I asked the kids where the ant was at 0 seconds, and everyone pointed to the top of the wheel, so I labeled it 0 sec. And then I asked them where the ant was at 3 seconds, and at 1.5 seconds, and we labeled those quickly as well since those were "obvious" to the kids. Then, less obviously, I asked them where on the wheel the ant was at 1 second and 2 seconds. To do this, they figured out that you have to divide the wheel up into thirds. And then we can further divide this wheel up to see where the ant is at 0.5, 2.5, 3.5, ... seconds.

By the end of our discussion, we got a diagram that looks something like this (on the board and also on their papers):

Each time, we referred horizontally over to the height scale that I had sketched on the board, and we estimated how high up the ant actually is. In doing so, the students noticed that they had to change the heights in their table. See below:

t = 0 --> h = 60
t = 0.5 --> h = 45
t = 1 --> h =15
t = 1.5 --> h = 0
t = 2 --> h =15
t = 2.5 --> h = 45
t = 3 --> h = 60
t = 3.5 --> h = 45
t = 4 --> h =15
t = 4.5 --> h = 0

One kid said, "But that doesn't make sense! I had divided it into equal parts before and that made sense." So, we had to discuss as a class that the ant is moving mostly vertically between certain parts of the wheel (ie. between 0.5 and 1 second, or between 2 and 2.5 seconds), and mostly horizontally between other parts of the wheel (ie. between 1 and 1.5 second, or from 1.5 to 2 seconds). Finally this kid is convinced that the height change is not a steady rate at all times.

So, with this consensus, I asked the kids to take out their graphing calcs and to create a scatterplot with this data (continuing the pattern all the way to 9 seconds, by 0.5 second increments, in order to reinforce their understanding of this circular pattern around the wheel). We looked at the graphs, and tada! It looks like a wave. Some kids were able to see it immediately, while others needed me to draw the scatterplot on the board and to connect it for them in order for them to see it.

Then, I asked them what type of function this was, and they were able to vaguely say sine or cosine (but they weren't sure which). We didn't get too far, but we started finding the equation of this function by hand for both sine and cosine functions and discussing the meaning of each part of the wave equation, which we will verify afterwards using the calculator's sine regression functionality. (Good time for the kids to practice all kinds of tech skills on the calculator, which they may or may not need for their internal assessment this year, depending on their internal assessment topics.)

Anyway, just thought I'd throw it out there. I didn't do super exploratory/introductory stuff this year in introducing waves, since the kids already have seen these equations the year before (in Grade 10), but I thought this lesson hook was really nice for re-introducing waves to them.

Addendum: I have decided that I'll be starting the next class by asking the kids where on the wheel the ant will likely raise two of his legs and say, "whee!" and then we draw it on the board on the wheel as well as on the wave graph, in order to connect this to rollercoasters and an itty bitty physics. On the other side, when the ant is rising rapidly, we can draw the ant hanging on barely, with two of his legs swinging in the wind.

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