A million thanks to Zoya Voskoboynikov and her two sections of “regular” calculus at Lick for letting me come litter their otherwise pure math class with actual data. Of course, it was after the AP exam, and these are last-quarter seniors, so my being there didn’t interfere with any learning they needed to get through.
It worked great. It had what I most wanted: the aha experience of arriving at the destination by another route. Fortunately (and unsurprisingly), none of these successful math students remembered the theorem from the geometry class they took as frosh.
What we did
- I set up the problem and had them predict, informally, what the function would look like. The main purpose of this is to orient students to what we’ll be measuring and to the idea that if you measure two quantities, you can see their relationship in a graph.
- Using circle handouts I made (so everybody would have the same-sized circles), students measured and plotted (using desmos and sliders) and came up with the formula.
- They used that insight—no matter what chord passes though a point, the product is the same—to find the diameter of the round tables they were sitting at without getting out of their chairs. That is, they used the 30cm rulers as a chord, splitting it 15-and-15; measured the “gap” in the middle (about 2 cm); and calculated the remaining big part of the diameter, which was (15)(15)/(2), or about 112 cm.
- I stepped them through the proof (I review it in the EGADs materials, where you will find other handouts (which we did not use) and the follow-up activity (which we did not do). This was where the epiphanic ahas were spoken; these relatively mature students gasped a little when they saw that we could arrive at the same result by plotting a function as we do by setting up proportions in similar triangles. It felt to me as if they were getting at least a momentary feeling that somehow it all hung together.
Unanticipated Cool Things that Happened
There were many; here are two I remember…
In one class, when students had started measuring and plotting, there was a question whether everyone at the same table should choose the same point for their chords to go through. I said, basically, decide for yourselves. One kid asked, “but if we choose different points, won’t we get different functions?” Great question, I said, and wrote it on the board. By the end, they had seen how some people had different values for the parameter in . So I asked the question: did you get different functions? Simultaneously, some students said “yes” and others said “no.” So we got to discuss whether, say, is a different function from . In the language of that school, they allowed as how they all had the same parent function, but had different values for k.
The other yummy moment happened when some students noticed that there was a minimum possible value for the two distances, namely, the shortest distance from the point to the circle. “That should be the value for the asymptote,” they said; then they adjusted their function by transforming it, as in, (for a 2-cm distance). The function did not work, of course, so we got to discuss the difference between the idea of an asymptote and the idea of a domain restriction.
And this was only the warm-up
Well, it took most of the 75-minute block. But the purpose of running this activity was partly to get students used to data, using desmos, and generally changing their way of looking at things. The payoff was our subsequent work on the Mercator Equation. Stay tuned!