An Anchor Problem for Riemann Sums
I like to start most new units in Calculus with an “anchor problem,” a common sense, every day problem that motivates new techniques and serves as a base that you can constantly refer back to. Some that I have used in the past, to varying degrees of success, are Infection for Inflection, Your Speedometer and the Intermediate Value Theorem, and Predicting Stock Prices with Differentials.
For Riemann Sums, and integration in general, I use the question that really inspired integration in the first place: how do you find the area of an irregular shape? I tell my students:
You work at the glass company. You are given the task of replacing all the glass on the front of this beautiful building, the Duxford Aviation Museum. How much glass do you need? All we know is that the building is 90 m long and 18.5 m tall in the very center.
(This task was partly inspired by this post from Shawn at ThinkThankThunk).
I have it printed on two sheets of printer paper for every group (so big enough to draw on and mark), and I give them 10 minutes to come up with an estimate. Every group writes it on a piece of paper, and then I put it in an envelope. About a week and a half later when we learn the definite integral, we calculate the actual area (using a parabola fitted to the top of the building) and the winner gets…. well nothing. But I announce it at least?
Most students struggle a bit at first and then eventually just start to try something. Some students try some sort of bizarre modified equation for the area of a circle (which I always find really interesting), some turn it into triangles, but most use the maybe-not-that-subtle hint that the window is broken up into square panes.
Right after they are finished making their predictions, we discuss. I ask them what their strategies were and how they could have made their predictions more accurate. I try to get them to come up a couple of points (that sounds manipulative):
We took an irregular shape that has no simple geometric area equation and turned it into a shape that does have a simple geometric equation.
We split up a larger shape into a bunch of smaller shapes to be able to do this.
The smaller our shapes are and the more of them there are, the more accurate our estimate would be. In fact, if we could use infinitesimally small shapes, we could be perfectly accurate.
I think that this activity really shows them how difficult the problem that we are trying to solve is, and primes them to know why we set up Riemann Sums the way that we do, but to be unsatisfied with this solution to the grand area problem. Prepped and primed for Riemann Sums, but with some foresight to know where we are going.