## How Long Is A Bad Ellipse Question?

Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.