One of my favorite things to do in Precalculus Honors is ask my students, “How do polynomials behave?” They started working on this deceptively simple question in early November and are still plugging away at it. In the process they’ve had to come up with a definition of polynomial, decide what we mean by “behave,” figure out a way to tackle the question, and then look for patterns from which they can develop conjectures that they can make convincing arguments for. In other words, they’re being mathematicians, not just doing math. Some people just stare at me in disbelief when I tell them that my honors class is in its seventh week of working with polynomials, but if didn’t spend this much time on it or do it this way, then we wouldn’t get classes like the one we had today.
A couple of groups in the class had been working on the patterns they were seeing for turning points on a graph, which are things like those labeled “A” and “B” in the graph below:
They had looked at a lot of graphs and were pretty convinced that for a polynomial of degree n, there should be n-1 turning points. (The truth is that there will be at most n-1 turning points, but I wanted them to explore their conjecture themselves, not just wait for the right answer from me.)
While this conjecture made sense to them, it led to some challenges in addressing curves like this:
This curve has no apparent turning points, but they knew that it needed to have some in order for their conjecture to hold, so they spent a lot of time trying to figure out where the turning points were. They come up with something like this:
When they looked at points “A” and “B” they knew that they were not maximum or minimum y-values, which they had figured out happened at turning points, but they felt like there was a distinct change in the shape of the graph at these points and so these must be the turning points.
As they were presenting their work to the class, there was a lot of discussion about these non-extreme-value turning points. Everyone was having trouble figuring out exactly where they were or how to define them. One member of the group said that she wanted to think of them as the places where the rate of change changes, but she knew that wasn’t correct because she knew the rate of change (slope) was continuously changing. This led a classmate to suggest that maybe they could pick a couple of points that were close together, find the slope between those points, and use that to get a sense of the rate of change. Prompted by another classmate, he went on to say that by doing that in a couple different places they’d be able to compare the rates of change, and by keeping the points really close together they should get at least a decent idea of what the rate of change was.
In other words, he was suggesting approximating the rate of change by calculating the slope of the secant line between points that were close together.
In other words, he was about 20 minutes from developing an intuitive definition of the derivative.
I do a lot of student-driven inquiry work with my classes because to me it’s like a road trip – we have a general sense of where we’re headed, but sometimes we take a detour because something interesting catches our eye. This was one of those times. What I really wanted to do was let three or four students starting playing around with this question of approximating the rate of change. To do so would have been an amazing, fun experience that would have let them continue to develop as mathematicians. Unfortunately, it also would have meant that I wouldn’t have gotten to something like sequences and series, plus I would have stolen the thunder of the AP Calculus teacher. Still, it does make me wonder – how flexible could our courses be so that we can follow a line of inquiry without worrying about going past what’s on our syllabus?