Today, for the first time in years, I included a related rates lesson in my calculus class. I had never liked related rates, and when I got my own class and could create my own syllabus, I dropped the topic. This fall I’m at a new school, though, and I decided while revamping my course plans to give related rates another shot.
Background: related rates didn’t sit well with me for a long time before I could enunciate why. Then I learned about the notion of “low-threshold, high-ceiling” tasks, which provide multiple levels of entry for students, as well as a lot of space for growth and exploration. I realized that the classic related rates problems fail both tests. They generally have a high threshold, because students have to understand the entire process of translating the word problem into symbols, then differentiating, then solving, before they have any measurable confidence that they can begin such a problem. And they generally also have a low ceiling, because one the immediate question has been answered, there is no enticement to do further analysis, or even any indication such analysis is possible.
As an example, consider the problem of the sliding ladder. This is included in almost every textbook section on related rates, in almost exactly the following form.
A 10-foot long ladder is leaning against a wall. If the bottom of the ladder is sliding away from the wall at 1 foot per second when it is six feet away from the wall, how quickly is the top of the ladder sliding downward at that instant?Now, that is an incredibly difficult problem to read. Some books may have slightly better phrasing (I decided not to quote any book in particular, so as not to single out just one malefactor), but the gist is the same. Before you’ve even gotten a sense of what the situation is and what’s changing, you’re asked a question involving bits of data that seem to come out of nowhere, and whose answer is completely uninspiring.
Like I said, for a few years my solution was to avoid these types of problems entirely. I had seen too many students struggle to set up these problems and go through the motions of solving them, only to get a single number at the end that showed nothing other than their ability to set up and solve a contrived problem. What I realized while preparing for today’s class is that, when the problem is done, you don’t feel like you’ve learned anything about how the world works. Calculus is supposed to be about change, yet the problem above feels static because it only captures a single moment in a process. A static answer is antithetical to the subject of calculus. Moreover, most related rates problems arise out of nowhere, flinging information at the reader willy-nilly to answer a single question, despite a decidedly unnatural feel to these questions. Unnatural questions are antithetical to mathematics. So I decided to remove these elements of antithesis as best as I could.
Here is the question I posed at the start of class today.
A 10-foot long ladder is learning against a wall. Suppose you pull the bottom away from the wall at a rate of 1 foot per second. At the same time, the top of the ladder slides down the wall. Does it:I claim this version is both more natural and easier to start discussing than the near-ubiquitous original. It almost seems like a question one might come up with on one’s own. It’s clear what quantities are changing, and that there is a relationship between them. The process itself can be demonstrated; I used a ruler and a book, rather than bringing a ladder to class. (How would you demonstrate that instantaneous rate of change in the original problem?) No overly specific information is given. And best of all, the answer is a bit surprising, at least to some. (When I asked my students what they thought after a couple minutes of discussion, about half thought the top would start slowly, then speed up, and about half thought it would slide at a constant rate.)
- slide down at a constant rate,
- start out slowly, then speed up, or
- start out quickly, then slow down?
I don’t claim any originality in this idea. Probably many other excellent math teachers have made exactly this change. I may have encountered it as one of Dan Meyer’s examples, or somewhere else, and it stuck in the back of my mind. I should emphasize that I really hated related rates problems, and I saw little chance of rehabilitating them. I’m glad to have realized that they can be interesting and reveal interesting things about the world, when they are restored to the state of natural questions.
I doubt today’s lesson was perfect. I still probably talked too much and introduced symbols too quickly. But it was good enough that I’m going to keep teaching related rates in my calculus classes from now on.
If you have other examples of this better type of related rates problem, please share in the comments!