Posted by Bowman Dickson
I always struggle a bit with teaching related rates, because the focus of all textbook/AP problems is to calculate how fast something is changing in one single moment in time, which short shrifts the beauty of the topic. The main idea seems more to be thinking about how two rates are, well, related to each other. What must happen to one rate as time goes on if another rate stays constant? That’s what makes that classic ladder problem even remotely interesting (why the hell would one side be moving at a constant rate while the other is speeding up?).
I saw this gif of an automatic pizza saucer a while ago and immediately thought it would be a fabulous discussion piece for this very idea:
Someone who designed this used some sort of calculus, even if it was the intuitive kind! We talked about this in class for 15 minutes or so, and the students that enjoy wrestling with not-so-obvious and applied situations really enjoyed thinking about these types of questions:
- If the pizza spins at a constant rate and the sauce comes out at a constant rate, what has to happen to the speed of the arm?
- If the pizza spins at a constant rate and the arm moves at a constant rate, what has to happen to the rate at which the sauce comes out?
- If the arm of the pizza moves at a constant rate and the sauce comes out at a constant rate what has to happen to the rate at which the pizza spins?
We came to some interesting conclusions about the above, including that one of the situations above is not possible (I think!), which we only figured out because one student and I were vehemently defending two different and opposite things.
Anyone want to try to throw some numbers on this to figure out these very questions?
Posted by Bowman Dickson
One of my favorite ways to start class is by putting out whiteboards with a problem paper-clipped at the top, and names of random groups. I love it most because every single person is engaged in mathematics within 30 seconds of class starting. In fact, students always ask me a minute or two before class starts “can we begin?” They can’t seem to resist the markers and the problem in front of them. Also, I found when I wanted to use whiteboards in the middle of class and put students in random groups that it just ate up a few minutes in each class, so this just feels more efficient (I’m kind of neurotic in terms of efficient use of class time).
Continuing my experiments with different modes of math whiteboarding, a great whiteboard warm up I tried was having them illustrate related rates type situations for objects that are changing in different ways. For example:
A pumpkin grows in a garden…
1. With a constant increase in the radius of the pumpkin
2. With a constant increase in the volume of the pumpkin
Then I had them describe what is happening to the rate of change of the important variables (so if dV/dt is constant, what is happening to dr/dt?). We then had a really good full class discussion where students explained their situation. I think this helped clarify for a lot of students the difference between “V” increasing and “dV/dt” increasing, or how just because “dV/dt” is decreasing it doesn’t mean the volume is decreasing.
This was part of a larger goal of mine to focus on big ideas and deep understanding this year – I’ve always asked students interpretation questions on tests (my final this past term had a crap-ton of writing) but I never felt like I actually directly taught them these sorts of things. For Related Rates, we solve all these problems and come up with all these numbers, but never actually talk about why they are interesting problems – the fact that as one aspect of a situation changes, another may change at a totally different rate, and that there is a relationship between all these rates that explain how things change the way they do. And honestly, I think this little activity made a huge difference – on the interpretation question on the Related Rates quiz, tons of students drew pictures to aid their explanations. 15 minutes well spent!