Category Archives: Related Rates
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
Category Archives: Related Rates
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!
Posted by Bowman Dickson
As the last big topic to end our first term, we explored Related Rates in our AP Calculus class, one of the first topics that involves really in-depth, complicated problem solving (which scares the bejesus out of a lot of students). I did some new things that worked and had some ideas on how to improve what I did, so I wanted to write them down now to reflect, especially because (for some odd reason I go into this topic in about 2 months in my very differently sequenced non-AP course).
1. Visualizing the BIG IDEA Behind Related Rates Problems
I think that one of the hardest things for students to do is to visualize related rates problems. And the big idea behind them that makes them interesting problems in the first place – mainly that there is this relationship between different aspects of an object or situation and because of this relationship, when one changes, others do too. However, these things often don’t have linear relationships, so even though they change relative to each other, one changing at a certain rate doesn’t mean that another will change at a constant rate too. Right? Kind of wordy, but this is what I’ve always views as the important idea. Students tend to see them as more static situations and because they don’t have the big (changing) picture, get totally lost in the calculations. This is totally the kind of thing that gets brushed aside in my AP class because “we don’t have time,” but it’s wholly unsatisfying and not good teaching – I find myself cutting things out that I would never cut out normally just to keep up with my curriculum map. Sigh.
So I was looking for good (and let’s be honest, quick) ways to help students visualize these problems. I introduced Related Rates with the simple GeoGebra applet above (click to go to the dynamic view). As time increases, both balloons blow up, but one blows up with a constantly increasing volume and one blows up with a constantly increasing radius. I gave them the applet and gave them about 10 minutes or so to figure out everything they could about rates with the two balloons and pick which one fits the actual situation of blowing up a balloon (I gave them actual balloons to blow up too). Through this, we talked about the idea of how in one object, many things can be changing with respect to time (so it’s 