Relating those Rates
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 possible to have dV/dt AND dr/dt and many other things), which was helpful to do before throwing notation up on the board. From this, we talked about how those relationships can be related and did some quick calculations to confirm their predictions of how when dV/dt is constant, dr/dt is decreasing etc. It ended being a great way to introduce the idea and gave us good language to talk about future problems. I spent the rest of the next 3 or 4 class periods just solving this monster packet of problems with them and giving them time to solve problems themselves.
This was a great introduction, but I think that I will do more big picture activities like this with my non-AP class (and less calculation/pen on paper problem-solving), but we did do a few other quick things to help visualize problems. I made some GeoGebra applets based on problems we were solving (see below, click for the dynamic applet). In the future, an assignment I want to do is have my students pick a problem to animate it – I think it would be a great way for them to deconstruct a problem and REALLY understand the relationship.
The last visualization exercise that we did was with the classic ladder sliding down a wall problem (I got this from my AP Workshop this summer). With WinPlot, I made an animation of this situation. To help students see the non-intuitive fact that if one end is moving at a constant rate, the other end moves at an increasing rate, I had the whole class clap every time the end of the ladder passed a tick mark. First we did that with the bottom, which was moving at a constant rate, and with the clap you could really see/hear that it was moving at a constant rate. Then, we did it with the top as it was falling, and with the claps speeding up and speeding up, it was easier to visualize that the rate was different. What I want to do with my non-AP class is get a ladder and model this situation by putting it against the wall and pulling it away 1 ft at a time and measuring how far the top goes down the wall.
2. Finding Ways to Go About Complex PROBLEM SOLVING
The other new thing that I did with Related Rates was that I found a great way to get students to both organize information and get started in the problem solving. For every problem, we simply made a table to organize all the variables and their rates (I know, not revolutionary), like this:
I found that this helped for a couple of reasons. First, the most obvious is that it keeps them organized with all their information, which can totally get jumbled during a Related Rate problems. Second, it helped them figure out what the relationship was that they needed to differentiate – by filling in all the variables in the table, they usually had to use some equation that related all of them and thus stumbled upon the relationship that they needed to use in the problem solving. Third, it helped them decide what variables were actually constants and what variables could be replaced by relationships with other variables (like the height and radius in a cone) – by forcing them to write down the rates for each variable, they are forced to think “Is this thing changing or not?” Then, it was easy to tell them to substitute anything that was constant (or could be related to another variable) into the expression BEFORE differentiating, as it makes the derivative so much easier. I just graded finals and it was really nice to see how many of the students took hold to this idea and made good use of the table (though overall the AP Free Response Question did NOT go well – I guess it’s still November, so that’s okay!).
I’d love to hear any good ideas for Related Rates – I find this to be a really interesting, but REALLY TRICKY topic to teach. I love Sam’s Related Rates Logger Pro Investigation, and might try something like that in the future with my class. I feel like I could spend a whole semester talking about these problems… if only there was more time!