On an utterly unrelated note: my family and I moved to a different house last summer. Because we have somewhat less space than we did before (much larger yard, but about 200 square feet less living space), we have been somewhat slow to unpack. However, I finally got around to pulling my bike out of the shed and getting it tuned up and ready to go. So, for the first time in almost a year, I was able to ride into class last night. It was quite pleasant, despite the higher than 100 degree temperatures.
What I Taught
As seems to be my preference when teaching computationally-heavy sections, I spent about 30 minutes going over homework problems, 20 minutes on a “quiz” (really a group exercise), and the remaining 45 minutes on new material.
The new material boiled down to two problems involving related rates. I really like these kinds of problems, as they are perfectly suited for an introductory calculus class. They drive home the point that the derivative can (and often should) be thought of as a rate of change, rather than the slope of a tangent line, and they provide an example of a computation that really cannot be done without calculus.
Both of the problems that I presented were pretty contrived: in the first, a perfectly spherical balloon is being inflated at a known rate, and the question asks how quickly the length of the radius is increasing. The second problem has two planes leaving an airport in orthogonal directions at the same time, and asks how quickly the distance between them is growing after a certain amount of time.
In both cases, we have some known variable (the length of a radius, how fast the volumen of the balloon is growing, the distance that the planes have traveled, how fast the planes are traveling, and so on), and are asked to find an unknown that can be expressed in terms of derivatives. Once the problems are set up, the computations are pretty straight-forward.
I had wanted to talk about the falling ladder paradox, but I didn’t get there today. Perhaps in the next lecture…
Once again, group work proved helpful. While I provided more guidance this time than last, most groups managed to get the intended answers quite quickly. There were some issues with rounding—while the problem was presented with only three significant figures, I had several students give answers to the full precision of their calculators—but this proved a useful “teachable moment” (man, I really hate that phrase for no rational reason).
The related rates problems generated quite a bit of good discussion, and (better yet) from people who do not normally speak up in class. While we have been building toward such problems since day one (multiple interpretations of the derivative are mentioned, computations of derivatives are rampant, and implicit differentiation gives the idea that the derivative is a variable that we can solve for), the pieces all come together to deal with related rates problems. In this section, we really treat the derivative as a mathematical object that can be manipulated algebraically and solved for like any other variable, and reinforce the idea that it can have real world meaning.
I saw a lot of wheels turning, and feel confident that this clicked for many students. They may still have difficulty with related rates problems, but they are more clear on some of the more fundamental concepts, such as implicit differentiation and the application of the chain rule.
What Didn’t Work
I rather regret stating my airplane problem in knots, rather than miles (or kilometers) per hour. My intention was to note that understanding the units is really important—for instance, dimensional analysis is an excellent way to check the sanity of an answer. However, while knots are a natural unit for measuring airspeed or groundspeed, they are unfamiliar to students, and the lack of a “per hour” dimension proved to be really confusing—”What is a knot? Distance?” I wanted to emphasize units, but it would have been better to stick with miles per hour. I could have had the same discussion in half the time and lost far fewer students.
In the same vein, the “quiz” that I gave asked students to estimate population based on an exponential model. As noted above, several groups of students worked with the full precision of their calculators and reported results to 12 significant figures. I noted that the problem was only stated with three significant figures, and so the final answers should not be more precise than that. Looking through the “quizzes” after the fact, it seems that these students have never been told about significant figures, and have no idea what they mean. Half the class took it to mean three decimal places, while the other half seems to have ignored me entirely. I am only exaggerating a little. I suppose that I should spend some time on this topic, but it is so far outside of the curriculum that I need to get through that I don’t know when it is going to happen… I kind of regret broaching the topic at all.