Presentation to Algebra students

Today, I presented to Ms. Colwell's algebra students - in two classes, actually. Normally I'm in her fourth hour, today I was in both her third and fourth hours. The presentation was on optimization, which ties into what they've done recently with linear programming. I showed what linear programming versus non-linear programming were by building a model with foamcore, hot-melt glue, and floral foam. The model is shown at the left - the blue and gold model is the linear programming model, and the blue and green model is the nonlinear one. (Unfortunately, they don't make floral foam in U of M colors!) I showed them that if you try to find the minimum, in the linear case, there's only one point - drop a marble in, and it always rolls to the same spot. If you drop a marble into the nonlinear one, it can go to different places, either internal or on the boundary.

The students had a worksheet that I'd made up, and the answers to the questions were all in the presentation that I gave. I explained how to set up an optimization problem - choose the system of interest, get an objective function, select variables, formulate constraints - and a little bit about how to solve them. Next, I gave them an example of a real optimization problem. Most problems I've done have been pretty technical, and a bit dry, so I got information from a lab-mate of mine. Tahira Reid and two other students in ME 555 last winter did a project that involved optimizing the hair of African-American women, involving length, styling, and damage from various sources. Tahira was generous enough to give me a copy of their slides and her permission to show them to the students, so I selected some of her slides and went over the goals of the project and some of the issues involved. Obviously, a lot of the math was too advanced for them, so I didn't dwell on it in any depth, just showed them how the problem was set up.

Next, they turned over their worksheet and there was an exercise we did together: make up a new optimization problem. I wasn't sure at first whether it would be better in small groups or as a class, but we ended up doing it as a class. In the third hour algebra class, they came up with several ideas - optimizing a guitar, for length of time between tuning; optimize life in general (that would be a rather complex objective function!); optimize a textbook, for minimum weight. We set up the optimization of the textbook. They selected quite a few possible design variables (I had asked for two) - thickness of the cover, thickness of paper, size of type used, size of margins. When we considered the constraints, they decided that making the textbook cover resistant to ripping was one constraint, and another big one was readability. In terms of margins, one of the constraints had to be that the margin couldn't be negative. Once the problem was formulated, I asked them whether they thought it would be hard or easy to solve. They decided - and I agree - that this would probably be a fairly easy problem, since a lot of things are linear.

In fourth hour, they decided to optimize popcorn quality. This was a little more difficult, since we had to talk about what makes quality popcorn, and what could be included. Some of the factors that could go into the "popcorn quality metric" were that it doesn't burn, it should all pop, big kernels, and taste. Variables could include the bag's dimensions, temperature, amount of popcorn in the bag, and amount of butter used. Some of the constraints were minimum and maximum temperature, that the bag should be pretty, and that it had to fit into the microwave. When the bag's attractiveness was brought up, I told them about some of the work being done on optimization and aesthetics, and how that could be expressed mathematically in terms of proportions, among other things. When I asked if the problem would be easy or hard to solve, they decided - and again, I agree - that it would probably be a hard problem. The size of the problem was one thing they mentioned. Another was that it was probably non-linear. One person also said that everything is kind of subjective, so how do you know you have the right function? Someone else might make a different function for the perfect popcorn. That brought up a really good point, which I hope they remember - that it's important to start with the right problem before you try to solve it.

Overall, they seemed to be interested. Some of the students really got into the exercise, but they all liked the change - and the fact that, aside from turning in the worksheet I gave them, Ms. Colwell didn't give any homework. The third hour said that if a presentation by me means no homework, could I come back again sometime?

Slippery roads today...

There were quite a few students either absent or late today, due to weather - there isn't much snow, but the freezing rain made the roads a challenge. Many of the buses were late, so attendance in first hour physics was rather low. Today's physics work was reviewing for their test tomorrow. The test is going to cover circular motion and universal gravitation. I looked over the test that Mr. Ambrose is planning to give, after class was over, and I think that it's fair; the students who have been working hard and doing the homework should do just fine on it. After the test, they're going to move on to energy.

In Ms. Colwell's algebra class, they learned how to use matrices to solve systems of linear equations; then, while most students took a quiz, I took one of the students aside who had been absent and wasn't taking the quiz, and went over some of the material he had missed. He hadn't been there when they learned how to solve equations by linear combination, so we went through those problems. First I set up and solved one, then I gave him a problem with slightly different numbers for him to do; then I set up a slightly different problem, more complex, and did the same thing. By the end of the class period, he seemed to be catching on pretty well. He'll still need practice, of course, but that's true of everything.

The students in both classes have gotten to know me fairly well by now, and they're more comfortable just talking. I had mentioned to them, several times, the need to do a "sanity check" and gave an example of how it had helped me to catch an error that could have haunted me down the road in my own work. I had to re-do a fairly big bunch of derivations, but at least I knew where the algebra started going wrong. One of the algebra students said that sounded frustrating, and asked how I kept wanting to do it. It was a serious question, so I told him the truth; that I did occasionally have days when I DIDN'T want to. Everyone has days when nothing goes right and it would be easy to just quit trying. Sometimes work doesn't go right, and the world as a whole seems terribly dark and pointless. But if I do my work right, it will make a difference - if I can develop a better method of doing something, that matters. I told him about the pleasure, when I was working in industry, of seeing a machine that was just sketches on paper at one point turn into a real, physical thing - and that in a special way, it was MINE, regardless of who it was actually sold to. I told him about the special feeling of realizing something that quite likely no one had seen before, looking up at the stars in the quiet night sky, and thinking that at that moment, you know some small thing that no one else does. The rewards are more than worth the difficulties, in the end. I don't know if he totally understood what I was trying to describe, but if he gets a little bit of it, maybe he'll gain more of a joy in learning.

Circular motion and Inequalities

On Friday in Mr. Ambrose’s physics class, the students did an experiment. They ran a string through a tube and connected one end to a weight; the other end was connected to a small rubber stopper. One person spun the stopper in a circle, and the other one timed it. The idea was to measure the period and the radius, calculate the speed and centripetal acceleration, and compare it with the force provided by the weight. Most of them had pretty good experimental data and relatively low errors, once they did the calculations properly. In the rush of doing the lab, there were quite a few careless errors, but they’re smart enough to see that if the results are THAT far off, something is wrong. I helped several of them find their mistakes, and many times it was a decimal point in the wrong place, or reading a number out of the wrong column. Running a “sanity check” on results is a good skill to get in any field, so I think making those errors actually had a use. Today, they moved on to doing problems on curves – things like how fast can a car go around a curve without skidding. They haven’t gotten through banked curves yet, but that will be the next topic, and then they’ll move on to universal gravitation.

In Ms. Colwell’s class, they’ve been working with inequalities, and graphing them on number lines. I’ve mentioned that inequalities are useful in linear programming, and it turns out they’re actually going to cover that. Since that falls into the area of optimization, Ms. Colwell is going to give me a class period next Friday to talk about optimization and non-linear programming. Obviously, I can’t teach them in one class what it takes multiple graduate classes to learn, but I can give them an idea of what you can do with this type of math and why it’s important.