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?
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?
1 comment:
Diane,
I like the way you allowed the class to construct the problems, and then had them evaluate how difficult it would be to solve the problem. It certainly gets away from the notion of right and wrong answers and puts the emphasis on the process and higher level mathmatical thinking. Choosing the right problem is important!
Carol Cramer
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