Those of us who teach AP classes in the northeastern United States face an interesting problem every year in May. The AP exams are administered, signaling the end of the course, but we still have another month of classes. So… what should we do?

This is one of those good problems in the sense that there are too many options. Many teachers will just truck on through into whatever seems like the logical next topic. Another approach is to get into some sort of enrichment activity. The appropriate choice will probably depend on more specific context, but I think it’s far more interesting to consider the enrichment opportunities. Which brings me to…

The Optimization Project!

One topic that gets shortchanged in my AP Calculus courses is optimization. While College Board’s curriculum outline does mention optimization, in terms of the test that means min-max problems for a given function. But in just about every calculus textbook that has ever been published, “optimization” refers to contextual problems where a situation is presented that requires the student to come up with an equation and possibly some constraints before the calculus kicks in.

These kinds of problems can be quite frustrating for students. It takes a fair amount of practice to learn how to create the mathematical model. And while I recognize a desperate need for more mathematical modeling in most American k-12 classrooms, I still struggle with finding ways to fit it in. The blank canvas of the post-AP exam classroom is a great opportunity to address this. I first saw the idea for this project on the AP Calculus Teacher’s group on facebook, which is an incredible resource and I encourage every AP Calc teacher to join.

The Details

Before starting the project, I usually spend a day or two discussing some classic optimization problems. At the end of that sequence, I give them a homework assignment: bring in some sort of rigid product container. Examples could be a cereal box, soup can, or pringles cylinder. Some bad examples would be a tube of toothpaste or bag of candy.

The next day, I provide them with rulers and tape measures to find the dimensions of their package. Their task is to find the optimal dimensions that would minimize surface area (e.g. packing material) and preserve the volume.

At this point, everyone embarks on their own adventure. Many students will choose some sort of rectangular prism, like a box of pasta, and quickly realize that their situation doesn’t have enough obvious constraints. With completely independent length, width, and height parameters, how can they ever convert the surface area into a single-variable function?

Now they have to come up with some more assumptions. This is where it can get pretty interesting. Perhaps the student who brought in the box of spaghetti decides that the box’s length should be equal to the length of a piece of spaghetti. And the cereal box should have a certain amount of area on two opposite sides to accommodate the artwork. The pringles can radius seems like it should be restricted to fit a single chip… unless they want to have four stacks along side each other. Not only do students with the same shaped package have good reasons to impose different constraints, but even students with the exact same package might eventually decide on totally different priorities.

This idea can really go as deep as the student is willing to take it. Maybe they want to create a cost-benefit analysis for the marketing advantage that would result from using more material on the cereal box to create those larger sides. Meanwhile, another student might not even want to think about the size of the hole in the tissue box they brought in.

Finally, each student writes a letter to the company that manufactures their product. In that letter, they explain their work and findings, which includes either a recommendation for an improved design or a congratulatory message on their remarkably optimal design. I mail them out using the school as a return address, and some of them even write back!

Buzzwords and stuff

One thing I love about this project is that it naturally tailors itself to each student’s abilities and interests. The differentiated instruction occurs naturally, as each student makes a decision for the detail in their model. And the best part is that they will all learn an important aspect of mathematical modelling, which is that the models always start rough and can be refined later.

I have also noticed a higher sense of ownership over their work because everyone has their own situation to work with. In discussing their work, they say things like “my box”. This seems to result in a more personal connection to their work than if I had just brought in a single container and had them all base their work on that.