Optimization: An Introductory Activity & Project

I switched things around with optimization in calculus this year, and I realized if I had the time, I would spend a month on it. [1] I wonder if this shouldn’t be a crux of the class. Not the stupid “maximization and minimization” problems but finding some real good ones — in economics, physics, chemistry, ordinary situations. There have got to be tons of non-crappy ones!

Anyway, I wanted to share with you two things.

First, how I introduced the idea of optimization to my kids. Instead of going for the algebra/calculus approach, I wanted them to toy with the idea of maxima and minima, so I had them spend 35-40 minutes working on this in class:


I thought it was pretty cool to see my kids engaged. I rarely do things like this, but I did it (I was being videotaped during this lesson… and I had never done it before… and I had the idea to create it the night before…). It was fun! And although I cut the debrief the next day short (ugh, why?), I enjoyed seeing kids engaged in problem solving through various strategies. And there was a healthy level of competition. (The winners for the 1st and 2nd tasks got a package of jelly beans, but they were so gross I threw them out! One student gave them to his rabbit who likes jelly beans, and even the rabbit didn’t like them!) But when it came down to it, it drove home the idea that optimization was something that trial and error is good for, sometimes we do it intuitively, sometimes our intuition is terrible and sometimes it is good, and sometimes we get an answer but we don’t know how to prove there isn’t a better answer (e.g. in problem #3). Some kids liked that this felt more “real world” than this world of algebra and graphing that we’ve been meandering in.

Second, I have allotted a few days for students to work on this project during class (it’s the week before Spring Break and kids are overburdened, so I didn’t want to have them do something which involved a lot of at-home time). They’ve been working on it this week, and I’ve heard some good conversations thus far. (They’re doing this in pairs, and I have one group of three.) The fundamental question is: with a given surface area, what are the dimensions of a cylinder with maximal volume?


Now I don’t quite know how their posters will turn out yet, or whether students will have truly gotten a lot of “mathematical” knowledge out of it. But each day, I’ve had a couple kids say things that indicate that this isn’t a terrible project. (I don’t do projects, so that’s why I’m very conscientious about it.) A few said something equivalent to “Wow, the companies could be giving me x% more creamed corn!” or how they like doing artsy-crafty things. At the very least, I can pretty much be assured that students — if I ask them if there is any question that calculus can answer at the grocery store — will be able to say yes.

Next year I will probably add the reverse component (for a given volume of liquid you want to contain, how can we package it in a cylinder to minimize cost… what about a rectangular prism… what about a cube… what about a sphere… etc.?).

[1] The one thing I found in this book my friend gave me (on science and calculus) was an experiment where you shoot a laser at some height at some angle into an aquarium, so that it hits a penny at the bottom (remember the laser beam will “change” angles as it hits the water) to minimize the time it takes for the photon to travel from the laser to the penny. I almost did it, but deciding to do it was too last minue.



  1. Optimization is a good use of calculus. Could you teach them constrained optimization with Lagrangian multipliers? I’m still pissed at all my math teachers (from high school through grad school) for never teaching me that very useful technique. I learned it from a student (once I was an engineering professor) who’d had a much more practical math education than mine.

  2. Sam-
    The can thing reminded me of something we looked at with my Algebra kids last year (although super-informal). Diet Pepsi had the ‘skinny cans’ out as a promotion for awhile (still 12 ouncers, but taller and skinnier). We had an impromptu discussion towards the end about packaging as well, because fitting different sized cans into 24 per case cardboards also affects the packaging cost, even though aluminum has to be more expensive per unit. Maybe you could add that on (the cardboard packaging as well as the ‘regular’ vs. ‘skinny’ discussion.

  3. Sam! I love that optimization sheet! What a good idea to present optimization from a variety of standpoints. I like how the problems are all hands-on and steadily increase in mathy-ness. I’m interested to hear how you’ll make the transition from this experimentation approach to algebra and calculus. I often find that it’s hard to get kids to work through a problem algebraically if they feel like they can just as easily (or more easily) do it with trial and error. That said, I think I’ll try doing some of this in my algebra 2 class! Thanks for the great idea!

  4. Hi Sam, I used your Cans Cans optimization project in my regular Calculus class this year.

    Every student only had to do one can, instead of 5, for time sake. I gave everyone in class a different can so that at the end we could pool the posters together and generalize.

    Something that I added to your project that I thought was useful was this:
    Instead of a picture of the can, students had to draw their can exactly to scale. Then right next to it they had to draw the optimized can. This way they could see how the can changed. (shorter and wider or taller and thinner). It was then easy to see what kind of cans were more optimized and what less.

    They also had a write a short conclusion and explain why the makers of their can decided to make it this way and not necessarily optimize for volume.

    Thanks for sharing your work!!

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