Long ago when I took Pre-Cal, there was this problem that appears to be the classical textbook way of trying to throw some "real life" in your face:

I've seen it in our Algebra II textbook as well. My first year I think I put something like this on an assignment when I was going to have a sub. Naturally, no one figured it out or found it amazing. The whole point is to attach some meaning to the pixelated graph of a polynomial and teach something about maximums. Coincidentally, I saw this a few weeks ago:

Problem solving workshop Q: "How do you incorporate this strategy into topics such as polynomial multiplication?PLEASE ANSWER THIS ONE!"

— Dan Meyer (@ddmeyer) April 12, 2013

I don't know and did not pursue more context to this, but I assume the thought was how do you jazz up something bland and skill based like polynomial multiplication. I remembered that in the archives of 3-ACT Math is the Popcorn Picker scenario. The scenario in popcorn picker is actually something we used to do when prepping 10th graders for their state tests. But I was a little inexperienced to exploit all the value out of the activity.

Anyway, those thoughts in mind I returned to the box building scenario. We were about to start polynomials in Algebra II and we were in the middle of 11th grade testing weeks. Having to see both my Algebra II classes immediately following the math test, I gave them a bit of a chaser rather than doing anything intense.

### Task

Presented with two pieces of colored paper, can YOU be the one that makes a box lid with the greatest volume? Other than telling them to cut a square out of the corners, the design was up to them. At the time, I'm not sure any of them knew they were being tricked into doing math.

Lots of contemplating. Lots of careful measuring. I gave them two pieces of paper to see if they had a better idea after trying it one time. I gave them 15 minutes or so to do the designing then instructing them to measure the resulting length, width, and height. Then jot the volume on the box for discussion making special note of the height.

### Analysis

About 25 minutes in at this point and I have a list on the board to compile results. A lot of pride from the ones who are putting up big numbers. Then a trend emerges. It seems the kids putting up the high numbers all cut squares that were around 1.5 inches on each side. That's weird...

Then comes the math where we talk about what they did was simply a brute force solution to the maximization of (11-2x)(8.5-2x)x:

A good discussion about how this kind of thing becomes a serious discussion in business boardrooms, how cereal companies keep prices the same but shave a few centimeters off the width. And why the heck does my chip bag look so empty when I open it?

And finally, how the math solves this problem quickly instead of having a whole lab full of students hack out box designs one by one. The next day we dove into the skills associated with multiplying polynomials.

Try the textbook version if you want.