What's the biggest box you can make out of an 8 1/2 x 11 piece of paper? Go.

That's the question I posed to my precalc classes yesterday. The first class kind of just stared at me for a minute, then thought of some questions.

1. What does biggest mean? (we decided volume, not surface area)

2. What's a box? (um....)

3. Does it have a lid? (we decided no for simplicity's sake... but today figured it out with a lid)

4. What shape? (we went with a rectangular prism, but one boy wanted to do a cylinder, so he did! love him.)

As a class we came up with a net of the box on the board, labeled some values, and came up with a formula for its volume. But how do you find the maximum volume? Someone suggested graphing the volume equation... but then what? What does a cubic look like? How does it have a maximum if it goes up forever?

That lead us to a discussion on the domain of our volume. The polynomial itself will go to all reals, but that's not applicable for our box. Once we had a good window, we talked about how to find the maximum value (without tracing on the calculator) and what that ordered pair represented for our box.

At this point, everyone grabbed a piece of cardstock (cut in different sizes so they had to work independently), measured, and proceeded to maximize their box. (Question - what do I do with 90+ small open-lidded boxes?)

Today I had them do some checking for me. Everyone grabbed a constructed box from yesterday, measured it, and entered their values into a google spreadsheet that I'd set up. It was a little crazy with 14 people trying to enter their numbers all at the same time (because they kept typing over each other) but it was fun to see all of the data come in. Then they were to come up with the maximum volume for a lidded box (after we drew a net and found a volume formula together)... though I'm not sure the net we came up with would maximize the paper, it was fun to see them think through what it would need to look like!

I actually did this project at the end of last year when we were doing limits and derivatives in precalc. They then had to find the derivative of their volume function and determine where it would have a maximum (by setting = 0). I'm going to keep one of the pages they filled out leading them through the process and give it back to them in May so we can do the limit and see that it's magically the same value that the calculator gave us! Crazy how that works.

(Much thanks to Luajean Bryan, who showed us this project at NCTM!)

Great idea to tie what usually is a middle school activity to something that is done in calculus. Did you ever follow up with your class last May?

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Last year I did this activity with them at the end of the year and tied it into limits. This year I did it at the very beginning of the year (so we could talk max/min, domain, etc) and saved their results to give them back when we get to limits in May. I just wonder how many of them will remember making the box in the first place?!

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