I don't know that I ever wrote a "The Box Project (part 1)" post, but whatever. It's May.
On one of the first few days of school I give my precalc kids a randomly sized piece of rectangular card stock and tell them to make the biggest box possible. We have the "what is a box" conversation and then talk about the algebra involved.
It's a great review of all kinds of algebra... polynomials, domain, extrema, etc.
I have them fill out a summary sheet of their findings. They include the dimensions of their original paper, the dimensions of their box, and the cubic expression representing the volume of their box.
Then I put their papers in a folder and file it away, hoping that I'll find it on this day in May. So far I've been lucky and have located them every year. But let's not jinx myself.
We've been finding limits and discussing the difference quotient (and how it will help us find slopes) for the past couple of weeks. Today I gave back their box summary sheets (which they were amazed to see) so they could apply the difference quotient to their volume functions.
Amazingly, after setting their result equal to 0 and solving, one of the values was eerily close to the value they'd determined would create the maximum height of their box.
(Yes, we talked about why we were setting the derivative (although they don't know that term yet) equal to 0.)
And then I let them talk me into completing just 4 problems on a too-long worksheet that I'd given them. Because it's May.