Over the years, I have managed this activity’s roles differently. Now I just divide groups of four into two pairs. One group does all the manual work (measuring and drawing, cutting, gluing, filling the rice, measuring the volume), while the other does the actual calculus. I insist that I will be helping the groups doing calculus exclusively, and I encourage them to join the group that they feel is their weak point. It is amazing (at my school) how well this works: students who need extra math accept the invitation and join the calculus group, and students with good understanding of concepts foresee the math part, decide they know it already, and volunteer to join the hands-on pair. Not every student needs “another optimization problem”, and those strong math students learn a lot by working on the more open problem of shaping the pyramid.

In the past, I used to ask students designing the physical boxes to “guess what the best dimensions are”. It appears that they have a very good intuition of that, which decreases the value of the project. Now I ask pairs of manual workers to “choose different dimensions” without focusing particularly on the greatest volume.

Sometimes, it is baffling to see how much excellent math students struggle with cutting and gluing. Some of the most pitiful pyramid boxes I’ve seen over the years were done by “straight-A” students. It is certainly not a waste of time for them to go through that task.