__Pyramid Box Activity__

## Links to Activity

Rubric for labs pdf (pyramid box and rolling ball were combined as one project grade)

## Main Concepts

Calculus

Derivatives

Optimization

## Duration

80 minutes in class + 200 min homework or classwork

(Can be extended if students work individually)

## Summary of Activity

This activity is a variation of the “maximize the volume of the box” problems we are used to see in textbooks since Algebra 1 or Geometry. It is a hands-on activity, and uses a pyramid shaped box, which leads to some very nice equations that can all be solved without a calculator. It brings students to put their learning of derivatives in practice. Students will have to build a pyramidal box and measure its volume using rice and a graduated cylinder. Then they will compare their result with the volume they find algebraically. Finally, they will apply their knowledge of calculus to find the dimensions of the box of maximum volume.

## Notes and Insights

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.