By Tim Hickey, posted June 20, 2016 —

The study of volumes in calculus (or any math class, for that matter) is ripe with potential for students to find joy and inspiration in a math setting through building and hands-on learning. Here are just a couple of the ways I have created fun opportunities for students to apply calculus during our study of volumes over the years.

The Donut problem: I set out a few dozen donuts, napkins, plastic knives, calculators, and graph paper. Students work in groups and are given a slip of paper that reads simply, “Estimate the volume of a donut using the washer method.” The deliberately concise ten-word assignment can take some groups a full 80-minute class period to complete. Groups must cut the donut in half, place one-half of the donut on a coordinate plane, and trace the cross-sections of the donut. Then they must use their calculators to find the equations for two curves that form the boundaries of one cross-section (typically, students will think to use a few points to do two quadratic regressions, one for the top boundary and one for the bottom boundary), find the points of intersection of their curves, find a reasonable axis of rotation determined by the location of the two traced cross-sections, and then apply the washer method to estimate the volume of the donut. Some groups actually think to do all of this on their own; others need hints along the way. But it is always a fun—and delicious—activity.

The Volume project: In Calculus BC, I task students with creating any solid they can dream up and using calculus to find its volume. This can include the disk method, washer method, shell method, or cross-sections on a base, and I strongly encourage students to create a solid that is complex enough to require a combination of methods. In my school, after the AP exam, we do not meet on a regular basis for several reasons: an annual senior trip, our policy of exempting students from school on any day that they take an AP exam, and graduation practice. Thus, because this project is largely self-directed, I tend to assign it after the AP exam; but it could fit nicely in a unit on volumes as well. The directions are intentionally vague, and creativity and fun are emphasized.

What my students have built over the years has simply amazed me. One group of students created a mini submarine on our 3D printer. Another student used the 3D printer to replicate part of her trumpet. Yet another student carved an entire airplane out of wood. These are just a few examples, but the possibilities are endless. The topic of volumes at any level of math, whether calculus or otherwise, provides a great opportunity for students to build. And building is a joyful, memorable, enriching process for kids of all ages.

Tim Hickey is a Nationally Board Certified Teacher and the math department chair at Monticello High School in Charlottesville, Virginia.