By Tim Hickey, posted May 9, 2016 —

Find the radius of a right circular cylinder with a volume of 100 cubic milliliters and a minimum surface area. Bored yet? I’ll bet many of my students were not terribly inspired by this problem during my first year of teaching calculus. For a math purist, the problem is interesting enough. It is an optimization problem with the surface area as a primary equation and volume as a secondary equation. But for teenagers trying to figure out why they are learning applications of derivatives, the problem is lacking. So, the second year I taught calculus, it went something like this: What is the radius of a lidless 16-ounce coffee cup in the shape of a perfect right circular cylinder that has a minimum surface area? A few more students were engaged, but the problem still lacked “punch.” So in year three, I asked them to build the cups. Here were the instructions:

You are starting a coffee cup company that produces 16-ounce coffee cups in the shape of perfect right cylinders (without lids). To save money on costs, you want to use the minimum amount of material to create your cups. Show how to use calculus to determine the dimensions of your coffee cup. You may use a calculator along the way, but you may use only the basic functions of addition, subtraction, multiplication, division, and exponents. Then build your cup. (In building your cup, the following conversion might be helpful: 1 ounce = 29.574 cm³.)

I handed out some poster board and scissors, and almost all the students became highly engaged. Interesting questions regarding units and the meaning of the mathematics arose in the process of building the cups.

I did this for another year or two, but then I decided to take things one step further. I allowed students to build the cup using any material of their choice rather than handing them poster board. Some went down to our “maker space” and used the 3D printer, some went to the woodshop and used the woodworking tools there, and some went to our art room and used ceramics. The joy and inspiration among my students when solving this problem has been palpable ever since. In working the problem, they create something that they can observe, touch, and own that demonstrates why they are studying calculus.

On the basis of this experience, my advice is this: Find and seize opportunities for your students to build and create. You’ll be surprised by what they—and you—learn, and joy and inspiration will abound in your classroom!

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