Key Design Project

Links to Activity

Main Concepts



Volumes of Revolution, Disk Method, Washer Method


option 1: 80 min in class + 260 min of classwork or homework

option 2: 80 min in class + 520 min of classwork or homework

(See Notes at the bottom for details)

Summary of Activity

The idea is to design a key that fits given requirements. All requirements about the key are based on an imaginary key lock that the key should fit in. The key lock has a defined volume and defined shapes. In particular, in terms of Calculus, students will need to decide on the shape of their key and use integrals to find volumes of revolutionStudents also graph each part of the key on a computer software. Finally they create the key itself to scale using cardboard or Styrofoam board. After the work is done, students will have to write a reflection about the process of creating the key and show what they have learned.

Notes and Insights

This project is designed to be presented to students after the unit on integrals. It is a long-term project which will enable students to demonstrate what they have learned about integrals and volumes of revolution and apply it. Beyond Calculus, this project will also improve students’ sense and understanding of geometry, and their skills with graphing softwares. Moreover, the constraints of the problem allow for infinitely many solutions. Each student/group will have to design a solution that satisfies the expectations. At the beginning, students may feel stumped at the problem because of the abundance of choice which is unusual in math exercises. In the end, they enjoy the openness of the problem and the unicity of their solution.

It is possible to let creative students create bit-shapes that are different from the prerequisites, but it does take a lot more time to grade.

Some types of Styrofoam are very hard to cut neatly. Choose your Styrophoam well or use good quality cardboard.

The time difference between the two “options” of duration accounts for group work or individual work. At KIS, we present this project after the AP exam and do it in the last weeks of school. For seniors, we form groups of 3 and they finish before graduation. What is interesting is that the members of the groups are entirely independent (see notes below for more information). Juniors work on the project alone and complete all tasks with extended time since school ends later for them. That works even if juniors and seniors are in the same class.

If this project is done in groups of three, the task assignment can be done as follows: (1) a mathematician who finds all the equations, makes sketches, calculates volume, then takes over the craftsman’s work and puts equations into Grapher, finally writes equations with Mathtype (2) a designer who starts by creating the key head in Grapher, buys the material and cuts the key in Styrofoam or cardboard (3) an editor who reads the rubric carefully, works on the final product (in our case an ibook), explains the process, takes pictures, interviews the other two students to write a reflection, etc. We have to agree that this project doesn’t have the same “calculus”-impact on students depending on their role. In that sense it is better that students work individually on all tasks. But it has value for all students. We can decide to build on their strengths, or to work on their weaknesses. At KIS, given that we do this project after the AP exam, I usually opt to build on their strengths.