Lesson 22

Volume As a Function of . . .

Lesson Narrative

The purpose of this optional culminating lesson is to give students more experience working with non-linear functions that arise out of the work students have been doing with the volume of cylinders, cones, and spheres. In the first activity, students reason about how scaling the radius affects the volume of a sphere, similar to their earlier work considering how volume is affected by changing one or two dimensions of cylinder or cone.

In the second activity, students work with three different functions (represented three different ways) showing the height of water in three different shapes as a function of the volume of water. They consider questions such as:

  • Which container is largest?
  • At what volume of water poured is the height of water in the containers the same?
  • For what range of water volume poured does a particular container have the greatest height of water?
  • How do the representations show the maximum height of each container?

Learning Goals

Teacher Facing

  • Describe (orally) how a change in the radius of a sphere affects the volume.
  • Interpret (orally and in writing) functions that represent the volume of a sphere, cone, and cylinder, using different representations.

Student Facing

Let’s compare water heights in different containers.

Required Materials

Required Preparation

Provide access to straightedges for the activity A Cylinder, a Cone, and a Sphere.

Learning Targets

Student Facing

  • I can compare functions about volume represented in different ways.

CCSS Standards