Volume as a Function of . . .

5 min

Narrative

In this Warm-up, students reason about the volume and dimensions of a cylinder based on information given about a sphere of the same height. The purpose of this Warm-up is for students to recognize when they do not have enough information to reach a single answer. While they can determine the height of the cylinder, the radius is unknown, which means the volume of the cylinder could be anything.

Launch

Give students quiet work time, and follow with a whole-class discussion.

Student Task

A cylinder and sphere have the same height.

  1. If the sphere has a volume of 36π36\pi cubic units, what is the height of the cylinder?
  2. What is a possible volume for the cylinder? Be prepared to explain your reasoning.

Sample Response

  1. 6 units
  2. Students could answer anything here, as the radius is undetermined. Knowing the height, though, creates a function with variables VV and rr. Students just need to pick an rr, and they have a volume that works. Going the other way is harder but possible, and working with square roots is left for a future unit.

Synthesis

The purpose of this discussion is to make sure students understand that the volume of the cylinder could be anything. Ask students to share how they calculated the height of the cylinder. If any students made a sketch, display these for all to see.

Select at least 5 students to give a possible volume for the cylinder, and record these for all to see. Ask students,

  • “Why do we not know what the volume of the cylinder is?” (We don’t know the radius, only the height, so the volume could be anything.)
  • “Is knowing the height of a sphere enough information to determine the volume?” (Yes. The volume of a sphere is based on the radius, which is half the height.)

Tell students that in the next activity, they will investigate how changes to the radius of a sphere changes the volume of the sphere.

Anticipated Misconceptions

Some students may not understand why the second problem talks about “possible” volumes if they assume the radius of the cylinder is the same as the sphere. Ask these students to explain how they found the radius of the cylinder so that they notice that the problem never gives that information.

Standards
Addressing
  • 8.G.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
  • 8.G.C.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

15 min

25 min