The purpose of this activity is for students to reason about the volume of a sphere with a specific radius using a cone and cylinder with matching dimensions.

In the Launch, students begin with an image of a sphere in a cylinder. The sphere and cylinder have the same radius, and the height of the cylinder is equal to the diameter of the sphere. Students reason about how the volumes of the two figures compare in order to get a closer estimate of the volume of the sphere.

Then students watch a video that shows a sphere inside a cylinder set up like the image. A cone with the same base and height as the cylinder is introduced, and its contents are poured into the cylinder, completely filling the empty space inside the cylinder not taken up by the sphere. Students are asked to record anything they notice and wonder as they watch the video, and a list is created as a class. They then consider what must be true about the relationship between the volumes of a cone, cylinder, and sphere without doing any calculations (MP1).

In the Task Statement, students consider a cone, sphere, and cylinder with matching dimensions. With a partner, they reason about the volume of the sphere using the volumes they calculate for the cone and cylinder along with what they learned from the video.

Monitor for students who discuss either method for calculating the volume of the sphere:

• Subtracting the volume of the cone from the volume of the cylinder
• Making the connection that a cone is of the cylinder, so the sphere must be the that fills up the rest of the cylinder

The goal of this discussion is to compare and contrast the different ways students calculated the volume of the sphere.

Display 2–3 approaches or representations from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?” (They both use the volume of the formulas for a cylinder. One is using the fact that the volume of the cone is of the volume of the cylinder, so the sphere’s volume must make up the other . The other subtracts the volumes we know in order to get the unknown volume of the sphere.)
• “Why do the different approaches lead to the same outcome?” (They are both ways of calculating the volume of the same shape using the fact that the volume of a cone is the volume of a cylinder with matching dimensions.)
• “Did anyone solve the problem the same way but would explain it differently?”

Display this expression for all to see:

Ask students: “What does this expression represent?” (the volume of the cylinder minus the volume of the cone)

Draw students’ attention back to the guesses they made at the start of the activity about how much bigger the cylinder’s volume is than the sphere. Ask students if we can answer that question now. (Note: If students do not explicitly make the connection that the sphere’s volume is the volume of the cylinder, they will have another chance to look at the relationship in the next activity.)

The purpose of this activity is to build from the concrete version in the previous activity to a generalized formula of a sphere with an unknown radius. The previous activity prepared students with strategies to work through this task in which they must manipulate the variables in the volume equations. Students first calculate the volumes of the cylinder and cone in the activity and use what they learned in the previous activity to calculate the volume of the sphere. Finally, they are asked about the relationship between the volumes of the cylinder and sphere, which connects back to the discussion of the previous activity.

As students rearrange the volume equations, they are making use of algebraic structure to reason about the formula for the volume of a sphere (MP7).

Use this discussion to help students see the different ways to reason about calculating the volume of a sphere.

Here is a summary of the two approaches.

The method:

Subtraction method:

Although either method works, there are reasons students may choose one over the other. The subtraction method is a bit more involved, as it requires the distributive property to combine like terms and the subtraction of from 2. It may make more sense to students, however, since it describes the video demonstrating that the volume of the sphere is the difference between the volumes of the cylinder and cone. The method is a bit simpler in terms of manipulating expressions, but students might not fully understand why the volume of the sphere is  the volume of the cylinder.

Invite previously selected students to share their methods. Display for all to see the two different strategies side by side, and ask students:

• “Which method did you use to calculate the volume of the sphere?”
• “Which method do you prefer? Why?”

If time allows, conclude the discussion by inviting students to explain to their partner why the method they didn't use worked.

Add the formula and a diagram of a sphere to your classroom display of the formulas being developed in this unit.