Building on work in previous lessons in which students investigated how changing dimensions affects the volume of a shape, in this activity, students scale the radius of a sphere and compare the resulting volumes. They use repeated reasoning from calculations in a table to predict how doubling or halving the radius of a sphere affects the volume (MP8).

Monitor for students using different reasoning to answer the last question. For example, to get the volume of the smaller sphere, some students may calculate the radius of the larger sphere in order to find of that value, while others may reason about the volume formula and how volume changes when the radius goes from to .

The purpose of this discussion is for students to understand that since the value of is cubed in the formula for volume, changing the radius of a sphere affects the volume by the cubed value of that change.

Display the completed table for all to see, and invite groups to share a pair they added to the table along with their responses to the first two problems.

Select previously identified students to share their responses to the last question. If students did not use the formula for volume of a sphere to reason about the question (that is, by reasoning that if the radius is as large, then ), ask students to consider their responses to the second part of the second question, but replace  with .

The purpose of this activity is for students to bring together several ideas they have been working with, in particular, calculating volume and dimensions of round objects, comparing functions represented in different ways, interpreting the slope of a graph in context, reasoning about specific function values, and reasoning about when functions have the same value.

While students are only instructed to add a graph of the cylinder to the given axes, monitor for students who also plot the values for the sphere to share their graphs during the Activity Synthesis.

The purpose of this discussion is for students to share how they compared the different representations or made a new representation to answer the questions. Begin by surveying students for their answers to the third part about which container can fit the largest volume of water and which container can fit the smallest. Record the results for all to see. Invite 2–3 to share their reasoning.

Next, select previously identified students who graphed the data from the table in order to complete the problems. Display 1–2 of these representations for all to see.

Here are some questions to further student thinking about the graphs of the three shapes:

• “If I showed you this graph without telling you which function represented each shape, how could you figure out which one represents the cone, the cylinder, and the sphere?” (The graph of the cylinder must be linear since the shape of the container never changes as the water level rises. The graph of the cone would first fill quickly but then fill more slowly as it got wider near the top. (Note: It helps to know the cone is tip down here.) The graph of the sphere would change partway through since it would start fast, slow down toward the middle where the sphere is widest, then speed up again as the sphere narrows, and the table data, or discrete points, is the only representation that does that.)
• “Can you use the information provided about each function to determine the radius of the sphere and cone? Use the fact that the actual volume of the cone is 127.23 in³.” (The radius of the sphere is half the height, so the radius is 3 in. The cone has a volume of 127.23 in³, so we can tell that is about 20.25. We can use guess and check to find since 20.25 is between 16 and 25. We can check what happens when , which gives a value of 20.25 for . So the radius of the cone is about 4.5 in.)