Lesson 18

The purpose of this warm-up is to remind students that if a solid is scaled by a factor of \(k\), the solid’s volume increases by a factor of \(k^3\). Also, the situation described in this activity sets the stage for the next task in the lesson. As students evaluate the accuracy of a cylindrical model, they’re modeling with mathematics (MP4).

Monitor for different unit conversion strategies. Some students may start by converting 36 inches to 3 feet. Others may calculate the volume in cubic inches, then convert the result to cubic feet.

Students may forget to find the radius measurement by dividing the diameter by 2 before they calculate the volume. Prompt them to think about what the letters in the formula for the area of a circle represent.

Students may not remember that volume scales by the cube of the scale factor. Invite them to scale the dimensions by a factor of 2 and calculate the new volume.

The goal of the discussion is to ensure all students understand why the correct formula for the dilated drum volume is \(V=7.07k^3\).

If possible, highlight different unit conversion strategies in the discussion. Ask students what the dimensions and volume would be if the drum were dilated by a scale factor of 2, making note they can calculate the new volume by either scaling the dimensions and calculating the volume directly, or by using the formula \(V=7.07k^3\).

In this activity, students use a cube root graph to analyze the relationship between a solid’s volume and its dimensions in a situational context. Before students begin working, they are explicitly asked to make an estimate. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).

Monitor for students who use the graph to solve each problem, and for students who use other methods for one or more of the problems.

Arrange students in groups of 3–4.

Remind students that in the warm-up, they looked at a drum with diameter 3 feet, height 1 foot, and volume of about 7 cubic feet. Ask students to estimate: “Before calculating anything, what do you think the volume would be if we dilated the drum using a scale factor of 5?” Poll the class for their estimates. Display the results of the poll for all to see.

Provide access to devices that can run Desmos or other graphing technology.

After students have worked on the task for several minutes, pause for a whole-class discussion to make sure that all groups have successfully graphed a correct equation. If many groups are struggling, solve and graph the equation together as a class. Verify that all groups know how to locate points on the graph using a trace feature or by tapping and dragging.

Students may struggle with solving the volume equation for \(k\). Prompt them to substitute a value for the volume and record their steps in solving for the scale factor. Remind them that the order of operations is reversed when solving equations—both sides of the equation must be divided by 7.07 before the cube root can be applied.

Some students may be unsure how to access the cube root function on their graphing tool. Demonstrate for these students how to do so.

The goal of the discussion is for students to view the graph as a tool. The shape of the graph helps clarify the relationship between the two variables, and the graph is useful for solving certain problems.

Ask students:

• “How did your initial estimate of the volume of the drum scaled by a factor of 5 line up with your answer to the last question?”
• “What does the shape of the graph tell us about the relationship between the amount of helium we have and the scale factor we can get?” (Volume increases in the 0 to 2,000 cubic feet range result in large scale factor increases. However, after that, we can add lots of helium without necessarily gaining a lot of balloon diameter.)
• “Did you need the graph to solve these problems? Was the graph more helpful for some than for others?” (Sample response: the last problem would be relatively easy to solve without using the graph, by multiplying the dimensions of the balloon by 5 and calculating the new volume directly.)

Students compare two cube root graphs to draw conclusions about scaled volumes.

This activity works best when each student has access to devices that can run the Desmos applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

The goal of the discussion is to pull together observations about the graph. Ask students:

• “Which balloon is more cost-effective in terms of helium use, the drum or the beach ball? How does the graph help you know?” (The ball is more efficient. Consider a scale factor of 8. The beach ball balloon’s volume is around 8,000 cubic feet for that scale factor, while the drum balloon’s volume is only about 4,000 cubic feet.)
• “For a scale factor of 10, how do the volumes of the two balloons compare?” (The drum balloon’s volume would be almost twice as large as the beach ball balloon’s volume.)