Lesson 18
Volume and Graphing
18.1: Parade Balloon: Part 1 (5 minutes)
Warm-up
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.
Student Facing
A company makes giant balloons for parades. They’re designing a balloon that will be a dilated version of a drum similar to the one in the image. The real-life drum’s diameter is 36 inches and it’s 1 foot wide.
- What’s the approximate volume of the real-life drum in cubic feet? Round to the nearest hundredth.
- Suppose the drum is dilated by scale factor \(k\). Write an equation that gives the volume, \(V\), of the dilated drum.
- What are some reasons the actual drum volume might be different from what you calculated?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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 \(V=7.07k^3\).
18.2: Parade Balloon: Part 2 (20 minutes)
Activity
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.
Launch
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.
Design Principle(s): Support sense-making
Supports accessibility for: Organization; Attention
Student Facing
A company makes giant balloons for parades. They’re designing a balloon that will be a dilated version of a drum with diameter 36 inches and height 1 foot. The balloon will be inflated with a gas called helium. The balloon designers want to be able to find the scale factor they can achieve with different volumes of helium.
In an earlier activity, you found the volume of the original drum, and you wrote an equation to describe the volume, \(V\), of a version of the drum that had been dilated by a factor of \(k\).
- Rearrange your equation to solve for \(k\).
- Use graphing technology to graph your rearranged equation. Set the viewing window to show a maximum of about 15,000 cubic feet on the \(x\)-axis.
- The point \((1,\!000, 5.21)\) is on the graph. What does this point mean?
- One tank of helium contains enough gas to fill 500 cubic feet of space. Suppose the company can afford 12 tanks.
- What scale factor can they use?
- What will the diameter of the balloon be?
- The company received a donation that will double the number of helium tanks they can afford. How will this change the diameter of the balloon they can create?
- The company learned that the parade route has size restrictions. The balloon can be no more than 15 feet in diameter.
- What scale factor would they need to get this diameter?
- What would the volume of the balloon be in this case?
- How many helium tanks would be required?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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.)
18.3: Beach Ball Balloon (10 minutes)
Activity
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.
Launch
Arrange students in groups of 3–4. Provide each group a device to access the applet embedded in the activity.
Supports accessibility for: Organization; Attention
Student Facing
The parade balloon company will make a second balloon, modeling a beach ball with radius 1.5 feet. The volume of the original beach ball is about 14.14 cubic feet.
- Explain why the equation \(k=\sqrt[3]{\frac{V}{14.14}}\) gives the scale factor \(k\) needed to achieve a dilated volume of \(V\).
- The applet contains graphs of the equations \(k=\sqrt[3]{\frac{V}{7.07}}\) and \(k=\sqrt[3]{\frac{V}{14.14}}\), which show the relationship between volume and scale factor for the drum and the beach ball. There is a horizontal line on the graph that can be moved up and down with a slider. What do points on the horizontal line represent?
- The company wants the two balloons to have the same scale factor.
- If the scale factor is 5, how much helium will they use in total?
- How much helium will they use if the scale factor is 10?
- Suppose the company purchases a total of 12,000 cubic feet of helium.
- About what scale factor should they use if they want to use all their helium and have the same scale factor for each balloon?
- What will be the approximate radius of the scaled beach ball in this case?
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
- Write an expression that gives the ratio of surface area to volume of a sphere with radius \(r\).
- Create a graph with the sphere’s radius, \(r\), on the \(x\)-axis and the ratio you wrote on the \(y\)-axis.
- Describe the shape of the graph for small radii.
- Describe the shape of the graph for large radii.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Arrange students in groups of 3–4. Provide students with access to rulers.
Supports accessibility for: Organization; Attention
Student Facing
The parade balloon company will make a second balloon in the shape of a beach ball that has radius 1.5 feet. The volume of the original beach ball is about 14.14 cubic feet.
- Explain why the equation \(k=\sqrt[3]{\frac{V}{14.14}}\) gives the scale factor \(k\) needed to achieve a dilated volume of \(V\).
- Two equations are graphed that show the relationship between volume and scale factor for the drum and the beach ball: \(k=\sqrt[3]{\frac{V}{7.07}}\) and \(k=\sqrt[3]{\frac{V}{14.14}}\). Use a ruler to draw a horizontal line on the graph at \(k=5\).
What do points on this line represent?
- The company wants the two balloons to have the same scale factor. If the scale factor is 5, how much helium will they use in total?
- Suppose the company purchases a total of 12,000 cubic feet of helium.
- About what scale factor should they use if they want to use all their helium and have the same scale factor for each balloon?
- What will be the approximate radius of the scaled beach ball in this case?
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
- Write an expression that gives the ratio of surface area to volume of a sphere with radius \(r\).
- Create a graph with the sphere’s radius, \(r\), on the \(x\)-axis and the ratio you wrote on the \(y\)-axis.
- Describe the shape of the graph for small radii.
- Describe the shape of the graph for large radii.
Student Response
For access, consult one of our IM Certified Partners.
Activity 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.)
Lesson Synthesis
Lesson Synthesis
In this lesson, students have used graphs to solve problems involving geometric contexts. Here are some questions for discussion:
- “Consider the volume formula for a sphere of radius \(r\): \(V=\frac43 \pi r^3\). Suppose you want to know the radius of different spheres with volumes 100, 200, 300, 400, and so on. What are some ways to find the radii that produce these volumes?” (Substitute each volume into the formula and solve for \(r\) repeatedly. Or, solve for \(r\) in terms of \(V\) and substitute each volume into that expression. Alternatively, solve for \(r\) and represent the relationship with a graph, then use the graph to estimate each radius.)
- “What are the advantages and disadvantages of these different techniques?” (Solving for radius repeatedly will yield more precise results than estimating with a graph, but it requires solving a new equation for each desired volume. Solving for radius once and then inputting different volumes is just as precise, and only requires solving one equation. However, it requires multiple computations with a calculator if decimal approximations are required. Estimating with the graph requires solving only one equation and can find solution estimates for many inputs easily. This may come at the cost of precision.)
18.4: Cool-down - Circular Logo (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Suppose a farm has a water tank shaped like a cone. Water is poured in from the top, and a valve can be opened to let the water flow out the bottom. There is currently a small volume of water in the tank. The section of the cone that is filled has radius 1 foot and height 1 foot.
Using the expression \(\frac{1}{3}\pi r^2 h\), we find that the volume of water in the tank is about 1.05 cubic feet because\(\frac{1}{3}\pi (1)^2 (1)\approx 1.05\). As more water is poured into the tank, the shape of the water will be a dilation of the original small cone. An equation that expresses the volume, \(V\), in terms of the scale factor of dilation, \(k\), is\(V=1.05k^3\). This equation can be rearranged, resulting in \(k=\sqrt[3]{\frac{V}{1.05}}\). Here is a graph of the rearranged equation.
The point \((100,4.57)\) on the graph tells us that if the farmer puts 100 cubic feet of water in the tank, the scale factor of dilation will be 4.57. That means that the height of the water would be the original height of 1 foot times the scale factor 4.57, or 4.57 feet.