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.
Activity
None
A cylinder and sphere have the same height.
If the sphere has a volume of cubic units, what is the height of the cylinder?
What is a possible volume for the cylinder? Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
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.
Activity 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.
22.2
Activity
Standards Alignment
Building On
Addressing
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.
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 .
Launch
Arrange students in groups of 2. Give 2–3 minutes of quiet work time to complete the first problem on their own, then time to discuss their solutions with a partner. Groups then finish the remaining problems together and follow with a whole-group discussion.
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “It looks like . . . .” “Is it always true that . . .?” and “We can agree that . . . .” Supports accessibility for: Language, Organization
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “What happens to the volume of this sphere if its radius is halved?” Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Activity
None
Fill in the missing volumes in terms of . Add two more radius and volume pairs of your choosing.
radius (cm)
1
2
3
100
volume (cm3)
How does the volume of a sphere with radius 2 cm compare to the volume of a sphere with radius 1 cm?
How does the volume of a sphere with radius cm compare to the volume of a sphere with radius 1 cm?
A sphere has a radius of length .
What happens to the volume of this sphere if its radius is doubled?
What happens to the volume of this sphere if its radius is halved?
Sphere Q has a volume of 500 cm3. Sphere S has a radius as large as Sphere Q. What is the volume of Sphere S?
Activity Synthesis
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.
Launch
Tell students to close their books or devices (or to keep them closed). Display the graph from the Task Statement. Tell students that the graph represents water filling a container. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph. Possible responses:
Students may notice:
The graph is not a linear function.
The graph is a piecewise function.
The maximum height of the function is 6 inches.
Students may wonder:
What is the shape of the container?
What is happening when the height stops at 6 inches?
Why does the volume increase but the height stays constant on the far right?
Discuss possible responses for questions that students wondered. If what would happen if someone kept pouring water into a container even though the water level had reached the top does not come up during the conversation, ask students to discuss this idea. Ensure students understand that the height stops at 6 inches because that is how tall the container is. Any more water poured into the container at that point just overflows.
Arrange students in groups of 2–3, and provide access to straightedges. Tell students to open their books or devices, then give groups work time, and follow with a whole-class discussion.
Activity
None
Three containers of the same height were filled with water at the same rate. One container is a cylinder, one is a cone, and one is a sphere.
As they were filled, the relationship between the volume of water and the height of the water was recorded in different ways, shown here:
Cylinder:
Cone:
Coordinate plane, horizontal, volume, inches cubed, 0 to 120 by twenties, vertical, height, inches, 0 to 7 by ones. Curve starts at origin, increases steeply through volume = 10, decreases less steeply to volume = 125, then remains constant at height = 6 to the edge of the graph.
Sphere:
volume (in3)
height (in)
0
0
8.38
1
29.32
2
56.55
3
83.76
4
104.72
5
113.04
6
120
6
200
6
The maximum volume of water the cylinder can hold is . What is the radius of the cylinder?
Graph the relationship between the volume of water poured into the cylinder and the height of water in the cylinder on the same axes as the cone. What does the slope of this line represent?
Which container can fit the largest volume of water? The smallest?
About how much water does it take for the cylinder and the sphere to have the same height? The cylinder and the cone? Explain how you know.
For what approximate range of volumes is the height of the water in the cylinder greater than the height of the water in the cone? Explain how you know.
For what approximate range of volumes is the height of the water in the sphere less than the height of the water in the cylinder? Explain how you know.
Student Response
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Building on Student Thinking
If students are not sure how to compare some of the different representations, consider asking:
“Tell me more about what the values in the table tell you.”
“How could you use a graph to think of the relationship between the volume of water and the height of the water for the sphere in a different way?”
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.)
MLR8 Discussion Supports. At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain how to answer the question “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?” Invite groups to rehearse what they will say when they share with the whole class. Advances: Speaking, Conversing, Representing
Standards Alignment
Building On
Addressing
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.
