standard_description: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
problem_activity: Activity
In this activity, students use different solid figures to estimate an upper and lower bound for the volume of a hemisphere. For the upper bound, the hemisphere fits snugly inside a cylinder whose height and radius are equal to the radius of the hemisphere. For the lower bound, the cone fits snugly inside the hemisphere, and its radius and height also equal the radius and height of the hemisphere.
Select students who use the reasoning from the first problem to assist them in answering the second problem to share during the Activity Synthesis. For example, since the cylinder and cone have the same dimensions, the volume of the cone must be
\(\frac13\)
that of the cylinder.
Launch
Keep students in the same groups. Display the image of a hemisphere of radius 5 that fits snugly inside a cylinder of the same radius and height.
[IMG]
Ask “How does the hemisphere affect the height of the cylinder?” and then give students one minute of quiet think time, then one minute to discuss their response with a partner. Ask partners to share their responses. If not mentioned by students, point out that the height of the cylinder is equal to the radius of the hemisphere.
Give students work time for the activity followed by a whole-class discussion.
Representation: Internalize Comprehension.
Provide examples of actual 3-D models of cylinders, hemispheres, and cones for students to view or manipulate. Ask students to use the 3-D models and the volumes calculated in each question to estimate the volume of the hemisphere.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height.
[IMG]
Calculate the volume of the cylinder.
Estimate the volume of the hemisphere. Explain your reasoning.
A cone fits snugly inside a hemisphere, and they share a radius of 5.
[IMG]
What is the volume of the cone?
Estimate the volume of the hemisphere. Explain your reasoning.
Compare your estimate for the hemisphere with the cone inside to your estimate of the hemisphere inside the cylinder. How do they compare to the volumes of the cylinder and the cone?
Student Response
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Student Facing
Are you ready for more?
Estimate what fraction of the volume of the cube is occupied by the pyramid that shares the base and a top vertex with the cube, as in the figure.
[IMG]
Student Response
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Anticipated Misconceptions
Students may not realize that the radius of the hemisphere determines the height of the cone and cylinder. If students struggle with identifying the needed information, remind them of the previous activity in which the radius of the cylinder was also the height of the box (rectangular prism). Ask students how that can help them determine the dimensions of the cone or cylinder.
Activity Synthesis
Ask previously identified students to share their responses to the first two problems. Draw attention to any connections made between the two problems.
The purpose of this discussion is for students to recognize how the upper and lower bounds for the volume of a hemisphere are established by the cylinder and cone. Consider asking the following question:
“What do the volumes of the cone and cylinder tell us about the volume of the hemisphere?” (The volume of the hemisphere has to be between the two volumes.)
“Did anyone revise their original estimate for the hemisphere based on the calculation of the volume of the cone?” (Answers vary. If students estimated low values after calculating the volume of a cylinder, some may have needed to adjust their estimates.)
“Compare the equations for volume of a cylinder and cone where radius and height are equal. If the volume of the hemisphere has to be between these two, what might an equation for the volume of a hemisphere look like?” (Cylinder volume:
\(V=\pi r^3\)
; Cone volume:
\(V= \frac 13 \pi r^3\)
. A possible hemisphere volume might be the average of these two, or
\(V= \frac 23 \pi r^3\)
.)
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask students to meet with 2–3 partners in a row for feedback. Provide students with prompts for feedback that will help them to strengthen their ideas and clarify their language (e.g., “Why do you think…?”, “How did you compare the volume of the two shapes?”, and “How does the first/second question help?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)


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