The purpose of this Warm-up is for students to review how to manipulate the formulas for volume of a cylinder and cone and consider what they look like when the height and radius are the same, which will be useful when students encounter these shapes throughout the lesson.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the fact that .
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Display the shapes for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
None
Here are two shapes. What do you notice? What do you wonder?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the shapes. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the “missing” variable for height, , does not come up during the conversation, ask students to discuss this idea. Ensure students understand that the two equations have no variable for height since the was replaced by due to the height and radius being the same for both shapes.
19.2
Activity
Standards Alignment
Building On
Addressing
8.G.C
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
In this activity, students think about a hemisphere fitting inside the smallest possible box (or rectangular prism). Using the structure of volume formulas, students reason that the smallest box in which a hemisphere can fit has a square base with an edge length that is the same as the diameter of the hemisphere and that the height of the box will be the radius of the hemisphere (MP7).
A key takeaway from this activity is that the volume of the box gives an upper bound for the volume of the hemisphere. This activity prepares students for later work determining a lower bound and a new upper bound for the volume of a hemisphere using cylinders and cones.
Launch
Begin the discussion by inviting students to share examples of spheres they have seen. Some examples they might come up with are ping pong balls, soap bubbles, bath bombs, paper lanterns, baseballs, volleyballs, and a globe. If students don’t come up with many examples, offer your own, or display results from an internet image search for “sphere.” If possible, acquire spheres to display or pass around the class. Explain to students that the radius of a sphere is the distance from the center of the sphere (the point in the exact middle) to any point on the sphere.
Next, ask students if they are familiar with the word “hemisphere” and if they can think of examples of hemispheres. Some examples are hemispherical (or “half”) balance boards, half of Earth, a dome or planetarium, and molds used in baking or to make bath bombs. Display an example, illustration, or diagram of a hemisphere for all to see, like the one shown here of hemisphere molds, pointing out how the radius is the distance from the center of the flat side of the sphere to any point on the curved surface of the sphere.
Tell students that in this activity, they will think about building a box (or rectangular prism) around a hemisphere. Arrange students in groups of 2. Ask students to quietly work through the first question and then share their reasoning with their partner. Pause for a whole-class discussion, and invite students to share their responses. The purpose of this problem is to see how the measurements of a hemisphere determine the dimensions of the prism. Ensure everyone understands the box must have edge lengths 6, 6, and 3 inches and that the hemisphere must have a volume that is less than cubic inches (since ). Discuss how much less students think the volume of the sphere is along with their reasoning, then ask students to move onto the second problem with their partner.
MLR6 Three Reads. Keep books or devices closed. Display only the first problem stem, without revealing the questions. Say, “We are going to read this statement 3 times.”
After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the questions. Ask, “What are some ways we might get started on this?”
Advances: Reading, Representing
Activity
None
Mai has a dome paperweight that she can use as a magnifier. The paperweight is shaped like a hemisphere and made of solid glass, so she wants to design a box to keep it in so it won't get broken. Her paperweight has a radius of 3 cm.
What should the dimensions of the inside of the box be so that the box is as small as possible?
What is the volume of the box?
What is a reasonable estimate for the volume of the paperweight?
Tyler has a different box with side lengths that are twice as long as the sides of Mai's box. Tyler's box is just large enough to hold a different glass paperweight.
What is the volume of the new box?
What is a reasonable estimate for the volume of this glass paperweight?
How many times bigger do you think the volume of Tyler’s paperweight is than the volume of Mai’s paperweight? Explain your thinking.
Student Response
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Building on Student Thinking
Students may believe that since this box is not a cube, but a rectangular prism, the volume will not increase by 8 when the side lengths are doubled. Consider asking:
“Tell me how you calculated the volume of the box.”
“What happens to the volume if you replace the length, width, and height with values that are doubled?”
Activity Synthesis
The purpose of this discussion is for students to see that since a rectangular prism’s volume gets larger by a factor of when edge lengths are doubled, it would make sense for a hemisphere’s volume to do the same.
Invite students to share their answers to the second question. Ensure that students understand the volume calculated for the box holding the hemisphere is greater than the actual volume of the hemisphere because of the space left around the paperweight in the box. This will be an upper bound of the volume of a sphere—that is, a value we know the volume of the hemisphere must be less than.
Tell students that in the next activity they are going to investigate a better way to estimate the volume of a hemisphere. If time allows, ask students to suggest shapes they are already familiar with that they could use to find the volume of a hemisphere.
19.3
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.
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.
The Activity Synthesis concludes by inviting students to compare the formulas for the volume of a cylinder and cone where and consider what the formula for the volume of a hemisphere might look like, which helps prepare them for making sense of upcoming work determining the formula for the volume of a sphere (MP1, MP7).
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.
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, and follow with a whole-class discussion. As students work, select groups 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 that of the cylinder.
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide examples of actual three-dimensional models of cylinders, hemispheres, and cones for students to view or manipulate. Ask students to use the three-dimensional models and volumes calculated in each question to estimate the volume of the hemisphere. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Activity Synthesis
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. Begin the discussion by asking previously selected groups to share their responses to the first two problems. Draw attention to any connections made between the two problems.
Here are some more questions for discussion:
“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?” (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 volume of a cone in which the radius and height of each solid 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: ; Cone volume: . A possible hemisphere volume could be the average of these two, or .)
Lesson Synthesis
To prompt students to describe some of the important understandings of the lesson, ask:
“How did we use a cylinder to overestimate the volume of a hemisphere?” (We found the volume of a cylinder that the hemisphere could fit inside without extra space.)
“How did we use a cone to underestimate the volume of a hemisphere?” (We found the volume of the largest cone that could fit inside of the hemisphere.)
“How did we get a closer estimate for the volume of a hemisphere?” (Earlier we figured out an upper bound using a prism. Here, we have a smaller upper bound using a cylinder and a lower bound using a cone.)
Explain to students that we used figures that we know how to find the volume of (a cone and cylinder) to try and estimate the volume of a figure we do not know how to find the volume of (a sphere). Next, we will use similar reasoning to find the volume of a sphere.
Student Lesson Summary
We can estimate the volume of a hemisphere by comparing it to other shapes for which we know the volume. For example, a hemisphere of radius 1 unit fits inside a cylinder with a radius of 1 unit and height of 1 unit.
Since the hemisphere is inside the cylinder, it must have a smaller volume than the cylinder, making the cylinder's volume a reasonable overestimate for the volume of the hemisphere.
The volume of this particular cylinder is about 3.14 cubic units since , so we know the volume of the hemisphere is less than 3.14 cubic units.
Using similar logic, a cone of radius 1 unit and height 1 unit fits inside of the hemisphere of radius 1 unit.
Since the cone is inside the hemisphere, the cone must have a smaller volume than the hemisphere, making the cone's volume a reasonable underestimate for the volume of the hemisphere.
The volume of this particular cone is about 1.05 cubic units since , so we know the volume of the hemisphere is more than 1.05 cubic units.
Averaging the volumes of the cylinder and the cone, we can estimate the volume of the hemisphere to be about 2.10 cubic units since . And since a hemisphere is half of a sphere, we can also estimate that a sphere with radius of 1 would be double this volume, or about 4.19 cubic units.
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.
A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height.
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.
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?
