Lesson 23

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. Students will encounter these shapes again later in the lesson.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Select students to share things they have noticed and things they have wondered. Encourage students to think about the relationship with the dimension \(r\) and the volume \(V\) of each of these shapes. Ensure students understand that the two equations have no variable \(h\) for height since the \(h\) was replaced by \(r\) due to the height and radius being the same for both shapes.

In this activity, students think about a hemisphere fitting inside the smallest possible box (or rectangular prism). Students reason that the smallest box in which a hemisphere can fit has a square base with edge length that is the same as the diameter of the hemisphere, and its height will be the radius of the hemisphere. Further, if we calculate the volume of the box, we have an upper bound for the volume of the hemisphere. This activity prepares students for the next, where they will calculate the upper and lower bounds for the volume of a hemisphere by considering cylinders and cones that fit outside or inside the hemisphere.

Ask students if they are familiar with the word sphere, and if they can think of examples of spheres. Some examples they might come up with are: ping pong balls, soap bubbles, baseballs, volleyballs, a globe. If students don’t come up with many examples, offer your own, or perhaps display an Internet image search for “sphere.” If you brought in any examples of physical spheres, display these or allow students to hold them. 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. If you brought in physical examples of spheres, ask students to rank them from smallest diameter to largest diameter.

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 convex security mirrors. Display an example, illustration, or diagram of a hemisphere for all to see, 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 \(108\) cubic inches (since \(6^2\boldcdot 3=108\)). Discuss how much less students think the volume of the sphere is and their reasoning, then have students move onto the second problem with their partner.

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. Since volume of a rectangular prism is the product of length, width, and height, the side lengths do not need to be equal; if every side length is doubled, then the volume will be increased by \(2 \boldcdot 2 \boldcdot 2 = 8\).

Invite students to share their answers to the second question. The purpose of this discussion is for students to see that since a rectangular prism’s volume gets larger by a factor of \(2^3\) when edge lengths are doubled then it would make sense for a hemisphere’s volume to do the same. 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 and that this will be the upper bound of the volume of a sphere.

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.

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.

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 followed by a whole-class discussion.

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.

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 \(V= \frac 23 \pi r^3\).)