Unit 6, Lesson 19
Estimating a Hemisphere
Accelerated 7
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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
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.
Here are two shapes. What do you notice? What do you wonder?
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,
MLR6: Three Reads
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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.
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
The purpose of this discussion is for students to see that since a rectangular prism’s volume gets larger by a factor of
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.
To Gather
Spherical objects
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
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
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:
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
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
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
Help us improve by sharing suggestions or reporting issues.
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:
A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height.
A cone fits snugly inside a hemisphere, and they share a radius of 5.