standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
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.
Launch
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.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: sphere and hemisphere.
Supports accessibility for: Memory; Language
Reading, Speaking, Listening: MLR6 Three Reads.
Use this routine to support comprehension of the situation before looking at the questions. In the first read, students read the problem with the goal of comprehending the situation (e.g., Mai has a paperweight that is a dome. She wants to design a box for it.). If needed, discuss the meaning of unfamiliar terms at this time. Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., the radius of the dome, the area of the base of the dome, the side lengths of the box). In the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​
Student Facing
[IMG]
Mai has a dome paperweight that she can use as a magnifier. The paperweight is shaped like a hemisphere 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 box be so 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 the paperweight in this box is than the volume of Mai's paperweight? Explain your thinking.
Student Response
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Anticipated Misconceptions
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
\(2 \boldcdot 2 \boldcdot 2 = 8\)
.
Activity Synthesis
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.


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