Middle School Math Lesson Using Diameter, Volume and Surface Area to Determine the Dimensions of PSA's Computer

The Main Problem:
NASA needs you to design a computer for the Personal Satellite Assistant, a robot helper that will assist the astronauts in space. This robot is shaped like a sphere. The computer must meet the following criteria:

1. a rectangular prism with a volume of 24 cubic inches
2. must fit inside a sphere with a diameter of 8 inches
3. must have the largest possible surface area to allow the computer to easily release heat so that it doesn't overheat

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Prerequisites:

• What is diameter?
• What is diameter of a sphere?
• How do you calculate volume?
• What is surface area?
• How do you calculate surface area?

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• Multiplying and dividing decimals.
• A rectangular prism is a three dimensional solid with six rectangular faces.
• A sphere is a round three-dimensional solid with all points of the surface at the same distance from the center.

Materials:
For teacher demonstrations:

• 1 basketball
• 1 sphere cut in half (optional)
• A large cube or rectangular prism
• 2 cylinders of equal volume but different shape (1 short and fat and 1 long and skinny. To ensure equal volume, you might use two cylinders made of clay dough)

For each group:

• 24 one-inch cubes
• 1 paper circle with an 8-inch diameter
• graph paper
• 1 calculator per student

Procedure
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1. Introduce the problem to students.
2. Have students come up with possible dimensions that would result in a volume of 24 cubic inches.
3. Ask students how many solutions they think there might be. Encourage discussion about this and guide students to conclude that using fractions or decimals results in many possible solutions.
4. Looking at the possible dimension solutions that students came up with, have them determine which dimensions would not fit inside a sphere with an 8-inch diameter.
5. Guide students in a discussion about what dimensions would work and why. Help students to conclude that because of the curvature of a sphere, a rectangular prism that is 8 inches wide may fit in the center of the sphere, but the corners would stick out at the top of the sphere. Also help them see that the computer needs to have enough room to release heat. As they work on the problem and through discussion, they will likely conclude that all computer dimensions will need to be less than 7.5 inches.
6. Discuss how students will meet the third guideline of maximized the surface area. Suggest that students try several different dimensions that result in a volume of 24 cubic inches and fit within a diameter of 8 inches to see which has the largest surface area. Allow students to use cubes, paper circles, calculators and graph paper to work on this problem.
7. As students come up with solutions, they will likely start with whole numbers. Discuss with students whether it would be possible to have decimals or fractions in their solutions. Also, discuss how they might figure out solutions using decimals, since their cubes are whole cubes.

Solutions and Conclusions

1. 1. List student volume solutions on the board with the smallest dimension last (as the height). List these in order from the largest height to smallest height. Write the corresponding surface area next to each area. Check each criteria for each answer to make sure that each solution will work. Also, ask students to explain why their answers work and make sure that classmates are convinced that each works. There are many possible solutions. See the PDF of this lesson for more details.
2. 2. Discuss with students the patterns that they see and the conclusions they can draw. What causes surface area to increase? What happens to the shape as the surface area increases?
   
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3. Ask students if they think there might be other possible solutions. Discuss that when using fractions there are many possible solutions. If they had more time, they would probably find more and may find one with more surface area.

Extension Problem
NASA's robot also uses fans to move around in microgravity. These fans are shaped like cylinders. The cylinders also need to have the maximum surface area in order to release heat. Should NASA use short and fat cylinder fans or long and skinny cylinder fans? Explain your answer.

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Use two cylinders of equal volume but different shape to illustrate this.

Guide students to apply what they learned about increasing the surface area of rectangular prisms to cylinders. Ask them how they can increase the surface area of a cylinder? This problem is also featured in Lesson 3.

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