The goal of this lesson is provide students hands on experience building nets and creating rectangular prisms. To help build this understanding, students will use hands-on or virtual manipulatives to explore the relationships between nets, 3-dimensional figures, volume, and surface area. Polydrons are plastics pieces with hinged edges, which connect to each other to form polyhedra, and should be used if available. Alternately, use Klikkos, a similar type of manipulative, or Mathigon Polypad, a virtual manipulative.
 Cube Net Built out of Polydrons |  Cube Built out of Polydrons |
Note: If Polydrons and Klikkos are unavailable, or you are unable to use a virtual manipulative, square pieces of paper and tape can be used. While it is possible to build the shapes out of paper, they will not be as easy to alter. Each group will need many squares, so cut these out ahead of time. Keep the squares relatively small with a side length of about 2 inches. Also, interlocking cubes, if available, can be used for the surface area analysis.
Building Blocks
Building Blocks Activity Sheet
Hand out the Building Blocks Activity Sheet. Divide students into groups of 3 and give each student in the group 6 Polydrons, or have them open Mathigon Polypad on their devices.
If you are using Polydrons:
- Ask students to connect their Polydrons to form the shape shown in Question 1. Display this net on the board so students can use this as a comparison when creating their additional 10 nets in the questions that follow.
- Have students fold the net up into a cube and have a discussion about nets and how they are used to create 3-dimensional objects. Do not mention surface area at this point.
- Challenge students to create 10 additional nets that create a cube.
- When a student suggests that they have new net, have them first ensure it folds up into a cube and then bring their net to the board rotating it to make sure it is not a duplicate. Then, have them draw their net on the board.
- Students should also sketch these nets on the graph in Question 2. After students believe they have the 11 nets, they can use the Cube Nets Interactive to check their solutions if computers are available.
If you are using Mathigon Polypad:
- Have students create the shape shown in Question 1 using the square polygon piece. Display this net on the board so students can use this as a comparison when creating their additional 10 nets in the questions that follow.
- Once it's complete, students can fold up the net by highlighting the entire image and selecting "fold net." Have a discussion about nets and how they are used to create 3-dimensional objects. Do not mention surface area at this point.
- Note: To unfold the 3D figure, click on the figure and select "unfold."
- Challenge students to create 10 additional nets that create a cube. They can test their net using the fold net feature. When students have successfully created new nets, call on volunteers to draw their ideas on the board.
- Students should also sketch these nets on the graph in Question 2.
After you've allowed time for students to provide a variety of different examples on the board, have them explore other possible solutions using the Cube Nets Interactive.
Cube Nets Interactive
Allow time for groups to complete the activity sheet. Walk among the groups, and observe how students answered Question 5. Did they use the formula and subtract the top or did they add up the areas of the sides? Ask students how they can use their Polydrons to answer the question. Students may also have other ideas. Use this example for a class discussion on how the surface area formula will not calculate the correct answer unless students understand its meaning. Ask students how they calculated surface area and use the opportunity to discuss how students can use different methods to solve the same problem.
If groups finish early, ask them to predict the ratio of two surface areas and the ratios of the two volumes for a cube in which they assume each side has length 3 and go through the same steps they used for the 2×2×2 cube analysis.
Have a classroom discussion about Question 12. Many students intuitively think the surface areas and volumes double. If students have not discovered it on their own, discuss that the ratio of the volumes is the cube of the ratio of the sides, and the ratio of the surface areas is the square of the ratio of the sides. This will be more apparent to groups that have time to consider the 3×3×3 situation.
Fish Tank 
Distribute the Fishing for the Best Prism Activity Sheet. As a class, students will suggest 3 different rectangular prisms that have a volume of 8 cubic units. Based on the total number of groups in the class, divide up the 3 configurations. Multiple groups can work on the same configuration if there are more than 3 groups. Each group should complete pages 1 and 2.
Fishing for the Best Prism Activity Sheet
Students may need help creating the net in Question 2 because they may start immediately to build the prism. Have models of both a net and an isometric drawing of a rectangular prism that students can reference as they work on page 1. Review students' answers for the volume, surface area, and surface area of a fish tank. If necessary, show student how to rotate their rectangular prism to find the various configurations for the fish tank.
As a class, review the answers to the questions on pages 1 and 2. The discussion should build on their discoveries from the Building Blocks activity. Students investigated a real-life open ended problem in Questions 5-9. A tropical fish company hired the students to build the most appealing fish tank. In Question 9, students may not choose the cheapest fish tank because it may not be the most appealing. It is an important question. Students apply math to a practical problem, which should help them realize that a mathematical solution is an option but there are other factors that influence decisions. It is important for this activity that you allow students to be creative.
To conclude the activity, write the questions on page 3 on the board, allowing ample space under each for answer. Assign each group to begin with a different question, and then have groups walk from question to question and write their answers underneath. If for a particular question their answer is the same as another group, they should be encouraged to create a picture or alternative explanation that helps illustrate the answer. Once all groups have contributed their answers to all questions, review the answers as a class.