## Fishing for the Best Prism

• Lesson
6-8,9-12
2

In this lesson, students use polydrons to create nets of rectangular prisms. They discover that there are many configurations for rectangular prisms with the same volume, and determine that certain configurations minimize surface area. The lesson continues in a discovery activity related to building the most cost-efficient and appealing fish tank.

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 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.

 Cube Net Built out of Polydrons Cube Built out of Polydrons

If Polydrons and Klikkos are unavailable, 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

Hand out the Building Blocks activity sheet. Divide students into groups of 3 and give each student in the group 6 Polydrons. Ask students to connect their Polydrons to form the shape shown in Question 1. Draw this net on the board so students can use this as a comparison when creating their additional 10 nets in the questions that follow.

 Building Blocks Activity Sheet

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 tool to check their solutions if computers are available.

 Cube Nets Tool

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.

Assessments

1. Have students role play, convincing a tropical fish tank company to pick their proposal. Choose a student from a group not presenting to play the fish tank company executive. You can have students vote on the best role play, rating each other based on mathematical arguments and examples. However, do not use student votes in your own assessment of the activity.
2. Assign students to create the different prisms with a volume of 24 cubic units. Students could then calculate surface areas, and compare these results to those from the activity. Challenge students to explore a general explanation of how to create a rectangular prism with minimal surface areas. Note: Due to the size of these prisms, using the Polydrons may not practical for this activity.

Extensions

1. Students can create other types of nets using the Polydrons and explain how they can calculate surface area with these new solids. Polydron sets also include include triangles, rectangles, pentagons, and hexagons.
2. Allow students to build with the Polydrons creatively and describe the geometric characteristics of the objects they are building.
3. Students can use interlocking cubes and contrast the use of 2-dimensional nets and 3-dimensional objects to represent solids and explore volume and surface area.

Questions for Students

1. What is a net?

[A 2-dimensional diagram representing an unfolded 3-dimensional solid.]

2. Does a net represent the volume or the surface area? Explain your reasoning.

[Students should realize that a net represents a 3-dimensional figure. It can be used to easily find the surface by adding up the area of each face of a rectangular prism.]

3. Why might a company care about the dimensions of different containers with the same volume?

[There are several reasons. The most apparent from this activity would related to surface area. Just because containers have the same volume, that doesn't mean they have the same surface area. Minimizing surface area would help them minimize the cost of producing the container. Other considerations might include the strength of the container or the aesthetic value of certain dimensions.]

4. How can a company minimize the cost of constructing a container?

[Minimize the surface area of the container.]

5. What is the difference between surface area and volume?

[Many students describe volume as what fits inside and relate surface area to the outside. Let students describe this is with a vocabulary in which they are comfortable. Relate their definitions to these:

A surface area is the total area of the surfaces of a 3-dimensional object measured in square units.

Volume is the amount of space occupied by a 3-dimensional solid and is measured in cubic units.]

Teacher Reflection

• What learning styles does this lesson address? Are there other learning styles that can be addressed by modifying the lesson?
• Did all students understand the difference between surface area and volume?
• What advantages did using Polydrons provide over using paper nets?
• Did students understand that objects with the same volume can look very different?
• What were some of the ways your students illustrated that they were actively engaged in the learning process?

### Learning Objectives

Students will:

• Use polydrons to create nets and 3-dimensional figures
• Create different nets which form the same rectangular prism
• Create nets and rectangular prisms that have that the same volume but different surface areas
• Discover the type of rectangular prism that can be built to minimize cost
• Apply real-life considerations to minimizing cost, such as aesthetic appeal

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

• CCSS.Math.Content.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

• CCSS.Math.Content.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.