Previously, students used a given net of a polyhedron to find its surface area. Here they use a given polyhedron to draw a net and then calculate its surface area. 

Use the provided polyhedra to differentiate the work for students with varying degrees of visualization skills. Rectangular prisms (A and C), triangular prisms (B and D), and square pyramids (F and G) can be managed by most students. Triangular Prism E requires a little more interpretive work (in that the measurements of some sides may not be immediately apparent to students). Trapezoidal Prism H and Polyhedron I (a composite of a cube and a square pyramid) require additional interpretation and reasoning. The work here offers opportunities for students to make sense of problems and persevere in solving them (MP1).

As students work, remind them of the organizational strategies discussed in previous lessons, such as labeling polygons, showing measurements on the net, and so on.

Ask students who finish their calculation to find another person in the class who has the same polyhedron and discuss the following questions (displayed for all to see):

• “Do your calculations match? Should they?”
• “Do your nets result in the same polyhedra? Should they?”
• “Do your models match the picture you were given? Why or why not?”

If time is limited, consider having the answer key posted somewhere in the classroom so students can quickly check their surface area calculations.

Reconvene briefly for a whole-class discussion. Invite students to reflect on the process of drawing a net and finding surface area based on a picture of a polyhedron. Ask questions such as:

• “How did you know that your net shows all the faces of your polyhedron?”
• “How did you know where to put each polygon or how to arrange all polygons so that, if folded, they can be assembled into the polyhedron in the drawing?”
• “How did the net help you find surface area?”

In this activity, students compare the surface areas and volumes of three rectangular prisms given nets that are not on a grid. To do this, they need to be able to visualize the three-dimensional forms that the two-dimensional nets would take when folded.

In grade 5, students had learned to distinguish area and volume as measuring different attributes. This activity clarifies and reinforces that distinction.

Students should have little trouble finding areas of rectangles but may have trouble keeping track of pairs of measurements to multiply and end up making calculation errors. Suggest that they label each polygon in the net and the corresponding written work and double-check their calculations to minimize such errors.

If students struggle to find the volume of their prism using information on a net, suggest that they sketch the prism that can be assembled from the net and label the edges of the prism. 

Students may need a reminder that area is measured in square units and that volume is measured in cubic units.

students who worked on the same prism if they agree or disagree about the calcualation. Record and display the results for all to see.

Invite students to share a few quick observations about the relationship between the surface areas and volumes for these three prisms, or between the amounts of material needed to build the boxes and the number of cubes that they can contain. Discuss questions such as:

• “If these prisms are boxes, which prism—B or C—would take more material to build? Which can fit more unit cubes?” (Prisms B and C would likely take the same amount of material to build because their surface areas are the same. Prism C has a greater volume than does Prism B, so it can fit more unit cubes.)
• “Which prism—A or B—would take more material to build? Which can fit more unit cubes?” (A and B can fit the same number of unit cubes but, B would require more material to build.)
• “If two prisms have the same surface area, would they also have the same volume? How do you know?” (No, Prisms A, B, and C are examples of how two figures with the same volume may not have the same surface area, and vice versa.)

Students will gain more insights into these ideas as they explore squares, cubes, and exponents in upcoming lessons. If students could benefit from additional work on distinguishing area and volume as different measures, do the optional lesson "Distinguishing Between Surface Area and Volume."