This activity serves two goals: to uncover the defining features of prisms and pyramids as well as to introduce nets as two-dimensional representations of polyhedra.

Students first analyze prisms and pyramids and try to define their characteristics. Next, they learn about nets and think about the polygons needed to compose the nets of given prisms and pyramids. They then use their experience with the nets of prisms and pyramids to sharpen and refine their definitions of these polyhedra.

As students discuss the features of prisms and pyramids, encourage them to use the terms face, edge, and vertex (vertices) in their descriptions.

Select groups to share their arrangements of polygons. If time permits and if possible, have students tape their polygons and fold the net to verify that it could be assembled into the intended polyhedron. Discuss:

• “What do the nets of prisms have in common?” (They all have rectangles. They have a pair of polygons that may not be rectangles.)
• “What do the nets of pyramids have in common?” (They all have triangles. They have one polygon that may not be a triangle.)
• “Is there only one possible net for a prism or a pyramid?” (No, the polygons can be arranged in different ways and still be assembled into the same prism or pyramid.)

Explain the following points about prisms and pyramids:

• A prism has two parallel, identical faces called bases and a set of rectangles connecting the bases.
• Prisms are named for the shape of the bases. For example, if the base of a prism is a pentagon, then the prism is called a “pentagonal prism.”
• A pyramid has one face called the base that can be any polygon and a set of faces that are all triangles. Each edge of the base is shared with an edge of a triangle. All of these triangles meet at a single vertex.
• Pyramids are named for the shape of their base. For example, if the base of a pyramid is a square, then the pyramid is called a “square pyramid.”

In this activity, students cut and assemble nets into polyhedra. They use nets to find surface area, applying what they learned earlier about areas of triangles and parallelograms. The presence of a grid supports students in their calculations. It also reinforces the idea of area as the number of unit squares in a region and the connection between area and surface area. 

As they find surface area, students have an opportunity to look for and make use of structure (MP7). For instance, they may  identify multiple copies of the same polygon and find the combined area with a single calculation. They may also group rectangles with a common side length into a larger rectangle and find the area of the latter. 

Monitor for the different approaches that students take. Here are some likely approaches, starting from less systematic to more systematic:

• Find the area of each face separately and add the areas.
• Identify unique shapes, find their areas, multiply the area of each unique shape by how many there are, and add the areas.
• Rearrange and group two or more faces (for instance, rearrange a pair of triangles into a rectangle or a parallelogram) before finding and adding the areas.

Also monitor for how students organize their work, such as whether they record the measurements of each face, label the shapes in the net and the associated calculations, and account for all the faces of their polyhedron. Encourage students with disorganized or scattered work to take a more methodical approach.

Invite previously selected students to share their responses and reasoning. For each polyhedron, sequence the discussion of the strategies in the order listed in the activity narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as: 

• “Before cutting out and assembling your net, how did you know what polyhedron it would create?”
• “We saw more than one strategy for finding the surface area of a polyhedron [letter]. How are the strategies alike? How are they different?”
• “How can we make sure that we’ve included the areas of all the faces of the polyhedron?”

If methods for organizing the work—such as listing measurements and labeling polygons and computations—are not mentioned, bring them up. If needed, demonstrate these organizational methods and discuss their benefits.

Connect the approaches used here to those used earlier in the unit by asking students: “How is finding the surface area of a polyhedron like finding the area of a polygon? How is it different?”

Highlight that even though polyhedra are three-dimensional figures, surface area is a two-dimensional measure, so the same strategies we used to find area—such as decomposing and rearranging—still apply here.