Lesson 11

In this warm-up, students analyze examples and counterexamples of polyhedra, observe their defining characteristics, and use their insights to sort objects into polyhedra and non-polyhedra. They then start developing a working definition of polyhedron.

Prepare physical examples of polyhedra and non-polyhedra for students to sort. These examples should be geometric figures rather than real-world objects such as shoe boxes or canisters. If such figures are not available, make some ahead of time using the nets in the blackline master.

As students work and discuss, notice those who can articulate defining features of a polyhedron. Invite them to share later.

Arrange students in groups of 3–4. Give students 1 minute of quiet time to study the examples and non-examples in the task statement. Ask them to be ready to share at least one thing they notice and one thing they wonder. Give the class a minute to share some of their observations and questions. 

Next, give each group a physical set of three-dimensional figures. The set should include some familiar polyhedra, some unfamiliar ones, and some non-polyhedra.

Ask groups to sort the figures into polyhedra and non-polyhedra (the first question). If groups members disagree about whether a figure is a polyhedron, discuss the disagreements. When the group has come to an agreement, give them 2–3 minutes of quiet time to complete the second question.

Invite students to share the features that they believe characterize polyhedra. Record their responses for all to see. For each one, ask the class if they agree or disagree. If they generally agree, ask if there is anything they would add or elaborate to make the description clearer or more precise. If they disagree, ask for an explanation or a counterexample.

Students will have a chance to refine their definition of polyhedra later in the lesson—after exploring prisms and pyramids and learning about nets, so it is not important to compile a complete or precise set of descriptions or features.

Use a sample polyhedron or a diagram as shown here to introduce or reinforce the terminology surrounding polyhedra.

• The polygons that make up a polyhedron are called faces.
• The places where the sides of the faces meet are called edges.
• The “corners” are called vertices. (Clarify that the singular form is "vertex" and the plural form is "vertices.")

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.

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

Adjust the timing of this activity to 15 minutes.

Tell students to skip the last question. Students will have time to practice using nets in the next activity.

Arrange students in groups of 3–4.

For the first question:

• Tell students that Polyhedra A–F are all prisms and Polyhedra P–S are all pyramids. (Display and pass around the prisms and pyramids in the task statement, if available.)
• Give students 2–3 minutes of quiet time for the first question and 2–3 minutes to discuss their observations in their groups. Ask them to pause for a class discussion before moving on.
• Solicit students' ideas about features that distinguish prisms and pyramids. Record students' responses in a two columns—one for prisms and the other for pyramids. It is not important that the lists are complete at this point.

Next, tell students that we are going to use nets to better understand prisms and pyramids. Explain that a net is a two-dimensional representation of a polyhedron.

Display a cube assembled from the net provided in the blackline master for the warm-up, as well as a cutout of an unfolded net (consider removing the flaps). Demonstrate how the net with squares could be folded and assembled into a cube, or use this book of digital applets, https://ggbm.at/rcu3Ka3j, created in GeoGebra by the GeoGebra DocuTeam. Point out how the number and the shape of the faces on the cube correspond to the number and the shape of the polygons in the net.

For the second question:

• Give groups a minute of quiet think time and a minute to discuss their response.
• To verify their answer, give each group one of the three nets from the first page of the blackline master. Ask them to try to assemble a triangular pyramid from their net.
• Invite groups to share with the class whether it can be done. Discuss why net 3 cannot be assembled into Pyramid P (two of the triangles would overlap).

For the last question, tell students that they will create a net of another prism or pyramid:

• Assign each group a prism or a pyramid from the task statement (except for Prism B and Pyramid P).
• Give each group a set of pre-cut polygons from the last two pages of the blackline master. 
• Tell students to choose the right kind and number of polygons that make up their polyhedron. Then, arrange the polygons so that, when taped and folded, the arrangement is a net and could be assembled into their prism or pyramid. Encourage them to think of more than one net, if possible.

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 and learn to use nets to find surface area. 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. Students apply what they learned earlier about areas of triangles and parallelograms to find surface area.

As students make calculations, monitor their processes. Note those who work systematically to find surface area (e.g., by organizing the measurements of each face, calculating the area of each face, and adding the areas together) and those who don't. Encourage students with disorganized or scattered work to take a more systematic approach. Demonstrate strategies such as labeling both the polygons on the net and portions of their work that pertain to those faces.

Also notice students who look for and use structure (MP7), for instance, by grouping certain polygons together and finding the area of the composite shape (e.g., a group of rectangles that have a common side length), or by identifying multiple copies of the same polygon and calculating the area once. Select them to share their work later.

Adjust the timing of this activity to 15 minutes.

As part of the launch, assign each group to one of the nets: A, B, or C. Tell the groups to follow the questions listed for their assigned net. For groups that complete their questions early, encourage them to find additional methods to get the surface area.

During the activity synthesis, select one group for each of the nets to share how the determined the surface area. After each group shares, invite other groups to share if they used a different method to find the surface area.

Arrange students in groups of 3. Give each group one of each net (A, B, and C), tape, and access to their geometry toolkits (especially scissors). Explain to students that they will cut some nets, assemble them into polyhedra, and calculate their surface areas. Remind students that the surface area of a three-dimensional figure is the sum of the areas of all of its faces. Ask students to complete the first question before cutting anything.

Point out that the net has shaded and unshaded polygons. Explain that only the shaded polygons in the nets will show once the net is assembled. The unshaded polygons are “flaps” to make it easier to glue or tape the polygons together. They will get tucked behind the shaded polygons and are not really part of the polyhedron. Tell students that creasing along all of the lines first will make it easier to fold the net up and attach the various polygons together. A straightedge can be very helpful for making the creases.

Tell students that it is easy to miss or  double-count the area of a face when finding surface area. Ask them to think carefully about how to record their calculations to ensure that all faces are accounted for, correct measurements are used, and errors are minimized.

When students have completed their calculations, ask them to compare and discuss their work with another student with the same polyhedron.

If students do not identity the specific type of prism or pyramid, remind them that they should also name each figure by the shape of their base.

For each polyhedron, select at least 2 students with correct calculations but different approaches to share their work, if possible.

For Polyhedron A, select students who took the following approaches, in this sequence:

• Found the area of each rectangle separately
• Found the areas of pairs of identical rectangles (3 pairs total)
• Calculated the area of a group of connected rectangles with the same length or width (e.g., the four rectangles on the net with side length 6 units)

For Polyhedron B, select students who:

• Found the area of each of the 5 polygons separately
• Found the area of the square, rearranged the 4 triangles into 2 parallelograms, and calculated the area of each parallelogram
• Calculated the area of the square and the area of 1 triangle, and multiplying the area of the triangle by 4

For Polyhedron C, select students who:

• Found the area of each of the 5 polygons separately
• Rearranged the 2 right triangles into a rectangle, and then found the area of each rectangle separately
• Calculated the area of each right triangle and doubled it, and found the area of the group of connected rectangles with a width of 4 units

Point out that the reasoning strategies we used earlier in the unit still apply here. Even though we are working with three-dimensional figures, surface area is a two-dimensional measure. 

Highlight the benefits of approaching the problems systematically, e.g., by labeling parts, listing measurements and computations in order, etc.