Lesson 14

This warm-up prompts students to match nets to polyhedra. It invites them to think about the polygons that make up a polyhedron and to mentally manipulate nets, which helps develop their visualization skills.

Give students 3 minutes of quiet think time to match nets to polyhedra and then another 2 minutes to discuss their response and reasoning with a partner. Encourage students to use the terminology they learned in prior lessons.

To support students who need more time or help in visualization, prepare physical models of the polyhedra and copies of the nets from the blackline master. Pre-cut the nets or have scissors available so that students can assemble the nets and test their ideas.

If students have trouble distinguishing between figures A, C, and D, remind them that prisms and pyramids can both contain faces that are triangles. In a pyramid, all triangular faces that are not the base meet at a one vertex and have shared edges. In a prism, there can be a triangular base, but the other faces are quadrilaterals.

Invite a few students to share their matching decisions and reasoning with the class. Ask students: “What clues did you use to help you match? How did you check if you were right?” If there is not unanimous agreement on any of the nets, ask students with differing opinions to explain their reasoning. Discuss to come to an agreement.

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.

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.