Students may struggle to draw the cone surface that’s in the shape of a sector of a circle. Ask them to consider snipping the cone in a straight line along this face and unrolling it.

Select groups to share their categories and how they sorted their solids. Discuss as many different types of categories as time allows, but ensure that one set of categories distinguishes between right and oblique solids, and another distinguishes between solids with an apex versus those without. Attend to the language that students use to describe their categories, giving them opportunities to describe their solids more precisely. Highlight the use of terms like “pyramid,” “cone,” “perpendicular,” and “slanted.”

Ask students to compare the cone on Card E and the cylinder on Card F. (The cone gets smaller at the top. The cylinder does not.) If need be, encourage students to rephrase their answer, using the language of cross-sections. (Cross-sections taken parallel to the base of prisms and cylinders are congruent throughout the solid. Cross-sections taken parallel to pyramid and cone bases are similar to each other, but are not congruent.)

Tell students that we can use the categories they created to define some characteristics of solids. A pyramid is a solid with one face (called the base) that’s a polygon. All the other faces are triangles that all meet at a single vertex, called the apex. A cone also has a base and an apex, but its base is a circle and its other surface is curved. Ask students to point to an apex on their cards.

Just like prisms and cylinders can be right and oblique, so can cones and some pyramids. For a cone, imagine dropping a line from the cone’s apex straight down at a right angle to the base. If this line goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids with bases that have a center, like a square, a pentagon, or an equilateral triangle, can also be considered right or oblique in the same way that cones can be.

Ask students to hold up a card with a right cone or pyramid. (Card E) Ask students to hold up a card with an oblique cone or pyramid. (Card A, Card D, Card G)

Point out that some mathematicians consider a cone to be a “circular pyramid,” others consider pyramids to be “polygonal cones,” and still others classify them in totally separate categories. Regardless of what we call them, the two kinds of solids share certain properties that will be explored in upcoming activities.

The goal of the discussion is to make observations about the relationships between the three pyramids and the prism. Here are some questions for discussion. It’s okay for the answers about triangle congruence to be informal.

• “Take a look at the pyramid marked P3. Which face would you consider its base? Is there only one possibility?” (Any face of this pyramid could be considered the base. No matter which face we choose to call the base, the remaining faces are all triangles. This is actually true for all three pyramids.)
• “Which faces of the prism are its bases?” (The faces marked P1 and P3 are the prism’s bases.)
• “Do the pyramids marked P1 and P3 have any congruent faces? If so, which are they, and how do you know?” (The faces marked P1 and P3 are congruent because they are the prism’s bases. The faces with the lines are also congruent, because together, they form a rectangle.)
• “Do the pyramids marked P2 and P3 have any congruent faces? If so, which are they, and how do you know?” (The gray-colored faces are congruent because together, they form a rectangle. The two faces that are unmarked are congruent to each other. When assembled into the prism, each line segment that forms the sides of the triangles is shared between the two shapes.)

To ensure the pyramids are available for an upcoming activity, collect the assembled pyramids or direct students to place them in a safe storage area.