Students may not be sure if the cylinder must be oblique or right. Point out that the directions aren’t specific, and they can choose which they prefer to draw.

Students may not be convinced that the pairs of pyramids have the same volumes. Remind them of the stack of coins activity, and the work they did in previous lessons with oblique figures.

The goal of this discussion is to help students extend these ideas to all triangular pyramids, not just the specific ones used in the lesson. Here are some questions for discussion:

• “Do any of these arguments depend on the specific triangular prism we used?” (No. If you sliced any triangular prism apart into three pyramids like this, you’d still have congruent bases and equal heights for pyramid pairs P2/P3 and P1/P3.)

Display this image for all to see.

• “This pyramid doesn’t look like the ones we used in the lesson. Is it possible to create 3 pyramids with the same volume as this one that could be assembled into a prism?” (The first step would be to shift the apex of the pyramid so that it’s directly over one vertex, resembling the P1 pyramid in earlier activities. This step does not change the volume of the pyramid. Then the pyramid can be duplicated, flipped upside down, and shifted to resemble the P3 pyramid. Finally, the pyramid could be duplicated and shifted to look like P2.)

Display this applet for all to see, moving the slider to show a similar process for a square prism.

The GeoGebra applet “Splitting a Prism into Pyramids—Synthesis” is available here: https://www.geogebra.org/m/wzm4xjhb.

• “How, then, could we find the volume of this pyramid?” (Just like in the activity, we could find the volume of the prism with congruent base and equal height, then multiply by .)
• “Could we use a formula to find the volume? What would it look like?” (Yes. The formula could be written or .)

Students generalize the process for finding the volume of a triangular pyramid, concluding it applies to all pyramids. As they compare cross-sections across different solids, they are looking for and making use of structure (MP7).

Some students may believe that the cross-sections have area 5 square units rather than 2.5 square units. Remind them that if a two-dimensional figure is dilated by a factor of , the area is multiplied by . However, the exact value of the area isn’t as important as the concept that the areas are the same for all 3 cross-sections.

The goal of this discussion is to help students use the reasoning developed in the task to extend the formula to all pyramids, not just triangular ones. Ask students:

• “How did you calculate the volume of the triangular prism?” (Multiplied the area of the base, 10 square units, by the height, 6 units, then multiplied by to get 20 cubic units.)
• “How did you find the volume of the remaining solids?” (The volumes of all the solids are the same.)
• “Suppose we didn’t know the actual values of the height or the area of the base, but we knew all the bases had the same area and the solids all had the same height . Would all 3 solids still have the same volume?” (Yes. Each set of cross-sections would still have the same areas. Instead of , each cross-section would have area .)
• “Does the expression give the volume for any pyramid or cone? Why or why not?” (Yes, it does. For any pyramid or cone, we can draw a triangular pyramid with identical volume, the same height, and a base with equal area. We know that works for this triangular pyramid, so it also works for the particular pyramid or cone we’re looking at, too.)