In this activity, students practice mentally decomposing, into simpler prisms, a more-complicated prism that has a non-rectangular base. The decomposition corresponds to a decomposition of a complicated two-dimensional figure into simpler two-dimensional figures. This expands on students’ ability to calculate the area of a base of a figure and to rely on the structure of a prism to find its volume (MP7). This prepares students to calculate the volume of this figure and other figures in future lessons.

As students work in their groups, monitor for the different ways in which students are decomposing or constructing the base of the figure into more familiar shapes.

The purpose of this discussion is to clarify decomposing shapes to find area, which can then be used to calculate the volume. Select students to share whose method they decided to use and why. Ask students: 

• “Whose method could not be used? Explain how you know.” (Lin’s, because we don’t know the base and height of the trapezoids.)
• “How did you find the area of the base?”
• “What was different about the base of this figure in comparison to other bases we have worked with?” (This base needed to be decomposed to calculate its area.)
• “What was your process to calculate the volume?” (I used the area of the base and multiplied it by the height of the figure.)
• “Why would a chocolatier want to know the volume of a heart shaped box like this?” (He may want to know how many candies can fit inside of a box.)

Explain to students that they might encounter figures that have non-rectangular bases in future activities or lessons. It will be important for them to think about different strategies to calculate the area of the base.

In this activity, students practice finding the volume of another prism with a non-rectangular base by applying the formula .

As students work on the task, monitor for students who decompose or compose the base of the figure into more familiar shapes.

The goal of this activity is to compare different methods of decomposing the base of the prism. Select previously identified students to share the different methods for calculating the area of base. If not brought up by students, explain to students that the base of this figure can either be decomposed into rectangles and triangles or be composed into a larger rectangle by adding two additional triangles. Ask students how they used the area of the base to calculate the volume of the figure (area of the base multiplied by the height).