The purpose of this activity is for students to find the volumes of figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts. There are different ways to decompose the figures. Monitor for students who break apart the figures differently and find the same total volume. To reinforce earlier work with cubic units of measure, ask students for the unit of measure in their response if they state the volume as only a number (MP2). If students finish early, give them isometric grid paper to draw a figure composed of two rectangular prisms for their partner to find the volume.

If students find a volume that does not represent the volume of Figure 1 or Figure 2, consider asking,

• “Where do you see rectangular prisms in this figure?”
• Show an expression that represents the volume of the figure. “How does this expression represent the volume of the figure?”

• Display Figure 2.
• “How did you break up this shape to find the volume?” (I cut off the overhanging piece vertically to make two rectangular prisms. I cut off the bottom piece that the shape is resting on, horizontally.)
• “How did you find the side lengths of the rectangular prisms?” (Some of the lengths are provided, and I found the others by subtracting.)
• “Did you get the same volume when you broke up the figure differently? Why?” (Yes. The calculations were different, but both tell me how many cubic inches it takes to fill the shape.)

The goal of this activity is to represent expressions as decompositions of a figure made of two non-overlapping right rectangular prisms. This gives students an opportunity to interpret parentheses in expressions while also checking their understanding of different ways to represent the volume of a rectangular prism; namely, length times width times height, and area of a base times the corresponding height.
Students work abstractly and quantitatively in this problem (MP2) as they relate abstract expressions to decompositions of figures composed of two rectangular prisms.

• Invite students to share how the expressions for the first prism represent its volume.
• “Why are there three factors in the expression  but only two factors in the expression ?” (The 2, the 3, and the 4 are the length, the width, and the height of the prism in the back. The 5 is the height of the prism on the bottom, the base area of which is 6.)
• “In the expression , what do the parentheses tell you?” (They tell me to first multiply 3 by 3, which gives me the area of the base, and then I multiply that by 2 to get the volume of the prism.)