The purpose of this activity is for students to find the volume of a figure in different ways. The given figure can be decomposed in two ways into rectangular prisms by making different cuts. The volume also can be found by removing a smaller rectangular prism from a single, larger rectangular prism. This provides an opportunity to express the volume of the figure as a difference of the volumes of rectangular prisms. Students may notice this feature, and it is highlighted in the Activity Synthesis.

When students decide whether or not they have the same expressions, they need to reason carefully about what “the same” means. They consider if the order of the factors is different, is it the same expression, and if the order of the addends is different, is it the same expression. Students use what they know about volume, geometric figures, and the properties of operations to justify the equivalence of the expressions and critique the reasoning of their peers (MP2, MP3, MP7).

• Invite students to share different expressions for the volume of the figure.
• Display the expression:
• “How does the expression represent the volume of the figure?” ( represents the prism that is 3-units-by-5-units-by-5-units tall and the represents the prism that is 2-units-by-2-units-by5-units tall.)
• If no student wrote the expression , display this expression.
• “How does the expression show the volume of the figure in cubic units?” (There is a 5-unit-by-5-unit-by-5-unit cube, and a piece has been taken away. The piece taken away measures 2 units by 3 units by 5 units.)

The purpose of this activity is for students to write equivalent expressions in order to find the volume of a figure composed of two right rectangular prisms. Students decompose the figure in two different ways, and write matching expressions to find the volume. For extra support, provide students with colored pencils to shade the two parts of the prism before they find the side lengths needed to calculate the volume.

Monitor and select a student, with each of the following approaches, to share in the Activity Synthesis:

• Decompose the figure into a prism, with the side lengths 4 feet by 4 feet by 3 feet and a prism with the side lengths 10 feet by 4 feet by 3 feet, and write this expression (or the equivalent) to represent the volume: .
• Decompose the figure into a prism with the side lengths 4 feet by 8 feet by 3 feet and a prism with the side lengths 6 feet by 4 feet by 3 feet, and write this expression (or the equivalent) to represent the volume: .
• Enclose the figure to create a prism with side lengths 10 feet by 8 feet by 3 feet, and subtract the part of the prism with side lengths 6 feet by 4 feet by 3 feet, using a subtraction equation like Mai’s in problem 3 to represent the volume.

The approaches will be displayed later, side by side with Mai’s approach, to help students connect how they decompose each figure and the expression they use to find the volume. If a student uses an approach like Mai’s in the first problem, then invite them to display their original thinking in the Activity Synthesis discussion. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently.

• Invite previously selected students to display their work side by side for all to see, without sharing their thinking. Also display Mai’s approach if no one in the class uses this subtraction strategy.
• “Take a minute to look at the diagrams and expressions.”
• Connect students’ approaches by asking:
  • “How are the diagrams the same?” (Each way to think about the prism has 2 other prisms.)
  • “How are they different?” (In each of two diagrams, the figure is decomposed into 2 rectangular prisms to find the volume of each, and then the volumes are added. Mai’s diagram has a larger rectangular prism, with the side lengths 10 feet by 8 feet by 3 feet. It shows subtracting a rectangular prism with the side lengths 6 feet by 4 feet by 3 feet.)
• Connect students’ approaches to the learning goal by asking:
  • “How do the expressions with addition relate to the diagrams?” (The prism is decomposed into 2 smaller prisms. Multiply the side lengths of each prism, and then add them together.)
  • “How does Mai’s expression represent the volume of the prism?” (Multiply the side lengths of the larger and smaller rectangular prisms, and then subtract the smaller volume from the larger volume to find the volume of the figure.)
  • “What is the value of ?” (168, I calculated it, but I also know this because it represents the volume. The value is the same as the other expressions for volume for this figure.)