In the previous lesson, students reasoned abstractly about the volume of a rectangular prism when they considered the volume in terms of layers or equal groups of unit cubes. This activity continues to develop the idea of decomposing rectangular prisms into layers. Students explicitly multiply the number of cubes in a base layer by the number of layers. Students can use any layer in the prism as the base layer as long as the height is the number of those base layers.

• Display the expressions:
• “How do these expressions represent the volume of Prism A?” (There are two layers of 12. We also can see 3 layers of 8.)
• “How does thinking about layers help us find the volume of prisms that are not completely filled?” (I know that all the layers have the same number of cubes even if they are not shown.)

In the previous activity, students saw that a rectangular prism is composed of layers and there are different ways to decompose a prism into layers, depending on how it is viewed. Students recognize that the volume remains the same, regardless of the orientation of the prism. The goal of this activity is for students to identify how different expressions represent the volume of the same prism and correspond to the organization of the layers. Students have worked with parentheses in previous grades, so the Lesson Synthesis provides an opportunity for students to revisit expressions with parentheses. Students will have more experience with evaluating expressions with grouping symbols in future lessons. Students go back and forth between numerical expressions and a geometric object, the volume of which is represented by the expression (MP2).

• “How does represent the volume of the prism?” (There are 5 layers if I cut the prism horizontally, and each layer has 24 cubes.)
• Display the expression:
• “How does this expression represent the volume of the prism?” (It shows the 5 layers and the 24 cubes as 4 rows of 6 cubes in each layer.)
• “How does  represent the volume of the prism?” (There are 6 layers if I cut vertically along the front face. Each layer has 5 rows of 4 or 20 cubes.)
• Display the expression: .
• “How does this expression represent the volume of the prism?” (It shows the 6 layers and 4 columns of 5 cubes in each layer.)