The purpose of this activity is for students to interpret expressions that represent the volume of a rectangular prism. This matching task gives students opportunities to analyze rectangular prisms and expressions closely and make connections between the structure in rectangular prisms and the symbols in their related expressions (MP2, MP7). If there is time and you would like to add student movement, have students make a poster to display the sorted cards. Students can walk around and add additional expressions to other posters to represent the volume of the prism.

• Invite previously selected groups to share their matches.

• “How do these expressions represent the volume?”
• Display:
  • =
• “How does the equation relate to Prism A?” (Both expressions show that the prism has a height of 6. One expression shows the side lengths of the base. The other expression shows the area of the base.)

The purpose of this activity is for students to compare and contrast two different ways to calculate the volume of a rectangular prism: multiplying the area of the base and its corresponding height, and multiplying all three side lengths. Students see that both strategies result in the same volume. It is a convention to consider a prism’s base the face on which it rests, however when calculating the volume of a rectangular prism, any face of the prism can be considered the base, as long it is multiplied by the corresponding height. Similarly, when calculating the volume of a rectangular prism, any edge can be considered the length, the width, or the height.

If students write numbers that don't correspond to the height for a given base, consider asking,

• “How did you decide which numbers to write in the table?”
• Display an expression that represents the height and the base of a prism. “How does this expression represent the volume of the prism?”

• Ask students to share responses to the second problem. Display the expression:
• “How does this expression represent the volume of Prism A?” (The prism's side lengths are 6 units, 3 units, and 4 units, and I multiply them to find the volume.)
• Display the expression:
• “How does this expression represent the volume of Prism A?” (One base has a length of 6 units and a width of 3 units and the height is 4 units.)
• “Which expression could you use to find the volume, using the 3-unit-by-4 unit base?” (We could use either  or .  They are equal, and both represent the volume of the prism.
• Display the equation: =
• “How do you know the equation is true?” (Both expressions represent the volume of the prism, and we can see both expressions in the prism. One expression represents a base with side lengths of 6 units and 3 units, and a height of 4 units. The other expression represents a base with side lengths of 3 units and 4 units, and a height of 6 units.)

This activity is optional if students need additional practice writing expressions to represent the volume of a rectangular prism. This activity also supports students in identifying the information they need to represent volume. Students are given the opportunity to write and interpret expressions that show that the volume is the same when multiplying the side lengths or multiplying the area of the base and the height. In the second part of the activity, students reason abstractly and quantitatively when they interpret the meaning of expressions in the context of volume (MP2).

• Display each prism.
• “Which expressions represent the volume of the prism in cubic units? Which does not?”
• “How did you decide that an expression did not represent the volume of a rectangular prism?” (Looking at the different bases and heights, and experimenting with expressions. Finding the product and checking that it does not match the volume of any of the prisms.)