In this activity, students revisit the meaning of volume and how to find it by building the largest cube possible from 32 snap cubes. Students also become familiar with two perfect cubes, 27 and 64, before the next activity introduces this term.

The activity prompts students to recall their work from grade 5—that the volume of a rectangular prism can be calculated in two different ways: by counting unit cubes that can be packed into the prism, and by multiplying the edge lengths of the prism.

Monitor the different approaches students take to find the volume of the built cube. Here are some likely ways, from less efficient to more efficient:

• Counting all of the snap cubes individually
• Counting the number of snap cubes per layer and then multiplying that by the number of layers
• Multiplying three edge lengths

This activity works best when each student has access to snap cubes. If physical cubes are not available, consider using the digital version of the activity. In the digital version, students use an applet that has 32 cubes with which to build a prism. For students who finish early, another applet with 64 cubes can be found in Are You Ready for More?

Focus the whole-class discussion on the ways in which students found the volumes of the two cubes they built. Invite previously selected students to share their strategies. Sequence the discussion of the strategies in the order listed in the Activity Narrative, starting from the less efficient (counting individual cubes) to the most efficient (multiplying side lengths).

Connect the last strategy to the learning goals by emphasizing how the side length of a cube determines its volume. Specifically, highlight that the number 27 is . If any group built a cube with 64 snap cubes, point out that the number 64 is . These observations prepare students to think about perfect cubes in the next activity and about a general expression for the volume of a cube later in the lesson.

In this activity, students think about examples and non-examples of perfect cubes and find the volumes of cubes given their edge lengths. They see that the edge length of a cube determines its volume, notice the numerical expressions that can be written when calculating volumes, and write a general expression for finding the volume of a cube (MP8).

Some students may be unsure about writing the answer to the last question symbolically because it involves a variable. Those students may prefer to write a verbal explanation. This is fine, because in an upcoming lesson they will learn to use exponential notation with numbers and variables.

Watch for students who use square units instead of cubic units. Remind them that volume is a measure of the space inside the cube and is measured in cubic units.

Students may multiply by 3 when finding the volume of a cube instead of multiplying three edge lengths (which happen to be the same number). Likewise, they may think a perfect cube is a number times 3. Suggest that they sketch or build a cube with that edge length and count the number of unit cubes. Or ask them to think about how to find the volume of a prism when the edge lengths are different (for instance, a prism that is 1 unit by 2 units by 3 units).

Direct students' attention to the reference created using Collect and Display. Ask students to share how they thought about the first question and decided if a number is or is not a perfect cube. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. 

Highlight the idea that multiplying three edge lengths allows us to determine volume efficiently, and that determining if a number is a perfect cube involves thinking about whether it is a product of three of the same number.

If a student asks about 0 being a perfect cube, wait until the end of the lesson, when exponent notation is introduced. 0 is a perfect cube because .

Make sure that students see the answers to the last three questions written as expressions:   

This activity introduces students to the exponents “2” and “3” and the language that we use to talk about them. Students use and interpret this notation in the context of both geometric squares and their areas, and geometric cubes and their volumes. Students are likely to have seen exponent notation for in their work on place values in grade 5. That experience would be helpful but is not necessary.

Note that the term “exponent” is deliberately not defined more generally at this time. Students will work with exponents in more depth in a later unit.

As students work, observe how they approach the last two questions. Identify a couple of students who approach the fourth question differently so they can share later. Also notice whether students include appropriate units, written using exponents, in their answers.

Direct students' attention to the reference created using Collect and Display. Ask partners to share disagreements in their responses, if any. Then focus the whole-class discussion on the last two questions. Select a couple of previously identified students to share their interpretations of the question about a cube with an edge length of 5 inches. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.

Highlight that a cube with a volume of cubic units has an edge length of 6 units, because we know there are unit cubes in a cube with that edge length.

In other words, we can express the volume of a cube using a number (216), a product of three numbers (), or an expression that uses exponent (). This idea can be extended to all cubes. The volume of a cube with edge length is:   

Students will have more opportunities to generalize the expressions for the volume of a cube in the next lesson.