Lesson 11

This warm-up reviews the volume work students had done previously to prepare for the work in this lesson. It reinforces the idea of using unit cubes and fractional-unit cubes as a way to measure the volume of a rectangular prism.

Give students 2–3 minutes of quiet work time. Follow with a class discussion.

Select several students to share their responses and reasoning. After each person explains, ask students to indicate whether they agree. To involve more students in the discussion, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree with the reasoning? Why?”

Tell students that they will use their understanding of the volume of rectangular prisms to solve other geometric problems.

This activity extends students’ understanding about the volume of rectangular prisms from earlier grades. Previously, students learned that the volume of a rectangular prism with whole-number edge lengths can be found by computing the number of unit cubes that can be packed into the prism. Here, they draw on the same idea to find the volume of a prism with fractional edge lengths. The edge length of the cubes used as units of measurement are not 1 unit long, however. Instead, they have a unit fraction (\(\frac 12\), \(\frac 14\), etc.) for their edge length. Students calculate the number of these smaller cubes in a prism and use it to find the volume in a standard unit of volume measurement (cubic inches, in this case).

By reasoning repeatedly with small cubes (with \(\frac12\)-inch edge lengths), students notice that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.

 Students may need a reminder to label volume in cubic units.  

Display a completed table for all to see. Give students a minute to check their responses. Invite a few students to share how they determined the volume of the prisms, and their observations about the relationships between the number of cubes (with \(\frac12\)-inch edge lengths) and the volume of prism in cubic inches. Highlight that the volume can be found by calculating the number of cubes and multiplying it by \(\frac 18\), because each small cube has a volume of \(\frac 18\) cubic inch. 

If students do not notice a pattern in their table, ask them what they notice about the edge lengths of the prisms in the second, fourth, and sixth rows of the table and their corresponding volumes. Make sure they see that the volume of each prism can also be found by multiplying its side lengths. In other words, we can find the volume of a rectangular prism with fractional edge lengths the same way we find that of a prism with whole-number edge lengths.

In this activity, students continue the work on finding the volume of a right rectangular prism with fractional edge lengths. This time, they do so by packing it with unit cubes of different unit fractions for their edge lengths—\(\frac13\), \(\frac12\), and \(\frac14\) of an inch. They use these cubes to find the volume of the prism in cubic inches, decide which unit fraction works better to accomplish this goal and why, and explain whether cubes of different fractional edge lengths would lead to the same volume in cubic inches (MP3).

As students work, notice those who are able to clearly explain why cubes with a particular fractional edge length are preferable as a unit of measurement and why the volume in cubic inches will be the same regardless of the cubes used. Invite them to share later.

Arrange students in groups of 3–4. Give them 8–10 minutes of quiet work time, and then 5 minutes to discuss their responses with their group. Ask groups to be sure to discuss the third question. Encourage students to draw a sketch to help with reasoning, if needed.

If students have trouble getting started, ask them to revisit their work with \(\frac12\)-inch cubes from a previous lesson. Remind them that a cube with \(\frac 12\)-inch edge length has a volume of \(\frac18\) in³ (because we can fit 8 of such cubes in a 1-inch cube). Ask them to think about how many \(\frac13\)-inch cubes can fit into a 1-inch cube, or think about what the volume of a \(\frac13\)-inch cube is in cubic inches.

Some students may not be able to visualize and keep track of the measurements of the boxes in this task. Encourage students to draw and label the measurements of the boxes described in these questions.

Select several students to share their responses and articulate their reasoning. Compare the different strategies students used for finding the volume of the rectangular prism. Ask students:

• “Does it matter which fractional-unit cubes we use to find the volume? Why or why not?” (As long as the unit fraction can fit evenly into all three edge lengths of the prism, it doesn’t matter what unit fraction we use.)
• “Do certain unit fractions work better as edge lengths of the small cubes than others?” (It helps to use as large a unit fraction as possible, since it means using fewer cubes and working with fractions that are closer to 1.)
• “Is there another way of finding the volume of a rectangular prism with fractional edge length besides using these small cubes?” (Multiply the fractional edge lengths.)

Point out that it is helpful to use a unit fraction that is a common factor of the fractional edge lengths of the prism. Make sure students also recognize that multiplying the edge lengths of the prism is a practical way to find the volume of such a rectangular prism.

In this activity, students solve word problems that involve finding the volume of rectangular prisms that have fractional edge lengths, and calculate unknown edge lengths given other measurements. The last question in the activity requires students to interpret how the same volume of liquid would fit in two different containers in the shape of rectangular prisms.

As they work, monitor for different representations students use to solve the problems.

Keep students in groups of 3–4. Give students 5 minutes of quiet work time for the first question and 2–3 minutes to discuss their responses with their group. Then, give students time to complete the second question either individually or with their group. Encourage students to draw a sketch to help with reasoning, if needed.

Invite a few students to share their solutions, explanations, and drawings (if any). Record and display their solutions for all to see. To involve more students in the discussion, ask students to indicate whether they agree or disagree with their classmate’s reasoning, if they approached it the same way but could explain it differently, or if they have an alternative path.