In this activity, students find the volume of a prism with fractional edge lengths. In an earlier grade, 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, the cubes have a unit fraction (such as or ) for their edge lengths. Students calculate the number of such cubes that can be packed into a rectangular prism and use that number to find the volume in a standard unit of volume measurement (cubic inches, in this case).

By reasoning repeatedly with small cubes (with -inch edge lengths), students notice regularity (MP8): that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.

This activity works best when each student has access to physical cubes because students will benefit from manipulating the cubes and building prisms by hand.

If physical manipulatives are not available, consider using the digital version of the activity. In the digital version, students use an applet to build rectangular prisms from cubes with edge lengths of unit. The applet allows students to measure the edge lengths of the resulting prisms.

The digital version is adapted from an applet made in GeoGebra by Susan Addington.

The purpose of the discussion is to highlight that the volume of a rectangular prism can be found by:

• Packing it with cubes with no gaps and multiplying the number of cubes by the volume of each cube.
• Multiplying the edge lengths of the prism.

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.

If no students mention using the number and volume of the -inch cubes, ask them to observe the relationship between the values in the last two columns of the table (the number of cubes with -inch edge lengths and the volume of prisms in cubic inches). Highlight that the volume can be found by calculating the number of cubes and multiplying it by , because each small cube has a volume of cubic inch.

If no students mention finding the product of the edge lengths in inches, draw students’ attention to the values in the first three columns of the table and in the last column. (Looking at the whole-number values in the second, fourth, and sixth rows of the table may be particularly helpful.) Make sure students see that the volume of each prism can also be found by multiplying its side lengths in inches.

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—, , and of an inch. They use these cubes to find the volume of the prism in cubic inches and explain whether cubes of different fractional edge lengths would lead to the same volume. In articulating their reasoning and considering others’, students practice constructing a logical argument and critiquing the reasoning of others (MP3).

Select 2–3 students to explain why Diego’s statement about the number of -inch cubes in the prism is correct and to share what the volume of the prism is.

Next, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “Lin and Noah will not find the same volume in cubic inches because Lin uses 8 cubes and Noah uses 64 cubes, which is 8 times as many cubes as Lin uses.”

  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2 minutes to work with a partner to revise the first draft. Invite students to include a labeled diagram if it could further clarify their revised explanation.
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time permits, 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.)
• “Is there another way of finding the volume of a rectangular prism with fractional edge lengths besides using these small cubes?” (YMultiply the fractional edge lengths.)

In this activity, students solve word problems that involve finding the volume of rectangular prisms given the area of the base of the prism and its height. Students also calculate an unknown edge length in a rectangular prism given other measurements.

The question in Are You Ready for More? requires students to interpret how the same volume of liquid would fit in two different containers in the shape of rectangular prisms. All questions offer students an opportunity to make sense of problems and persevere in solving them (MP1).

Students might not recall that the volume of a rectangular prism can also be found by multiplying the area of the base of the prism by the height of the prism. If so, ask them how the area of the base is related to the edge lengths of the prism. If needed, remind students that the area of the base is the product of the length and the width of the prism.

Focus the discussion on how students use known volume measurements to find the height of the water in the last question. Invite 1-2 students to share their solution, explanation, and drawing (if any). Record and display their reasoning for all to see.

To involve more students in the conversation, consider asking:

• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Do you agree or disagree? Why?”