## IM Commentary

The purpose of this task is to introduce students to fractional units for volume. This task provides opportunities to

- extend the formula $V=lwh$ for non-whole number side lengths;
- practice fraction multiplication and verify the result geometrically;
- highlight the fact that a volume can be a numerically smaller number than the lengths of its sides.

A note about "plenty of snap cubes" for planning: the largest rectangular solid that a student would build for this task consists of 48 snap cubes. It would be appropriate for students to work in pairs for this task, so you don't necessarily need 48 snap cubes for every student. It may be helpful, ahead of time, to cut up and provide students with lengths of straws that are each the length of two snap cube edges (one dubsnap).

We chose the word "dubsnaps" in place of "unit" to avoid any misreadings, since "unit" often implies "one." But it would be mathematically fine, if desired, to replace all instances of "dubsnap" in the task with "unit" (or even with a new word that the class likes better.)

We suggest students be given time to work and make sense of the task with intermittent pauses for whole-class clarification and discussion. It will be necessary for students to attend to precision (MP6) as they communicate about the problem and their solutions, because if they neglect units of measure while speaking, this is likely to lead to confusion. Once students have had a chance to struggle a bit on their own, the teacher should listen for the language students are using and encourage precision where needed.

As a part of whole-class discussion, the teacher should highlight the geometric interpretation of the volume alongside the result of using a formula like $V=lwh$ that students should know from previous work with whole numbers. For example, part (a)(ii) asks about the volume of a single snap cube. A single snap cube is geometrically $\frac18$ of a dubsnap cube, but also $\frac18$ is the product $\frac12 \times \frac12 \times \frac12$.

One idea might be to preface this task with discussion about the statement, "Always, Sometimes, or Never: The volume of a rectangular prism is a larger number than any of its side lengths." Many students may initially conjecture "always" and give some examples, and then the statement could be revisited after this task.

We highly recommend that students work with physical snap cubes. This Desmos sketch is provided as a resource, perhaps for demonstration purposes. (Created by Nathan Kraft.)

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