The purpose of this activity is for students to review the concept that volume is the number of unit cubes required to fill a space without gaps or overlaps. Students are asked to find all of the different ways that they can arrange 126 sugar cubes to create a rectangular prism. This provides practice with factoring since the side lengths will be factors of 126. If students struggle to find factors of 126, it may be worthwhile to pause early on in the task and discuss the different strategies students are using to find factors of 126.

A variety of different suggestions for how to pack the cubes should be anticipated and encouraged with the focus on how students decide on a particular shape of rectangular prism. In practice, many concerns influence the actual choice, such as the amount of packaging material needed and how the package fits on a store shelf. When students interpret the meaning of the numbers in the context, they reason abstractly and quantitatively (MP2).

The goal of the Activity Synthesis is to share ideas about predictions for how the cubes are packaged and how students decided they should be packaged.

• Invite students to share different ways they suggested packing the cubes and their reasoning for the choices.
• “How did you find the different side lengths?” (I divided 126 by different numbers and used multiplication facts I knew to find other combinations.)
• “Why is a 1 by 1 by 126 arrangement not useful for packaging the sugar cubes?” (It would be too long. It would break easily. It would be difficult to handle, ship, and display in stores.)

The purpose of this activity is for students to solve problems about the volume of different structures. While students can find products of the given numbers, those products do not represent the actual volumes of the structures because neither the Great Pyramid of Giza nor the Empire State Building is a rectangular prism. The pyramid steadily decreases in size as it gets taller, while the Empire State Building also decreases in size at higher levels but not in the same regular way as the pyramid. With not enough information to make a definitive conclusion, students can see that both structures are enormous and that their volumes are roughly comparable, close enough that more studying would be needed for a definitive conclusion (MP1).

• Invite students to share their calculations for the volumes of the two structures.
• “Why is it hard to find the exact volume of the Great Pyramid?” (It’s not a rectangular prism. It has slanted sides.)
• “Is the product of the area of the base and the height larger than the volume of the pyramid or smaller? How do you know?” (It is larger, because the pyramid does not fill all of that space. It gets more and more narrow toward the top.)
• “Why is it hard to find the exact volume of the Empire State Building?” (It’s also not a rectangular prism. It also gets narrower toward the top.)
• “Which do you think has greater volume?” (I think it’s too close to tell. I think the Great Pyramid is bigger because it looks like the base of the Empire State Building does not go up very far. It gets a lot narrower quickly. The Great Pyramid gets narrower more gradually.)