Lesson 6

The purpose of an Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. In this activity, students estimate the number of sugar cubes in the bowl. During the synthesis, revisit what students know about rectangular prisms and volume and connect it to the image of the sugar cubes.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “What would make it easier to find the exact number of cubes?” (If the cubes were organized in a way that we could count groups. If we could see them all.)

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 synthesis is to share ideas about predictions for how the cubes are packaged and how students decided they should be packaged.

• Groups of 2
• Display the image from the student workbook:

  

  

• “These are sugar cubes. They are used to sweeten coffee and tea. Has anyone ever seen or tasted a sugar cube?”
• “You are going to investigate different ways you can arrange 126 cubes to make a rectangular prism and then look at a particular example.”

• 2 minutes: independent work time
• 5–7 minutes: partner work time

• Invite students to share different ways they suggested packing the cubes and their reasoning for the choice.
• “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.)

The purpose of this activity is for students to solve problems about the volume of different buildings. While students can find products of the given numbers, those products do not represent the volume of the structure. In both cases, the Great Pyramid of Egypt and the Empire State Building, neither structure 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).

• Groups of 2
• To help understand how large the Great Pyramid and the Empire State Building are, consider estimating the size of the classroom. Estimates will vary but should be a few hundred cubic meters (versus several million for these huge structures).

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who
  • use the standard algorithm for multiplication
  • make estimates rather than using the standard algorithm for multiplication
  • identify that it is not possible with the given information to find the exact volume of either structure

• Invite students to share their calculations for the volumes of the 2 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?” (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.)