Students will find how many sugar cubes are in the box by finding the volume of the box using sugar cubes as the unit of measure.

In this task, students use a nonstandard unit of measure (a sugar cube) in order to determine the volume of a box of sugar. This task serves to illustrate the relationship between addition and multiplication and how the two operations can be used to determine the volume of a rectangular prism.

Some of the big ideas that will likely emerge in this task include:

• Volume is an attribute of a three-dimensional space;
• Volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps;
• The volume of a rectangular prism is related to the edge lengths;
• The volume of a rectangular prism can be found by counting the cubes in one layer and multiplying it by the number of layers;
• The volume of a rectangular prism can be determined by finding the area of the base and multiplying by the number of layers;
• The volume of a rectangular prism can be determined by multiplying length, width and height; and,
• The product remains unchanged, no matter how the numbers being multiplied are ordered (commutative property of multiplication).

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A variety of tools for students to use to think through the problems, such as:

• Linking cubes or Omnifix cubes
• Isometric dot paper & colored pencils or markers
• Grid paper
• Whiteboards & markers
• Teaching remotely? NCTM’s Isometric Drawing Tool can be used to build and manipulate cube structures

Show students the following video:

Then, ask students:

What do you notice?

What do you wonder?

Have students do a Think-Pair-Share routine:

1. Students have individual think time to jot down ideas on paper or whiteboard (minimum 1 minute). Students may benefit from watching the video a second time or having a still image from the video to refer to.
2. Students share their observations and questions with a partner.
3. Students share as a whole group, either their own noticings and wonderings, or a meaningful observation or question they heard from their partner during the share (while giving credit to their partner). All contributions are acknowledged and recorded on an anchor chart on the board.

Possible points that may come up include:

• I notice there’s a box of sugar with a single cube next to it.
• I wonder how many sugar cubes are in the box?
• I wonder how much the box costs?
• I wonder how much a single cube costs?
• I notice the box hasn’t been opened.
• I wonder how many calories are in the box?
• I wonder what the dimensions of the box are?
• I wonder how much the box weighs?
• I wonder how much a single cube weighs?

Some wonderings can be answered immediately and crossed off the list:

• The box weighs 1kg.
• The box costs $1.95.

Spending time to acknowledge and address specific thoughts that students shared, whether a notice or a wonder, is crucial to building a culture in your classroom where students know that their voice is being valued and thus encourages them to continue sharing their thoughts and opinions later in this lesson and in future lessons.

Once student noticings and wonderings have been acknowledged and noted, the class can settle on a first question to explore:

How many sugar cubes are in the box?

Make an estimate.

Students begin by making an estimate before they are provided with all the information they need to answer the question, thus requiring them to consider what would be a reasonable number of cubes.

Encouraging students to begin by considering estimates that are “too high” and “too low” will provide a range within which more likely estimates can be framed. It can also provide students who may be reluctant to share with a safe entry point. Students can then make a “reasonable estimate”, which they will have a chance to revise.

Be sure not to skip over asking students to make an estimate using only their spatial reasoning skills as this is a very important step in the Curiosity Path. Providing students an opportunity to make an estimate and try to articulate their thinking with their peers provides a very low floor opportunity for them to not only better understand the context, but to also begin nudging them to think about what will be important to make their estimate more precise as we continue through the lesson.

Students should be given an opportunity to share their estimates at this point, but refrain from sharing their rationale just yet in order to give everyone a chance to develop their own thinking.

Have students turn to a partner and generate questions they could ask that would provide them with information they could use to answer the question. Ask students how they would use the information to answer the question.

Prompt:

What information would you need to answer the question?

How would you use that information?

Reveal to the students the image of a single layer of sugar cubes:

At this point, students may wish to update their estimates. Provide them with time to do so before revealing the next image. A simple prompt might be:

You’ve seen how many cubes are in a single layer. Does this affect your estimate? Would you say your estimate is still reasonable, or will you revise it?

Provide students with time to revise their estimates.

Listen and observe as students revise their estimates. Are students aware that the layered structure of a rectangular prism allows them to think additively or use multiplication?

How are students approaching the problem? What strategies and mathematical models are they leveraging? Are they:

• Counting cubes one by one?
• Making use of the additive nature of volume?
• Using the layered structure in a rectangular prism to think about volume using multiplication?
• Using area models to find products?
• Using an algorithm?

This is an opportunity to make note of where students are developmentally. Prompt students to explain their thinking and how they plan to use their model/strategy to determine the total number of sugar cubes in the box.

Proceed to reveal the number of layers by showing the following image:

To begin the sense making portion or the lesson, provide students with the following prompt:

There are 9 layers of sugar cubes in the box. If each layer has 4 rows of 8 cubes, how many sugar cubes are in the box?

Use a mathematical model of your choice to convince your mathematical community.

Be sure to explicitly state that calculators are not to be used to determine the number of sugar cubes in the box. However, do consider making the following tools available to students without explicitly directing students to use them:

• Linking cubes or Omnifix cubes
• Isometric dot paper & colored pencils or markers
• Grid paper
• Whiteboards & markers
• Teaching remotely? NCTM’s Isometric Drawing Tool can be used to build and manipulate cube structures

Remind students to refrain from shouting out answers, and be sure to emphasize the importance of creating a viable argument – a model that can be used to convince others.

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Consider sharing the following reveal video with your students:

Alternatively, you could share the reveal image:

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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