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## CAN'T GET ENOUGH CEREAL

### VOLUME AND SURFACE AREA OF RECTANGULAR PRISMS

Explore concepts relating to volume and surface area of right prisms.

## Intentionality & Unit Overview

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Students will compare the volume of right rectangular prisms with fractional side lengths as they decide which cereal box has the most cereal.

In this task, students will compare the volume of rectangular prisms. This task serves to illustrate the relationship between multiplication and volume, as students develop the formula for volume of right rectangular prisms.

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 right rectangular prism is related to the edge lengths.
• The volume of a right rectangular prism can be determined by finding the area of the base and multiplying by the height.
• The volume of a rectangular prism can be determined by multiplying length, width and height.
• The area of a rectangle having fractional side lengths can be found by tiling unit squares of the appropriate unit fraction.

What You’ll Need…

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.

## What Do You Notice? What Do You Wonder?

Show students the following image:

Which option would you rather?

Have students do a Think-Pair-Share routine:

1. Students have individual think time to jot down ideas on paper or whiteboard.
2. Students share their initial ideas with a partner.
3. Students share as a whole group, either their own preference, or a meaningful observation 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.

Students may have different opinions and perspectives about their preferred option. Possible points that may come up include:

• Option 3 because it looks like it’s less packaging.
• Option 2 because the cereal won’t go stale.
• Option 1 because it looks like it would have the most cereal.

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.

## Estimation: Prompt

Once students’ initial ideas have been acknowledged and noted, the class can settle on a question to explore:

Which option has the most cereal?
Make an estimate.

Students begin by making an estimate before they are provided with all the information they need to answer the question, thus providing students who may be reluctant to share with a safe entry point.

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:

How might you use that information?

Which option has the most cereal?

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:

Here are some dimensions. 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.

## Crafting A Productive Struggle: Prompt

Proceed to reveal the dimensions of the cereal box by showing the following image:

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

The dimensions of each box (along with the number of boxes) are shown in the diagram. Which option would have the most cereal?

Use a mathematical model of your choice to construct a viable argument that clearly shows which option has the most cereal.

Be sure to explicitly state that calculators are not to be used to determine the number of volume of each cereal 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

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.

Facilitator’s Note:

There may be students who are familiar with the algorithm who may arrive at a solution rather quickly. This can be an opportunity to reinforce flexibility as you encourage students to solve the problem in a different way.

## While Students Are Productively Struggling…

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## Student Approach #1: Representational Model & Partitioning Strategy

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## Student Approach #2: Direct Modeling and Area Model

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## Student Approach #3: Decomposing Strategies

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## Student Approach #4: Fraction/Decimal Equivalencies and Use of an Algorithm

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## Reveal

Consider showing the following image to reveal the volume of cereal offered by each option.

## Consolidation

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## Reflect and Consolidation Prompts

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

### Consolidation Prompt #1:

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Consolidation Prompt #2:

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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.