# Lesson 4

Use Layers to Determine Volume

## Warm-up: Estimation Exploration: How Many Cubes? (10 minutes)

### Narrative

The purpose of this warm-up is for students to consider the information they need to find the volume of a rectangular prism and use the structure of a rectangular prism to think about a reasonable estimate. Students can see the 9 cubes on the front layer, but it is difficult to see how many layers there are.

The purpose of an Estimation Exploration is to think about reasonableness based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

### Launch

• Groups of 2
• Display image
• “What is an estimate that’s too high?” “Too low?” “About right?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses

### Student Facing

About how many cubes were used to build this prism?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Why are multiples of 9 good estimates?” (We can see the layer of 9.)
• “What information would help you to find the exact number of cubes in the prism?” (How many layers of 9 cubes there are. How deep the prism goes.)
• “Based on this discussion does anyone want to revise their estimate?”
• Optional: Reveal a picture that shows the number of layers, 10.

Math Community

• Ask students to reflect on both individual and group actions while considering the question “What norms, or expectations, were we mindful of as we did math together in our mathematical community?”
• Record and display their responses under the “Norms”​ ​header.

## Activity 1: Layers in Rectangular Prisms (20 minutes)

### Narrative

In the previous lesson, students reasoned abstractly about the volume of rectangular prisms when they considered the volume in terms of layers or equal groups of unit cubes. This activity continues to develop the idea of decomposing rectangular prisms into layers. Students explicitly multiply the number of cubes in a base layer by the number of layers. Students can use any layer in the prism as the base layer as long as the height is the number of those base layers.

Action and Expression: Internalize Executive Functions. Synthesis: Invite students to plan a strategy, including the tools they will use, for finding volume of partially filled prisms. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Memory

### Required Materials

Materials to Gather

### Required Preparation

• Have connecting cubes available for students who need them.

### Launch

• Groups of 2
• Display first image from student workbook
• “What do you know about the volume of this prism?”
• “What would you need to find out to find the exact volume of this prism?”
• “You are going to work with prisms that are only partially filled in this activity.”

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• As students work, monitor for:
• students who notice that prisms A and D and prisms B and C are “the same” but they are sitting on different faces so the layers might be counted in different ways.
• students who reason about the partially filled prisms by referring to the cubes in one layer they would see if all of the cubes were shown.
• students who recognize that there are several different layers they can use to determine the volume of a prism, all of which result in the same volume.

### Student Facing

1. Complete the table. Be prepared to explain your reasoning.

prism number of cubes in one layer number of layers volume
A $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$
B $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$
C $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$
D $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

$$\phantom{\hspace{2.5cm}}$$

2. Find the volume of each prism. Explain or show your reasoning.

$$\phantom{\hspace{2.5cm}}$$

3. How can you find the volume of any rectangular prism?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display the expressions:
• $$2 \times 12$$
• $$3 \times 8$$
• “How do these expressions represent the volume of prism A?” (There are two layers of 12. We can also see 3 layers of 8.)
• “How does thinking about layers help us find the volume of prisms that are not completely filled?” (I know that all the layers have the same number of cubes even if they are not shown.)

## Activity 2: Finding Volume in Different Ways (15 minutes)

### Narrative

In the previous activity, students saw that a rectangular prism is composed of layers and there are different ways to decompose a prism into layers, depending on how students view the prism and decompose the prism. Students recognize that the volume remains the same, regardless of the orientation of the prism. The goal of this activity is for students to identify how different expressions represent the volume of the same prism and correspond to the organization of the layers. Students have worked with parentheses in previous grades, so the lesson synthesis provides an opportunity for students to revisit expressions with parentheses. Students will have more experience with evaluating expressions with grouping symbols in future lessons. Students go back and forth between numerical expressions and a geometric object whose volume is represented by the expression (MP2).

MLR7 Compare and Connect. Synthesis: Ask, “what is the same and what is different about the strategies used to calculate the volume of the rectangular prism?” Add labels or annotations to a visible display to support connections between approaches, and to amplify language such as layers, horizontal, vertical, volume and expression.

### Required Materials

Materials to Gather

### Required Preparation

• Have connecting cubes available for students who need them.

### Launch

• Groups of 2
• “You are going to analyze different ways to find the volume of a rectangular prism.”

### Activity

• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for students who identify the factor 5 or 6 as the number of layers and the second factor is then the number of cubes in each layer.

### Student Facing

1. Explain or show how the expression $$5\times24$$ represents the volume of this rectangular prism.
2. Explain or show how the expression $$6\times20$$ represents the volume of this rectangular prism.
3. Find a different way to calculate the volume of this rectangular prism. Explain or show your thinking.
4. Write an expression to represent the way you calculated the volume.

### Student Response

For access, consult one of our IM Certified Partners.

If students do not explain how the expression represents the volume of the prism, ask “Can you explain how you would find the volume of this prism?” Then, connect the student’s description to the numbers in the expressions.

### Activity Synthesis

• “How does $$5 \times 24$$ represent the volume of the prism?” (There are 5 layers if I cut the prism horizontally and each layer has 24 cubes in it.)
• Display expression: $$5\times(4\times6)$$
• “How does this expression represent the volume of the prism?” (It shows the 5 layers and the 24 cubes as 4 rows of 6 cubes in each layer.)
• “How does $$6\times20$$ represent the volume of the prism?” (There are 6 layers if I cut vertically along the front face. Each layer has 5 rows of 4 or 20 cubes.)
• Display expression: $$6\times(4\times5)$$.
• “How does this expression represent the volume of the prism?” (It shows the 6 layers and 4 columns of 5 cubes in each layer.)

## Lesson Synthesis

### Lesson Synthesis

Display the image from the warm-up showing all the layers of the prism.

“Describe the layers in the prism to a partner. What is a multiplication expression that would represent the volume of the prism? How does the expression represent the volume of the prism?” ($$10 \times 9$$, there are 9 cubes in each layer and I can see 10 layers.)

Math Community

Revisit the “Norms”​ ​list. Ask students to discuss with a partner when a norm was helpful as they did math. Add any missing ideas or revise earlier ones.

## Cool-down: Use Expressions (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

We call the amount of space an object takes up volume. This prism has a volume of 120 cubes.

To find the volume of any prism, we can find the number of cubes in one layer and multiply that number by the number of layers. We can describe this prism as having 6 layers of 20 cubes, 4 layers of 30 cubes, or 5 layers of 24 cubes. We can use all of these expressions to represent the volume of the prism:
$$5\times24$$, $$5\times(6\times4)$$
$$6\times20$$, $$6\times(5\times4)$$
$$4\times30$$, $$4\times(5\times6)$$