# Lesson 11

Volume of Prisms

## 11.1: A Box of Cubes (5 minutes)

### Warm-up

This warm-up reviews the volume work students had done previously to prepare for the work in this lesson. It reinforces the idea of using unit cubes and fractional-unit cubes as a way to measure the volume of a rectangular prism.

### Launch

Give students 2–3 minutes of quiet work time. Follow with a class discussion.

### Student Facing

1. How many cubes with an edge length of 1 inch fill this box?

2. If the cubes had an edge length of 2 inches, would you need more or fewer cubes to fill the box? Explain your reasoning.
3. If the cubes had an edge length of $$\frac 12$$ inch, would you need more or fewer cubes to fill the box? Explain your reasoning.

### Activity Synthesis

Select several students to share their responses and reasoning. After each person explains, ask students to indicate whether they agree. To involve more students in the discussion, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree with the reasoning? Why?”

Tell students that they will use their understanding of the volume of rectangular prisms to solve other geometric problems.

## 11.2: Volumes of Cubes and Prisms (20 minutes)

### Activity

This activity extends students’ understanding about the volume of rectangular prisms from earlier grades. Previously, students learned that the volume of a rectangular prism with whole-number edge lengths can be found by computing the number of unit cubes that can be packed into the prism. Here, they draw on the same idea to find the volume of a prism with fractional edge lengths. The edge length of the cubes used as units of measurement are not 1 unit long, however. Instead, they have a unit fraction ($$\frac 12$$, $$\frac 14$$, etc.) for their edge length. Students calculate the number of these smaller cubes in a prism and use it to find the volume in a standard unit of volume measurement (cubic inches, in this case).

By reasoning repeatedly with small cubes (with $$\frac12$$-inch edge lengths), students notice that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.

Teacher Notes for IM 6–8 Accelerated
This activity is optional. Consider including this activity if students struggled with the pre-unit assessment item about volume of a rectangular prism or if they struggled with this lesson’s warm-up activity.

### Launch

Tell students to look at the image of the 1-inch cube in the task (or display it for all to see). Ask students:

• “This cube has an edge length of 1 inch. What is its volume in cubic inches?” (1 cubic inch.) “How do you know?” ($$1 \boldcdot 1 \boldcdot 1 = 1$$)
• “How do we find the volume of a cube with an edge length of 2 inches?” ($$2 \boldcdot 2 \boldcdot 2 = 8$$. Or, since we can pack it with eight 1-inch cubes, we can tell its area is $$8 \boldcdot 1$$ or 8 cubic inches.)

If no students mentioned using the 1-inch cube to find the volume of the 2-inch cube, bring it up. Consider telling students that we can call a cube with edge length of 1 inch a “1-inch cube.”

Arrange students in groups of 3–4. Give each group 20 cubes and 5 minutes to complete the first set of questions. Ask them to pause for a brief class discussion afterwards.

Invite students to share how they found the volume of a cube with $$\frac12$$-inch edge length and the prism composed of 4 stacked cubes. For the $$\frac12$$-inch cube, if students do not mention one of the two ways shown in the Possible Responses, bring it up. For the tower, if they don’t mention multiplying the volume of a $$\frac12$$-inch cube, which is $$\frac18$$ cubic inch, ask if that is a possible way to find the volume of the prism.

Next, give students 7–8 minutes to complete the rest of the activity.

For classes using the digital materials, an applet is provided, but using physical cubes is preferred and recommended. Adapted from an applet made in GeoGebra by Susan Addington.

Conversing, Representing: MLR2 Collect and Display. As students work on the first set of questions, listen for the words and phrases students use as they discuss how to find the volume of each prism (length, width, height, and volume). Display collected language for all to see, and include any diagrams that students use to represent their thinking. Continue to update the display throughout the lesson. Remind students to borrow language from the display as needed, as this will help students produce language related to volume.
Design Principle(s): Support sense-making

### Student Facing

1. Here is a drawing of a cube with edge lengths of 1 inch.

1. How many cubes with edge lengths of $$\frac12$$ inch are needed to fill this cube?
2. What is the volume, in cubic inches, of a cube with edge lengths of $$\frac12$$ inch? Explain or show your reasoning.
2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of $$\frac12$$ inch. Sketch the prism, and find its volume in cubic inches.
3. Use cubes with an edge length of $$\frac12$$ inch to build prisms with the lengths, widths, and heights shown in the table.

1. For each prism, record in the table how many $$\frac12$$-inch cubes can be packed into the prism and the volume of the prism.

prism
length (in)
prism
width (in)
prism
height (in)
number of $$\frac12$$-inch
cubes in prism
volume of
prism (in3)
$$\frac12$$ $$\frac12$$ $$\frac12$$
1 1 $$\frac12$$
2 1 $$\frac12$$
2 2 1
4 2 $$\frac32$$
5 4 2
5 4 $$2\frac12$$
2. Examine the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?
4. What is the volume of a rectangular prism that is $$1\frac12$$ inches by $$2\frac14$$ inches by 4 inches? Show your reasoning.

### Launch

Tell students to look at the image of the 1-inch cube in the task (or display it for all to see). Ask students:

• “This cube has an edge length of 1 inch. What is its volume in cubic inches?” (1 cubic inch.) “How do you know?” ($$1 \boldcdot 1 \boldcdot 1 = 1$$)
• “How do we find the volume of a cube with an edge length of 2 inches?” ($$2 \boldcdot 2 \boldcdot 2 = 8$$. Or, since we can pack it with eight 1-inch cubes, we can tell its area is $$8 \boldcdot 1$$ or 8 cubic inches.)

If no students mentioned using the 1-inch cube to find the volume of the 2-inch cube, bring it up. Consider telling students that we can call a cube with edge length of 1 inch a “1-inch cube.”

Arrange students in groups of 3–4. Give each group 20 cubes and 5 minutes to complete the first set of questions. Ask them to pause for a brief class discussion afterwards.

Invite students to share how they found the volume of a cube with $$\frac12$$-inch edge length and the prism composed of 4 stacked cubes. For the $$\frac12$$-inch cube, if students do not mention one of the two ways shown in the Possible Responses, bring it up. For the tower, if they don’t mention multiplying the volume of a $$\frac12$$-inch cube, which is $$\frac18$$ cubic inch, ask if that is a possible way to find the volume of the prism.

Next, give students 7–8 minutes to complete the rest of the activity.

For classes using the digital materials, an applet is provided, but using physical cubes is preferred and recommended. Adapted from an applet made in GeoGebra by Susan Addington.

Conversing, Representing: MLR2 Collect and Display. As students work on the first set of questions, listen for the words and phrases students use as they discuss how to find the volume of each prism (length, width, height, and volume). Display collected language for all to see, and include any diagrams that students use to represent their thinking. Continue to update the display throughout the lesson. Remind students to borrow language from the display as needed, as this will help students produce language related to volume.
Design Principle(s): Support sense-making

### Student Facing

Your teacher will give you cubes that have edge lengths of $$\frac12$$ inch.

1. Here is a drawing of a cube with edge lengths of 1 inch.

1. How many cubes with edge lengths of $$\frac12$$ inch are needed to fill this cube?
2. What is the volume, in cubic inches, of a cube with edge lengths of $$\frac12$$ inch? Explain or show your reasoning.

2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of $$\frac12$$ inch. Sketch the prism, and find its volume in cubic inches.
3. Use cubes with an edge length of $$\frac12$$ inch to build prisms with the lengths, widths, and heights shown in the table.

1. For each prism, record in the table how many $$\frac12$$-inch cubes can be packed into the prism and the volume of the prism.

prism
length (in)
prism
width (in)
prism
height (in)
number of $$\frac12$$-inch
cubes in prism
volume of
prism (in3)
$$\frac12$$ $$\frac12$$ $$\frac12$$
1 1 $$\frac12$$
2 1 $$\frac12$$
2 2 1
4 2 $$\frac32$$
5 4 2
5 4 $$2\frac12$$
2. Examine the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?

4. What is the volume of a rectangular prism that is $$1\frac12$$ inches by $$2\frac14$$ inches by 4 inches? Show your reasoning.

### Student Facing

#### Are you ready for more?

A unit fraction has a 1 in the numerator.

• These are unit fractions: $$\frac13, \frac{1}{100}, \frac11$$.

• These are not unit fractions: $$\frac29, \frac81, 2\frac15$$.

1. Find three unit fractions whose sum is $$\frac12$$. An example is: $$\frac18 + \frac18 + \frac14 = \frac12$$ How many examples like this can you find?

2. Find a box whose surface area in square units equals its volume in cubic units. How many like this can you find?

### Anticipated Misconceptions

Students may need a reminder to label volume in cubic units.

### Activity Synthesis

Display a completed table for all to see. Give students a minute to check their responses. Invite a few students to share how they determined the volume of the prisms, and their observations about the relationships between the number of cubes (with $$\frac12$$-inch edge lengths) and the volume of prism in cubic inches. Highlight that the volume can be found by calculating the number of cubes and multiplying it by $$\frac 18$$, because each small cube has a volume of $$\frac 18$$ cubic inch.

If students do not notice a pattern in their table, ask them what they notice about the edge lengths of the prisms in the second, fourth, and sixth rows of the table and their corresponding volumes. Make sure they see that the volume of each prism can also be found by multiplying its side lengths. In other words, we can find the volume of a rectangular prism with fractional edge lengths the same way we find that of a prism with whole-number edge lengths.

## 11.3: Cubes with Fractional Edge Lengths (20 minutes)

### Activity

In this activity, students continue the work on finding the volume of a right rectangular prism with fractional edge lengths. This time, they do so by packing it with unit cubes of different unit fractions for their edge lengths—$$\frac13$$, $$\frac12$$, and $$\frac14$$ of an inch. They use these cubes to find the volume of the prism in cubic inches, decide which unit fraction works better to accomplish this goal and why, and explain whether cubes of different fractional edge lengths would lead to the same volume in cubic inches (MP3).

As students work, notice those who are able to clearly explain why cubes with a particular fractional edge length are preferable as a unit of measurement and why the volume in cubic inches will be the same regardless of the cubes used. Invite them to share later.

### Launch

Arrange students in groups of 3–4. Give them 8–10 minutes of quiet work time, and then 5 minutes to discuss their responses with their group. Ask groups to be sure to discuss the third question. Encourage students to draw a sketch to help with reasoning, if needed.

Representation: Internalize Comprehension. Activate or supply background knowledge about measurement. Students may benefit from watching a quick demonstration or video of packing cubes into a box. Review terms such as dimensions, volume, and cubic units.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

1. Diego says that 108 cubes with an edge length of $$\frac13$$ inch are needed to fill a rectangular prism that is 3 inches by 1 inch by $$1\frac13$$ inch.
1. Explain or show how this is true. If you get stuck, consider drawing a diagram.

2. What is the volume, in cubic inches, of the rectangular prism? Explain or show your reasoning.
2. Lin and Noah are packing small cubes into a larger cube with an edge length of $$1\frac12$$ inches. Lin is using cubes with an edge length of $$\frac12$$ inch, and Noah is using cubes with an edge length of $$\frac14$$ inch.

1. Who would need more cubes to fill the $$1\frac12$$-inch cube? Be prepared to explain your reasoning.
2. If Lin and Noah each use their small cubes to find the volume of the larger $$1\frac12$$-inch cube, will they get the same answer? Explain or show your reasoning.

### Anticipated Misconceptions

If students have trouble getting started, ask them to revisit their work with $$\frac12$$-inch cubes from a previous lesson. Remind them that a cube with $$\frac 12$$-inch edge length has a volume of $$\frac18$$ in3 (because we can fit 8 of such cubes in a 1-inch cube). Ask them to think about how many $$\frac13$$-inch cubes can fit into a 1-inch cube, or think about what the volume of a $$\frac13$$-inch cube is in cubic inches.

Some students may not be able to visualize and keep track of the measurements of the boxes in this task. Encourage students to draw and label the measurements of the boxes described in these questions.

### Activity Synthesis

Select several students to share their responses and articulate their reasoning. Compare the different strategies students used for finding the volume of the rectangular prism. Ask students:

• “Does it matter which fractional-unit cubes we use to find the volume? Why or why not?” (As long as the unit fraction can fit evenly into all three edge lengths of the prism, it doesn’t matter what unit fraction we use.)
• “Do certain unit fractions work better as edge lengths of the small cubes than others?” (It helps to use as large a unit fraction as possible, since it means using fewer cubes and working with fractions that are closer to 1.)
• “Is there another way of finding the volume of a rectangular prism with fractional edge length besides using these small cubes?” (Multiply the fractional edge lengths.)

Point out that it is helpful to use a unit fraction that is a common factor of the fractional edge lengths of the prism. Make sure students also recognize that multiplying the edge lengths of the prism is a practical way to find the volume of such a rectangular prism.

## 11.4: Fish Tank and Baking Pan (20 minutes)

### Optional activity

In this activity, students solve word problems that involve finding the volume of rectangular prisms that have fractional edge lengths, and calculate unknown edge lengths given other measurements. The last question in the activity requires students to interpret how the same volume of liquid would fit in two different containers in the shape of rectangular prisms.

As they work, monitor for different representations students use to solve the problems.

### Launch

Keep students in groups of 3–4. Give students 5 minutes of quiet work time for the first question and 2–3 minutes to discuss their responses with their group. Then, give students time to complete the second question either individually or with their group. Encourage students to draw a sketch to help with reasoning, if needed.

Representation: Access for Perception. Read all problems aloud. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

### Student Facing

1. A nature center has a fish tank in the shape of a rectangular prism. The tank is 10 feet long, $$8\frac14$$ feet wide, and 6 feet tall.

1. What is the volume of the tank in cubic feet? Explain or show your reasoning.

2. The nature center's caretaker filled $$\frac45$$ of the tank with water. What was the volume of the water in the tank, in cubic feet? What was the height of the water in the tank? Explain or show your reasoning.
3. Another day, the tank was filled with 330 cubic feet of water. The height of the water was what fraction of the height of the tank? Show your reasoning.

2. Clare’s recipe for banana bread won’t fit in her favorite pan. The pan is $$8\frac12$$ inches by 11 inches by 2 inches. The batter fills the pan to the very top, and when baking, the batter spills over the sides. To avoid spills, there should be about an inch between the top of the batter and the rim of the pan.

Clare has another pan that is 9 inches by 9 inches by $$2\frac12$$ inches. If she uses this pan, will the batter spill over during baking?

### Student Facing

#### Are you ready for more?

1. Find the area of a rectangle with side lengths $$\frac12$$ and $$\frac23$$.
2. Find the volume of a rectangular prism with side lengths $$\frac12$$, $$\frac23$$, and $$\frac34$$.
3. What do you think happens if we keep multiplying fractions $$\frac12\boldcdot \frac23\boldcdot \frac34\boldcdot \frac45\boldcdot \frac56...$$?
4. Find the area of a rectangle with side lengths $$\frac11$$ and $$\frac21$$.
5. Find the volume of a rectangular prism with side lengths $$\frac11$$, $$\frac21$$, and $$\frac13$$.
6. What do you think happens if we keep multiplying fractions $$\frac11\boldcdot \frac21 \boldcdot \frac13\boldcdot \frac41\boldcdot \frac15...$$?

### Activity Synthesis

Invite a few students to share their solutions, explanations, and drawings (if any). Record and display their solutions for all to see. To involve more students in the discussion, ask students to indicate whether they agree or disagree with their classmate’s reasoning, if they approached it the same way but could explain it differently, or if they have an alternative path.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After a student explains their solution to the class, call on students to use mathematical language to restate and/or revoice the strategy presented (volume, cubic feet, cubic units, rectangular prism, edge lengths). Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. This will provide additional opportunities for all students to produce language that describes strategies for finding the volume of rectangular prisms.
Design Principle(s): Support sense-making; Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we used fraction multiplication and division to solve several kinds of problems about the volume of rectangular prisms. Consider using this time to help students reflect on their problem-solving process and asking questions such as:

• “How was finding the volume of a prism with fractional edge lengths like finding the volume of a prism with whole-number edge lengths? How is it different?”
• “When calculating volume, did you find it harder to work with mixed numbers than with fractions less than 1? Why or why not?” (Working with mixed numbers is a little harder since it often involves an extra step of converting them into fractions. If an error is made then, the work that follows is affected. It is easier, however, to make sense of the size of a quantity when it is written as a mixed number.)
• “How was the process of finding an unknown length of a rectangle the same or different than finding an unknown length of a prism?” (In both cases, there is one missing factor. When working with area, there are 3 quantities to keep track of: area, base, and height. When working with volume, there are 4 quantities to consider: volume, length, width, and height.)
• “Were there certain parts of calculating a volume or an unknown length that you found challenging or were prone to making mistakes? If so, which parts?”

## Student Lesson Summary

### Student Facing

If a rectangular prism has edge lengths of 2 units, 3 units, and 5 units, we can think of it as 2 layers of unit cubes, with each layer having $$(3 \boldcdot 5)$$ unit cubes in it. So the volume, in cubic units, is: $$\displaystyle 2\boldcdot 3\boldcdot 5$$

To find the volume of a rectangular prism with fractional edge lengths, we can think of it as being built of cubes that have a unit fraction for their edge length. For instance, if we build a prism that is $$\frac12$$-inch tall, $$\frac32$$-inch wide, and 4 inches long using cubes with a $$\frac12$$-inch edge length, we would have:

• A height of 1 cube, because $$1 \boldcdot \frac 12 = \frac12$$.
• A width of 3 cubes, because $$3 \boldcdot \frac 12 = \frac32$$.
• A length of 8 cubes, because $$8 \boldcdot \frac 12 = 4$$.

The volume of the prism would be $$1 \boldcdot 3 \boldcdot 8$$, or 24 cubic units. How do we find its volume in cubic inches? We know that each cube with a $$\frac12$$-inch edge length has a volume of $$\frac 18$$ cubic inch, because $$\frac 12 \boldcdot \frac 12 \boldcdot \frac 12 = \frac18$$. Since the prism is built using 24 of these cubes, its volume, in cubic inches, would then be $$24 \boldcdot \frac 18$$, or 3 cubic inches.

The volume of the prism, in cubic inches, can also be found by multiplying the fractional edge lengths in inches: ​​​​​​$$\frac 12 \boldcdot \frac 32 \boldcdot 4 = 3$$

If a rectangular prism has edge lengths $$a$$ units, $$b$$ units, and $$c$$ units, the volume is the product of $$a$$, $$b$$, and $$c$$. $$\displaystyle V = a \boldcdot b \boldcdot c$$

This means that if we know the volume and two edge lengths, we can divide to find the third edge length.

Suppose the volume of a rectangular prism is $$400\frac12$$ cm3, one edge length is $$\frac{11}{2}$$ cm, another is $$6$$ cm, and the third edge length is unknown. We can write a multiplication equation to represent the situation: $$\displaystyle \frac{11}{2} \boldcdot 6 \boldcdot {?} = 400\frac12$$

We can find the third edge length by dividing: $$\displaystyle 400\frac12 \div \left( \frac{11}{2} \boldcdot 6 \right) = {?}$$