# Lesson 6

Expressions for Volume

## Warm-up: True or False: Parentheses or No Parentheses (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate strategies and understandings they have for determining equivalence of numerical expressions. These understandings help students deepen their understanding of the properties of operations and are helpful as students interpret expressions for volume. In this activity, students have an opportunity to notice and make use of structure (MP7) when they use the properties of operations to determine equivalence without having to calculate.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$(4\times2)\times5=4\times(2\times5)$$
• $$(2\times5)\times4=2\times20$$
• $$5\times4\times2=10\times40$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Focus Question: “How can you justify your answer without evaluating both sides?” (I could see on the first equation that all of the factors are the same so it is true.)
• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Can we make any generalizations based on the statements?”

## Activity 1: Card Sort: Match the Expression (15 minutes)

### Narrative

The purpose of this activity is for students to interpret expressions that represent the volume of a rectangular prism. Students connect the structure in rectangular prisms to the symbols in their related expressions (MP2, MP7). If there is time and you would like to add student movement, have students make a poster to display the sorted cards. Students can walk around and add additional expressions to other posters to represent the volume of the prism.

MLR8 Discussion Supports. Invite students to take turns finding a match and explaining their reasoning. Display the following sentence frames for all to see: “I noticed _____ , so I matched . . .” and  “_____ and _____  match/do not match because . . . .” Encourage students to challenge each other when they disagree.

### Required Materials

Materials to Gather

Materials to Copy

• Matching Prisms and Expressions

### Required Preparation

• Create a set of cards from the blackline master for each group of 2.
• Have connecting cubes available for students who need them.

### Launch

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.
• “What do you notice about the prisms on these cards?” (They don’t have any cubes, It says “units”.)
• “When the measurements are in units, the cubes we use to fill the prism are called cubic units.”

### Activity

• “In this activity, you will sort some cards into categories of your choosing. When you sort the cards, you should work with your partner to come up with categories.”
• 4 minutes: partner work time
• Select groups to share their categories and how they sorted their cards.
• “Now work with your partner to match each prism with the expressions that represent the volume.”
• 3 minutes: partner work time

### Student Facing

1. Match each rectangular prism with the expression(s) that represents its volume in cubic units. Be prepared to explain your reasoning.
2. For each prism write one additional expression, not in the card sort, that represents its volume in cubic units.

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students do not correctly match expressions to the prisms, ask:

“How can we use the connecting cubes to help you match the expressions to the prisms?”

### Activity Synthesis

• Select groups to share their matches.
• Display Prism A:
• “How do these expressions represent the volume?”
• $$6\times(5\times3)$$
• $$(6\times5)\times3$$
• $$15\times6$$
• Display:
• $$(5\times3)\times6$$ = $$15\times6$$
• “How does the equation relate to Prism A?” (Both expressions show that the prism has a height of 6. One expression shows the side lengths of the base. The other expression shows the area of the base.)

## Activity 2: A Tale of Two Tables (10 minutes)

### Narrative

The purpose of this activity is for students to compare and contrast two different ways to calculate the volume of a rectangular prism: multiplying the area of the base and its corresponding height, and multiplying all three side lengths. Students see that both of these strategies result in the same volume. It is a convention to consider a prism’s base the face it is resting on, however when calculating the volume of a rectangular prism, any face of the prism can be considered a base as long it is multiplied by the corresponding height. Similarly, when calculating the volume of a rectangular prism, any edge can be considered the length, width, or height.

Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to various tools that can be used to solve the problem. For example, colored pencils can be used to shade the base and different layers of the prisms.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing

• Groups of 2

### Activity

• 1 minute: independent work time
• 8 minutes: partner work time

### Student Facing

1. Work with your partner to complete the tables. One partner completes Table 1 and the other completes Table 2.

Table 1

length (units) width (units) height (units) volume (cubic units)
Prism A
Prism B

Table 2

area of the base (square units) height (units) volume (cubic units)
Prism A
Prism B
2. Compare your tables and discuss:
1. What do the tables have in common?
2. What is different about the tables?

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If a student does not write the correct corresponding height for a given base, ask “How do the numbers in the table relate to the prism?” or “How did you decide which numbers to write in the table?”

### Activity Synthesis

• Ask students to share responses to the second problem. Display the expression: $$6\times3\times4$$
• “How does this expression represent the volume of prism A?” (The prism's side lengths are 6, 4, and 3 and I multiply them to find the volume.)
• Display expression: $$(6\times3)\times4$$
• “How does this expression represent the volume of prism A?” (One base has a length of 6 units and a width of 3 units and the height is 4 units.)
• “Which expression could you use to find the volume using the 3 unit by 4 unit base?” (We could use either $$(3\times4)\times6$$ or $$6\times(3\times4)$$. They are equal and they both represent the volume of the prism.
• Display equation: $$(6\times3)\times4$$ = $$(3\times4)\times6$$
• “How do you know the equation is true?” (Both expressions represent the volume of the prism and we can see both expressions in the prism. One of them represents a base with the side lengths 6 and 3 and a height of 4. The other expression represents a base with the side lengths 3 and 4 cubes and a height of 6.)

## Activity 3: Two Truths and a Lie [OPTIONAL] (10 minutes)

### Narrative

This activity is optional if students need additional practice writing expressions to represent the volume of a rectangular prism. This activity also supports students in identifying the information they need to represent volume. Students are given the opportunity to write and interpret expressions that show that the volume is the same when multiplying the edge lengths or multiplying the area of the base and height.  In the second part of the activity, students reason abstractly and quantitatively when they interpret the meaning of expressions in the context of volume (MP2).

### Launch

• Groups of 2
• “You and your partner are going to play 2 truths and a lie with rectangular prisms.”
• “You will each write expressions, 2 true and one false, to represent the volume of two prisms and then trade to answer some questions.”
• “One partner writes 2 truths and a lie for Prisms A and C and the other partner writes about Prisms B and D.”

### Activity

• 5 minutes: independent work time (create expressions)
• “Switch papers with your partner and see if you can figure out the expression that is a lie for each of their prisms.”
• 5 minutes: independent work time on partner’s problems (analyze expressions)

### Student Facing

Your teacher will assign you and your partner two prisms.

For each of your assigned prisms:

• Write 2 expressions to represent the volume in cubic units.
• Write 1 expression that does NOT represent the volume in cubic units.

1. For each prism, which expression does not represent its volume in cubic units? How do you know?
2. What other expressions represent the volume of this prism in cubic units?

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students do not write any correct expressions that represent the volume of the prism, refer to an expression that does represent the volume of the prism and ask, “Can you explain how this expression represents the volume of the prism?”

### Activity Synthesis

• Display each of the prisms.
• “Which expressions represent the volume of the prism in cubic units? Which do not?”
• “How did you decide the expressions that did not represent the volume of a rectangular prism?” (Looking at the different bases and heights and experimenting with expressions. Finding the product and checking that it does not match the volume of any of the prisms.)

## Lesson Synthesis

### Lesson Synthesis

Display Prism C from activity 1:

“Which expressions could we write to represent the volume of this prism in cubic units?”

For each expression, ask students to explain how it represents the volume of the prism. As students explain, record expressions on a poster for all to see. Use parentheses to show which factors represent the area of a base and which factor represents the corresponding height. If not mentioned by students, display and discuss these expressions.

• $$(7 \times 4) \times 3$$
• $$28 \times 3$$
• $$(7 \times 3) \times 4$$
• $$21 \times 4$$
• $$7 \times (3 \times 4)$$
• $$7 \times 12$$

Math Community

After the Cool-down, ask students to individually reflect on the following question: “Which one of the norms did you feel was most important in your work today, and why?” Students can write their responses on the bottom of their Cool-down paper, on a separate sheet of paper, or in a math journal.

Tell students that as their mathematical community works together over the course of the year, the group will continually add to and revise its “Doing Math” and “Norms” actions and expectations.

## Cool-down: Choose the Expression (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.