The purpose of this activity is for students to interpret expressions that represent the volume of a rectangular prism. This matching task gives students opportunities to analyze rectangular prisms and expressions closely and make connections between the structure in rectangular prisms and 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.

• Invite previously selected groups to share their matches.

• “How do these expressions represent the volume?”
• Display:
  • =
• “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.)

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 strategies result in the same volume. It is a convention to consider a prism’s base the face on which it rests, however when calculating the volume of a rectangular prism, any face of the prism can be considered the 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, the width, or the height.

If students write numbers that don't correspond to the height for a given base, consider asking,

• “How did you decide which numbers to write in the table?”
• Display an expression that represents the height and the base of a prism. “How does this expression represent the volume of the prism?”

• Ask students to share responses to the second problem. Display the expression:
• “How does this expression represent the volume of Prism A?” (The prism's side lengths are 6 units, 3 units, and 4 units, and I multiply them to find the volume.)
• Display the expression:
• “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  or .  They are equal, and both represent the volume of the prism.
• Display the equation: =
• “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 expression represents a base with side lengths of 6 units and 3 units, and a height of 4 units. The other expression represents a base with side lengths of 3 units and 4 units, and a height of 6 units.)

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 side lengths or multiplying the area of the base and the 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).

• Display each prism.
• “Which expressions represent the volume of the prism in cubic units? Which does not?”
• “How did you decide that an expression 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.)

This Warm-up is the start of Part 2.

The purpose of this Notice and Wonder is to elicit the idea that all whole numbers (except 1) are either prime or composite, which will be useful when students express numbers as products of prime factors in a later activity. While students may notice and wonder many things about this chart, reviewing the definitions of prime and composite numbers from grade 4 and the fact that the number 1 is neither prime nor composite are the most important discussion points.

• Display the chart with the labels filled in.
• “The chart shows some prime and composite numbers. A prime number is a whole number that is greater than 1 and has exactly one factor pair: the number itself and 1. A composite number is a whole number that has more than 1 factor pair.”
• “[Student’s name] noticed that the number 1 is not listed on either side of the chart. Since 1 only has one factor, it doesn’t have any factor pairs. So 1 is neither a prime number nor a composite number.”
• “What are some other numbers that we could add to each side of the chart?” (17, 19, 23, and 37 are prime numbers, and 14, 18, 25, and 39 are composite numbers.)

The purpose of this activity is for students to represent volume by writing equations and to identify equations that contain only prime number factors. Students find the side lengths of different rectangular prisms with the same volume. They may choose to use connecting cubes to build their prisms as they find side lengths and write equations.

It is important to note that the order in which students write the numbers (side lengths) in each row does not matter and that students should list the same combination of three numbers only once for each volume. For example, in this activity, a prism with side lengths 3, 4, and 5 units is the same as a prism with side lengths 4, 5, and 3 units.

If students identify only side lengths that include at least one composite number, consider asking:

• “Can you explain how you determined the side lengths of your prisms?”
• “Is this side length a prime number or a composite number?”
• “How could you use the composite side length to help you find different side lengths that are prime numbers?”

“Today we identified different rectangular prisms with the same volume. We wrote equations to represent the different prisms. We learned that we can represent the volume of those two prisms using equations with only prime factors. In the next activity, we will explore if it’s possible to create a rectangular prism with only prime number side lengths for any given volume.”

The purpose of this activity is for students to further apply what they’ve learned about prime factors to their understanding of volume. Students are given volumes of rectangular prisms, and they have to determine if the volume could be created with side lengths that are all prime numbers.

Students discover that all of the numbers in this activity can be expressed as a product of prime factors. However, the constraint of using exactly three prime factors makes it impossible for some of the volumes to be created with only prime number side lengths. This constraint is discussed in the Activity Synthesis.

• Invite students to share volumes that can be created with side lengths that are only prime numbers and to share their reasoning.
• “Why didn’t 32 cubic units work for this challenge?” (It can be decomposed into prime factors, but there are more than three. The only way to decompose 32 into three factors is to use one or more factors that aren’t prime, like .)
• “Why didn’t 35 cubic units work for this challenge?” (It can be decomposed into three factors (1, 5, and 7), but 1 is not a prime factor.)
• Display: 
• “In this activity, we explored representing different numbers using prime factors. We found that we can use different amounts of prime factors to express some numbers. We will keep exploring this idea in the next activity.”

The purpose of this activity is for students to express given numbers as products of only prime numbers. Students write expressions to represent a product using only prime factors.

• Invite previously selected students to share how they decomposed one of the numbers into factors that are prime numbers.
• “What do all of the numbers in this activity have in common?” (They are all composite numbers. They all can be expressed as a product of prime factors.)
• “Why can't you express a prime number as a product of prime factors?” (A prime number only has two factors, 1 and itself, and 1 is not a prime number.)