This activity serves two goals. One goal is to allow students to practice drawing a net and finding the surface area and volume of a cube. The other goal is to encourage students to write expressions to represent the surface area and volume of a cube. 

In the first set of questions, the edge length of the cube is a small number (5), enabling students to compute the surface area and volume numerically. Use students’ work here to check that they are drawing a net correctly.  In the next set of questions, the edge length of the cube is a larger number (17), making computation more cumbersome and prompting students to build expressions rather than evaluating them.

As students work on the questions about a cube with a 17-inch edge length, monitor for students who write different expressions for the second question. Here are some expressions they may write, from longer (more expanded) to shorter (more succinct):

• Products, such as , or
• Sums of products, such as
• Combination of like terms, such as
• Exponents, such as , or
• Completed calculations, such as 1,734 or 4,913

A note about notation:
In a later unit, students will learn that means the same as . At this point, expect them to write instead of . It is not critical that they understand that a number placed next to a variable (or a number placed next to an expression in parentheses) are being multiplied.

The purpose of this discussion is to emphasize that exponents can be used to concisely express calculations for surface area and volume. Briefly discuss the answers to the questions about the cube with a 5-inch edge length. Then, focus the discussion on the larger cube with a 17-inch edge length.

Invite previously selected students to share their expressions for the last two questions. Sequence the responses in the following order to help students see how the expressions and come about. If any expressions are missing but are needed to illustrate the idea of writing succinct expressions, add them to the lists. Refer to parts of expressions using terms such as “sum,” “product,” and “factor” to support students in using mathematical terms when working with expressions.

Surface area:

• 1,734

Volume:

• 4,913

Connect the expressions to the learning goals by asking questions such as:

• “In each expression, where do you see the area of one face of the cube?” ( , , 289)
• “In one expression, the number 17 is written twelve times. In another expression, it is written six times. How do they both represent the same surface area?” (They both show the sum of the areas of the six square faces. The second expression uses the exponent 2 to represent the product of two 17s.) 
• “Some expressions show addition but others don’t. How do they all represent the surface area?” (Multiplication is used for repeated addition. Adding six of the same expression is the same as multiplying it by 6.) 
• “Which expression doesn't require a lot of computation and is the most efficient way to represent the surface area of the cube?” ( )
• “Which expression doesn't require a lot of computation and is the most efficient way to represent the volume of the cube?” ()

To further encourage students to see regularity in the expressions, consider offering another example, for instance: “Suppose the edge length of a cube is 38 cm. How can we express its surface area and volume?” ( cm² and cm³, respectively) Using a large number for the edge length will discourage calculation and prompt students to focus on building an expression and using exponents.

If time permits, consider directing students’ attention to the units of measurements. Remind students that, rather than writing square units, we can write units², and instead of cubic units, we can write units³. Unit notations will appear again later in the course, so it can also be reinforced later.

In this activity, students develop the formulas for the surface area and the volume of a cube in terms of a variable edge length .

Students should be encouraged to refer to their work in the preceding activity as much as possible and to generalize from it. As before, monitor for different ways of writing expressions for surface area and volume. The expressions are likely to be in the following forms, listed from longer (more expanded) to shorter (more succinct):

• Products, such as   or
• Sums of products, such as 
• Combination of like terms, such as 
• Expressions with exponents, such as  , or 

As they write and discuss expressions for volume and surface area, students practice looking for and making use of structure (MP7) and seeing regularity through repeated reasoning (MP8).

To support students in seeing structure and generalizing, discuss the problems in as similar a fashion as was done in the earlier activity involving a cube with an edge length of 17 units. 

Ask previously selected students to share their responses with the class. If possible, sequence the responses in the following order (as shown in the activity narrative) to help students see how the expressions and come about. If any expressions are missing but are needed to illustrate the idea of writing succinct expressions, add them to the lists. Refer to parts of expressions using terms such as “sum,” “product,” and “factor” to support students in using mathematical terms when working with expressions.

Surface area:

• or

Volume

If students have trouble understanding where the most concise expression of surface area comes from, refer back to the example involving a numerical side length (a cube with an edge length of 17 units).

Present the surface area as .