Lesson 6

The purpose of this Math Talk is to elicit strategies and understandings students have for finding volumes of cubes. These understandings help students develop fluency and will be helpful later in this lesson when students analyze the effect of scaling on cube volumes.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they relate the word “cube” to both exponents and geometric solids.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the task. Follow with a whole-class discussion.

Students may be unsure how to cube \(\frac12\). Ask them to visualize a tape diagram. First, what is \(\frac12\) of \(\frac12\)? Next, what’s \(\frac12\) of the result? Alternatively, students may visualize a cube with edge length 1 inch, and compare its volume to the pictured cube with edge length \(\frac{1}{2}\) inch. How many of the small cubes fit in the large cube?

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Remind students that when we multiply a number by itself 3 times, that’s called cubing the number. We can use an exponent to indicate cubing. For example, \(2 \boldcdot 2 \boldcdot 2 = 2^3\). Ask students why we call it “cubing.”

The purpose of this activity is to explore the effect of dilation on the surface area and volume of a cube. Just like in two dimensions, the area (now surface area) is multiplied by the square of the scale factor. Volume, however, is multiplied by the cube of the scale factor.

Students may struggle to write an expression for the surface area of a unit cube dilated by scale factor \(k\). Ask them how they found areas of dilated two-dimensional figures and ask them if that relates to this situation.

The goal of the discussion is to conclude that dilating a cube by a factor of \(k\) multiplies the surface area by \(k^2\) and the volume by \(k^3\).

• “What does the 6 in \(6k^2\) represent?” (It represents the surface area of a single unit cube before dilation.)
• “Why doesn‘t the \(k^3\) expression have a coefficient?” (The original volume is 1, so technically the expression is \(1\boldcdot k^3\).)
• “How does the work with area in 2 dimensions relate to this work with surface area in 3 dimensions?” (In both cases, dilating by \(k\) changes the result by \(k^2\). We can think of surface area as the area of a two-dimensional net of the solid, so the concept is really the same.)
• “How do the exponents in our expressions relate to dimensions?” (When we work with area, which is a two-dimensional measurement, we use squaring or an exponent of 2. When we work with volume, which is a three-dimensional measurement, we use cubing or an exponent of 3.)

Students create an informal argument to explain why all solids have the property that if they’re dilated by a factor of \(k\), the volume is multiplied by \(k^3\). Then, students practice calculating the surface area and volume of a dilated solid.

Students aren’t expected to use formal language or symbolic representations in their arguments. The important part is to imagine a solid getting filled by cubes of different sizes similar to how an irregular two-dimensional figure was filled with rectangles in an earlier activity.

As students create their explanations, they are constructing viable arguments (MP3).

If students question why they’re asked to explain why volume changes by a factor of \(k^3\) but they’re not asked why surface area changes by a factor of \(k^2\), remind them that we can think of surface area as a two-dimensional net. Therefore, the properties that applied to area also apply to surface area.

The purpose of the discussion is to make sure students understand how to calculate the surface area and volume of a dilated solid. Here are some questions for discussion:

• “In the triangular prism problem, what is the value of \(k\)? How about \(k^2\)? \(k^3\)?” (4; 16; 64)
• “How do the values of \(k\), \(k^2\), and \(k^3\) relate to the dilated prism?” (All the side lengths will be multiplied by \(k\). The area of each face, and the total surface area, get multiplied by \(k^2\). The volume is multiplied by \(k^3\).)