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.

In the digital version of the activity, students use an applet to complete a table relating scale factor, surface area, and volume. The applet allows students to visualize how the cube changes after they update the scale factor. Use the digital version if physical manipulatives are not available.

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing that students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including, especially, everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

The goal of this discussion is to conclude that dilating a cube by a factor of multiplies the surface area by and the volume by .

Direct students’ attention to the reference created using Collect and Display. Ask students to share their expressions for surface area and volume. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. 

Here are questions for discussion:

• “What does the 6 in represent?” (It represents the surface area of a single unit cube before dilation.)
• “Why doesn‘t the expression have a coefficient?” (The original volume is 1, so technically the expression is .)
• “How does the work with area in two dimensions relate to this work with surface area in three dimensions?” (In both cases, dilating by changes the result by . 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.)

The purpose of this discussion is to make sure that 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 ? How about ? ?” (4; 16; 64)
• “How do the values of , , and relate to the dilated prism?” (All the side lengths will be multiplied by . The area of each face, and the total surface area, get multiplied by . The volume is multiplied by .)

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