Lesson 6

Scaling Solids

Lesson Narrative

Students previously learned that when a two-dimensional shape is dilated by a scale factor of $$k$$, the area is multiplied by a factor of $$k^2$$. Here, students continue their analysis of the connections between geometric measurement and dimension by studying the results of dilating a three-dimensional solid.

In grade 6, students learned that the formulas $$V=\ell wh$$ and $$V=Bh$$ can be used to find the volume of a right rectangular prism. In this lesson’s warmup, students revisit that concept and apply it to cubes. Then, they extend their knowledge of dilated areas to surface areas, and they investigate the relationship between scale factor and volume to find that dilating by a scale factor of $$k$$ multiplies the volume by $$k^3$$. This will lead to students creating and interpreting a graph representing $$y=\sqrt[3]{x}$$ in an upcoming lesson.

Students have the opportunity to construct viable arguments (MP3) when they explain why the property that dilating by a factor of $$k$$ multiplies volume by $$k^3$$ applies to all solids.

Learning Goals

Teacher Facing

• Comprehend that when a solid is dilated by a scale factor of $k$, its surface area is multiplied by $k^2$ and its volume is multiplied by $k^3$.

Student Facing

• Let’s see how the surface area and volume of solids change when we dilate them.

Required Preparation

Devices are required for the digital version of the activity How Do Surface Area and Volume Change with Scaling?. Be prepared to display an applet for all to see.

If using the print version of the activity How Do Surface Area and Volume Change with Scaling?, each group of 3–4 will need access to about 100 small cubes.

Student Facing

• I know that when a solid is dilated by a scale factor of $k$, its surface area is multiplied by $k^2$ and its volume is multiplied by $k^3$.

Building On