In this lesson, students work backwards from the volumes of original and scaled solids to calculate scale factors. To illustrate the relationship between volume and scale factor, students create a graph of the cube root equation \(y=\sqrt[3]{x}\) based on a situation arising from a geometric context. They use the graph to analyze rates of change in scale factor for different volume inputs. Then, they solve a design problem, using cube roots and square roots to find particular scale factors. This builds on work from grade 8, in which students evaluated cube roots of small perfect cubes and used rational approximations of irrational numbers.
As students discuss how the average rate of change on the graph representing \(y=\sqrt[3]{x}\) differs across input values and relate the rates of change to a situation, they are reasoning abstractly and quantitatively (MP2).
Lesson overview
 7.1 Warmup: The Number That Cubes (5 minutes)

7.2 Activity: Thinking Inside the Box (20 minutes)
 Digital applet in this activity

7.3 Activity: Satellite Scale Factors (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 7.4 Cooldown: Finding a Scale Factor (5 minutes)
Learning goals:
 Create and interpret (orally and in writing) cube root graphs to represent the relationships between scale factors and volumes.
 Generate a scale factor that produces a desired volume or surface area dilation.
Learning goals (student facing):
 Let’s look at relationships between volumes, areas, and scale factors using graphs and situations.
Learning targets (student facing):
 I can create and describe graphs that show relationships between volumes and scale factors.
 I can work backwards from a volume or surface area scaling to find a scale factor.
Required materials:
 Graph paper
 Scientific calculators
Required preparation:
 Devices are required for the digital version of the activity Thinking Inside the Box.
 Acquire devices that can run the applet, 1 per student.
Glossary:
 cube root  The cube root of a number \(x\), written \(\sqrt[3]{x}\), is the number \(y\) whose cube is \(x\). That is, \(y^3 = x\).
 Access the complete Geometry Course glossary.
Standards:
 This lesson builds on the standards:CCSS.8.EE.A.2MS.8.EE.2MO.8.EEI.A.2aMO.8.EEI.A.2bCCSS.8.NS.A.2MS.8.NS.2MO.8.NS.A.2CCSS.HSGGMDMS.GGMDMO.G.GMD.AMO.G.GMD.B
 This lesson builds towards the standard:CCSS.HSFIF.C.7.bMS.FIF.7bMO.A1.IF.C.7
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