In this lesson, students work backward 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 , 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. 

As students discuss how the average rate of change on the graph representing differs across input values and relate the rates of change to a situation, they are reasoning abstractly and quantitatively (MP2).