The purpose of this activity is for students to examine how changing the input of a nonlinear function changes the output. In this activity, students consider how the volume of a rectangular prism with a square base and a known height of 11 units changes if the edge lengths of the base triple. By studying the structure of the equation representing the volume function, students see that tripling the input leads to an output that is 9 times greater (MP2). Students conclude the activity by creating a graph of the situation and notice that it, unlike previous graphs, is nonlinear.

Identify students who make sketches of the two rectangular prisms or write expressions of the form or to describe the volume of the tripled rectangular prism.

The purpose of this discussion is for students to create a graph representing the relationship between the volume and side , noticing that unlike previous graphs, this one is nonlinear.

Select previously identified students to share whether they think Han is correct. If possible, begin with students who made sketches of the two rectangular prisms to make sense of the problem. If not brought up by students, connect Han’s reasoning to the equation for the volume of the prism, .

Tell students that they are now going to think about what the graph of this volume equation looks like. Ask students: “If this equation were graphed with edge length on the horizontal axis and the volume of the prism on the vertical axis, what would the graph look like?” Give 3–5 minutes for students to make a graph. If needed, suggest students first make a table showing the volume of the rectangular prism when equals 1, 2, 3, 4, and 5 units (the corresponding values of volume are 11, 44, 99, 176, and 275 cubic units) and then sketch a graph using these points. 

Select 1–2 students to display their table and graph for all to see. Ask students what they notice about the graph when compared to the graphs from previous lessons (the graph is nonlinear—the volume increases by the square of whatever the base edgelength increases by).

In this activity, students continue working with function representations to investigate how changing dimensions affects the volume of a shape. Students start by representing the relationship between volume and radius for cones with a fixed height with an equation and graph. They use these representations to justify what they think will happen when the radius of the cylinder is tripled.

Identify students who use the equation to answer the last question and students who use the graph to answer the last question.

In the digital version of the activity, students use an applet to make the graph. The applet allows students to graph accurately and quickly. Use the digital version if time is limited and students can access the applet readily on a device.

The purpose of this discussion is for students to use the graph and equation to see that when the radius is tripled, the result is a volume that is 9 times as large.

Ask previously identified students to share their graphs and equations. Display both representations for all to see, and ask students to point out where in each representation it can be seen that the volume is 9 times as large. Ask students:

• “If the radius was quadrupled (made 4 times as large), how many times as large would the volume be?” (The volume would be 16 times as large since .)
• “If the radius was halved, how many times as large would the volume be?” (The volume would be times as large since .)
• “If the radius was scaled by an unknown factor , how many times as large would the volume be?” (The volume would be times as large since .)

If students do not see the connection between scaling the radius length with a known value, like 4, and an unknown value , use several known values to help students generalize that scaling the radius by  scales the volume by .

If time allows, ask students to compare this activity to the previous. How do the equations compare? How do the graphs compare? (In the last activity, the graph was sketched during the discussion.)