The purpose of this Warm-up is for students to explore how scaling the addends or factors in an expression affects their sum or product. Students determine which statements are true and then create one statement of their own that is true. This will prepare students to see structure in the equations they will encounter in the lesson. Identify students who:

• Choose the correct statements (b, c).
• Pick numbers to test the validity of statements.
• Use algebraic structure to show that the statements are true.

Ask these students to share during the discussion.

Ask previously identified students to share their reasoning about which statements are true (or not true). Display any examples (or counterexamples) for all to see, and ask students to refer to them while sharing. If using the algebraic structure is not brought up in students’ explanations, display for all to see:

• If , , and are all tripled, the expression becomes , which can be written as by using the distributive property to factor out the 3. So if all the addends are tripled, their sum, , is also tripled.
• Looking at the third statement, if is tripled, the expression becomes , which, by using the associative property, can be written as . So if just is tripled, then , the product of , , and  is also tripled.

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.

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.