Lesson 22

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.

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time followed by time to discuss their chosen statements with their partner. Follow with a whole-class 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 have students refer to them while sharing. If using the algebraic structure is not brought up in students’ explanations, display for all to see:

• If \(a\), \(b\), and \(c\) are all tripled, the expression becomes \(3a +3b+3c\), which can be written as \(3(a+b+c)\) by using the distributive property to factor out the 3. So if all the addends are tripled, their sum, \(m\), is also tripled.
• Looking at the third statement, if \(a\) is tripled, the expression becomes \((3a)bc\), which, by using the associative property, can be written as \(3(abc)\). So if just \(a\) is tripled, then \(n\), the product of \(a\), \(b\), and \(c\) is also tripled.

This activity is optional.  The purpose of this activity is for students to examine how changing the input of a non-linear 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. In the following activity, students will continue this thinking with the volume function for cylinders.

Identify students who make sketches of the two rectangular prisms or write expressions of the form \(99s^2\) or \(11(3s)^2\) to describe the volume of the tripled rectangular prism.

Give students quiet work time. Leave 5–10 minutes for a whole-class discussion and follow-up questions.

Some students might think that because you triple \(s\) then you are only tripling one dimension. Encourage these students to make a sketch of the prism before and after the doubling and to label all three edge lengths to help them see that two dimensions are tripling.

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.

Ask students: “If this equation was graphed with edge length \(s\) on the horizontal axis and the volume of the prism on the vertical axis, what would the graph look like?” Suggest that they complete a table showing the volume of the rectangular prism when \(s\) 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. Give students quiet work time and then select a student to display their table and graph for all to see. Ask students what they notice about the graph when compared to the graphs from the previous lesson (the graph is non-linear—the volume increases by the square of whatever the base edge-length increases by).

This activity is optional.  In this activity, students continue working with function representations to investigate how changing the radius affects the volume of a cone with a fixed height. Students represent the relationship between the volume of the cone and the length of its radius with an equation and graph. They use these representations to justify what they think will happen when the radius of the cylinder is tripled. The work students did in the previous activity prepared them to use the equation to see the effect that tripling the radius has on the volume of the cone.

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

While students calculate the volume (or write an equation) they might mistake \((3r)^2\) as \(2\boldcdot 3 r\), remind students what squaring a term involves and encourage them to expand the term if they need to see that \((3r)^2\) is \(3r\boldcdot 3r\).

If students struggle to use the equation to see how the volume changes, encourage students to make a sketch of the cone.

The purpose of this discussion is for students to use the graph and equation to see that when you triple the radius you get 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 we see 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 \(\frac13 \pi (4r)^2=\frac13 \pi r^2 4^2=16\boldcdot \frac13 \pi r^2\).)
• “If the radius was halved, how many times as large would the volume be?” (The volume would be \(\frac14\) times as large since \(\frac13 \pi \left(\frac12 r\right)^2=\frac13 \pi r^2 \left(\frac12\right)^2=\frac14 \frac13 \pi r^2\)).
• “If the radius was scaled by an unknown factor \(a\), how many times as large would the volume be?” (The volume would be \(a^2\) times as large since \(\frac13 \pi (ar)^2=\frac13 \pi r^2 a^2=a^2 \frac13 \pi r^2\).

If students do not see the connection between scaling the radius length with a known value like 4 and an unknown value \(a\), use several known values to help students generalize that scaling the radius by \(a\) scales the volume by \(a^2\).

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.)