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.
Student Lesson in Spanish
Launch
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.
Activity
None
Student Task Statement
, , , , and all represent positive integers. Consider these two equations:
Which of these statements are true? Select all that apply.
If is tripled, is tripled.
If , , and are all tripled, then is tripled.
If is tripled, is tripled.
If , , and are all tripled, then is tripled.
Create a true statement of your own about one of the equations.
Student Response
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Building on Student Thinking
Activity Synthesis
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.
18.2
Activity
15 mins
A Square Base
Standards Alignment
Building On
Addressing
8.F.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
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.
Launch
Give students quiet work time. Leave 5–10 minutes for a whole-class discussion and follow-up questions.
Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames such as “Han is because ” or “I agree with because .” Supports accessibility for: Language, Organization
MLR8 Discussion Supports. Prior to solving the problem, invite students to make sense of the situation and take turns sharing their understanding with their partner about how the shape of the rectangular prism changes when the value is tripled. Listen for and clarify any questions about the context. Advances: Reading, Representing
Activity Synthesis
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).
18.3
Activity
15 mins
Playing with Cones
Standards Alignment
Building On
Addressing
8.F.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
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.
Launch
Arrange students in groups of 2. Give students 4–7 minutes to work with their partner. Follow with a whole-class discussion.
Activity
None
Student Task Statement
There are many cones with a height of 7 units. Let represent the radius and represent the volume of these cones.
Write an equation that expresses the relationship between and . Use 3.14 as an approximation for .
Predict what happens to the volume if the value of is tripled.
Graph this equation.
What happens to the volume if you triple ? Where do you see this in the graph? How can you see it algebraically?
Activity Synthesis
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.)
Lesson Synthesis
Display these graphs for all to see, and give students 1 minute to consider what they represent.
Ask students:
“What do these graphs represent? How are these graphs similar? Different?” (The first graph shows the relationship between the height and volume of all the cylinders with a fixed radius. The second graph shows the relationship between the radius and volume of all the cylinders with a fixed height. The first is linear, and the second is nonlinear.)
“Think about what happens when a cube’s edge lengths are doubled or tripled. What happens to the volume?” (The volume is increased by or by .)
“Why do you think changing the radius of a cylinder results in a graph that is not proportional?” (Two dimensions change when the radius of a cylinder is changed.)
Student Lesson Summary
There are many rectangular prisms that both have a length of 4 units and width of 5 units but have different heights. If represents the height, then the volume of such a prism is
The equation shows us that the volume of a prism with a base area of 20 square units is a linear function of the height. Because this is a proportional relationship, if the height gets multiplied by a factor of , then the volume is also multiplied by a factor of :
What happens if we scale two dimensions of a prism by a factor of ? In this case, the volume gets multiplied by a factor of twice, or .
For example, think about a prism with a length of 4 units, width of 5 units, and height of 6 units. Its volume is 120 cubic units since . Now imagine the length and width each get scaled by a factor of , meaning the new prism has a length of , width of , and a height of 6. The new volume is cubic units since .
A similar relationship holds for cylinders. Think of a cylinder with a height of 6 and a radius of 5. The volume would be cubic units since . Now, imagine the radius is scaled by a factor of . Then the new volume is , or cubic units. So scaling the radius by a factor of has the effect of multiplying the volume by !
Why does the volume multiply by when only the radius changes? This makes sense if we imagine how scaling the radius changes the base area of the cylinder. As the radius increases, the base area gets larger in two dimensions (the circle gets wider and also taller), while the third dimension of the cylinder, height, stays the same.
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Clare sketches a rectangular prism with a height of 11 and a square base and labels the edges of the base . She asks Han what he thinks will happen to the volume of the rectangular prism if she triples .
Han says the volume will be 9 times bigger. Is he right? Explain or show your reasoning.
