Lesson 17

Scaling One Dimension

17.1: Driving the Distance

CCSS Standards


Addressing

  • 8.F.A.1

Warm-up: 5 minutes

The purpose of this warm-up is for students to jump back into recognizing functions and determining if two quantities are a function of each other. Students are asked questions similar to these throughout this lesson, and the discussion of this warm-up is meant to get students using the language of functions, which continues throughout the rest of the activities.

Launch

Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.

Visual-Spatial Processing: Visual Aids.Provide handouts of the representations for students to draw on or highlight.

Student Facing

Here is a graph of the amount of gas burned during a trip by a tractor-trailer truck as it drives at a constant speed down a highway:

  1. At the end of the trip, how far did the truck drive, and how much gas did it use?
  2. If a truck traveled half this distance at the same rate, how much gas would it use?
  3. If a truck traveled double this distance at the same rate, how much gas would it use?
  4. Complete the sentence: ___________ is a function of _____________.
     
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Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Invite students to share their answers and their reasoning for why gas burned is a function of distance traveled. Questions to further the discussion around functions:

  • “Looking at the graph, what information do you need in order to determine how much gas was used?” (We need to know the number of miles traveled.)
  • “What is the independent variable? Dependent variable? How can you tell from the graph?” (The independent value is the distance traveled. The dependent value is the gas burned. By convention, the independent is on the $x$-axis and the dependent is on the $y$-axis.)
  • “What are some ways that we can tell from the graph that the relationship between gas burned and distance traveled is proportional?” (The graph is a line that goes through the origin, we can see a constant ratio between $y$ and $x$ in some points like $(80,10), (160, 20)$, and $(240, 30)$.)

17.2: Double the Edge

CCSS Standards


Addressing

  • 8.F.A.1
  • 8.F.B

Optional activity: 10 minutes

The purpose of this activity is for students to apply what they know about functions and their representations in order to investigate the effect of a change in one dimension on the volume of a rectangular prism. Students use graphs and equations to represent the volume of a rectangular prism with one unknown edge length. They then use these representations to investigate what happens to the volume when one of the edge lengths is doubled. Groups make connections between the different representations by pointing out how the graph and the equation reflect an edge length that is doubled. Identify groups who make these connections and ask them to share during the discussion.

Launch

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, followed by time to discuss the last question with their partner. Follow with a whole-class discussion.

Students using the digital activity can generate the graph using the digital applet.

Conceptual Processing:
  • Eliminate Barriers. Assist students to see the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
  • Processing Time. Review an image of a right rectangular prism in order to activate prior knowledge of the context of the problem.

Student Facing

There are many right rectangular prisms with one side of length 5 units and another side of length 3 units. Let $s$ represent the length of the third side and $V$ represent the volume of these prisms.

  1. Write an equation that represents the relationship between $V$ and $s$.

  2. Graph this equation.

  3. What happens to the volume if you double the side length $s$? Where do you see this in the graph? Where do you see it algebraically?

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

The goal of the discussion is to ensure that students use function representations to support the idea that the volume doubles when $s$ is doubled.

Select previously identified groups to share what happens to the volume when you double $s$. Display the graph and equation for all to see and have students point out where they see the effect of doubling $s$ in the graph (by looking at any two edge lengths that are double each other, their volume will be double also). Ask students:

  • “Which of your variables is the independent? The dependent?” (The side length, $s$, is the independent variable, and volume, $V$, is dependent variable.)
  • “Which variable is a function of which?” (Volume is a function of side length.)

If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the edge length $s$. Display volume equation $V=15(2s)$ for all to see. Ask, “How can we write this equation to show that the volume doubled when $s$ doubled?” (Using algebra, we can rewrite $V=15(2s)$ as $V=2(15s)$. Since the volume for $s$ was $15s$, this shows that the volume for $2s$ is twice the volume for $s$).

17.3: Halve the Height

CCSS Standards


Addressing

  • 8.F.A.1
  • 8.F.B
  • 8.G.C.9

Activity: 10 minutes

This activity is similar to the previous (optional) one from a function point of view, but now students investigate the volume of a cylinder instead of a rectangular prism. Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height. The exploration and representations resemble what was done in the previous activity, and students continue to identify the effect of the changing dimension on the graph and the equation of this function. As groups work on the task, identify those who make the connection between the graph and equation representations, and encourage groups to look for similarities and differences between what they see in this activity and what they saw in the previous activity. Invite these groups to share during the whole-class discussion.

Launch

Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.

Students using the digital activity can graph their equations using the applet. 

Conceptual Processing: Processing Time.Review an image of a cylinder in order to activate prior knowledge of the context of the problem.

Student Facing

There are many cylinders with radius 5 units. Let $h$ represent the height and $V$ represent the volume of these cylinders.

  1. Write an equation that represents the relationship between $V$ and $h$. Use 3.14 as an approximation of $\pi$.

  2. Graph this equation.

  3. What happens to the volume if you halve the height, $h$? Where can you see this in the graph? How can you see it algebraically?

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Ask previously identified groups to share their graphs and equations. 

If students completed the optional activity, display student graphs from both activities and ask:

  • “Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?”
  • “Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?”
  • “How can you tell that this is a linear relationship?”

For the last question, make sure students understand that “looks like a line” is insufficient evidence for saying a relationship is linear since the scale of the axes can make non-linear graphs look linear. It is important students connect that the equation is linear (of the form y=mx+b) or relate back to the relationship between the height and volume: that the volume is $25\pi$, or about 78.5, times the height.

If students did not complete the optional activity, ask:

  • “Which of your variables is the independent? The dependent?” (The height, $h$, is the independent variable, and volume, V, is dependent variable.)
  • “Which variable is a function of which?” (Volume is a function of height.)
  • “How can you tell that this is a linear relationship?”

See notes above on the last question in this list. If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the height $h$. Display volume equation $V=78.5(2h)$ for all to see. Ask how we can write this equation to show that the volume doubled when $h$ doubled (using algebra, we can rewrite $V=78.5h$ as $V=2(78.5h)$. Since the volume for $h$ was $78.5h$, this shows that the volume for 2$h$ is twice the volume for $h$).

17.4: Figuring Out Cone Dimensions

CCSS Standards


Addressing

  • 8.F.A.1
  • 8.F.B
  • 8.G.C.9

Activity: 10 minutes

The purpose of this activity is for students to use representations of functions to explore more about the volume of a cone. This activity differs from the previous ones because students are given a graph that shows the relationship between the volume of a cone and the height of a cone. They use the graph to reason about the coordinates and their meaning. Students use the equation for the volume of a cone and coordinates from the graph to determine the radius of this cone and discuss the answer to this last question with their partners. Identify students to share their strategies during the discussion.

Launch

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by time to discuss the answer and their strategies for the last question with their partner. Follow with a whole-class discussion.

Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

Conceptual Processing: Eliminate Barriers. Allow students to use calculators to ensure inclusive participation in the activity.

Student Facing

Here is a graph of the relationship between the height and the volume of some cones that all have the same radius:

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  1. What do the coordinates of the labeled point represent?
  2. What is the volume of the cone with height 5? With height 30?
  3. Use the labeled point to find the radius of these cones.  Use 3.14 as an approximation for $\pi$.
  4. Write an equation that relates the volume $V$ and height $h$.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Ask previously identified students to share their answers and their strategies for finding the radius in the last question.

To further the discussion, consider asking some of the following questions:

  • “How did you deal with the $\frac13$ in the equation?”
  • “How did you deal with $\pi$ when you were trying to solve for the radius?”
  • “How do you know that the relationship between volume and height for these cones is a function? How is this shown in the graph? In the equation?”
  • “Identify the independent and dependent variables in this relationship. If they were switched, would we still have a function? Explain how you know.”

17.5: A Missing Radius

CCSS Standards


Addressing

  • 8.F.A.1
  • 8.G.C

Cool-down: 5 minutes

Launch

Student Facing

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Here is a graph of the relationship between the height and volume of some cylinders that all have the same radius, $R$. An equation that represents this relationship is $V=\pi R^2 h$ (use 3.14 as an approximation for $\pi$).

What is the radius of these cylinders?

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Student Lesson Summary

Student Facing

Imagine a cylinder with a radius of 5 cm that is being filled with water. As the height of the water increases, the volume of water increases.

We say that the volume of the water in the cylinder, $V$, depends on the height of the water $h$. We can represent this relationship with an equation: $V= \pi \boldcdot 5^2h$ or just

$$V = 25\pi h$$

This equation represents a proportional relationship between the height and the volume. We can use this equation to understand how the volume changes when the height is tripled.

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The new volume would be $V = 25 \pi (3h) = 75 \pi h$, which is precisely 3 times as much as the old volume of $25\pi h$. In general, when one quantity in a proportional relationship changes by a given factor, the other quantity changes by the same factor.

Remember that proportional relationships are examples of linear relationships, which can also be thought of as functions. So in this example $V$, the volume of water in the cylinder, is a function of the height $h$ of the water.