Lesson 5

Equations and Their Graphs

5.1: Which One Doesn't Belong: Hours and Dollars (5 minutes)

This warm-up prompts students to carefully analyze and compare features of graphs of linear equations. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and learn how they talk about characteristics of graphs.

The work here prepares students to reason about solutions to equations by graphing, which is the focus of this lesson.

Arrange students in groups of 2–4. Display the graphs for all to see.

Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular graph does not belong, and together find at least one reason each item doesn't belong.

Which one doesn’t belong?

A line graphed on a set of axes, origin O. Hours versus dollars. The line starts at O and extends up and to the right at a somewhat steep angle.

Graph of a line. Vertical axis, hours. Horizontal axis, dollars.

Graph of 12 plotted points. Vertical axis, dollars. Horizontal axis, hours.

Graph of a line. Vertical axis, dollars. Horizontal axis, hours.

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Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as "$y$-intercept" or "negative slope." Also, press students on unsubstantiated claims. (For example, if a student claims that graph A is the only one with a slope greater than 1, ask them to explain or show how they know.)

5.2: Snacks in Bulk (10 minutes)

Previously, students saw that an equation in two variables can have many solutions because there are many pairs of values that satisfy the equation. This activity illustrates that idea graphically. Students see that the coordinates of all points on the graphs are pairs of values that make the equation true, which means that they are all solutions to the equation.

They also see that, because the given equation models the quantities and constraints in a situation, not all points on the graph are meaningful. For example, only positive $x$ or $y$ values on the graph (that is, only points in the first quadrant of the coordinate plane) have meanings in this context, because almonds and figs cannot have negative values for their weight.

During the activity, look for students who perform numerical computations straightaway and those who first write a variable equation and then use it to answer the first two questions.

Arrange students in groups of 2 and provide access to calculators. Give students a few minutes of quiet work time, and then time to share their responses with a partner.

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (Clare purchased some snacks: salted almonds and dried figs). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for, and amplify, the important quantities that vary in relation to each other in this situation: the cost per pound of each snack food, the amount of each snack food purchased, and the total amount of money spent before tax. After the third read, ask students to brainstorm possible strategies to determine the amount of one snack food purchased if the amount of the other snack food is known. This helps students connect the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

To get snacks for a class trip, Clare went to the “bulk” section of the grocery store, where she could buy any quantity of a product and the prices are usually good.

Clare purchased some salted almonds at $6 a pound and some dried figs at $9 per pound. She spent $75 before tax.

If she bought 2 pounds of almonds, how many pounds of figs did she buy?
If she bought 1 pound of figs, how many pounds of almonds did she buy?
Write an equation that describes the relationship between pounds of figs and pounds of almonds that Clare bought, and the dollar amount that she paid. Be sure to specify what the variables represent.
Here is a graph that represents the quantities in this situation.
1. Choose any point on the line, state its coordinates, and explain what it tells us.
2. Choose any point that is not on the line, state its coordinates, and explain what it tells us.

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Some students may say that the points not on the line are impossible given that Clare spent $75. Encourage these students to think about what those points would mean if we didn’t know how much money Clare spent.

Display the graph for all to see. Invite students to share their equation for the situation and their interpretations of the points on and off the graph. Make sure students understand that a point on the graph of an equation in two variables is a solution to the equation. Discuss questions such as:

"What does the point $(10, 3)$ mean in this situation?" (Clare purchased 10 pounds of almonds and 3 pounds of figs.)
"Is that a possible combination of pounds of figs and almonds? Why or why not?" (No. It doesn't lie on the graph. Also, if Clare bought 10 pounds of almonds and 3 pounds of figs, it would cost her $87, not $75.)
"From the graph, it looks like $(7,3.5)$ might be a solution, but it is hard to know for sure. Is there a way to verify?" (Substitute the values into the equation and see if they make the equation true.)
"Suppose we extend the two ends of the graph beyond the first quadrant. Would a point on those parts of the line—say, $(\text-1,9)$—be a solution to the equation $6a+9f=75$? Why or why not?" (It would still be a solution to the equation, but it wouldn't make sense in this context. The weight of almonds or figs cannot be negative.)

5.3: Graph It! (20 minutes)

In the previous activity, students analyzed and interpreted points on a graph relative to an equation and a situation. In this activity, they write a linear equation to model a situation, use graphing technology to graph the equation, and then use the graph to solve problems. Each given situation involves an initial value and a constant rate of change.

Before students begin the activity, introduce them to the graphing technology available in the classroom. Offer a quick tutorial on how to graph equations, adjust the graphing window, and plot points. This tutorial could happen independently of the activity as long as it precedes the activity.

Digital

Give all students access to graphing technology. Tell students that in this course they will frequently use technology to create a graph that represents an equation and use the graph to solve problems. Demonstrate these instructions for all to see.

Open the Desmos graphing calculator from Math Tools or go to desmos.com and click “Start Graphing.”

Look for these features:
- On the left: blank rows for listing expressions or equations
- On the right: a blank coordinate plane
- At the bottom: a keyboard
Experiment with these actions:
1. Figure out how to hide the keyboard and the list of expresisons. Then, show both features again.
2. In a blank row on the left, type $y=2x+3$. Notice that a graph appears.
3. Click the $y$-intercept of the graph to reveals its coordinates.
4. Click elsewhere on the graph to see the coordinates of the clicked point.
5. Drag the point along the graph to see how the coordinates change.
Follow these instructions:
1. Delete the first equation and type $y=100x+200.$
2. Can you see the $y$-intercept? If not, click the button with a “$-$” sign (on the right side of the graphing window) to zoom out. Repeat until the $y$-intercept is visible.
3. Does it look like the graph overlaps with the vertical axis? If so, click the wrench button in the upper right corner.
4. Experiment with the scales for the $x$- and $y$-axes until the graph seems more useful and the intercepts can be seen more clearly. (For example, these boundaries produce a helpful graphing window: $\text-10<x<10$ and $\text-50<y<250$.)
5. In a blank row on the left, type $(\text-1,100)$. Do you see a point plotted? Click this point to reveal its coordinates, or check the “Label” box that appears in the expression list.

Arrange students in groups of 2–4. Assign one situation to each group. Ask students to answer the first few questions, including writing an equation, and then graph the equation and answer the last question.

Representation: Access for Perception. Provide students with a physical copy of written directions for using graphing technology and read them aloud. Include step-by-step directions for how to enter equations, adjust the graphing window, and plot a point.
Supports accessibility for: Language; Memory

A student has a savings account with \$475 in it. She deposits \$125 of her paycheck into the account every week. Her goal is to save \$7,000 for college.
1. How much will be in the account after 3 weeks?
2. How long will it take before she has \$1,350?
3. Write an equation that represents the relationship between the dollar amount in her account and the number of weeks since she started depositing \$125 each week.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks, and also the week she has \$1,350. Write down the coordinates.
5. How long will it take her to reach her goal?
A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.
1. How many gallons will be in the tank after 7 minutes?
2. How long will it take for the tank to have 200 gallons?
3. Write an equation that represents the relationship between the gallons of water in the tank and minutes the tank has been draining.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the gallons after 7 minutes and the time when the tank has 200 gallons. Write down the coordinates.
5. How long will it take until the tank is empty?

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Demonstrate how to use the technology available in your classroom to create and view graphs of equations. Explain how to enter equations, adjust the graphing window, and plot a point. If using Desmos, please see the digital version of this activity for suggested instructions.

A student has a savings account with $475 in it. She deposits $125 of her paycheck into the account every week. Her goal is to save $7,000 for college.
1. How much will be in the account after 3 weeks?
2. How long will it take before she has $1,350?
3. Write an equation that represents the relationship between the dollar amount in her account and the number of weeks of saving.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks and the week she has $1,350. Write down the coordinates.
5. How long will it take her to reach her goal?
A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.
1. How many gallons will be in the tank after 7 minutes?
2. How long will it take for the tank to have 200 gallons?
3. Write an equation that represents the relationship between the gallons of water in the tank and minutes the tank has been draining.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the gallons after 7 minutes and the time when the tank has 200 gallons. Write down the coordinates.
5. How long will it take until the tank is empty?

For access, consult one of our IM Certified Partners.

Are you ready for more?

Write an equation that represents the relationship between the gallons of water in the tank and hours the tank has been draining.
Write an equation that represents the relationship between the gallons of water in the tank and seconds the tank has been draining.
Graph each of your new equations. In what way are all of the graphs the same? In what way are they all different?
How would these graphs change if we used quarts of water instead of gallons? What would stay the same?

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Select a group who analyzed the first situation and one group who analyzed the second situation to share their responses. Display their graphs for all to see.

Focus the discussion on two things: the meanings of the points on the graph, and how the graph could be used to answer questions about the quantities in each situation. Discuss questions such as:

“How did you find the answers to the first two questions?” (By calculation, for example, computing $475 + 125(3)$, or finding $1,\!350 - 475$, and then dividing by 125.)
“How did you find the answer to the last question?” (By calculation, for instance, finding $7,\!000 - 475$, and then dividing by 125. Or, alternatively, by using the graph.)

Highlight how the graph of the equations could be used to answer the questions. If not already mentioned by students, discuss how the graph of $y=475-125x$ can be used to find the answers to all the questions about the student's savings account, and the graph of $y=450-20x$ can help us with the questions about draining the water tank.

Keep the graphs of the two equations displayed for the lesson synthesis.

Representing, Conversing: MLR7 Compare and Connect. As students share their responses with the class, call attention to the different ways the quantities are represented graphically and within the context of each situation. Take a close look at both graphs to distinguish what the points represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

To help students sum up the key ideas of the lesson, display dynamic graphs of the two equations from the "Graph It!" activity. Also, display these questions for all to see:

In the first situation, will the student have \$5,000 after saving for 20 weeks?
In the second situation, after how many minutes should we stop the draining if we want to leave 150 gallons of water in the tank?

Discuss with students:

"How can we use the graph of $y=475+125x$ to answer the first question?" (See if the point $(20, 5000)$ is on or below the graph. It is not, so the answer is no.)
"How can we use the graph of $y=450-20x$ to answer the second question?" (Find the point on the graph where the $y$-value is 150, and see what the $x$-value is.)
"On the graph for the first situation, what does the point $(15, 3000)$ mean?" (After 15 weeks, there is \$3,000 in the bank account.)
"Is that ordered pair a solution to the equation $y=475+125x$? How can we tell?" (No. The point $(15, 3000)$ is above the graph, not on the graph. After 15 weeks, there is less than \$3,000 in the account.)
"Is $(25, \text-50)$ a solution to the equation $y=450-20x$? Why or why not?" (No. The point is not on the graph. A negative $y$-value also has no meaning in this situation. The point $(25, \text-50)$ means that after 25 minutes, there are -50 gallons of water in the tank, which doesn't make sense.)
In general, how can a graph helps us find solutions to two-variable equations?" (Any point on the graph of the equation is a solution to that equation.)

5.4: Cool-down - A Spoonful of Sugar (5 minutes)

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Student Lesson Summary

Like an equation, a graph can give us information about the relationship between quantities and the constraints on them.

Suppose we are buying beans and rice to feed a large gathering of people, and we plan to spend \$120 on the two ingredients. Beans cost \$2 a pound and rice costs \$0.50 a pound.

If $x$ represents pounds of beans and $y$ pounds of rice, the equation $2x + 0.50y = 120$ can represent the constraints in this situation.

The graph of $2x + 0.50y = 120$ shows a straight line.

Graph of a line. Vertical axis, pounds of rice. Horizontal axis, pounds of beans.

Each point on the line is a pair of $x$- and $y$-values that make the equation true and is thus a solution. It is also a pair of values that satisfy the constraints in the situation.

The point $(10,200)$ is on the line. If we buy 10 pounds of beans and 200 pounds of rice, the cost will be $2(10) + 0.50(200)$, which equals 120.
The points $(60,0)$ and $(45,60)$ are also on the line. If we buy only beans—60 pounds of them—and no rice, we will spend \$120. If we buy 45 pounds of beans and 60 pounds of rice, we will also spend \$120.

What about points that are not on the line? They are not solutions because they don't satisfy the constraints, but they still have meaning in the situation.

The point $(20, 80)$ is not on the line. Buying 20 pounds of beans and 80 pounds of rice costs $2(20) + 0.50(80)$ or 80, which does not equal 120. This combination costs less than what we intend to spend.
The point $(70,180)$ means that we buy 70 pounds of beans and 180 pounds of rice. It will cost $2(70)+0.50(180)$ or 230, which is over our budget of 120.

Lesson 5

5.1: Which One Doesn't Belong: Hours and Dollars (5 minutes)

Warm-up

Launch

Student Facing

Student Response

Activity Synthesis

5.2: Snacks in Bulk (10 minutes)

Activity

Launch

Student Facing

Student Response

Anticipated Misconceptions

Activity Synthesis

5.3: Graph It! (20 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

5.4: Cool-down - A Spoonful of Sugar (5 minutes)

Cool-Down

Student Lesson Summary

Student Facing