# Lesson 10

Connecting Equations to Graphs (Part 1)

## 10.1: Games and Rides (5 minutes)

### Warm-up

Throughout this lesson, students will use a context that involves two variables—the number of games and the number of rides at an amusement park—and a budgetary constraint. This warm-up prompts students to interpret and make sense of some equations in context, familiarizing them with the quantities and relationships (MP2). Later in the lesson, students will dig deeper into what the parameters and graphs of the equations reveal.

### Launch

Arrange students in groups of 2. Give students a couple of minutes of quiet work time and then another minute to share their response with their partner. Follow with a whole-class discussion.

## 10.2: Graphing Games and Rides (20 minutes)

### Activity

This activity is the first of several that draw students' attention to the structure of linear equations in two variables, how it relates to the graphs of the equations, and what it tells us about the situations.

Students start by interpreting linear equations in standard form, $$Ax+By=C$$, and using them to answer questions and create graphs. They see that this form offers useful insights about the quantities and constraints being represented. They also notice that graphing equations in this form is fairly straighforward. We can use any two points to graph a line, but the two intercepts of the graph (where one quantity has a value of 0) can be quickly found using an equation in standard form.

Students then analyze the graphs to gain other insights. They determine the rate of change in each relationship and find the slope and vertical intercept of each graph. Next, they rearrange the equations to isolate $$y$$. They make new connections here—the rearranged equations are now in slope-intercept form, which shows the slope of the graph and its vertical intercept. These values also tell us about the rate of change and the value of one quantity when the other quantity is 0.

### Launch

Tell students that they will now interpret some other equations about games and rides. They will also use graphs to help make sense of what combinations of games and rides are possible given certain prices and budget constraints.

Read the opening paragraph in the task statement and display the three equations for all to see. Give students a minute of quiet time to think about what each equation means in the situation and then discuss their interpretations. Make sure students share these interpretations:

• Equation 1: Games and rides cost $1 each and the student is spending$20 on them.
• Equation 2: Games cost $2.50 each and rides cost$1 each. The student is spending $15 on them. • Equation 3: Games cost$1 each and rides cost $4 each. The student is spending$28 on them.

Arrange students in groups of 3–4. Assign one equation to each group (or ask each group to choose an equation). Ask them to answer the questions for that equation.

Give students 7–8 minutes of quiet work time, and then a few minutes to discuss their responses with their group and resolve any disagreements. Ask groups that finish early to answer the questions for a second equation of their choice. Follow with a whole-class discussion.

Conversing: MLR2 Collect and Display. During the launch, listen for and collect language students use to describe the meaning of the three equations. Record a written interpretation next to each of the three equations on a visual display. Use arrows or annotations to highlight connections between specific language of the interpretations and the parts of the equations. This will provide students with a resource to draw language from during small-group and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

### Student Facing

Here are the three equations. Each represents the relationship between the number of games, $$x$$, the number of rides, $$y$$, and the dollar amount a student is spending on games and rides at a different amusement park.

Equation 1: $$x + y = 20$$

Equation 2: $$2.50x + y = 15$$

Equation 3: $$x + 4y = 28$$

Your teacher will assign to you (or ask you to choose) 1–2 equations. For each assigned (or chosen) equation, answer the questions.

First equation: $$\underline{\hspace{50mm}}$$

1. What’s the number of rides the student could get on if they don’t play any games? On the coordinate plane, mark the point that represents this situation and label the point with its coordinates.
2. What’s the number of games the student could play if they don’t get on any rides? On the coordinate plane, mark the point that represents this situation and label the point with its coordinates.

3. Draw a line to connect the two points you’ve drawn.
4. Complete the sentences: “If the student played no games, they can get on $$\underline{\hspace{.75in}}$$ rides. For every additional game that the student plays, $$x$$, the possible number of rides, $$y$$$$\underline{\hspace{1.5in}}$$ (increases or decreases) by $$\underline{\hspace{.75in}}$$.”
5. What is the slope of your graph? Where does the graph intersect the vertical axis?
6. Rearrange the equation to solve for $$y$$.
7. What connections, if any, do you notice between your new equation and the graph?

Second equation: $$\underline{\hspace{50mm}}$$

1. What’s the number of rides the student could get on if they don’t play any games? On the coordinate plane, mark the point that represents this situation and label the point with its coordinates.
2. What’s the number of games the student could play if they don’t get on any rides? On the coordinate plane, mark the point that represents this situation and label the point with its coordinates.
3. Draw a line to connect the two points you’ve drawn.
4. Complete the sentences: “If the student played no games, they can get on $$\underline{\hspace{.75in}}$$ rides. For every additional game that a student plays, $$x$$, the possible number of rides, $$y$$,  $$\underline{\hspace{1.5in}}$$ (increases or decreases) by $$\underline{\hspace{.75in}}$$.”
5. What is the slope of your graph? Where does the graph intersect the vertical axis?
6. Rearrange the equation to solve for $$y$$.
7. What connections, if any, do you notice between your new equation and the graph?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may not know how to interpret the phrase “for every additional game that a student plays.” Suggest to students that they compare how many rides they could take if they played 3 games, to the number of rides they could take if they played 4 games. What about if they played 5 games? Ask them to notice how the number of rides changes when one more game is played.

### Activity Synthesis

Select students to briefly share the graphs and responses. Keep the original equations, the rearranged equations, and their graphs displayed for all to see during discussion.

To help students see the connections between linear equations in standard form and their graphs, ask students:

• “How did you find the number of possible rides when the student plays no games?” (Substitute 0 for $$x$$ and solve for $$y$$.)
• “How did you find the number of possible games when the student gets on no rides?” (Substitute 0 for $$y$$ and solve for $$x$$.)
• “Where on the graph do we see those two situations (all games and no rides, or all rides and no games)?” (On the vertical and horizontal axes or the $$y$$- and $$x$$-intercepts.)
• “The three equations are all given in the same form: $$Ax + By = C$$. What information can you get from an equation in this form? What do the $$A$$, $$B$$, and $$C$$ represent in each equation?” ($$A$$ is the price per game, $$B$$ is the price per ride, and $$C$$ is the amount of money the student spends on games and rides.)

To help students see that an equivalent equation in slope-intercept form reveals other insights about the situation and the graph, discuss:

• “If we rearrange the first equation and solve for $$y$$, we get the equation $$y = 20-x$$. Is the graph of this equation different from that of the original equation?” (No, the equations are equivalent, so they have the same graph.)
• “You were asked to complete some sentences about what would happen if the student played more games. How did the graph help you complete the sentences?” (The graph shows how many rides the student can get on if they played no games. The line slants downward, which means that the more games are played, the fewer rides are possible. The graph shows how much the $$y$$-value (number of rides) drops when the $$x$$-value (number of games) goes up by 1.)
• “Would you have been able to see the trade-offs between games and rides by looking at the original equations in standard form?” (No, not easily.)
• “Do the rearranged equations still describe the same relationships between games and rides?” (Yes. They are equivalent to the original.)
• “What new insights does this form of equation give us?” (Isolating $$y$$ gives an equation in the form of $$y=mx+b$$, which reveals the slope of the graph and where it intersects the $$y$$-axis. The slope tells us how the number of rides changes if the student plays additional games. The $$y$$-intercept tells us the possible number of rides when no games are played.)

Highlight that each form of equation gives us some insights about the relationship between the quantities. Solving for $$y$$ gives us the slope and $$y$$-intercept, which are handy for creating or visualizing a graph. Even without a graph, the slope and $$y$$-intercept can tell us about the relationship between the quantities.

Representation: Internalize Comprehension. Demonstrate, and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to illustrate where the slope appears in each equation and corresponding graph. Continue to use colors consistently as students discuss “What do the $$A$$, $$B$$, and $$C$$ represent in each equation?”
Supports accessibility for: Visual-spatial processing

## 10.3: Nickels and Dimes (10 minutes)

### Activity

This activity serves two practice goals: writing and graphing linear equations of the form $$Ax + By=C$$ to represent a constraint, and interpreting points on a graph in terms of the situation it represents. In this case, only whole-number values are meaningful for both variables (number of dimes and number of nickels). Students need to consider whether decimal solutions are reasonable in the situation.

Graphing the equation involves some decisions. The axes of the blank coordinate plane are not labeled, so students need to decide which quantity goes on which axis (and to recognize that the decision affects what each point on the graph represents). Students could also choose to draw a continuous graph (a line) or a discrete graph (points at whole-number values of one variable or both variables).

As students work, notice the graphing decisions students make. Identify students who draw a discrete graph so they could share their rationale during class discussion.

Students engage in quantitative and abstract reasoning (MP2) as they think about the solutions and graph of an equation in context. They practice aspects of modeling (MP4) as they write an equation for a constraint, decide on representations for the model, and reflect on whether the mathematical results make sense in the given situation.

### Launch

Consider keeping students in groups of 3–4.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to provide opportunities for students to analyze how different mathematical forms and symbols can represent different situations. Display only the problem statement without revealing the questions that follow. Invite students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that address quantities of each type of coin.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Internalize Comprehension. Activate or supply background knowledge about generalizing a process to create an equation for a given situation. Some students may benefit by first calculating how many nickels Andre would have if there were 0, 1, 5, or 10 dimes in the jar, and then how many dimes if there were 1, 5, or 10 nickels in the jar. Invite students to use what they notice about the processes they used to create an equation.
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

Andre’s coin jar contains 85 cents. There are no quarters or pennies in the jar, so the jar has all nickels, all dimes, or some of each.

1. Write an equation that relates the number of nickels, $$n$$, the number of dimes, $$d$$, and the amount of money, in cents, in the coin jar.
2. Graph your equation on the coordinate plane. Be sure to label the axes.
3. How many nickels are in the jar if there are no dimes?
4. How many dimes are in the jar if there are no nickels?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

What are all the different ways the coin jar could have 85 cents if it could also contain quarters?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students who wish to change their equation from standard form to slope-intercept form may get stuck because they are not sure whether to solve for $$n$$ or $$d$$. Either choice is acceptable, but this is a good opportunity for students to think through the implications of their choice. Ask students: “In $$y=mx+b$$, which variable goes on the horizontal axis? Which goes on the vertical?”

Other students might wish to graph using the equation in standard form without first rewriting it into another form. Ask if they could identify two points on the graph. Alternatively, ask them to think about how many nickels there would be if there were 0 dimes, 1 dime, 2 dimes, and so on, and plot some points accordingly.

### Activity Synthesis

Select previously identified students to share their graphs. For each graph, ask if anyone else also drew it the same way. If no one drew discrete graphs and no one mentioned that fractional values of $$d$$ or $$n$$ have no meaning or are not possible in the situation, ask students about it.

Display the following graphs (or comparable graphs by students) for all to see.

Make sure students understand that all of these graphs are acceptable representations of the relationship between the quantities. A graph showing only points with whole-number coordinate values represents the solutions to the equation accurately but may be time consuming to draw. A line may be a quicker way to see the possible solutions and can be used for problem solving as long as we are aware that only points with whole-number values make sense.

For example, when reasoning about the last question, students who used a continuous graph might see that the jar would contain 8.5 dimes if it has no nickels. It is important that they recognize that this is impossible. The same reflection about the context is also necessary if students answered the question by solving the equation for $$d$$ when $$n$$ is 0.

If time permits, discuss these questions to reinforce the connections to earlier work on equivalent equations and their graphs:

• "Suppose you were to express the relationship between the same quantities but in dollars instead of in cents. What would the equation look like?" ($$0.05n + 0.1d = 0.85$$)
• "What would the graph of this equation look like? Try graphing it on the same coordinate plane." (It'd be the same line as the graph for $$5n+10d=85$$.)
• "Why would the graph of this equation be identical to the other one?" (The two equations are equivalent. Dividing the first equation—representing the relationship in cents—by 100 gives the second equation—representing the relationship in dollars. The same combinations of nickels and dimes make both equations true.)

## Lesson Synthesis

### Lesson Synthesis

To help students consolidate their work in this lesson, discuss questions such as:

• "We saw equations in different forms representing the same constraint. For example, $$x+4y=28$$ and $$y=\text-\frac14 x+ 7$$ both represent the games and rides that a student could do with a fixed budget. What information about the situation and about the graph can we gain from the standard form, $$Ax+By=C$$?" (In this example, the standard form allows us to see the cost per ride, the cost per game, and the budget.)
• "What information does the slope-intercept form give us?" (It gives us the slope and $$y$$-intercept of the graph. The slope tells us what is given up in terms of rides for each additional game played. The $$y$$-intercept tells us how many rides are possible when no games are played.)
• "What might be an efficient way to graph an equation of the form $$Ax+By=C$$?" (Substituting 0 for $$x$$ or for $$y$$ in the equation. Doing so gives us $$(x,0)$$ and ($$0,y)$$, which are the horizontal and vertical intercepts of the graph. We could choose two other points, as well, but using 0 eliminates one of the variables, simplifying the calculation. Alternatively, we could isolate $$y$$ and rearrange the equation into slope-intercept form, which shows us the $$y$$-intercept and the slope.)

## 10.4: Cool-down - Kiran at the Carnival (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Linear equations can be written in different forms. Some forms allow us to better see the relationship between quantities or to predict the graph of the equation.

Suppose an athlete wishes to burn 700 calories a day by running and swimming. He burns 17.5 calories per minute of running and 12.5 calories per minute of freestyle swimming.

Let $$x$$ represents the number of minutes of running and $$y$$ the number of minutes of swimming. To represent the combination of running and swimming that would allow him to burn 700 calories, we can write:

$$17.5x + 12.5y = 700$$

We can reason that the more minutes he runs, the fewer minutes he has to swim to meet his goal. In other words, as $$x$$ increases, $$y$$ decreases. If we graph the equation, the line will slant down from left to right.

If the athlete only runs and doesn't swim, how many minutes would he need to run?

Let's substitute 0 for $$y$$ to find $$x$$:

\begin {align} 17.5x + 12.5(0) &= 700\\ 17.5x &= 700\\ x&= \dfrac {700}{17.5}\\ x&=40 \end{align}

On a graph, this combination of times is the point $$(40,0)$$, which is the $$x$$-intercept.

If he only swims and doesn't run, how many minutes would he need to swim?

Let's substitute 0 for $$x$$ to find $$y$$:

\begin {align} 17.5(0) + 12.5y &= 700\\ 12.5y &= 700\\ y&= \dfrac {700}{12.5}\\ y&=56 \end{align}

On a graph, this combination of times is the point $$(0,56)$$, which is the $$y$$-intercept.

If the athlete wants to know how many minutes he would need to swim if he runs for 15 minutes, 20 minutes, or 30 minutes, he can substitute each of these values for $$x$$ in the equation and find $$y$$. Or, he can first solve the equation for $$y$$:

\begin {align}17.5x + 12.5y &= 700\\ 12.5y &= 700 - 17.5x\\ y &= \dfrac {700-17.5x}{12.5}\\ y &=56 - 1.4x \end{align}

Notice that $$y=56 - 1.4x$$, or $$y=\text-1.4x + 56$$, is written in slope-intercept form.

• The coefficient of $$x$$, -1.4, is the slope of the graph. It means that as $$x$$ increases by 1, $$y$$ falls by 1.4. For every additional minute of running, the athlete can swim 1.4 fewer minutes.
• The constant term, 56, tells us where the graph intersects the $$y$$-axis. It tells us the number minutes the athlete would need to swim if he does no running.

The first equation we wrote, $$17.5x + 12.5y = 700$$, is a linear equation in standard form. ​In general, it is expressed as $$Ax + By = C$$, where $$x$$ and $$y$$ are variables, and $$A, B$$, and $$C$$ are numbers.

The two equations, $$17.5x + 12.5y = 700$$ and $$y=\text-1.4x + 56$$, are equivalent, so they have the same solutions and the same graph.