Lesson 10

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. 

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.

Invite students to share their interpretations of the equations.

Most students are likely to associate the 20 in the equation with the $20 that Jada has, but some students may interpret it to mean the combined number of games and rides Jada enjoys. (This is especially natural to do for \(x+y=20\).) If this interpretation comes up, acknowledge that it is valid. 

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.

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.

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.

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.

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.

Consider keeping students in groups of 3–4.

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.

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.)