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 the discussions.

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 , and solve for .)
• “How did you find the number of possible games when the student gets on no rides?” (Substitute 0 for , and solve for .)
• “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 - and -intercepts.)
• “The three equations are all given in the same form: . What information can you get from an equation in this form? What do the , , and represent in each equation?” ( is the price per game, is the price per ride, and 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 , we get the equation . 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 that games are played, the fewer rides that are possible. The graph shows how much the -value (number of rides) drops when the -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 gives an equation in the form of , which reveals the slope of the graph, , and where it intersects the -axis, . The slope tells us how the number of rides changes if the student plays additional games. The -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 gives us the slope and -intercept, which are handy for creating or visualizing a graph. Even without a graph, the slope and -intercept can tell us about the relationship between the quantities.

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 or . Either choice is acceptable, but this is a good opportunity for students to think through the implications of their choice. Ask students: “In , 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 or have no meaning or are not possible in the situation, ask students about it.

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

Make sure that 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 when 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?" ()
• "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 .)
• "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.)