Intent Students practice the calculator method for curve fitting. The class discussion explores the effects on the graph of changing the coefficients of the linear function.

Mathematics Students use the graphing calculator to fit a line to data and then use the line to make predictions. In the activity, students must use a negative linear term, which they connect to the context of the situation. They also identify the effects on the graph of ax + b as a and b take on different values.

Progression Students work individually and then come together to share results and to make connections between the symbolic form of the function, the situation, and the graph.

Approximate Time 5 minutes for introduction 25 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Materials Graphing calculators

Doing the Activity Remind students of their work in The Basic Student Budget, and ask them to restate Cal’s, Doc’s, and Bernie’s main concern. Some students might remember wanting to go to the concert; others will remember the need to have enough money at the end of the month for rent. Tell students that in this activity they will reexamine this question using their new graphing calculator techniques.

Discussing and Debriefing the Activity The discussion can begin with volunteers showing what they have done so far in their work on this activity. After one or two presentations, give students an opportunity to confirm their own results in fitting a line to the data using the calculator, and then use the results to make predictions about who will be able to pay the rent. Now direct students’ attention to the symbolic form of the linear functions. Ask, What is different about the lines in this graph compared to the lines in Sublette’s Cutoff? What did you have to do in the equations to make the graphs turn downward? Encourage students to justify why the subtraction, or negative, sign is necessary, based on the situation. Take this exploration a bit further by selecting a starting value from one of the student presentations—representing the amount of money one of the students had at the beginning of April 1—and ask how students think this value affects the graph and the rule. Again, summarize student observations about the starting value and how it shows up in the graph and the rule.

Key Questions What is different about the lines in this graph compared to the lines in Sublette’s Cutoff ? What did you have to do in the equations to make the graphs turn downward?