Intent

This activity extends students’ work with graphing and predicting to writing rules, setting the stage for building a contextually meaningful understanding of symbolic representations of linear functions.

Mathematics

In a linear function of the form f(x) = ax + b, the y-intercept, b, may be thought of as the starting value (when considering first-quadrant data such as in applied contexts like those in this unit), and the slope, a, is the rate of change of y with respect to the change in x. This activity treats these concepts and connections informally.

Progression

Students will work in groups to plot data, sketch a line of best fit, and make predictions based on the line in order to answer questions about the data. The class discussion will involve some analysis of the numbers used in the rules and their association with the situation.

Approximate Time

25 minutes

Classroom Organization

Groups, followed by whole-class discussion

Doing the Activity

Tell students that this activity is similar to their recent work, but placed in a modern-day context.

As groups work, watch for opportunities to highlight students’ use of the connections among the various representations of the situation—graphical, tabular, and symbolic.

If groups are having difficulty getting started, help them to see that one useful approach is to estimate the daily amount Cal, Bernie, and Doc each spend and to work from there. After making such an estimate, they are in a good position to find a rule. Essentially, what they need to do is to repeatedly subtract the amount each person uses each day from his starting amount. The formula that results is something like this, in which the value of x tells how many times to subtract the daily spending amount.

Amount on day x = initial amount – (amount spent daily) · x

Keep in mind that “amount spent daily” is an estimate or average, as it is not the same each day.

Discussing and Debriefing the Activity

Questions 1 and 2 are similar to the prediction process used in Sublette’s Cutoff and Who Will Make It?

Responses to Question 3 can be interesting. Some students will identify who cannot afford the concert; others may suggest they can share their money so all three can go. Some students may neglect to pay attention to having enough money at the end of the month for rent.

Focus attention in this discussion on Question 4—finding rules for Cal, Bernie, and Doc. Ask, How did you find a rule for the amount of money each person would have? This is more complicated than finding rules for many previous In-Out tables because no linear function fits the data perfectly.

Make sure students took into account the various starting amounts as well as the amounts in the table; there are four data points—not just three—for each roommate. For example, the information for Cal should include the amount of $1100 for March 31.

Ask students what connections they see between the rule and the situation. Some may identify the constant term as being equal to (or approximately) the amount of money the student began the month with. Some may identify the coefficient of the linear term as representing the amount spent, on average, per day.

If these observations are not forthcoming, you might draw them out by asking, How can you see in the graph that Cal started with the most money? Can you see that in the rule as well?

Address the idea of rate with the question, Who spends the most money on average per day? Can you see this in the rule? How?Someone may even recognize the nature of the subtraction (or negative rate) and its effect on the rule. How does each student’s graph tell you who is spending the most per day?

Simply asking these questions to raise the awareness of starting values and rates is enough, as these ideas are more intentionally developed in subsequent activities.

Key Questions

How did you find a rule for the amount of money each person would have?

What connections do you see between the rule and the situation?

How can you see in the graph that Cal started with the most money? Can you see that in the rule as well?

Who spends the most money on average per day? Can you see this in the rule? How?

How does each student’s graph tell you who is spending the most per day?