Intent

In this activity, students examine linear situations in which they need to determine the starting value and the rate. Rather than developing a formalized method to derive rules from each type of situation, the goal in this activity is developing fluency in reasoning about linear functions and their representations.

Mathematics

Students use various kinds of information to find equations for linear functions. In one situation, they are given the rate and one data point and need to find the starting value in order to write the rule. In another situation, they are given two data points and must determine the rate, then the starting value, and finally the rule. In each case, they examine how the rule is related to the data and see the crucial role of the assumption that the rate is constant.

Progression

Students work individually and then compare results and methods in a class discussion.

Approximate Time

25 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Let students know that “they have arrived” at their next milepost, and point out Fort Hall on the map. This activity begins by bringing up a choice many travelers faced: whether to continue on their journey or to make a home somewhere short of California. Many travelers did choose to settle along the trail.

Discussing and Debriefing the Activity

After students have worked individually on the activity, ask several groups to prepare presentations of either Question 1 or 2.

Have at least two presentations of each question, with at least one using a graph. Encourage the audience to ask questions as needed in order to understand how the presenters thought about the problem.

Ask the class, How are these situations different from those in Following Families on the Trail? For emphasis, follow with, What didn’t you know? There are two key differences. Neither of these scenarios told students the starting value, and Question 2 did not explicitly give a rate. In the earlier problems, both values were given, which made finding a rule fairly straightforward.

Help students articulate the importance of the starting value in finding a rule for a situation. As students begin to appreciate its importance, they might give it a name of their own, such as initial value, 0-value, or y-intersect.Introduce the term y-intercept and connect the idea of a starting value, where x is equal to zero, with the place where the graph hits the vertical axis.

In Question 1 students use the given information—$360 after 4 months and $50 per month—to find some other values. To find how much the Winstons started with they might use repeated subtraction or subtract four months’ worth of deposits (4 . $50) all at once, to get $160.

For Question 1c, elicit various approaches. In a graphical approach, students might simply extend the line to where the y-value is $1000 and read down to find the x-value. In an algebraic approach, they could set up an equation, such as 160 + 50x = 1000, and use a guess-and-check method to solve the equation. With a table, they could use a “month-by-month growth” method, repeatedly adding $50 to the account until the Out reaches, and exceeds, $1000.

If students get the exact, non-integer answer of 16.8, they will need to decide what to do. One approach is to point out that the Winstons make their deposits at the end of each month, meaning the account reaches and passes $1000 at the end of the seventeenth month.

For Question 2, be sure students clearly articulate how they found the ticket price. For instance, they might say they divided the change in the amount of money in the register by the number of tickets sold between the manager’s two inquiries. Be sure they show the process with numeric examples as well as state the method.

Once students have found the individual ticket price, they might use either method described for Question 1, or some other approach, to get a rule. For example, if they use the fact that there was $70 in the register after the sale of 60 tickets, they will probably get the expression 70 + 0.75(p – 60). If they use the fact that there was $40 in the register after the sale of 20 tickets, they will probably get 40 + 0.75(p – 60). As with Question 1, use the variety of resulting rules to reinforce ideas about equivalent expressions and the distributive property.

If students see that the expressions in their rules can be simplified to 25 + 0.75p, you could help them connect the starting amount and the individual ticket price to their roles in this simplified rule and in the graph.

Key Question

How are these situations different from those in Following Families on the Trail ?

Supplemental Activities

Keeping Track, A Special Show, and Keeping Track of Sugar (reinforcement) offer additional opportunities for students to find starting values, rates of change, and rules for linear situations. All three involve constructing and using equations given two data points.