Intent This activity continues to engage students in solving problems involving four representations of linear functions. They are specifically asked to write equations and to identify rates and starting values, with the goal of strengthening connections between the situation and the rule for linear functions.

Mathematics Students continue to draw upon their understanding of the relationships among the four views of a function in order to develop equations for situations in which two data pairs are provided. The roles of rate and starting value are emphasized, setting the stage for formalization of the standard form of a linear function.

Progression After students work individually on the activity, they come together as a class to discuss methods for determining the rate and starting value and for writing the equation.

Approximate Time 5 minutes for introduction 20 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Materials Students’ work on Previous Travelers, Sublette’s Cutoff, and Who Will Make It?

Doing the Activity This activity will be quite familiar to those who have taught a traditional algebra I course; students are determining the equation of a line given two points. The goal of this activity, however, is not simply for students to be able to write the equation of a line given two points. Rather, these scenarios are opportunities for students to reason about linear functions, drawing upon the connections among the four representations to identify processes that will help them write equations. With that in mind, encourage students to use what they know to help them to think about these scenarios rather than being concerned that they “discover” a tried-and-true shortcut procedure.

Discussing and Debriefing the Activity Students may benefit from having a few minutes to share responses and methods in their groups. During this initial discussion phase, pass around a few transparencies for volunteers to record their methods. Focus the discussion on clarifying the presenter’s methods, including the mechanics and thinking behind getting the equations and how the numbers in the equation are related to the situation. Although students were not explicitly asked to make graphs of the lines, you may want to ask for them as part of the discussion. In finding the equation for Question 1, students may have answered the first part of Question 1b (finding the rate) before Question 1a (getting the equation). To get the rate, students might use just a single point and the common-sense approach of dividing the amount of beans by the number of people. For instance, the point (4, 48) represents needing 48 pounds of beans for 4 people, so each person needs 12 pounds. If a presenter demonstrates this approach, try to also identify someone who used both points, (4, 48) and (10, 120), and reasoned that the additional 6 people (the difference between 4 and 10) required an additional 72 pounds of beans (the difference between 48 and 120), which again gives 12 pounds per person. Students will then likely intuitively understand that they can simply multiply the number of people by the amount per person to get a rule like B = 12N. For the second part of Question 1b, students need only note that the number indicating the rate is the same as the coefficient of the independent variable. Elicit a variety of approaches for Questions 2 and 3, emphasizing the connections to the contexts. A few students may recognize that the rate always shows up in the equation as the coefficient of the independent variable and work from that. For Question 3, have one or two students present their contextual settings and describe what the two points mean in that context. Talk about how to develop the equation from the coordinates of the points and the contextual significance of the numbers in the equation. To conclude this activity, ask groups to write out a general process—an algorithm—to determine the equation of a line if they are given two points.