Intent

Students make predictions and estimates based on limited data. This begins the development of their awareness of the connections between rate and starting point and the equation of a linear graph. The activity also gives students their first, informal experience reasoning about a system of linear functions.

Mathematics

Students continue to develop awareness of and flexibility with connections among situations, tables, and graphs. Though no rules are asked for per se in this activity, students will make predictions from the data, possibly employing some sort of curve-fitting technique. They also interpret graphs to answer informal, context-based questions related to intercepts, intersections, and rate.

Progression

After some time working in their groups, students share responses in a class discussion.

Approximate Time

30 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Colored pencils

Grid transparencies

Doing the Activity

Sublette’s Cutoff started just west of South Pass, which was considered the halfway point on the trip to California. Point out on a map where the wagons are on the journey. South Pass is near present-day Highway 28, south of Lander, Wyoming, along the Continental Divide. Introduce the decision to be made at the cutoff: on the one hand, saving time, and on the other hand, the danger of little water or grass. Tell students that they will be given some information about the water supplies of three families who traveled the cutoff and will be asked to predict who will have enough water to complete the journey safely.

As students work, continue to assess how well they are graphing data and interpreting their graphs and to monitor the productiveness of group interactions. This activity lends itself to separate work, but students should also stay connected to what their group members are doing so that everyone is successful with the activity.

You might choose two or three groups—perhaps based on their use of alternative approaches—to give presentations. Select at least one group that settled on a linear model. Provide these groups with grid transparencies and pens. Remind them that they will talk about the key decisions they made in setting up their graphs and to be thoughtful about how they will explain the reasoning behind their answers for the other questions.

Discussing and Debriefing the Activity

If students bring up the unreliability of using the given data to make a prediction, have the class discuss the issue. Perhaps a student will make the point that people often need to make decisions when they do not have all the data and that an educated guess is better than a random decision.

One common error in setting up the scale is marking Day 2, Day 5, and Day 9 at equal intervals along the horizontal axis, forgetting to take into account that these time intervals are not all the same. Students may use a variety of approaches to answer Questions 2, 3, and 4. Linear and nonlinear models can be equally valid, provided students explain their estimates. Many students may reason numerically to answer these questions, rather than using the graph. This offers an opportunity to discuss the merits of each approach. Question 4 is part of the ongoing focus on rates and starting values and their connection with data and graphs. Encourage students to discuss what they had to do to arrive at their solutions.

Students might ask how long someone can go without water in dry conditions and while exerting great amounts of energy. Others may suggest that the three families share their water once one family’s supply runs out. Encourage this sort of real-life discussion of the situation. It is important for students to see that mathematics is a tool that can help answer such questions, but does not necessarily tell the whole story.