Intent

Students practice plotting data and then using the graph to make predictions. In the process, they discover that in some situations, a linear approximation is not very useful.

Mathematics

Students create a set of data by estimating areas. They plot the data and then interpret and make predictions from the resulting graph. That the data involve unusual spacing and very large numbers challenges students to be thoughtful about the scales they select. That the data don’t lend themselves to a linear model will encourage students to consider when and why to use a line of best fit.

Progression

Students work in their groups to help one another complete the process of plotting and interpreting the data, followed by a class discussion of the nonlinearity of the data.

Approximate Time

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Grid paper

Enlarged copies of the five maps [link to Broken Promises pdf, p. 8–9]

Doing the Activity

Little introduction to the activity will be necessary. Students may need to be reminded of how to measure area. You may want to point out the suggestion in Question 1 that students trace each map onto grid paper and use one square as the unit of area. Students will have difficulty estimating the shaded areas of the maps: you may want to mention that rough estimates will suffice.

Discussing and Debriefing the Activity

Let students compare results—especially their graphs for Question 2—in their groups, and then bring students together as a whole class. You may want to take a few minutes to talk about students’ techniques for estimating the areas and for setting up the scales on the axes.

Then turn to Questions 3 and 4.What are your predictions for the future of Native American land?Based on the graph, one might expect all Native American land to have been gone by early in the twentieth century. In fact, though, there was comparatively little net change in the amount of land possessed by Native Americans during the twentieth century.

One conclusion to draw from this activity is that looking at a graph out of context can be fairly meaningless in predicting real-world events. A related conclusion is that using a linear approximation, or a line of best fit, is not particularly valid unless there is a good reason to expect the data to follow a linear pattern. You might connect the nonlinear nature of these data with the idea that the rate at which the Native Americans lost land was not constant.

Key Questions

How did you estimate the areas?

What are your predictions for the future of Native American land?

Supplemental Activity

Movin’ West (extension) deals with questions about migration patterns in the United States and whether one can expect such patterns to continue. In addition to developing a general algebraic formula for a rate-of-change situation, students must think about comparative rates of change and do some commonsense reasoning to decide how well the model suggested by their formula might work in the future.