Intent This activity will give you a sense of how well students can interpret ideas about the relationship between two variables as communicated in a graph. The activity also sets the stage for assigning numbers to the axes of a graph.

Mathematics To interpret the “story” that a graph tells, and to create a graph to represent a story, students must focus on how a relationship between two quantities can be expressed in this visual form. Along the way, they are encouraged to continue to employ the language of graphing—in particular, independent variable and dependent variable.

Progression Following their explanation of Wagon Train Sketches and Situations, students work individually to interpret graphs and then to create their own graphs. In a class discussion, they share ideas and interpret each other’s work.

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

Classroom Organization Individuals, then small groups, followed by whole-class discussion

Materials Transparencies of the graphs [link to Graph Sketches pdf, p. 3–4] Index cards (optional)

Doing the Activity When you introduce the activity, you may want to suggest that students put their work for Part II on index cards to facilitate an exchange of problems in the discussion.

Discussing and Debriefing the Activity Allow time for students to share their ideas in their groups about the graphs in Part I. You may want to introduce the term step function for the situation in Question 3. This term describes a function whose graph jumps from one horizontal segment to the next. You might use Questions 2 and 4 to distinguish between linear and nonlinear situations. As needed, help students to articulate that in Question 2, the graph indicates that the student’s “rate of work” increases as the POW deadline gets closer, whereas in Question 4, the amount of money grows at the same rate with each ticket sold. You could also use these examples to illustrate discrete versus continuous variables and their resulting graphs. The independent variables in Questions 2 and 4 take on only whole-number values, whereas the dependent variables can be interpreted in different ways, depending on the degree of precision. In contrast, Question 3 has a continuous independent variable and a discrete dependent variable, while Question 1 is continuous in both variables. This idea will arise again in Question 4 of The Issues Involved. In Part II, students may have imaginative descriptions to accompany their sketches. You might have each group pass their papers (or index cards) to the next group, with the descriptions face up. Each group should attempt to make sketches that illustrate the descriptions and then compare their sketches with those provided by the creators of the problems. If disagreement about whose answer is “correct” arises, emphasize that there can often be more than one correct graphical interpretation of a verbal description.

Supplemental Activity More Graph Sketches (reinforcement) provides a variety of contexts for which students can create additional graph sketches.