Intent

In this activity, students examine issues related to the scaling of graphs.

Mathematics

This activity raises some general issues about the scaling of graphs, in particular that scales should be consistent—evenly spaced—and that axes are usually assumed to begin at zero unless otherwise marked. The discussion of the activity offers opportunities to emphasize important vocabulary—such as continuous graph, discrete graph, first coordinate, second coordinate, coordinate axes, and ordered pair—and to relate graphs back to In-Out tables.

Progression

After students work individually, the teacher leads a discussion to clarify ideas, using student observations to communicate the conventions associated with scaling graphs.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

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

Materials

Transparency of the graph [link to The Issues Involved pdf, p. 5]

Doing the Activity

Tell students that in this activity they will consider the issues involved in making good decisions about scaling the axes of a graph.

Discussing and Debriefing the Activity

Have students share their questions and problems from Question 1 in their groups and try to answer each other’s questions. At the same time, assign each group one of Questions 2 through 4 to present to the class.

After a short time, bring the class together and ask each group to state one of the questions or difficulties relating to the scaling of axes that they encountered. If the group resolved the question, ask that they share the answer or solution.

In the discussion of Question 2a, explain that there is no absolute rule for whether the vertical or horizontal axis should begin at zero; it is really a judgment call. In some cases starting a scale other than at zero gives a wrong impression of the data, while in others it does not. It is also true that for some data sets, starting the scale at zero might give the wrong impression. You may want to mention the ethical issues of using misleading scales, which are discussed in the next unit, The Pit and the Pendulum, in the activity A Picture Is Worth a Thousand Words.

In Question 2b, elicit the idea that there needs to be enough information shown to give a sense of scale, but beyond that, it is a judgment call as to what would make a graph more readable and what would clutter it up. Make sure students realize that they do not have to include every integer value along a particular axis.

In Question 3, students should be able to articulate that the conclusion that boys grow at a constant rate through age five is faulty; the uneven labeling along the vertical axis distorts the graph. In simple Cartesian graphs like those in this unit, the scales should be marked so that equal distances on a given scale correspond to equal amounts, just as on a simple number line. This is a different issue from whether the scale on the vertical axis should be the same as the scale on the horizontal axis.

You might ask explicitly how much an average boy grows in his first year, his second year, and so on (based on this graph), both to focus on students’ graph-reading skills and to clarify that the amount decreases from year to year.

A correct graph for Question 3 might look like this. This graph makes it clear that a child’s growth generally slows down as the child gets older.

Figure 1

In Question 4, students should become aware that the issue of whether a graph is discrete or continuous usually depends on the context of the problem. Use this occasion to review the two terms, if necessary, and have students share their reasoning about whether this graph should be continuous or discrete. They may be able to draw larger generalizations, such as, “If the independent variable represents something that is counted with whole numbers, the graph will consist of individual points.”

Caution students that sometimes graphs that should be discrete are presented as continuous. This may be because the scale used makes it impossible to draw so many dots, because the graph is more readable without dots, or just because it simplifies the problem.

During the course of the discussion, important mathematical vocabulary can arise naturally as students need or use a term. As needed, remind students that the two numbers associated with a point on a graph are called its coordinates, with the number from the horizontal axis called the first coordinate and the number from the vertical axis called the second coordinate. The vertical and horizontal axes are sometimes referred to as coordinate axes.

Explain that we often represent a point on a graph by giving its two coordinates. For instance, in the “height versus age” graph, it appears that at 1 year old, an average boy is 30 inches in height. There is a point on the graph that corresponds to this information, and the conventional way to represent that point is (1, 30). Tell students that this notation is called an ordered pair. Ask whether they can think of a reason why the word ordered is used in this expression.

Also bring out that looking at a graph is like looking at an In-Out table: each point on a graph represents a pair of numbers, as does each row of an In-Out table. Remind students that we generally associate the horizontal axis with the In and the vertical axis with the Out.

Key Questions

What problems or questions did you have scaling the axes? How did you resolve them?

According to this graph, how much does an average boy grow in his first year? In his second year?