As a beginning math teacher I viewed graphs as an endpoint. When students had gathered their data and constructed a graph, they were done. Now I view graphs as a starting point for discussion and reflection, leading to increased mathematical literacy. To be mathematically literate, students must be able to both construct and interpret graphs.
Books, newspapers, magazines, advertisements, and web sites use graphs as an information source. Graphs also play an important role in the study of mathematics (NCTM, 2000), science (Rogers, 1995), and social studies (Garofalo, 1999). However, graphs can be used to give differing impressions or even distorted visual images of information. Changing the vertical or horizontal scales (or both) can either highlight or conceal relationships among the data values.
Research has shown that students often have little understanding of how changing the scaling unit changes the appearance of graph data (McMillen, 1993; Rogers, 1995). Yet, Principles and Standards for School Mathematics states "students in grades six to eight should begin to compare the effectiveness of various types of displays" (p. 49). Clearly, students need to understand how the choice of a scaling unit impacts the appearance of information in graphs. They need to develop the ability to work with more than one scaling unit.
Simply exposing students to different graphs with various scaling units will not illustrate the effect of scaling units. Because each graph represents a different situation, the students do not easily see the effect of the scaling unit. Rather, students need experience in representing the same data on differently scaled axes. This allows them to observe, reflect on, and make conjectures about the effect that various scale choices would have.
The activities described here were used with students in grades 4 through 12. Changing scaling units on both coordinate graphs and bar graphs led the students to a better understanding of the visual impact of scale choices.
Coordinate Graph Activities
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As a beginning exercise, fourth- and fifth-grade students plotted the same points on the same size grid, but used different scaling units. For example, the points (10, 6) and (2, 8) were plotted on the three grids shown in Figure 1. (The students plotted the points by hand.)
The students were both intrigued and surprised to see how different the graphs looked, although the coordinates of the points were the same on all three graphs. I asked them to tell me what was the same and what was different on the graphs. Here are some of their responses:
Sasha (pointing to the upper portion of the y-axis in Figure 1c): The points on the last graph don't have to go up high.
James (referring to Figure 1c): How can the points be so much closer to each other on the third graph when the numbers are the same?
Keisha (referring to Figure 1c): I don't like the last graph because it is too empty.
Marta: The first point is always a little higher than the second.
James: It's further left, too.
In other classrooms, I have asked the students to predict what the second and third graphs would look like before they drew them. For other graphing exercises, I allowed the students to choose their scaling unit as long as all the points could be plotted on the grid. Class discussion of this activity revealed that some students had developed an accurate understanding of the visual impact of changing the scales on the axes.
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Daunte drew the graph in Figure 2a, labeling both axes by tens. When I asked him what would happen to his points if he chose fives as a scaling unit, he replied, "They would go up higher." Moving his pencil from the lower left to the upper right of the grid, he said, "They would go up higher and move over more" (McMillen, 1993, p. 197).
Antonio drew the graph in Figure 2b, labeling both axes by fives. While discussing his graph, he mentioned that he could have labeled it by tens. Asked why he might have chosen tens, he answered, "In case you want to go further or more on the bottom (moving his hand to the far right of the grid). Or you want to go longer (spreading his hands apart in a horizontal direction)" (McMillen, 1993, p. 271).
Bar Graph Activities
I used a similar activity with students in grades 4 through 12 who were constructing bar graphs. The students in a fifth-grade class constructed the bar graph in Figure 3a (see p. 34) to show class members' favorite color for M&M's™ candies using twos as the scale. Next they redrew the graph using a smaller scaling unit (Figure 3b) and a larger scaling unit (Figure 3c), but using the same number of grid lines. Again we discussed similarities and differences. The students observed that the bar for blue was always the tallest bar, but sometimes its height was closer to the height of the other bars. I emphasized that the numbers represented by each bar did not change from graph to graph, but the appearance of the bars changed because of the change in scale.
Among the comments made by the fifth graders were:
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Benito: "The spaces between the bars look like they are still the same."
Mila: "The tallest bar is still for blue."
Nathan: "On the third one (3c), blue is only a little ahead of the others."
Matthew: "Sometimes the bar for blue is a lot taller than the rest. But, sometimes it's only a little taller."
Amanda: "Green and yellow are always the same."
We then discussed which graph would appeal to a person who liked blue M&M's best and which would be preferred by a person who liked green best. The children noted that Figure 3b might be preferred by persons liking blue because the blue bar "is really tall." They decided that the person who liked green M&M's would choose Figure 3c "because the bars are almost all the same."
Finally, I asked the fourth and fifth graders to construct a graph that would please someone whose favorite M&M's color was brown. As a class, they predicted this assignment would be "pretty hard, because only one person picked brown." After working a while, one group constructed the bar graph shown in Figure 3d, which uses units of ten. After discussion, the class conjectured that a "very large" scaling unit would make all of the bars "really low." As each group searched for the largest scaling unit, they came to realize that a scaling unit of 500 or 1,000 would make the heights of the bars hard to distinguish. They also discovered that it was too hard to draw a bar of height 10 when the scaling unit was so large. In spite of this, they were quite pleased with their newfound sense of control, which came from stretching or shrinking a bar graph by changing the scaling unit.
With high school-age students, I carried this idea further. These students found and analyzed bar graphs from newspapers and magazines. I asked them to identify the group or organization that prepared the graph. Then the students redrew the graph choosing a scaling unit that would favor a different group or point of view.
For example, a bar graph might show increases in a school district's budget each year for the past five years. A taxpayers' group, opposed to higher taxes, might choose small scaling units for a bar graph, thus accentuating the increases. The school board might be more likely to choose a larger scaling unit to make the increases appear less dramatic.
Truncated Graphs
Middle school and high school students also worked with graphs whose axes had been cut off, or truncated. When part of the vertical axis is missing, the differences among the bars of a bar graph or histogram are exaggerated. For example, suppose the bar graph in Figure 4a represents the favorite M&M's colors for all the students in a large school. While 444 students chose blue, 415 chose brown. Figure 4b shows a truncated version of the graph, displaying only the portion of the bars from 400 to 450.
Truncated graphs are commonly rescaled and elongated as in Figure 4c, but this exaggerates the difference among the heights of the bars. In Figure 4c, it appears that about seven times as many students prefer blue as prefer brown, since the height of the blue bar is about seven times that of the brown bar. Yet, the actual number of those preferring blue is not even one-and-a-quarter times the number preferring brown.
Students were asked to find examples of truncated graphs, write about why the graphs may have been truncated, and redraw them showing the entire y-axis. At first the students were amazed by how different the graphs looked with the entire y-axis. Their immediate reaction was to condemn the use of truncated graphs as misleading. However, after being prompted to look for legitimate uses of truncated graphs, they reconsidered. For example, they found that a nontruncated graph of the Dow-Jones averages could not show small day-to-day changes because the averages were all in the ten thousands. The students concluded that although truncated graphs are frequently used to mislead the reader, there are also legitimate reasons for creating truncated graphs.
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If you want to try this activity, be aware that it is often time-consuming for students to locate examples of truncated graphs. You may want to use the examples provided in the textbooks listed in the reference section.
Activities that involve changing the scales of graphs naturally lead to discussion and conjecture, as students share their scaling choices and the resulting graphs. The process of choosing a real-life graph and then using the same data to construct a graph with a different appearance gives students a feeling of mathematical power. They enjoy the feeling of confidence that comes with successfully controlling the visual impact of a graph.