In addition to visual representations, students can organize information by making organized lists and tables. From the lists, they can find, describe, and extend patterns and make generalizations for a variety of cases. Their tables can be transformed into visual representations by drawing graphs. The information in the graph then shows the relationships between the elements in the table. If an element increases at a constant rate, the graph will be linear. If an element increases exponentially, the graphic representation will be much different.

It is important for students to use representations that are meaningful. At first, students may choose to put data into a bar graph, although, as we saw in the Interpreting Graphs and Stories problem, a line graph might actually be more appropriate. However, if students can gain understanding from a bar graph, they later can extend their thinking to other graphic representations and transfer that meaning to those graphs as well.

Why do many students in the middle grades start with bar graphs? Mathematics in the elementary grades gives students many opportunities to collect data and represent it in bar graphs. For this reason, students are comfortable with this representation. In the middle grades, they will be given a variety of opportunities to extend their experience with bar graphs to linear graphs. As algebraic concepts become an integral part of the middle-grades curriculum, the transition of understanding from simple to more complex graphs will have meaning for students if we build on their earlier experiences. Remember, we want to use representations to support students in developing a deeper understanding of the mathematical concepts -- not as an end to themselves.

  Look at additional strategies for using representation