Connected Representations: From Proportion to Linear Functions

Mathematics may be inconceivable without its diagrams and symbols--its representations. Mathematical representations help individuals organize their thinking; they bring a visual component to abstract ideas and serve as templates for computation with understanding. But the inevitability of representations is no guarantee that they are used effectively. This article presents a sequence of problems that suggests how a cohort of related representations can trace a path from proportion to linear functions. The related representations include two-sided number lines, t-tables, graphs, and function rules. The linked representations can begin with the study of proportion in middle school mathematics; the study of proportion and its representations at the beginning of the first high school algebra course leads directly into linear functions, the heart of the course. In setting out a path of representations from proportion to linear functions, the author suggests a central role for t-tables, tied closely to graphs and function rules. (Contains 4 tables, 9 figures, and a bibliography.)