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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.)

Describes experience of teaching truth tables and problems that exercise students' critical reasoning. (KHR)

Truncation error bounds are developed for continued fractions K( a/1) where | a|≦1/4 for all n sufficiently large. The bounds are particularly suited (some are shown to be best) for the limit-periodic case when lim a=0. Among the principal results is the following: If | a|≦α/ n for all n sufficiently large (with constants α>0, p>0), then | f−f|≦ C[ D/( m+2)] for all m sufficiently large (for some constants C>0, D>0). Here f denotes the limit (assumed finite) of K( a/1) and f denotes its mth approximant. Applications are given for continued fraction expansions of ratios of Kummer functions F and of ratios of hypergeometric functions F. It is shown that p=1 for F and p=2 for F, where p is the parameter determining the rate of convergence. Numerical examples indicate that the error bounds are indeed sharp. [ABSTRACT FROM AUTHOR]