Mathematical objects admit a variety of representations. Using multiple (graphical, verbal, tabular, algebraic) representations of a function can bring out hidden properties of the objects being considered. For example, the graphs of two intersecting, parallel, or coincident lines reveal why linear systems of equations may have a unique solution, none at all, or infinitely many. This course focuses on the real (number) line and coordinate systems in two and three dimensions, as systems for representing certain mathematical objects (real numbers and correspondences, including functions) and for modeling relations among quantities. The primary goal is exploring and increasing the understanding of algebraic and geometric representations in these systems. Teachers discuss selected examples from educational research to gain insight into students’ reasoning about number lines, coordinate systems, and the mathematical objects they represent. Part-whole representations of fractions (pizza models) are critically assessed and contrasted with representations of fractions (1) on the real line and (2) in the plane. They are discussed as models of rational numbers as well as models of relations among physical quantities. Examples from educational research on students’ understanding and misunderstandings about fractions and division (e.g. the belief that multiplication makes bigger, division makes smaller, etc.) are also studied.