ABSTRACT
This chapter focuses on relationships between variables, as represented by equations and their graphs. Equations describe invariant relationships between two variables as they vary. Understanding these relationships requires students to develop covariational reasoning. The cartesian plane leverages this kind of reasoning to represent relationships between variables with graphs. However, students often see graphs as static shapes without ever considering how those graphs emerge from co-varying quantities. This chapter introduces Carlson’s framework for understanding how students develop covariational reasoning by coordinating their own mental actions. By coordinating covarying quantities that they construct, students can meaningfully represent their invariant relationships with equations and graphs. The chapter also introduces relevant history, from the Pappus problems of the 4th century to Emmy Noether’s study of algebraic invariants in the 20th century.
