Patterning tasks engage students in a core aspect of algebraic thinking-generalization (Kaput 2008). The National Council of Teachers of Mathematics (NCTM) Algebra Standard states that students in grades 9-12 should "generalize patterns using explicitly defined and recursively defined functions" (NCTM 2000, p. 296). Although educators frequently use patterns that can be modeled using linear relationships, using patterns that can be modeled by nonlinear relationships, such as quadratic relationships, is important because they help students develop flexibility with their pattern recognition and modeling abilities. Additionally, considering a task's level of cognitive demand (Smith and Stein 1998) is important because it can influence what information students draw upon. The purpose of this article is to present and distinguish students' thinking that arises from tasks using quadratic relationships and also to provide some teaching strategies for using these types of patterning tasks.