Intent The activities in The Importance of Patterns introduce a crucial mathematical idea—functions and their representations—that will weave its way through the entire curriculum. Students begin to build their ability to tackle novel mathematical problems, a way of doing mathematics that permeates IMP. In addition, these activities begin to establish expectations for students’ classroom interactions—as a whole class, in small groups, and individually—and written work throughout the course. Finally, students have their first experiences using graphing calculators.

Mathematics The central mathematical idea in The Importance of Patterns is the concept of function, one of the fundamental unifying principles in mathematics. Functions are introduced using numeric and nonnumeric examples, with an emphasis on looking for patterns and describing those patterns verbally. The term function is introduced in the context of describing the Out as “a function of” the In. The discussion introduces the principle that a function cannot have more than one output for a given input as well as the concepts of the domain and range of a function. Students use variables and algebraic expressions to describe numeric functions. They also apply In-Out tables to mathematical problems and see the distinction between tables arising from context and tables that are simply collections of number pairs. In-Out tables are a standard method for representing functions and are central to this unit and the curriculum. Variables play a vital role throughout mathematics, and they represent a major step toward mathematical abstraction. One major goal in Patterns is for students to use the symbolic language of algebra as shorthand to describe patterns, particularly arithmetic patterns in In-Out tables. Students use both their verbal description of a table’s rule and the pattern of arithmetic for finding specific outputs in order to develop an algebraic expression that describes the rule. As part of this work, they begin to use the terms variable and algebraic expression. The concept of equivalent expressions is introduced in the context of seeing that different expressions give the same results for an In-Out table.