Algebra and Functions

The algebra and functions strand of Core-Plus Mathematics develops student ability to recognize problems involving relations among quantitative variables, to use symbolic expressions and equations to represent those relationships, and to use a variety of reasoning methods to solve the problems. Central to the CPMP algebra development is use of mathematical functions as models for exact numerical relationships and data patterns. The key models are linear, exponential, power, polynomial, logarithmic, rational, and periodic functions and systems of linear and nonlinear relations. Each type of function is investigated in four linked representations - verbal, graphic, numeric, and symbolic - with appropriate use of graphing calculators as learning and problem-solving tools.

Algebraic concepts and skills are developed in 12 primary units of the CPMP curriculum, with significant connections in almost every unit of the other strands as well. Over the four CPMP courses, students are asked to develop increasingly sophisticated understanding and skill in use of symbolic expressions. Early units introduce basic algebraic ideas like variables, equations, inequalities, and functions in realistic problem contexts with strong links among tabular, graphic, and symbolic images. Building on that intuitive foundation, students in later units are asked to work more and more often with symbolic expressions and relationships that are independent of specific contextual cues.

Since algebra is the dominant strand of traditional high school mathematics curricula, there is particular interest in the effects of our new approach to the subject. A variety of evaluation and research efforts have assessed the performance of CPMP students on traditional symbolic manipulation tasks, algebraic problem-solving, and conceptual understanding of key algebraic ideas. Results suggest that CPMP students generally acquire stronger algebraic understanding and problem-solving skills than students in more traditional programs. On tasks that emphasize traditional symbol manipulations, CPMP students hold their own, although we focus on that kind of abstract reasoning somewhat later than the scope and sequence of some traditional algebra curricula would indicate. Thus, CPMP students will encounter some algebraic ideas (like functions and realistic application problems) earlier than students in traditional curricula and other ideas (like formal solution of quadratic and rational equations by symbolic manipulation) somewhat later than students in traditional curricula.

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