Algebra and Functions Strand

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 16 primary units of the Core-Plus Mathematics curriculum, with significant connections in almost every unit of the other strands as well. In addition, there are two units developing trigonometric functions and equations listed under the Geometry and Trigonometry strand. 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.

Course 1

Unit 1 - Patterns of Change develops student ability to recognize and describe important patterns that relate quantitative variables, to use data tables, graphs, words, and symbols to represent the relationships, and to use reasoning and calculating tools to answer questions and solve problems.

Topics include: variables and functions, algebraic expressions and recurrence relations, coordinate graphs, data tables and spreadsheets, and equations and inequalities.

Unit 3 - Linear Functions develops student ability to recognize and represent linear relationships between variables and to use tables, graphs, and algebraic expressions for linear functions to solve problems in situations that involve constant rate of change or slope.

Topics include: linear functions, slope of a line, rate of change, modeling linear data patterns, solving linear equations and inequalities, equivalent linear expressions.

Unit 5 - Exponential Functions develops student ability to recognize and represent exponential growth and decay patterns, to express those patterns in symbolic forms, to solve problems that involve exponential change, and to use properties of exponents to write expressions in equivalent forms.

Topics include: exponential growth and decay functions, data modeling, growth and decay rates, half-life and doubling time, compound interest, and properties of exponents.

Unit 7 - Quadratic Functions develops student ability to recognize and represent quadratic relations between variables using data tables, graphs, and symbolic formulas, to solve problems involving quadratic functions, and to express quadratic polynomials in equivalent factored and expanded forms.

Topics include: quadratic functions and their graphs, applications to projectile motion and economic problems, expanding and factoring quadratic expressions, and solving quadratic equations by the quadratic formula and calculator approximation.

Course 2

Unit 1 - Functions, Equations, and Systems reviews and extends student ability to recognize, describe, and use functional relationships among quantitative variables, with special emphasis on relationships that involve two or more independent variables.

Topics include: direct and inverse variation and joint variation; power functions; linear equations in standard form; and systems of two linear equations with two variables, including solution by graphing, substitution, and elimination.

Unit 2 - Matrix Methods develops student understanding of matrices and ability to use matrices to represent and solve problems in a variety of real-world and mathematical settings.

Topics include: constructing and interpreting matrices, row and column sums, matrix addition, scalar multiplication, matrix multiplication, powers of matrices, inverse matrices, properties of matrices, and using matrices to solve systems of linear equations.

Unit 5 - Nonlinear Functions and Equations introduces function notation, reviews and extends student ability to construct and reason with functions that model parabolic shapes and other quadratic relationships in science and economics, with special emphasis on formal symbolic reasoning methods, and introduces common logarithms and algebraic methods for solving exponential equations.

Topics include: formalization of function concept, notation, domain and range; factoring and expanding quadratic expressions, solving quadratic equations by factoring and the quadratic formula, applications to supply and demand, break-even analysis; common logarithms and solving exponential equations using base 10 logarithms.

Course 3

Unit 1 - Reasoning and Proof develops student understanding of formal reasoning in algebraic contexts and of basic principles that underlie those reasoning strategies.

Topics include: inductive and deductive reasoning strategies; principles of logical reasoning—Affirming the Hypothesis and Chaining Implications; rules for transforming algebraic expressions and equations.

Unit 2 - Inequalities and Linear Programming develops student ability to reason both algebraically and graphically to solve inequalities in one and two variables, introduces systems of inequalities in two variables, and develops a strategy for optimizing a linear function in two variables within a system of linear constraints on those variables.

Topics include: inequalities in one and two variables, number line graphs, interval notation, systems of linear inequalities, and linear programming.

Unit 5 - Polynomial and Rational Functions extends student ability to represent and draw inferences about polynomial and rational functions using symbolic expressions and manipulations.

Topics include: definition and properties of polynomials, operations on polynomials; completing the square, proof of the quadratic formula, solving quadratic equations (including complex number solutions), vertex form of quadratic functions; definition and properties of rational functions, operations on rational expressions.

Unit 7 - Recursion and Iteration extends student ability to represent, analyze, and solve problems in situations involving sequential and recursive change.

Topics include: iteration and recursion as tools to model and analyze sequential change in real-world contexts, including compound interest and population growth; arithmetic, geometric, and other sequences; arithmetic and geometric series; finite differences; linear and nonlinear recurrence relations; and function iteration, including graphical iteration and fixed points.

Unit 8 - Inverse Functions develops student understanding of inverses of functions with a focus on logarithmic functions and their use in modeling and analyzing problem situations and data patterns.

Topics include: inverses of functions; logarithmic functions and their relation to exponential functions, properties of logarithms, equation solving with logarithms; and inverse trigonometric functions and their applications to solving trigonometric equations.

Course 4

Unit 1 - Families of Functions extends student understanding of linear, exponential, quadratic, power, and trigonometric functions to model data patterns whose graphs are transformations of basic patterns; and develops understanding of operations on functions useful in representing and reasoning about quantitative relationships.

Topics include: linear, exponential, quadratic, power, and trigonometric functions; data modeling; translation, reflection, and stretching of graphs; and addition, subtraction, multiplication, division, and composition of functions.

Unit 3 - Algebraic Functions and Equations reviews and extends student understanding of properties of polynomial and rational functions and skills in manipulating algebraic expressions and solving polynomial and rational equations, and develops student understanding of complex number representations and operations.

Topics include: polynomials, polynomial division, factor and remainder theorems, operations on complex numbers, representation of complex numbers as vectors, solution of polynomial equations, rational function graphs and asymptotes, and solution of rational equations and equations involving radical expressions.

Unit 5 - Exponential Functions, Logarithms, and Data Modeling extends student understanding of exponential and logarithmic functions to the case of natural exponential and logarithmic functions, solution of exponential growth and decay problems, and use of logarithms for linearization and modeling of data patterns.

Topics include: exponential functions with rules in the form f(x) = Ae^kx, natural logarithm function, linearizing bivariate data and fitting models using log and log-log transformations.

Unit 7 - Concepts of Calculus develops student understanding of fundamental calculus ideas through explorations in a variety of applied problem contexts and their representations in function tables and graphs.

Topics include: instantaneous rates of change, linear approximation, area under a curve, and applications to problems in physics, business, and other disciplines.