Course 2 Unit 1 - Functions, Equations, and Systems

In Course 1, students have developed a robust understanding of the patterns of change for linear, exponential, and quadratic functions. Solving of equations based on these functions was accomplished by analyzing tables, graphs, and, in some cases, symbolic representations. This unit reviews and extends the algebra and functions strand to direct and inverse variation, multivariable functions, and systems of linear equations.

Unit Overview
In this unit, students review patterns of change that are modeled well by single-variable functions, with special attention to direct and inverse variation relations. Students then extend the notions of direct and inverse variation functions to two-variable functions. For example, if z = k • x/y, we say that z is directly proportional to x and inversely proportional to y. Then a different kind of two-variable function, z = ax + by, is examined with special focus on the set of solutions (xy) for linear equations in the form ax + by = c and their graphs. Finally, students develop the understanding and skill required to set up and solve systems of two linear equations in two variables using a graphical method, a substitution method, and an elimination method.

Objectives of the Unit
  • Review familiar families of single variable functions (especially linear, exponential, and quadratic functions)
  • Recognize direct and inverse variation functions with one or more independent variables, express those relationships in symbolic form, and manipulate those expressions into equivalent useful forms
  • Recognize and represent graphically and symbolically relationships in which one variable is a linear function of two independent variables and graph solutions of equations in the form "ax + by = c"
  • Set up and solve systems involving two linear equations with two variables by use of graphing, substitution, and elimination methods. Recognize whether systems have 0, 1, or 2 solutions by inspecting the equations

Sample Overview
The sample investigation below is the first investigation in Lesson 2. In this investigation, students are asked to think about the relationship of current to voltage and resistance in simple circuits as an example of a multivariable function that combines direct and inverse variation. Then they revisit the inclined plane experiment data (from Lesson 1) and work on a model of the combined effect of ramp length and platform height on roll time. Finally, they are asked to generalize these function ideas to consider general patterns in relationships of the form z = xy and z = x/y.
     This investigation makes use of the public-domain CPMP-Tools computer software. You will notice reference to the "Light It Up!" custom tool.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Algebra and Functions Strand Continues
In Course 2 Unit 2 and Unit 5, students develop matrix and linear combination methods for solving systems of two linear equations. They are introduced to function notation, review and extend their 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 are introduced to common logarithms and algebraic methods for solving exponential equations.
     In Course 3, students extend their understanding of formal reasoning in contexts, study linear inequalities and linear programming, polynomial and rational functions, sequences and series, and inverse functions.
     Course 4: Preparation for Calculus extends student algebraic skills and understandings in equations and functions in algebra units but also in geometry units such as Unit 2, Vectors and Motion, and Unit 6, Surfaces and Cross Sections. (See the CPMP Courses 1-4 descriptions.)

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