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Module Overview
Linear Functions
The purpose of this module is to explore the concept of linear functions and relate their use to a real-world application.
Many students may shrug their shoulders and say, “Why do we need to know this?”. Our response is, “linear functions are very useful in everyday life.” For example, a person can study a relationship between the number of minutes used and the cost on a cell phone bill. Introducing real-life examples may help students take concepts of linear functions out of the classroom and into their everyday lives. For example, a person can study a relationship between the number of minutes used and the cost on a cell phone bill. Using linear piecewise-defined functions allows us to look at a water bill and determine the cost based upon the number of gallons used. Bringing real-life examples into the classroom to show students how the concepts of linear functions can be used in their everyday lives is important.
The exploration of linear functions will cross levels from Elementary Algebra into College Algebra. Throughout this module we will provide activities that can be used in the classroom to encourage active participation of students with the concepts. The activities provided will address the levels beginning with plotting of points in a coordinate plane and culminating in class projects based on real-life applications of linear functions. From the beginning, students will be encouraged to actively participate in both a mental and physical exploration of point placement, the relationship between points and linear graphs, and interpretation of slope. Additional activities incorporating the use of technology will be provided for use by instructors and students. Some activities are intended for students to complete outside of class and/or for use by students in an online learning environment.
The following are the expected outcomes of this module:
  • Provide an introduction to student understanding of ordered pairs on the coordinate plane.
  • Explore of the relationship between ordered pairs and a linear function.
  • Explore various forms of an equation of a line.
  • Connect linear functions with a real-world application.
  • Extend the linear function concept to a piecewise-defined function application.
The study of linear functions or linear equations encompasses a chapter in Introductory Algebra, two to three sections in Intermediate Algebra, and one section in College Algebra. Although our focus is developmental math, this module should provide a smooth transition into college level algebra. Specific subtopics can be used to supplement classroom instruction or as a stand-alone unit. Activities using technology can easily be incorporated into an online developmental course to provide futher exploration and initiate discussion within the group.
Each unit contains the following:
  • Pre-test
  • Lecture notes
  • A physical activity--which is best suited for classroom instruction
  • Paper-pencil activities which can be adapted to other formats (Blackboard, WebCT, Classroom Participation Systems, etc.)
  • Post-test
Each unit will contain suggestions for extension of the unit topic. Instructors may select from the material to supplement lessons or replace current curriculum. Many of these activities will be easily adaptable for online courses. The module will culminate in a project based on real-life applications of linear functions.
We anticipate completion of the module to require six to seven, 50-minute class periods for an Introductory Algebra, if the entire module is utilized. For Intermediate Algebra, we anticipate four to five 50-minute class periods for completion.
Although we will use a variety of sofware and technological support, alternative forms of presentation will be provided within each unit. Expense to the institution and instructor can be minimal.
Future modules can explore related topics such as:
Students completing the module can expect to gain an appreciation of linear functions as a useful tool that can be applied to real-life situations. In addition, students will gain experience in cooperative learning and the use of technology to analyze patterns in linear functions.
Lecture notes include detailed step-wise processes, strengthened through physical and paper-and-pencil/computer activities. Through lecture notes and activities, students will become proficient in plotting points, graphing linear functions, and determining equations of lines.
Before we jump into applications of linear functions, letís look at some definitions. Students need to know about a coordinate system, placing points in the coordinate system, connecting points to equations and/or functions, and recognizing important concepts related to linear functions. Once a solid foundation is established, then we can begin to work with applications.
Definitions
Cartesian Coordinate System (rectangular coordinate system): a plane created by intersecting two number lines perpendicularly at the origin and used to plot the location of an ordered pair
y-axis: Vertical number line in the rectangular coordinate system
x-axis: Horizontal number line in the rectangular coordinate system
Ordered pair: Point on the rectangular coordinate system whose horizontal distance is x and its vertical distance is y from the origin
Origin: Point on the rectangular coordinate system where the horizontal and vertical axis intersect at zero
Quadrants: Regions on the rectangular coordinate system
Linear equations: Equation with two variables of degree one
Solutions to a linear equation: Two numbers that form a true statement when substituted into the equation
Relation: Set of ordered pairs
Function: Special type of relation where each x-value corresponds to exactly one y-value
x-intercept: Location on the rectangular coordinate system where the graph of an equation crosses the x-axis
y-intercept: Location on the rectangular coordinate system where the graph of an equation crosses the y-axis
Slope: Steepness of a line
Sources