More Complicated Functions: Introduction to Linear Functions

Abstract

This lesson is designed to introduce students to the idea of functions composed of two operations, with specific attention to linear functions and their representations as rules and data tables, including the mathematical notions of independent and dependent variables.

This lesson assumes that the student is already familiar with the material in the Introduction to Functions Lesson. These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to functions
  • have learned the terminology used with linear functions
  • have practiced describing linear functions in English sentences, data tables, and with simple algebraic expressions

Standards Addressed

Grade 6

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 7

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 8

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Eighth Grade

  • Expressions and Equations

    • Understand the connections between proportional relationships, lines, and linear equations.
    • Analyze and solve linear equations and pairs of simultaneous linear equations.

  • Functions

    • Define, evaluate, and compare functions.
    • Use functions to model relationships between quantities.

Functions

  • Building Functions

    • Build a function that models a relationship between two quantities
    • Build new functions from existing functions

  • Interpreting Functions

    • Understand the concept of a function and use function notation
    • Interpret functions that arise in applications in terms of the context
    • Analyze functions using different representations

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems
    • Interpret expressions for functions in terms of the situation they model

Grades 6-8

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will demonstrate an understanding of linear relations and fundamental algebraic concepts.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

  • Algebra

    • COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.
    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Algebra I

  • Algebra

    • Competency Goal 4: The learner will use relations and functions to solve problems.

6th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities.

8th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • Standard EA-2: The student will demonstrate through the mathematical processes an understanding of the real number system and operations involving exponents, matrices, and algebraic expressions.
    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
    • Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.

7th Grade

  • Probability and Statistics

    • 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions; line plots; histograms; stem-and-leaf plots; box-and-whisker plots; and scattergrams.

Secondary

  • Algebra II

    • AII.09 The student will find the domain, range, zeros, and inverse of a function; the value of a function for a given element in its domain; and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions.
    • AII.12 The student will represent problem situations with a system of linear equations and solve the system, using the inverse matrix method. Graphing calculators or computer programs with matrix capability will be used to perform computations.
    • AII.13 The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.
    • AII.9
    • AII.12
    • AII.13

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer and fractional arithmetic
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities
  • Algebraic: Students must be able to:
    • work with simple functions having one operation

Teacher Preparation

Key Terms

intercept

See x-intercept or y-intercept

linear function

A function of the form f(x) = mx + b where m and b are some fixed numbers. The names "m" and "b" are traditional. Functions of this kind are called "linear" because their graphs are straight lines

slope of a linear function

The slope of the line y = mx + b is the rate at which y is changing per unit of change in x. The units of measurement of the slope are units of y per unit of x (cf. Linear Functions Discussion).

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson.

    • Who remembers what a function is?
    • Can someone give me an example of a function?
    • Can someone give me an example of something that is not a function?

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn more about functions.We are going to use the computers to learn more about functions, but please do not turn your computers on until I ask you to.
    • I want to show you a little about this activity first.

  3. Teacher Input

    • Lead a discussion on building more complicated functions using composition.

  4. Guided Practice

    • Have the students practice "pumping" a few of these more complicated functions by hand by filling in a few tables. Give them some functions in English, some as tables and some as algebra. Have them write the functions in all the forms. For Example:
      1. Find the function that adds one and then multiplies the result by 2
      2. y = 4 - x/2
      3. x -2 -1 0 1 2
        y -7 -4 -1 2 5
        Note: The function rule for these more complicated functions can be much harder to guess from just the data table.
    • Lead a discussion on functions of the special form y = ___ * x + ___ .

  5. Independent Practice

    • Have students practice their linear function skills by using the Linear Function Machine . Be sure to have students record how many numbers they needed to look at before correctly guessing the function structure. Have them write the functions they worked with in three ways:
      • English Sentence
      • Table of Values
      • Algebraic Rule
    • Have them try to think of situations in their lives that might be governed by some of the functions they worked with

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

  • Omit the information on more complicated functions, discussing only functions of the form y = mx + b.
  • Add a "name that function" contest (modeled on name that tune) in which teams of students compete to figure out the function. Here is a set of possible rules for such a game:
    • Show two input/output pairs to both teams - two students on a team works very well.
    • Have each team state how many more pairs they think that they would need to see to "name that function." The team who claims the fewest needed pairs goes first.
    • If a team guesses wrong the other team gets to try, after seeing one more pair. Teams alternate turns until one guesses correctly.
  • Introduce more complicated non-linear functions by allowing exponentiation (whole numbers to start) and division by x.

Suggested Follow-Up

After these discussions and activities, students will have an intuitive understanding of functions and will have seen many examples of linear functions. The next lesson Graphing and the Coordinate Plane will introduce students to plotting points on the coordinate plane.