This lesson is designed to introduce students to the idea of functions and their representations
as rules and data tables, including the mathematical notions of independent and dependent
variables.

Objectives

Upon completion of this lesson, students will:

have been introduced to functions

have learned the terminology used with functions

have practiced describing functions with one operation in English sentences, data tables, and with simple algebraic expressions

Standards Addressed:

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.

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 3

Algebra and Functions

2.0 Students represent simple functional relationships

Grade 5

Algebra and Functions

1.0 Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results

Number Sense

1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers

Grade 6

Algebra and Functions

1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results

Grade 7

Algebra and Functions

3.0 Students graph and interpret linear and some nonlinear functions

Eighth Grade

Expressions and 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

Interpreting Functions

Understand the concept of a function and use function notation

Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems

Seventh Grade

Expressions and Equations

Use properties of operations to generate equivalent expressions.

Sixth Grade

Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

Grades 3-5

Algebra

Represent and analyze mathematical situations and structures using algebraic symbols

Grades 6-8

Algebra

Understand patterns, relations, and functions

Grades 9-12

Algebra

Represent and analyze mathematical situations and structures using algebraic symbols

Algebra 1

Algebra

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

Algebra I

Algebra

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

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.

4th grade

Algebra

Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of an unknown.

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-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.

Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.

Intermediate Algebra

Algebra

The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

Grade 5

Patterns, Relationships, and Algebraic Thinking

6. The student describes relationships mathematically.
The student is expected to select from and use diagrams and equations such as y = 5 + 3 to
represent meaningful problem situations.

Grade 6

Patterns, Relationships, and Algebraic Thinking

4. The student uses letters as variables in
mathematical expressions to describe how one quantity changes when a related quantity changes.

Grade 7

Patterns, Relationships, and Algebraic Thinking

3. The student solves problems involving direct
proportional relationships.

8th Grade

Computation and Estimation

8.4 The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Problems will be limited to positive exponents.

Patterns, Functions, and Algebra

8.14a The student will describe and represent relations and functions, using tables, graphs, and rules; and

8.14 The student will

Secondary

Algebra II

AII.04 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used as a primary method of solution and to verify algebraic solutions.

AII.05 The student will identify and factor completely polynomials representing the difference of squares, perfect square trinomials, the sum and difference of cubes, and general trinomials.

AII.06 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.07 The student will solve equations containing rational expressions and equations containing radical expressions algebraically and graphically. Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.08 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.

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.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.

AII.17 The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i.

AII.18 The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from
his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.

Reason for Alignment: The Introduction to Functions lesson is a great beginning and should be extra practice for the textbook. The terms, such as input and output, are explained and align well with the key concepts and terms of Section 4. The Function Machine activity used in the lesson is explained and used.

Student Prerequisites

Arithmetic: Student 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

Teacher Preparation

Access to a browser

pencil and paper

Copies of supplemental materials for the activities:

A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important

input

The number or value that is entered, for example, into a function machine. The number that goes into the machine is the input

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

output

The number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output

Lesson Outline

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.

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 about functions and their representations

We are going to use the computers to learn about functions and their representations, but
please do not turn your computers on until I ask you to. I want to show you a little about
this activity first

Teacher Input

Introduce the idea of functions as machines, by leading a class
discussion on functions.

Guided Practice

Have the students construct machines to test each other with. Start them with single operation
machines, and suggest that they build tables for the input/output pairs. Reinforce the
convention that mathematicians avoid confusion by always putting the input first in an
ordered pair enclosed in parentheses and separated by commas:

(x,y)

Give them one or two tables with a few outputs for them to fill in. Ask them to describe in
words what the function does. For Example:

INPUT

OUTPUT

INPUT

OUTPUT

5

-1

-5

-15

3

-3

2

6

-1

-7

4

12

4

-2

0

0

-5

-11

3

9

2

-4

-2

-6

10

10

-9

-7

After they practice describing functions in English sentences, discuss the convention of
letting a letter (often
but not always x) stand in for the input and another (often
but not always y) stand in for the output. Have them write all their earlier functions as algebra rules with x
as input and y as output.

Formalize the terminology:

Variable A letter standing in for an unknown or changeable number

Independent Variable The input into a function, often represented by x.

Dependent Variable The output from a function, often represented by y.

Functions A process that takes one or more numbers as input and produces a single number as output

Independent Practice

Have the students practice their new function building and pattern recognition skills with the
Function Machine Game . 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

Algebra Rule

Have them try to think of situations in their lives that might be governed by some of the
functions they worked with. For example,

y = x + 1

might be the function describing growing one year older on your birthday.

y = 2 * x

might be the function "everything tastes twice as good during the holiday."

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

This lesson can be rearranged in several ways.

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.

This game can be played in about 10 minutes per pair of teams, making it time consuming if
the entire class is to have a turn.

Introduce 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
More Complicated Functions will introduce students more general linear functions.