More Complicated Functions:
Introduction to Linear Functions
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.
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.
The activities and discussions in this lesson address the following
Understand patterns, relationships and functions
Represent and analyze mathematical situations and structures using algebraic symbols.
- represent, analyze, and generalize a variety of patterns with tables, graphs, words, and,
when possible, symbolic rules;
- relate and compare different forms of representation for a relationship;
- identify functions as linear or nonlinear and contrast their properties from tables, graphs,
use mathematical models to represent and understamd quantitative relationships
- develop an initial conceptual understanding of different uses of variables;
- explore relationships between symbolic expressions and graphs of lines, paying particular
attention to the meaning of intercept and slope;
- use symbolic algebra to represent situations and to solve problems, especially those that
involve linear relationships;
- recognize and generate equivalent forms for simple algebraic expressions and solve linear
Analyze change in varius contexts
- model and solve contextualized problems using various representations, such as graphs,
tables, and equations
- use graphs to analyze the nature of changes in quantities in linear relationships.
- Arithmetic: Students must be able to:
- perform integer and fractional arithmetic
- Algebraic: Students must be able to:
- work with simple functions having one operation
- Technological Students must be able to:
- perform basic mouse manipulations such as point, click and drag
- use a browser such as Netscape for experimenting with the activities
Students will need:
- Access to a browser
- pencil and paper
- Copies of supplemental materials for the activities:
This lesson introduces students to the following terms through the included discussions:
This lesson assumes that the student is already familiar with
the material in the Introduction to Functions
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.
- 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?
Let the students know what it is they will be doing and learning today. Say something like
- 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.
- Teacher Input
- Lead a discussion on building more complicated
functions using composition.
- 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:
Note: The function rule for these more complicated functions can be much harder to guess
from just the data table.
- Find the function that adds one and then multiplies the result by 2
- y = 4 - x/2
- Lead a discussion on functions
of the special form y = ___ * x + ___.
- 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 w
orked 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.
- 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
This lesson can be rearranged in several ways.
- 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:
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.
- 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.
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
The Coordinate Plane will introduce students
to plotting points on the coordinate plane.