Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 97
Standard Indicators
7.3.9, 7.3.10
Linear or Nonlinear − That Is the Question!
Purpose
Students will identify and compare linear and nonlinear functions
in a variety of ways.
Materials
Activity
A. Introduction
1. Place the transparency of the BLM Examples of Functions on
the overhead projector. Give each student a copy of the BLM
Examples of Functions.
2. Make comparisons of the first two graphs and equations. Compare
such things as finding slope by using the rise over run concept,
straight or curved, exponents or none, etc.
3. Discuss what it means to say that a line is straight. [It has a
constant slope.]
4. Tell students that a function that does not have constant slope
(has varying slope) is called a nonlinear function.
5. Have students look at the other two graphs on the BLM Examples
of Functions. Determine if the functions are linear or nonlinear
by examining the slope using the rise over run concept to compare
the slope at different points along the graph.
6. Tell students that they can also look at the equations to determine
whether a function is linear or nonlinear. Compare the equations
on the BLM Examples of Functions. Ask students how the
equations of the nonlinear functions differ from the equations
of the linear functions.
7. Have students assist you in creating a table by placing the
x-values and y-values from three or four points in separate
columns for one of the linear functions and one of the nonlinear
functions on the BLM Examples of Functions. Notice differences
between the tables. [y-values in nonlinear function get larger
more quickly, etc.]
(continued)
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002
ACTIVITY
Have students look
through newspapers
and periodicals to find
examples of graphs,
tables, equations,
and verbal descriptions
that represent either
linear or nonlinear
functions.
incorporating
TECHNOLOGY
Use a graphing
calculator or software
program and projector
instead of the overhead
projector to introduce
the concept of linear
and nonlinear functions.
Allow students to also
use a graphing
calculator during the
activity. Have them
practice graphing linear
and nonlinear functions
by visiting a Web site,
such as the QuickMath
section of www.algebraonline.com.
Standards Links
7.7.1, 7.7.6, 7.7.12
page 97
Standard 3
For the teacher: transparency of Black Line Master (BLM) Examples
of Functions, overhead projector
For each student: copies of the BLMs Examples of Functions and Linear
or Nonlinear?
extending
THE
Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 98
Activity (continued)
8. Discuss situations that could determine whether a function
is linear or nonlinear. Use phrases such as steady growth,
accelerates or speeds up, exponential growth, and constant rate
of change.
Standard 3
B. Student Activity
1. Distribute a copy of the BLM Linear or Nonlinear? to
each student.
2. Review briefly which characteristics of a graph, equation, table,
or verbal description determine whether a function is linear
or nonlinear.
3. Instruct students to evaluate each graph, equation, table,
or verbal description to determine if it represents a linear
or nonlinear function and tell why they chose their answer.
4. Discuss answers as a class.
Classroom Assessment
Basic Concepts and Processes
As students complete the BLM Linear or Nonlinear? and during
the discussion, ask students the following questions to gauge their
understanding of the Standard Indicator:
What about the slope of a line determines whether the line
is straight?
How can you tell if an equation is linear or nonlinear?
How might you tell if a table represents a linear
or nonlinear function?
Why do you think the y-values of a nonlinear function change
so quickly?
An experiment shows that a river runs faster as it approaches
a waterfall and slows at the bottom of the waterfall. Will the
function that represents the speed of the water along its course
be linear or nonlinear?
How did you determine your answer?
page 98
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002
Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 99
Name:
Examples of Functions
y = x2
y=x
y = 1/3 (x - 2)2
y = 2/3 (x + 2)
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002
Black Line Master 1
page 99
Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 100
Examples of Functions
Teacher Directions
Distribute the BLM Examples of Functions to students at the beginning of the activity. Have them
compare the graphs and equations to find the differences between the linear and nonlinear
functions.
Prepare a table on the chalkboard listing coordinates from the functions. Discuss the
characteristics of the tables. Discuss situations that can identify linear and nonlinear functions
using phrases such as steady growth, accelerates or speeds up, exponential growth, and constant
rate of change.
Answer Key
Not applicable.
Black Line Master 1
page 100
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002
Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 101
Name:
Linear or Nonlinear?
Look at the graphs, equations, table, and written description below and determine
if each represents a linear or nonlinear function. Circle the correct answer. Explain
why you chose your answer. Use more space on the back of the paper, if necessary.
1.
3.
5.
2.
Number of
Calendars Sold
Money to Collect
2
$12.00
3
$18.00
5
$30.00
10
$60.00
(Circle the correct answer below.)
(Circle the correct answer below.)
Linear
or
Nonlinear ?
Explain why you chose your answer:
Linear
or
Nonlinear ?
Explain why you chose your answer:
3x + y = 8
4.
y = 5x 2 - 6
(Circle the correct answer below.)
(Circle the correct answer below.)
Linear
or
Nonlinear ?
Explain why you chose your answer:
Linear
or
Nonlinear ?
Explain why you chose your answer:
A daisy grows 1 inch the 1st week of
June. It grows 3/4 inch the 2nd week
and its growth continues to slow
down. A function represents the
growth of the daisy over the summer.
Is that function:
6.
(Circle the correct answer below.)
Linear
or
Nonlinear ?
Explain why you chose your answer:
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002
(Circle the correct answer below.)
Linear
or
Nonlinear ?
Explain why you chose your answer:
Black Line Master 2
page 101
Math 7 Standard 3 08-06
8/21/02
4:31 PM
Page 102
Linear or Nonlinear?
Teacher Directions
Distribute the BLM Linear or Nonlinear? to the students at the end of the activity. Have them
look at the graphs, equations, table, and written description and determine if each represents a
linear or nonlinear function. Instruct them to circle the correct answers and explain why they
chose their answers. Tell them that they can use more space on the back of the paper, if
necessary. When students have completed the BLM, discuss answers as a class.
Answer Key
More than one explanation is correct for each item. One correct explanation is provided below:
1. Nonlinear; The graph has varying slope.
2. Linear; The rate of change is constant.
3. Linear; There are no exponents in the equation.
4. Nonlinear; There is an exponent in
the equation.
5. Nonlinear; The rate of growth changes.
6. Linear; The graph has constant slope.
Black Line Master 2
page 102
Standard 3 / Activity 7
Indiana Mathematics Grade 7 Curriculum Framework, October 2002