Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 60551
Nonlinear Functions
Students are asked to provide an example of a nonlinear function and explain why it is nonlinear.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, linear function, nonlinear function, slope, rate of change, initial value, y-intercept, slopeintercept form
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_NonlinearFunctions_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Nonlinear Functions worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to provide an example of a nonlinear function.
Examples of Student Work at this Level
The student:
Sketches a graph of a diagonal line and indicates that it is nonlinear.
Writes a linear equation that is not in slope-intercept form.
Writes a linear equation in one variable.
page 1 of 3 Sketches a graph of a nonlinear relation that is not a function.
Questions Eliciting Thinking
What does linear mean? What does the graph of a linear function look like? What does the equation of a linear function look like?
What does nonlinear mean? How might the graph of a nonlinear function look?
Does your equation represent a function? What are the inputs? What are the outputs?
What must be true of the graphs of all functions?
Instructional Implications
Review the concept of a linear function. Describe the equation of a linear function as one that can be written in the form ax + by = c (where b
0), but explain that
linear functions may be written in other forms, some of which are quite useful (such as slope-intercept form). Explain to the student that a distinguishing feature of a linear
function in two variables is that the variables are both raised to the first power. Provide many examples of equations of functions and ask the student to categorize the
functions into those that are linear and those that are nonlinear.
Be sure the student understands that the graph of a linear function is always a nonvertical line and every nonvertical line represents a linear function. Explain that a
distinguishing feature of a linear function is that the rate of change is constant [e.g., any two values of x (or inputs) that differ by the same amount will have y values (or
outputs) that differ by the same amount]. Use tables of values and graphs of both linear and nonlinear functions to compare rates of change. Ask the student to draw
graphs of nonlinear functions and identify equal intervals of the domain for which the corresponding intervals of the range are unequal.
Consider implementing other MFAS tasks for standard 8.F.1.3.
Making Progress
Misconception/Error
The student is unable to justify his or her example.
Examples of Student Work at this Level
The student provides an equation, graph, or table of a nonlinear function. To justify the example, the student:
Refers to the definition of a function or the vertical line test.
States that the function is “not increasing or decreasing.”
Provides an unclear explanation.
States that input and output are not consistent.
Questions Eliciting Thinking
What makes a function linear? What do you know about the equations and graphs of linear functions?
This is useful for telling me that this is a function, but how can you explain that your example is nonlinear?
What do you mean by “not increasing or decreasing?”
Instructional Implications
Explain to the student that a distinguishing feature of a linear function in two variables is that the variables are both raised to the first power. Consequently, when a variable
in the equation of a function is raised to a power other than one, the function is not linear. Use a graphing utility to graph examples of both linear and nonlinear functions.
Assist the student in relating features of the equations of the functions (e.g., the power of the variables) to the graphs. Provide additional examples of equations of
functions and ask the student to categorize the functions into those that are linear and those that are nonlinear and explain the reasoning for the determination.
Consider implementing other MFAS tasks for standard 8.F.1.3.
page 2 of 3 Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student provides an example of a nonlinear function using either an equation, graph, or table. The student:
Sketches a parabola or the graph of another nonlinear function and explains that it is nonlinear because it is not a line.
Creates a table of values that includes ordered pairs such as [(1, 1), (2, 4), (3, 9), (4, 16)] and explains that equal differences in x-values do not correspond to equal
differences in y-values.
Provides an equation such as y =
and explains that it is nonlinear because the variable is raised to the second power or its graph is not a line.
Questions Eliciting Thinking
What do you know about linear functions? What does the equation of a linear function look like? What does its graph look like? What can you say about the rate of change
as you move from one point to another on the graph?
Instructional Implications
Ask the student to provide examples of nonlinear functions in forms that he or she did not use on this task (graph, table, or equation).
Describe real-world examples of nonlinear relationships (e.g., comparing distance to time for an accelerating vehicle). Ask the student to create a graph, chart, or equation
for a nonlinear function with a real-world context. Ask the student to explain the significance of the different rates of change within the context.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Nonlinear Functions worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.F.1.3:
Description
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of
functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side
length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
page 3 of 3