Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 59833
What Is the Function Rule?
Students are asked to write function rules for sequences given tables of values.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, arithmetic sequence, geometric sequence, common difference, common ratio, function rule
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_WhatIsTheFunctionRule_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 problems on the What Is the Function Rule? worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student cannot write a function rule for arithmetic and geometric sequences.
Examples of Student Work at this Level
The student may observe common differences and common ratios in successive terms but is unable to write function rules that can be used to generate the terms of the
sequences.
Questions Eliciting Thinking
I see that you found a common difference of four for the first sequence. What does that tell you about the type of sequence? What kind of function describes this type of
sequence?
I see that you found a common ratio of two for the second sequence. What does that tell you about the type of sequence? What kind of function describes this type of
sequence?
page 1 of 4 Instructional Implications
Assist the student in identifying a sequence as arithmetic by observing a common difference between pairs of successive terms. Explain that function rules for arithmetic
sequences are linear functions. Assist the student in applying methods for writing linear functions given two ordered pair solutions to writing function rules for arithmetic
sequences. After providing additional examples, guide the student to observe the relationship between the common difference and the coefficient of the term number, n,
in the function rule. Also, help the student observe that the term associated with n = 0 is not only the constant in the function rule but also corresponds to the y-intercept
of the graph of the function.
Assist the student in identifying a sequence as geometric by observing a common ratio between pairs of successive terms. Explain that function rules for geometric
sequences are exponential functions. Assist the student in applying methods for writing exponential functions given two ordered pair solutions to writing function rules for
geometric sequences. After providing additional examples, guide the student to observe the relationship between the common ratio and the base of the exponential factor
in the function rule. Also, help the student observe that the term associated with n = 0 is not only the coefficient in the function rule but also corresponds to the yintercept of the graph of the function.
Provide additional examples of arithmetic and geometric sequences and ask the student to write function rules.
Moving Forward
Misconception/Error
The student is only able to write a linear function to represent the arithmetic sequence.
Examples of Student Work at this Level
The student correctly writes a linear function to represent the arithmetic sequence, but is not able to correctly write an exponential function to represent the geometric
sequence.
Questions Eliciting Thinking
Is the second sequence arithmetic? Why or why not?
What do you know about the second sequence?
I see that you found a common ratio of two for the second sequence. What does that tell you about the type of sequence? What kind of function describes this type of
sequence?
Instructional Implications
Assist the student in identifying a sequence as geometric by observing a common ratio between pairs of successive terms. Explain that function rules for geometric
sequences are exponential functions. Assist the student in applying methods for writing exponential functions given two ordered pair solutions to writing function rules for
geometric sequences. After providing additional examples, guide the student to observe the relationship between the common ratio and the base of the exponential factor
in the function rule. Also, help the student observe that the term associated with n = 0 is not only the coefficient in the function rule but also corresponds to the yintercept of the graph of the function.
If needed, assist the student in developing algebraic strategies (rather than trial and error) for writing function rules for arithmetic sequences. Provide additional examples of
arithmetic and geometric sequences and ask the student to write function rules.
Almost There
Misconception/Error
The student makes a minor error.
Examples of Student Work at this Level
The student correctly writes each function but:
Does not use notation given in the problem and/or uses terminology appropriate for exponential functions rather than geometric sequences.
page 2 of 4 Questions Eliciting Thinking
Is the notation you used consistent with the notation given in the problem? What problems might arise from introducing new notation in your work and answer?
What kind of sequence is the second sequence? When working with geometric sequences, what is the name for the factor by which one term can be multiplied to get the
next term?
Instructional Implications
If the student referred to the common ratio as the growth factor, commend him or her on observing the relationship between geometric sequences and exponential
functions. Remind the student of the terminology used in the context of sequences (e.g., arithmetic sequence, geometric sequence, term, term number, common
difference, common ratio) and model using terminology appropriate to the context.
If needed, assist the student in developing algebraic strategies for writing function rules (rather than using trial and error).
Consider implementing other MFAS exponential tasks Writing an Exponential Function From a Table (F-LE.1.2), Writing an Exponential Function From a Description (F-LE.1.2),
and Writing an Exponential Function From Its Graph (F-LE.1.2).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
For the arithmetic sequence, the student writes the rule f(n) = 4n + 1 and for the geometric sequence the student writes the rule g(n) =
or g(n) =
.
Questions Eliciting Thinking
How can you tell if a sequence is arithmetic or geometric?
What kind of function can be written for an arithmetic sequence? What kind of function can be written for a geometric sequence?
Instructional Implications
Challenge the student to write a function rule for a sequence represented by a quadratic function such as 2, 6, 12, 20, 30…. Consider implementing other MFAS exponential tasks Writing an Exponential Function From a Table (F-LE.1.2), Writing an Exponential Function From a Description (F-LE.1.2),
and Writing an Exponential Function From Its Graph (F-LE.1.2).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
What Is the Function Rule? worksheet
SOURCE AND ACCESS INFORMATION
page 3 of 4 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.912.F-LE.1.2:
Description
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description
of a relationship, or two input-output pairs (include reading these from a table). ★
page 4 of 4