Unit Design
for
Analyzing Functions
Developed by
Patrice Baxter
Cesar Chavez Academy Middle School
UBD Unit Design Worksheet / Saginaw Valley State University
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Understanding by Design
Unit Design Worksheet
Unit Title: Analyzing Functions
Subject/Course: Mathematics
Topic: How we can analyze, design, and
construct functions
Grade: 8
Staff Name: Patrice Baxter
Stage 1 - Desired Results
Established Goals:
Content Goals:
1. 8. F. 1 - Define, evaluate, and compare functions…Understand that a function is a rule that assigns to
each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and
the corresponding output.
Function notation is not required in Grade 8
2. 8. F. 2 - Define, evaluate, and compare functions….Compare properties of two functions each represented
in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a linear function represented by a table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.
3. 8. F. 3 - Define, evaluate, and compare functions…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 = s^2 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.
4. 8. F. 4 - A function to model a linear relationship between two quantities. Determine the rate of change and
initial value of the function from a description of a relationship or from two (x, y) values, including reading
these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of
the situation it models, and in terms of its graph or a table of values.
5. 8. F. 5 - Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g.,
where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative
features of a function that has been described verbally.
Literacy Goals (CCSS):
RST.8.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they
are used in a specific scientific or technical context relevant to grades 6-8 texts and topics.
SL.8.4 Present claims and findings, emphasizing salient points in a focused, coherent manner with relevant
evidence, sound valid reasoning, and well-chosen details
RST.8.7 Integrate quantitative or technical information expressed in words in a text with a version of that
information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).
RST.8.9 Compare and contrast the information gained from experiments, simulations, video, or multimedia
sources with that gained from reading a text on the same topic.
WHST.8.1 Write arguments focused on discipline-specific content.
̶ Introduce claim(s) about a topic or issue, acknowledge and distinguish the claim(s) from alternate or
opposing claims, and organize the reasons and evidence logically.
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̶ Provide a concluding statement or section that follows from and supports the argument presented.
SL.8.5 Integrate multimedia and visual displays into presentations to clarify information, strengthen claims
and evidence, and add interest.
Understandings:
Students will understand
1. a relationship between a function exists when each
input is assigned to exactly one unique output.
2. how to reason and compare functions from context,
a graph, or a table.
3. the properties of linear and non linear equations.
4. how to write a model for a linear function,
determine rate of change, and the initial value from
tables, graphs, equations or verbal descriptions.
Essential Questions:
1. What are the different properties and roles of
functions?
2. How can you determine the rate of change of
functions displayed in different ways?
3. What are similarities and differences of linear and
non linear functions?
4. How can you model linear functions from story
problems?
5. how to analyze, explain, and sketch graphs.
5. In what ways can you use a mathematical
expression, such as a graph or verbal description,
or table, and apply it to real-life situations?
Students will know
Students will be able to
1. inputs may be shared as long as each output
corresponds with only one input.
1. understand and graph a function.
2. how to display functions represented algebraically,
graphically, numerically in tables, or by verbal
descriptions.
2. rate of change.
3. the differences between linear and non linear
equations.
4. how to break down story problems and identify
different components of functions.
4. set up linear functions from story problems, tables
and graphs.
5. how to analyze a graph.
5. how to discuss characteristics of functional
relationships.
2. compare properties of functions represented
differently. (For example, given a linear function
represented by a table of values and a linear
function represented by an algebraic expression)
2. determine which function has a greater rate of
change.
3. interpret the equation y = mx + b as a linear
function, whose graph is a straight line.
3. provide 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).
4. determine the initial value and rate of change of a
function from a story problem, (x,y) values, tables
and graphs.
5. sketch a graph that exhibits the qualitative features
of a function that has been described verbally.
5. use mathematical expressions to describe real-life
situations.
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Unit Enduring Understanding:
Unit Question:
Students will be able to define, evaluate, and compare
functions and use functions to model relationships
between quantities.
What impact does changes in variables such as data,
minutes used, initial value, rate of change, and
representations have on functions?
Stage 2 - Assessment Evidence
Performance Tasks:
Goal: Your task is to show mastery of functions and the way they apply in the real-world from mathematics
class and gain parents approval to work.
Role: You are an 8th grade senior class of 2014. You and your three best friends are each working to make this
the best senior year from junior high ever! You are in charge of entertainment.
Audience: You are giving sales pitch in the form of a PowerPoint presentation to your parents. Your goal is to
convince them to allow you to get a job.
Situation: You and your friends want to go to the Eminem and Rihanna MONSTER concert at Comerica Park
to celebrate your senior year. Your parents agreed for you to get a job with strings attached. You are looking
for a job to purchase tickets for yourself and a group of 3 friends (4 tickets total). You need to find a job that
you are able to work after school and on the weekends and does not interfere with you completing school work
(stipulation from your parents). You need to look through the different job ads and pitch the best job to your
parents.
Product/Performance: Your parents want you to develop a presentation showing the different jobs in various
forms (text, graphs, tables, & equations). You then need to select the best job for your individual situation and
explain thoroughly why you should be able to work this job and why the other jobs do not work for you. The
presentation needs to be thorough, convincing, and accurate.
Standards: Your final decision will be judged using a rubric.
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Key Criteria: Rubric for GRASPS
The maximum points are 10 for each category, points will be awarded based off complete description, accurate
portrayal of mastery, and details for each category.
CATEGORY
Graphically
10-9
8-7
6-5
4-3
<2
Graph of all four jobs
are accurately
analyzed, shows the
initial value (how
much you get as a
signing bonus), and
the rate of change
(how time and rate
affect payment).
Most of the jobs are
accurately analyzed, a
shows the initial value
(how much you get as
a signing bonus), and
the rate of change
(how time and rate
affect payment).
Some of the jobs are
accurately analyzed,
shows the initial value
(how much you get as
a signing bonus), and
the rate of change
(how time and rate
affect payment).
A few of the jobs are
accurately analyzed,
shows the initial value
(how much you get as
a signing bonus), and
the rate of change
(how time and rate
affect payment).
Graphs are not
accurately analyzed,
do not show the initial
value (how much you
get as a signing
bonus), and the rate of
change (how time and
rate affect payment).
Most of the jobs are
correctly represented
using equations and
appropriate tables.
Some of the jobs are
correctly represented
using equations and
appropriate tables.
A few of the jobs are
correctly represented
using equations and
appropriate tables.
Jobs are not correctly
represented using
equations and
appropriate tables.
Algebraically All four jobs are
correctly represented
using equations and
appropriate tables.
Textually
All four jobs are
accurately represented
textually. The
different variables are
discussed and how
they will impact your
life.
Most of the jobs are
accurately represented
textually. The
different variables are
discussed and how
they will impact your
life.
Some of the jobs are
accurately represented
textually. The
different variables are
discussed and how
they will impact your
life.
A few of the jobs are
accurately represented
textually. The
different variables are
discussed and how
they will impact your
life.
Jobs are not
accurately represented
textually. The
different variables are
discussed and how
they will impact your
life.
Comparison
All four jobs are
compared and the
pro’s and con’s of
each job are
addressed.
Explanation is
detailed about which
job is ideal for your
situation.
Most of the jobs are
compared and the
pro’s and con’s of
jobs are addressed.
Explanation has
details about which
job is ideal for your
situation.
Some of the jobs are
compared and the
pro’s and con’s of
jobs are addressed.
Explanation has
details about which
job is ideal for your
situation.
Jobs are compared
and benefits are listed
but explanation is
vague and lacks
detail.
Jobs are not
accurately compared
and explanation is
confusing.
Presentation
Presentation includes
the use of all
appropriate
vocabulary, sound
valid reasoning, and
well-chosen details.
Presentation is clear,
accurate, and shows
understanding of
technology
(PowerPoint, video,
slideshow…)
Presentation includes
the use of most
vocabulary.
Reasoning is
comprehensive with
minor mistakes.
Presentation is mostly
clear, accurate, and
shows understanding
of technology
(PowerPoint, video,
slideshow…)
Presentation includes
the use of some
vocabulary, reasoning
has minor mistakes
and details are not
well chosen.
Presentation is
somewhat, accurate,
and shows some
understanding of
technology
(PowerPoint, video,
slideshow…)
Use of inappropriate
vocabulary, reasoning
has major mistakes
and details are not
well chosen.
Presentation is not
clear and does not
show understanding
of technology
(PowerPoint, video,
slideshow…)
Presentation does not
have any vocabulary,
reasoning and details
are not valid.
Technology is not
used.
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Other Evidence: Warm-ups, vocabulary activity, tiered assignment-(differentiation), notes, worksheets, class
work/homework, calculator TI-83
Before
Pretest- Students will be given an
assessment to understand their
knowledge on the unit before any
instruction is given.
Journal - Write an argument
Select a picture and write a
hypothesis of an argument stating if
the picture represents a function
Sorting -Sort representations of
pictures, tables, graphs, and
equations as either functions or
non-functions.
Matching- Match picture of
functions to the appropriate
equation.
What Do You Know About
Functions?- Discussion and visual
discovery of different components
of functions.
Visual Discovery/PresentationVisual discovery of a textual
situation represented as an
equation, table, and graph.
What Do You Know About- What
is an equation? What steps would
you take to solve an equation? How
do variables change equations?
What Do You Know About- How
does variables and the change in
variables effect an equation? What
are some real-world situations you
can think of and variables that
change the outcome?
During
Spiraled Warm Ups- Students
complete warm ups that review
problems learned from previous
units and review problems taught
in current unit
Quizzes- Complete quick multiple
choice and short answer quizzes
on Socrative app. Should be used
as a formative assessment with
quick results so teacher can
modify and adjust instruction
Matching- Match pictures of
functions graphically to the best
algebraic expression or written
situation.
After
Posttest- Students will be given an
assessment to show mastery of
contents taught during the unit.
GRASP Activity- Complete
PowerPoint Presentation following
rubric above
Journal - Modify your argument
from before section. After
inspecting your picture revise and
re-write your argument stating if
the picture represents a function
with details to defend your
statement.
Quick Writes- Given a table or
graph write a verbal situation to
accurately express the table.
Exit Cards- Answer review
questions based off lesson via
paper/pencil or Socrative app
Journal - Modify your argument
from before section. After
inspecting your picture revise and
re-write your argument stating if
the picture represents a function
with details to defend your
statement.
Class Work/Group ActivitiesStudents should complete class
work assignments and activities
Learn Zillion Videos- Watch
videos to reinforce lesson concepts
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Describe the assessment/s and state the prompt if
applicable.
xF xS
What type of scoring tools will be used for evaluation?
x Analytic rubric
□ Holistic rubric
x Criterion rubric
□ Checklist
□ Answer Key
□ Other
Student Self-Assessment and Reflection:
Students should complete reflection as directed for homework and warm-up. Students need to reflect on how
much they understand, what can be changed about lesson delivery and assignment. At the end of the unit
student will complete self assessment providing feedback on how they feel they did regarding, effort, time
spent on unit and understanding
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Stage 3 - Learning Plan
Differentiated Instruction:
Section I: Level C. Complete first assignments marked with *, Select either option 1 and option 2, or just
option 3 (50 pts.)
*Complete Pre-Assessment* mandatory (10 pts)
*Notes & Participation - be involved by taking notes and discussing the lessons.* (20 pts)
1. Write down 5 examples of functions and 5 examples of non-functions , explain why. (10 pts.)
2. Translate verbal situations to a table, graph, and algebraic expression. (10 pts.)
3. What do you know about- In mathematics, you have learned about equations. What is an equation? What
steps would you take to solve an equation? How does variables change equations? How does variables and
the change in variables effect an equation? What are some real-world situations you can think of and
variables that change the outcome? (20 pts)
Section II: Level B. Complete first assignments marked with *, Select either option 1 or option 2, and then
choose three of six options #3-6 (100 pts.)
*Complete daily spiraled warm ups* (10 pts) mandatory
*Complete quizzes and exit tickets on Socrative as assigned* (10 pts) mandatory
1. Complete class work assignments for each lesson as assigned by teacher (20 pts)
2. Create your own class work assignment with a key for each lesson, use the teacher’s assignment as an
example. Include vocabulary and steps involved to solve problems. (20 pts)
3. Create tables, graphs, expressions and verbal descriptions for each picture given. (20 pts)
4. Use results from teacher and write a linear equation on computing class average on test scores, can this be
represented as a function, why or why not? Explain in detail. (20 pts)
5. Journal - Write an argument - Modify your argument from before section. After inspecting your picture
revise and re-write your argument stating if the picture represents a function with details to defend your
statement. (20 pts)
6. Watch Learn Zillion Videos, take notes and summarize using math vocabulary and your own words. (20
pts)
7. Matching - Match pictures of functions graphically to the best algebraic expression or written situation. (20
pts)
8. Quick writes - Given a table or graph write a verbal situation to accurately express the table. (20 pts)
Section III: Level A. Complete first two assignments marked with *, then choose only one. (300 pts.)
*Complete Unit Post Test* (100 pts) mandatory
*Complete GRASP Assignment* (100 pts) mandatory
1. Compare cell phone plans from three different companies. Select the (1) plan from each that are most
similar and display graphically, algebraically, with a table and write a verbal description. (100 pts)
2. Create iBook lessons and notes explaining how to determine if an equation is a function and the different
ways you can express a function. Include background knowledge and challenge information. (100 pts)
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Grades: 100-92: A+
91-88: A
87-75: B
74-62: C
61-49: D
Good websites:
http://www.charleston.k12.il.us/cms/Teachers/math/Algebra/aunit1/L1-3.PDF
Learning Activities:
Where: Where are we going? After students take a pre-assessment test we journey through the concepts
students will learn throughout the unit. Students will also become aware of the final project and be given
the list of expectations and rubric to appreciate a deeper understanding of how to express functions in
different ways.
Why? Students will better understand how to use functions to describe real-life situations.
What is expected? Students will have a complete understanding of how to define, evaluate, and compare
functions and how they are used to model relationships between quantities
Hook: I will hook student interest by showing them different phone plans from various phone companies at the
beginning of the unit, and posing the question: “How can we use mathematics to help us make best
decisions in our day to day life?”
I will hold student interest through interactive activities, such as: artistic expression and creating iBooks
and PowerPoint presentations, student choice in layered curriculum, and alternative methods of the
student “showing what they know.”
Equip: Students will be equipped to do well on this unit by building on prior knowledge of linear equations,
making connections to previous knowledge and current content, and a variety of lessons and assignments,
class discussions, and their end of unit GRASP.
Revise/Rethink: Students will be asked to rethink and revise their work through determining the
characteristics of a function, mathematically reasoning before, during, and after the unit is completed, and
writing sound justifications to defend or refute scenarios.
Evaluate: Students will self-evaluate through quick checks, warm ups, and exit tickets throughout the unit.
Student should complete ongoing quizzes and other formative assessment before their cumulative PostAssessment on Common Core Standards. Student should complete pre-assessment and post-assessment
reflection activity and GRASP activity.
Tailor: Learning will be tailored by using the differentiated instruction and layered assignments above. Also,
by using modifications according to student ILP or BIP in addition to ESL and SIOP strategies.
Organize: The unit will be organized so that key concepts build upon one another, assignments tailored to the
learning goals, and an overall aligned unit. Students should write and briefly discuss the content objective
for the day and complete daily spiraled warm ups, leveled activities and exit tickets. See calendar below
for day-by-day schedule.
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Essential Vocabulary
Equation – Mathematical statement containing an equal sign, to show that two expressions are equal.
Functions – A function is a special relationship between valuables. Each of its input values gives back exactly
one output value. It is often written as “f(x)”, where x is the value you give it.
Initial Value- Where a function starts initially. The value of the function when the input is zero.
Input – The value you substitute in a function or equation, also known as the x value.
Linear function –A function that can be graphically represented in the Cartesian coordinate plane by a straight
line. A first degree polynomial of the form F(x) = mx+c, where m (slope) and c (y-intercept) are constants and x
is a real variable.
Linear Equation – An equation of the form Ax + By = C, where A ≠0 and B ≠ 0. The graph of a linear
equation is a straight line.
Non-Linear function – A function that cannot be graphically represented on the Cartesian coordinate plane by
a straight line.
Non-Linear Equation – An equation whose graph does not form a straight line (linear).
Output – The value you receive out of the function or equation, also known as the y value.
Rate of Change – The change in the value of a quantity divided by the elapsed time. For a function, this is the
change in the y-value divided by the change in the x-value for two distinct points on the graph. (This is the same
thing as the slope of the secant line that passes through the two points).
Slope – The ratio of the vertical change to the horizontal change between any two points on the line. (m =
rise/run)
Vertical Line Test – A test used to determine whether a relation is a function or not. A graph is said to be a
function if the vertical line drawn does not intersect the graph at more than one point.
x-value – The horizontal value in a pair of coordinates. How far along the point is. Always written in an
ordered pair of coordinates (x,y), such as (12,5), 12 is the x value.
y-intercept – The y-coordinate of a point where the graph crosses the y-axis.
y-value – The vertical value in a pair of coordinates. How far up or down the point is. Always written second
in an ordered pair of coordinates (x,y) such as (12,5), 5 is the y value.
Sequencing the Learning
UBD Unit Design Worksheet / Saginaw Valley State University
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Monday
Tuesday
Wednesday
Thursday
Friday
Sorting/Matching
Activity
Student will be
given vocabulary
terms to complete.
Spiraled Warm Up
Vocabulary
Spiraled Warm Up
Spiraled Warm Up
Notes + Level C
Activities
Notes + Level C
Activities
Exit Card
Exit Card
Lesson: Identify
functions from
equations, graphs,
and tables/ordered
pairs.
Lesson: Recognize
graphs as a
function passing
the vertical line
test.
Unit PreAssessment
Vocabulary- Visual
Discovery
Quick Write-Journal
Notes +
Level C Activities
Exit Card
Level C Activity
Exit Card
Lesson: Work with
function rules,
exactly one y-value
associated to any xvalue.
Monday
Tuesday
Wednesday
Thursday
Friday
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Review from last
week
Quick Review
Notes
Notes + Level B
Activities
Level B Activities
Formative
assessment-quiz
Exit Card
Notes + Level C
Activities
Exit Card
Exit Card
Exit Card
Lesson: Recognize
equations as
functions.
Exit Card
Lesson: Identify
functions as
equations, graphs,
tables/ordered pairs,
and verbal
scenarios.
Lesson: Compare
functions from
different
representations
Lesson: Compare
functions from
different
representations
Lesson: Write
verbal scenarios to
match given
functional
relationship
Sequencing the Learning
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Monday
Tuesday
Wednesday
Thursday
Friday
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Learn Zillion Video
+
Level B Activities
Journaling
+
Level B Activities
Level B Activities
Level B Activities
Quick Review
Exit Card
Exit Card
Formative
assessment-quiz
Exit Card
Exit Card
Lesson: Determine
if functions are
linear or non-linear
explain your
reasoning
Understand linear
functions have a
constant rate of
change between ant
two points.
Lesson: Write
verbal scenarios to
match given
functional
relationship
Exit Card
Lesson: Determine
the greater rate of
change
Lesson: Compare
linear functions
(rate of change,
slope and initial
value)
Monday
Tuesday
Wednesday
Thursday
Friday
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Learn Zillion Video
+
Level B Activities
Notes
+
Level B Activities
Level B Activities
Level B Activities
Level B Activities
Exit Card
Exit Card
Exit Card
Exit Card
Exit Card
Lesson: Determine
rate of change
(slope) initial value
(y-intercept) from
tables, graphs,
equations or verbal
descriptions.
Lesson: Determine
rate of change
(slope) initial value
(y-intercept) from
tables, graphs,
equations or verbal
descriptions.
Lesson: Write a
function (linear
equation)
Lesson: Write a
function (linear
equation)
Lesson: Write a
function (linear
equation)
Lesson: Define,
evaluate, and
compare functions
Sequencing the Learning
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Monday
Tuesday
Wednesday
Thursday
Friday
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Learn Zillion Video
+
Level B Activities
Notes
+
Level B Activities
Level B Activities
Level B Activities
Level B Activities
Exit Card
Exit Card
Exit Card
Exit Card
Exit Card
Lesson: Sketch a
graph to relate to
verbal description
Lesson: Sketch a
graph to relate
linear equation
Lesson: Provide a
verbal description
given a graph of
linear equation
Lesson: Use
functions to model
relationships
between two
activities
Lesson: Use
functions to model
relationships
between two
activities
Monday
Tuesday
Wednesday
Thursday
Friday
Spiraled Warm Up
Spiraled Warm Up
Spiraled Warm Up
Presentation Date
Unit Warm Up
Review
Review
Conferencing
Conferencing
Lesson: Describe
functions by
analyzing and
building graphs
Level A Activities
Level A Activities
Level A Activities
Exit Card
Exit Card
Exit Card
Lesson: Describe
functions by
analyzing and
building graphs
Lesson: Describe
functions by
analyzing and
building graphs
Lesson: Describe
functions by
analyzing and
building graphs
UBD Unit Design Worksheet / Saginaw Valley State University
Posttest
Lesson: Describe
functions by
analyzing and
building graphs
13