Unit 5 Proportional and Non-Proportional Functions
Grade 8
5E Lesson Plan Math
Grade Level: 8
Subject Area: Math
Lesson Title: Unit 5: Proportional and Lesson Length: 17 days
Non-Proportional Functions
THE TEACHING PROCESS
Lesson Overview
This unit bundles…
During this unit, students extend their previous understandings of slope and yintercept to represent proportional and non-proportional linear situations with
tables, graphs, and equations. These representations are used as students
distinguish between proportional and non-proportional linear situations. Students
specifically examine the relationship between the unit rate and slope of a line that
represents a proportional linear situation. Problem situations involving direct
variation are included within this unit as they are also proportional linear situations.
Graphical representations of linear equations are examined closely as students
begin to develop the understandings of systems of equations. Students are
expected to identify the values of x and y that simultaneously satisfy two linear
equations in the form y = mx + b from the intersections of the graphed equations.
Students must also verify these values algebraically with the equations that
represent the two graphed linear equations. The study of proportional and nonproportional linear situations allows students to enrich their understanding of
financial situations by explaining how small amounts of money, without interest,
invested regularly grow over time. Students also examine how periodic savings
plans can be used to contribute to the cost of attending a two-year or four-year
college after estimating the financial costs associated with obtaining a college
education. Students are formally introduced to functions as a relation in which
each element of the input (x) is paired with exactly one element of the output (y).
Students must identify functions using sets of ordered pairs, tables, mappings, and
graphs. Examining proportional and non-proportional linear relationships is
extended to include identifying proportional and non-proportional linear functions in
mathematical and real-world problems. A deep understanding of the
characteristics of functions is essential to future mathematics coursework beyond
Grade 8.
Unit Objectives:
Students will…
 examine the relationship between the unit rate and slope of a line that
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
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represents a proportional linear situation.
identify the values of x and y that simultaneously satisfy two linear
equations in the form y = mx + b from the intersections of the graphed
equations.
verify these values algebraically with the equations that represent the two
graphed linear equations
enrich their understanding of financial situations by explaining how small
amounts of money, without interest, invested regularly grow over time.
examine how periodic savings plans can be used to contribute to the cost of
attending a two-year or four-year college after estimating the financial costs
associated with obtaining a college education.
identify functions using sets of ordered pairs, tables, mappings, and graphs.
Standards addressed:
TEKS:
8.1A- Apply mathematics to problems arising in everyday life, society, and the
workplace.
8.1B- Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution.
8.1C- Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation, and
number sense as appropriate, to solve problems.
8.1D- Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language as
appropriate.
8.1E- Create and use representations to organize, record, and communicate
mathematical ideas.
8.1F- Analyze mathematical relationships to connect and communicate
mathematical ideas.
8.1G- Display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication.
8.4B- Graph proportional relationships, interpreting the unit rate as the slope of the
line that models the relationship.
8.5A- Represent linear proportional situations with tables, graphs, and equations in
the form of y = kx.
8.5B- Represent linear non-proportional situations with tables, graphs, and
equations in the form of y = mx + b, where b ≠ 0.
8.5E- Solve problems involving direct variation.
8.5F- Distinguish between proportional and non-proportional situations using
tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.
8.5G- Identify functions using sets of ordered pairs, tables, mappings, and graphs.
8.5H- Identify examples of proportional and non-proportional functions that arise
from mathematical and real-world problems.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
8.9A- Identify and verify the values of x and y that simultaneously satisfy two linear
equations in the form y = mx + b from the intersections of the graphed equations.
8.12C- Explain how small amounts of money invested regularly, including money
saved for college and retirement, grow over time.
8.12G- Estimate the cost of a two-year and four-year college education, including
family contribution, and devise a periodic savings plan for accumulating the money
needed to contribute to the total cost of attendance for at least the first year of
college.
ELPS:
ELPS.c.1A use prior knowledge and experiences to understand meanings in
English
ELPS.c.2D monitor understanding of spoken language during classroom
instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm
understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio
tape, video, DVD, and CD ROM to build and reinforce concept and language
attainment
ELPS.c.2G understand the general meaning, main points, and important details of
spoken language ranging from situations in which topics, language, and contexts
are familiar to unfamiliar
ELPS.c.3D speak using grade-level content area vocabulary in context to
internalize new English words and build academic language proficiency
ELPS.c.3J respond orally to information presented in a wide variety of print,
electronic, audio, and visual media to build and reinforce concept and language
attainment
ELPS.c.4F use visual and contextual support and support from peers and
teachers to read grade-appropriate content area text, enhance and confirm
understanding, and develop vocabulary, grasp of language structures, and
background knowledge needed to comprehend increasingly challenging language
ELPS.c.4H read silently with increasing ease and comprehension for longer
periods
ELPS.c.4J demonstrate English comprehension and expand reading skills by
employing inferential skills such as predicting, making connections between ideas,
drawing inferences and conclusions from text and graphic sources, and finding
supporting text evidence commensurate with content area needs
ELPS.c.5B write using newly acquired basic vocabulary and content-based gradelevel vocabulary
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns,
and connecting words to combine phrases, clauses, and sentences in increasingly
accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to
fulfill content area writing needs as more English is acquired
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Misconceptions:
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Some students may not relate the constant rate of change or unit rate to m
in the equation y = mx + b.
Some students may not relate the constant of proportionality or unit rate as
k in the equation y = kx or m in the equation y = mx + b, when b = 0.
Some students may think that a constant rate of change always means the
situation is always proportional.
Some students may not associate slope represented as whole number as a
rational number that can be represented as .
Some students may think that a function can have multiple outputs (y) for
the same input (x).
Some students may think that a function cannot have multiple inputs (x) that
correspond to the same output (y).
Underdeveloped Concepts:
 Some students may think that the slope in a linear relationship is
, since the x-coordinate (horizontal) always comes before
the y-coordinate (vertical) in an ordered pair, instead of the correct
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representation that slope in a linear relationship is
.
Some students may think that the intercept coordinate is the zero term
instead of the non-zero term, since intercepts are associated with zeros. In
other words, students may think (0, 4) would be the x-intercept because the
0 is in the x coordinate.
Students may not graph lines correctly on the coordinate plane
Students may use (y,x) as the ordered pair instead of (x,y)
Vocabulary:
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401(k) – a set amount of money, or percentage of pay, that is set aside
from an employee’s pay check by their employer, before the employee’s
wages are taxed. The employer may or may not contribute as well to the
employee’s 401(k) fund depending on employer’s policy. The money is
taxed when it is withdrawn at retirement age. In addition, if withdrawn prior
to retirement age, an additional penalty tax is assessed.
403(b) – a set amount of money, or percentage of pay, that is set aside
from an employee’s pay check by their employer, before the employee’s
wages are taxed. The money is taxed when it is withdrawn at retirement
age. In addition, if withdrawn prior to retirement age, an additional penalty
tax is assessed.
529 account – educational savings account managed by the state, and is
usually tax-deferred
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
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Annuity – deductible and non-deductible contributions may be made, taxes
may be waived if used for higher education
Direct subsidized federal student loan – a loan issued by the U.S.
government in an amount determined by the college available to
undergraduate students who demonstrate a financial need where the U.S.
Government pays the interest on the loans while the student is enrolled at
least half-time, up to six months after leaving school, or during a requested
deferment period
Direct unsubsidized federal student loan – a loan issued by the U.S.
government in an amount determined by the college available to
undergraduate or graduate students where the interest is paid by the
borrower from the time the loan is initiated, even during requested
deferment or forbearance periods
Direct variation – a linear relationship between two variables, x
(independent) and y (dependent), that always has a constant unchanged
ratio, k, and can be represented by y = kx
Function – relation in which each element of the input (x) is paired with
exactly one element of the output (y)
Grant – money that is awarded to students usually based on need with no
obligation to repay this money
Individual retirement account (IRA) – a set amount of money, or
percentage of pay, that is invested by an individual with a bank, mutual
fund, or brokerage
Inflation – the general increase in prices and decrease in the purchasing
value of money
Linear relationship – a relationship with a constant rate of change
represented by a graph that forms a straight line
Principal – the original amount invested or borrowed
Private student loan – a loan issued by a lender other than the U.S.
Government
Retirement savings – optional savings plans or accounts to which the
employer can make direct deposits of an amount deducted from the
employee's pay at the request of the employee
Savings account – a bank or credit union account in which the money
deposited earns interest so there will be more money in the future than
originally deposited
Scholarship – money that is awarded to students based on educational
achievement with no obligation to repay this money
Slope – rate of change in y (vertical) compared to the rate of change in x
(horizontal),
or
or
, denoted as m in y = mx + b
Social Security – a percentage of an employee's pay required by law that
the employer withholds from the employee's pay for social security savings
which is deposited into the federal retirement system; payment toward that
employee's eventual retirement; the employer also is required to pay a
matching amount for the employee into the federal retirement system
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Student loan – borrowed money that must be paid back with interest
Taxable investment account – many companies will create an investment
portfolio with the specific purpose of saving and building a strong portfolio to
be used to pay for college
Traditional savings accounts – money put into a savings account much
like paying a monthly expense such as a light bill or phone bill
U.S. savings bond – money saved for a specific length of time and
guaranteed by the federal government
Unit rate – a ratio between two different units where one of the terms is 1
Work study – programs that allow students to work in exchange for a
portion of their tuition
y-intercept – y-coordinate of a point at which the relationship crosses the
y-axis meaning the x-coordinate is equal to zero, denoted as b in y = mx + b
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Related Vocabulary:
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Constant of proportionality
Interest
Intersection
Investment
Linear
Mapping
Non-proportional
Origin
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Ordered pair
Portfolio
Proportional
Rate of change
Retirement
Rise
Room and board
Run
List of Materials:
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Tables
Milk Foldable
Finding Slope From Two Points
Anchor Chart Input Output
Direct Variation
Anchor Chart y=mx+b
Graphing Lines in Slopes Intercept Form
Music Download Investigation
Music Download Data Collection
Proportional versus Non-proportional Exercises
Proportional and Non-proportional Comparison
Systems of Equations Graphing
Unit 5 Assessment
INSTRUCTIONAL SEQUENCE
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Tuition
x-axis
x-coor
x-valu
y-axis
y-coor
y-valu
Unit 5 Proportional and Non-Proportional Functions
Grade 8
Phase One: Engage the Learner
Day 1
Day 1 Activity #1:
Activity: Slope – Colin Dodds - Slope (Math Song)
https://www.youtube.com/watch?v=qE463XcV1Ro
(video is on you tube)
What’s the teacher doing?
What are the students doing?
Shows the video
Watching and listening to the video
What is slope?
Students should be able to demonstrate
what slope is. Rise over run or the
change in y over the change in x.
Asks students to demonstrate slope
Try to correct any misconceptions–
Some students may think that the slope
in a linear relationship is
, since the x-coordinate (horizontal)
always comes before the y-coordinate
(vertical) in an ordered pair, instead of
the correct representation that slope in
a linear relationship is
Phase One: Engage the Learner
.
Day 1
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Day 1 Activity #2
Slope Demonstration Activity
Students rise up from their seats and then walk/run across the room.
What’s the teacher doing?
What are the student’s doing?
Teacher says:
Students rise and then walk/run across
the run
Slope – students rise from your seat
and walk/run across the room.
(Rise up 2, run over 3)
(Rise up 1, run over 2)
Students should start to realize they are
demonstrating slope (Rise over run)
The teacher has the students do this a
few times.
Encourage the students to make the
rise over run connection to them rising
out of their seat and running to the wall;
hence, the students are doing slope
Phase Two: Explore the concept
Day 1-2
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Day 1-2 Activity #1:
Create Some Anchor Charts
Focus on Patterns: Point out to students they can use a pattern to complete the
tables (Example tables)
Example 1:
Distance
(leagues)
Distance
(miles)
Example 2:
1
Number of
hours
Amount
earned ($)
1
2
6
3
36
2
40
15
45
1
7
6
Example 3:
Time (weeks)
Time (days)
21
105
Example 4:
Weeks
Number Sold
1
32
2
6
608
Extend the activity – have the students write a verbal description, draw a graphical
picture, and write an algebraic equation for each example.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Below is a sample graphic organizer for the extension activity:
The following is a teacher checklist. (Make sure students are exposed to all
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
bullets (vocabulary, examples, representations, etc))
What’s the teacher doing?
What are the students doing?
Asks: How can you use tables, graphs,
and equations to represent proportional
situations?
Sample answer: If the ratio between
one quantity and another is constant,
you can use tables, graphs, and
equations of the form y=kx to represent
a proportional relationship between
quantities
Teacher assigns groups to create
different tables to be displayed.
The slope of a line that represents a
linear proportional relationship is
equivalent to the unit rate of the
situation.
 How is the process to determine
the slope of a line similar to the
process to determine the unit
rate of a problem situation?
 What is the relationship between
the slope of a line, m =
,
the constant of proportionality,
Students complete tables.
Student creates tables to be displayed
in classroom.
Students answer the questions with
assistance from the teacher.
Unit rate compares two quantities
Slope has a numerator and a
denominator
, and the unit rate of a
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
situation that represents a linear
proportional relationship?
Phase Three: Explain the Concept
and Define the Terms
Day 3
Day 3 Activity #1:
Warm Up Activity
Activity #2
Connect Vocabulary (Unit Rate, Proportional, Non-proportional, origin, rate of
change, intersection, linear, input, output) and Identify the parts of a table
Bicycle Activity with Technology
(Students need graphing boards or graph paper as well as a graphing calculator.)
Display the following problem:
Bicycles are produced at a constant rate of 12 per hour. Why can the relationship
be described by the equation y=12x?
Students should create a table and graph that would work with the problem.
Encourage students to use different numbers. Have the students enter their
numbers in a list in the calculator and graph it. Next, have the students type the
equation in the calculator. The students should explore the functions of the
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
calculator and compare graphs with their elbow partner.
Calculator information can be found on the following WebPages:
http://mathbits.com/MathBits/TISection/Openpage.htm or
http://education.ti.com/en/us/home
What’s the teacher doing?
What are the students doing?
Remind students that a proportional
relationship is a relationship between
two quantities in which the ratio of one
quantity to the other quantity is constant
What do you look for in order to decide
whether the relationship is proportional?
Linear proportional situations have a
constant slope and y-intercept that is
zero.
 What are the characteristics of a
linear proportional situation?
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How can the equation of a linear
proportional situation be
manipulated to prove that the
constant of proportionality exists
within the relationship?
Phase One: Engage the Learner
Sample answer might be: The amount
earned divided by the hours is the same
for each pair of values.
Students answer the questions with
assistance from the teacher.
Goes through origin, straight line, in the
form y=mx+b
K = y/x
Day 4
Day 4 Activity #1:
Activity – Recall previous knowledge of Slope, then play one or both of the
Slope videos to the class. (Students might enjoy watching both videos)
Video Option 1:Slope (Rise over Run) – Slope Music Video on YouTube:
https://www.youtube.com/watch?v=qnMaWTmdbKk
Video Option 2: Slope (Rise up Run Out) – Slope Video on YouTube:
https://www.youtube.com/watch?v=FmIhlc1bJuA
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
What’s the teacher doing?
What are the students doing?
Shows the video
Watching and listening to the video
Asks students to demonstrate slope
Students should be able to demonstrate
what slope is. Rise over run or the
change in y over the change in x.
Teacher continues to try to correct any
misconceptions...
Some students may think that the slope
in a linear relationship is
, since the x-coordinate (horizontal)
always comes before the y-coordinate
(vertical) in an ordered pair, instead of
the correct representation that slope in
a linear relationship is
Phase Two: Explore the concept
.
Day 4
Day 4 Activity:
Worksheet: Slope from Two Points
(Students will need their reference material.)
Foldable Activity: Create a foldable using the worksheet titled Milk Foldable.
Below is an example of the foldable:
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
What’s the teacher doing?
What are the students doing?
Linear proportional situations have a
constant slope and y-intercept that is
Students answer the questions with
guidance from their teacher.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
zero.
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What are the characteristics of a
linear proportional situation?
What does a linear proportional
situation look like in a table,
graph, and equation?
How can the equation of a linear
proportional situation be
manipulated to prove that the
constant of proportionality exists
within the relationship?
Phase One: Engage the Learner
Goes through origin
straight line
in the form y=mx+b
K = y/x
Day 5
Activity:
Direct Variation -MATH DUDE Unit3-2 Slope and Direct Variation
https://www.youtube.com/watch?v=SVsXLwGab4g
(video is on you tube)
What’s the teacher doing?
What are the student’s doing?
Restate to students the graph of a direct
variation always passes through the
origin and represents a proportional
relationship. You can find the equation
of a direct variation from a graph and
use it to make a prediction.
What is the formula for direct variation
and where can it be found?
Connect vocabulary: Students may be
puzzled by an apparent contradiction in
name constant of variation since
variation and constant seem to be
opposites. Explain that although there
is a variation, or change, from one value
to the next, the ratio of output values
does not change.
The formula for direct variation is on the
reference material. It is y=kx.
K is a nonzero constant called the
constant of variation. The value of k is
the same as the constant of
proportionality in the equation y/x=k. If
there is a direct variation between x and
y, y varies directly with x.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Phase Two: Explore the concept
Day 5
Activity: Direct Variation Activity
Have students build and measure heights of several stacks of identical objects
such as chips, blocks, or pennies. Have them record a count of how many items
are in a stack and its height in inches or centimeters as an ordered pair. The ratio
is constant, so the height varies directly with the count.
What’s the teacher doing?
What are the students doing?
Walking around and monitoring that the
students are really creating a table of
data
Students are seeing what happens as
they add objects to their stacks.
Phase Three: Explain the Concept
Day5
Activity:
Create a table and discuss what happens. Use worksheet Anchor Chart Input
Output as a guide. Each student should be given the anchor chart handout.
The following is a sample Anchor Chart:
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
What’s the teacher doing?
What are the students doing?
In this relationship what is the input
variable and what is the output
variable?
Students should be able to answer with
teacher guidance.
Input – x coordinate
Problems involving direct variation are
linear proportional situations.
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What is the relationship between
the constant of proportionality
and problems involving direct
variation or linear proportional
situations?
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How is the process of identifying
the slope or y-intercept from a
problem involving direct variation
similar to the process of
identifying the slope or yintercept from a problem
involving a linear proportional
situation?
Phase Four: Elaborate on the
Concept
Output – y coordinate
Day 6
Activity:
Worksheet Direct Variation
(Assign each group a couple of problems to work out and present to the class.)
Students will create anchor charts with their problems.
What’s the teacher doing?
What are the students doing?
Problems involving direct variation are
linear proportional situations.
Answering assigned problems
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What is the relationship between
the constant of proportionality
and problems involving direct
variation or linear proportional
Creating anchor charts from problems
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
situations?
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How is the process of identifying
the slope or y-intercept from a
problem involving direct variation
similar to the process of
identifying the slope or yintercept from a problem
involving a linear proportional
situation?
Listening for math conversations about
direct variation and encouraging
students to stay focused and work
together to solve the problems.
Phase One: Engage the Learner
Day 7
Activity:
Slope – Positive, Negative, Zero, Undefined – Slope Dude
https://www.youtube.com/watch?v=ZcSrJPiQvHQ
(video is on you tube)
What’s the teacher doing?
What are the students doing?
Shows the video
Watching and listening to the video
Asks students to demonstrate positive,
negative, zero, and undefined slope
Students should be able to demonstrate
positive, negative, zero, and undefined
slope.
Try to correct any misconceptions–
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Phase Two/Three: Explore /Explain
the Concept
Day 8
Activity:
Create anchor chart for y=mx+b.
(Each student gets a copy of the anchor chart for their binder. Worksheet: Anchor
Chart y=mx=b)
Students practice writing equations of a line in the correct form using white boards.
What’s the teacher doing?
Help students make the connection
between the slopes of the graphed lines
and the coefficient of x in the equations.
If the slope is rise/run and the slope is 2/3,
then the rise is 2 and the run is 3.
Have students write out the slope and yintercept for each equation, or underline
the slope and circle the y-intercept.
What are the students doing?
Students correctly distinguish the slope
and the y-intercept.
Students need to study the anchor chart
and be able to answer all equations
about lines.
Asks questions about the anchor chart
What does the m represent?
What does the b represent?
M represents slope
B represents the y-intercept.
Phase Four: Elaborate on the
Concept
Day 9
Activity:
Math Talk – How can you graph a line using the slope and y-intercept? You plot
the y-intercept on the y-axis. Then you use the slope to find another point and
draw a line through the two points. Remember rise over run.
Worksheet: Graphing Lines in Slope Intercept Form
(Students work alone and then compare answers.)
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Below is an example of the Anchor Chart:
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
What’s the teacher doing?
What are the students doing?
Teacher is walking around making sure
students are completing the worksheet
and answering any questions
Students are completing the
worksheet.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Phase One/Two: Engage/Explore the
Concept
Day 10
Activity:
Display Worksheet - Music Download Investigation
Students use experience and reasoning skills to discuss a proportional and nonproportional problem situation
Pass out Music Download Data Collection Worksheet
(Students could work in groups or alone to complete the assignment.)
Homework: Worksheet Proportional versus Non-proportional Exercises
What’s the teacher doing?
What are the students doing?
Asks:
What are the two quantities involved
with this problem?
(number of songs downloaded and cost
per month)
How does the payment work if you use
E–Tunes? Bliss?
(E-Tunes requires a fee per song
downloaded without a monthly fee:
$0.98. Bliss requires a monthly fee to
listen plus a fee to download each song:
monthly fee is $4.90 and fee per song is
$0.49.)
If you download 5 songs in one month,
approximately how much would it cost if
you use ETunes?
Bliss? Justify your response.
(ETunes: $0.98 per song, use $1.00 →
5 songs would be $5.00 for one month.
Bliss: $4.90 plus $0.49 per song. Use
$5.00 plus $0.50 per song → 5 songs
would be $5 + 5 x $0.50 = $5 + $2.50 =
$7.50.)
How are the charges from the two
(Both have a set fee per song.
Bliss has a set fee per song plus a
monthly fee. ETunes’ set fee is more
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
companies similar? Different?
per song than the Bliss set fee per
song.)
Do you think E-Tunes will always have
a better monthly cost than Bliss?
Answers may vary. Yes, because you
will always have to pay the monthly
charge of $4.90 per month for Bliss. No,
because ETunes charges more per
song to download than Bliss; etc.
Day 11
Phase Four: Elaborate
Activity:
Students complete worksheet: Proportional and Non-Proportional Comparison
What’s the teacher doing?
What are the students doing?
What are the characteristics of a
proportional relationship?
There is a constant of proportionality.
The equation can be written in the form:
y = kx. The data is linear shaped and
contains the origin (0, 0).
Does this situation have those
characteristics?
Answers may vary. Yes, there is a
constant of proportionality and the data
set does contain the origin; etc.
Which one of these characteristics
would be most helpful when working
with the graph?
The graph of a proportional relationship
goes through the origin and the data is
linear shaped.
Which of these characteristics would
be most helpful when working with
the table?
Which of these characteristics would
be most helpful when working with
the equations?
There is a constant of proportionality for
each pair of data in the table.
Equation can be put in the form y = kx.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
What is the relationship between the
two quantities in the problem?
Answers may vary. For every hour
worked, Sally earns $8; etc.
How can you use the table of values
to create a graph?
Answers may vary. You plot the number
of hours on the x-axis and the amount
earned on the y-axis; etc.
Phase Two/Three: Explore /Explain
the Concept
Day 12-13
Activity:
Investigate Systems of equations by completing worksheet titled Systems of
Equations Graphing
The following is an example of systems:
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
What’s the teacher doing?
What are the students doing?
Elicit from students that the solution is
the ordered pair represented by the
point where the two lines intersect.
Why is the point of intersection a
solution?
It is the only ordered pair whose x- and
y-values satisfy both equations.
What is the solution if two lines are
parallel?
There is no solution because they never
intersect.
Phase Two/Three: Explore /Explain
the Concept
Day 14
Activity:
Create an anchor chart using the worksheet titled: Anchor Chart Input Output
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
The following is a teacher checklist. (Make sure students are all forms.)
What’s the teacher doing?
How do you know which ovals shows
the input values?
What are the students doing?
The ovals on the left show the input
values because the arrows go from that
oval to the one on the right.
What do you look for in a mapping
diagram to determine whether the
relationship is a function?
Each value in the input is paired with
only one of the output values.
For the relationship to be a function,
does it matter how many times a
number I repeated in the output
column?
No
Avoid common errors: Some students
think that there is an output value that
corresponds to more than one input
value; the relationship is not a function.
Remind students that a function is a rule
that assigns exactly one output to each
input, but two or more input values can
give the same output value.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Phase Three: Explain the Concept
Day 15 - 16
Activity: Describing Functions
y=2x+8
y=x2
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
What’s the teacher doing?
What are the students doing?
How can you show that the relationship
between x and y given in the form of an
equation is linear relationship and also a
proportional relationship?
First make a table of values for the
equation and graph the ordered pairs; if
the rate of change in the table is
constant and the graph is a line through
(0, 0), then the relationship is a
proportional linear relationship.
Remind students that a graph of a
proportional relationship is a line that
goes through the origin.
Phase Two/Three: Explore /Explain
the Concept
Day 15 - 16
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
Activity: Estimating College Costs and Payments
Look up the cost of attending a four year university verses a two year college.
Students will need the internet.

Various considerations for each college
o School related costs
 Tuition (in state or out of state)
 Fees
 Room and board
 Books
 Cost of living in location (various costs of living depending on
the city and state of college)
 Inflation – the general increase in prices and decrease in the
purchasing value of money
 When planning ahead of time for college savings, the
increase in all expenses based on inflation must be
considered (e.g., tuition, room and board, etc.)
o Family contribution
The following is a teacher checklist. (Make sure students are exposed to all
bullets (vocabulary, representations, etc)):
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
What’s the teacher doing?
What are the students doing?
How much money do you think you will
need to attend college for four years?
Students guess
How will you pay for college?
Answers will vary
What are some benefits to living on
campus?
Room and board usually includes an
on-campus room to live in and a meal
plan, which are both a convenience and
a predictable cost
What other expenses might you have if
living at home?
Transportation, meals while on campus
Phase Five: Evaluate students’
Understanding of the Concept
Day 17
Activity:
Option #1 - Unit 5 Assessment
Option #2 - Divide the class up into groups. Assign half the class Performance
Assessment #1 and the other half of the class Performance Assessment #2.
Performance Assessment #1
Provide students with access to the Internet or the costs associated with attending a Texas
college for one year.
Analyze the problem situation(s) described below. Organize and record your work for each
of the following tasks. Using precise mathematical language, justify and explain each
solution process.
1) From the day she was born, Elle’s parents began saving money for her to attend a Texas
college when she graduated from high school. In preparation for that event, her parents
deposited $750 a year since she was born into a special savings account for her education.
a) Without the consideration of interest earned on the account, generate a table, graph, and
equation that can be used to represent this situation in terms of x, number of years, and y,
the balance of the savings account.
b) Using your generated graph, describe how the unit rate of the amount of money
deposited annually is related to the slope of the line.
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
c) Describe if the relationship represented in the table, graph, and equation represents a
proportional or non-proportional situation.
d) Determine how much money will be contained in the account once the annual amount is
deposited in the savings account when Elle turns 18.
e) Describe how the number of years and amount of money Elle’s parents regularly deposit
affects the total amount of money invested for her college education.
2) Using the Internet or another resource, research the costs associated with attending a
Texas two-year versus four-year college.
a) Determine if Elle will have enough money saved for the first year of college at either
type of college if her parents deposit $750 in a savings account each year for 18 years.
b) Devise a periodic savings plan that Elle and her family can use to ensure she will have
enough money to attend the first year of attendance at a Texas four-year college.
Performance Assessment #2
For this Performance Assessment, students will need to create their graphical
representations on the same coordinate plane.
Analyze the problem situation(s) described below. Organize and record your work for each
of the following tasks. Using precise mathematical language, justify and explain each
solution process.
1) From the day she was born, Elle’s parents began saving for her to attend a Texas college
when she graduated from high school. In preparation for that event, her parents deposited
$750 a year since she was born into a special savings account for her education. Her
grandparents also deposited $27,250 into the savings account on the day that Elle was
born.
a) Without the consideration of interest earned on the account, generate a table, graph, and
equation that can be used to represent this situation in terms of x, number of years, and y,
the balance of the savings account.
b) Describe if the relationship represented in the table, graph, and equation represents a
proportional or non-proportional situation.
c) Use a set of ordered pairs, table, mapping, or graph to determine if the relationship
represents a function.
2) From the day he was born, Brandon’s parents began saving for him to attend a Texas
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Unit 5 Proportional and Non-Proportional Functions
Grade 8
college when he graduated from high school. In preparation for that event his parents
deposited $2,250 a year since he was born into a special savings account for his education.
His grandparents deposited $1,750 into the savings account on the day that Brandon was
born
a) Without the consideration of interest earned on the account, generate a table, graph, and
equation that can be used to represent this situation in terms of x, number of years, and y,
the balance of the savings account.
b) Describe if the relationship represented in the table, graph, and equation represents a
proportional or non-proportional function.
3) At one point, the age of Elle and Brandon as well as the total amount of money
deposited in the savings account will be equivalent.
a) Use the graphs of your equations to represent Elle and Brandon’s savings account
balances to determine the point of intersection that simultaneously satisfies both linear
equations.
b) Describe the meaning of the point of intersection in the context of the situation.
c) Describe the relationship between the number of years money is deposited into the
savings account and the amount of money deposited into the savings account.
What’s the teacher doing?
What are the students doing?
Monitor students as they complete Unit
5 Assessment.
Students will complete Unit 5
Assessment.
Monitor students as they work on the
performance indicator to determine if
any re-teaching is necessary prior to the
unit assessments.
Display understanding of the topics and
skills taught in this unit by completing
the performance indicator.
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