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WWW.MANEUVERINGTHEMIDDLE.COM
LINEAR RELATIONSHIPS UNIT
Table of Contents
PAGE
4
TOPIC
RESOURCE
Sample Pacing Guide
5-6
Ideas for Implementation and Helpful Hints
7-15
Binder Covers, Dividers and Spine Labels
17-18
Slope and Rate of Change
Student Handout 1
19
Slope and Rate of Change
Homework 1
21-22
The Slope Formula
Student Handout 2
23-24
The Slope Formula
Homework 2
25-26
Slope-Intercept Form: Part I
Student Handout 3
27
Slope-Intercept Form: Part I
Homework 3
29-30
Slope-Intercept Form: Part II
Student Handout 4
31-32
Slope-Intercept Form: Part II
Homework 4
33-34
Quiz: Slope and Slope-Intercept Form
Quiz 1
35-36
Graphing Linear Equations
Student Handout 5
37-38
Graphing Linear Equations
Homework 5
39-40
Multiple Representations
Student Handout 6
41
Multiple Representations
Homework 6
43-44
Proportional and Non-Proportional Relationships
Student Handout 7
45
Proportional and Non-Proportional Relationships
Homework 7
47-49
Linear Relationships Study Guide
Review
51-54
Linear Relationships Unit Test
Test
©Maneuvering the Middle LLC, 2016
LINEAR RELATIONSHIPS UNIT
PAC ING GU I DE
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Slope and Rate of
Change
The Slope Formula
Slope-Intercept
Form: Part I
Slope-Intercept
Form: Part II
Slope and SlopeIntercept Form Quiz
Student Handout 1
Homework 1
Student Handout 2
Homework 2
Student Handout 3
Homework 3
Student Handout 4
Homework 4
Quiz 1
DAY 6
DAY 7
DAY 8
Graphing Linear
Equations
Multiple
Representations
Proportional and
Non-Proportional
Relationships
Student Handout 5
Homework 5
Student Handout 6
Homework 6
Student Handout 7
Homework 7
DAY 9
DAY 10
Linear Relationships
Unit Study Guide
Linear Relationships
Unit Test
Review
Unit Test
NOTES
©Maneuvering the Middle LLC, 2017
LINEAR RELATIONSHIPS
Student Handouts
**This file has been organized for double-sided printing. Blank pages are left intentionally.**
STANDARDS
8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare
two different proportional relationships represented in different ways.
8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a
non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the
equation y = mx + b for a line intercepting the vertical axis at b.
8.F.4 Construct 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.
Included in this unit you will find the following:
Unit Overview - a sample pacing calendar, ideas and tips for teaching/introducing the
concepts, unit vocabulary, big ideas, vertical alignment, and common misconceptions
Student Handouts - student-friendly notes and practice problems, homework/independent
practice, quizzes, unit review, and unit assessment
Student Handouts as Google Slides - a Google Slide version of the unit (excluding
assessments)
Answer Keys - an answer key for each page of the unit
Editable Unit Assessment - a PPT file of the unit test has been provided for you to make
modifications
Need to get in touch? Please direct all questions to contact@maneuveringthemiddle.com.
©Maneuvering the Middle LLC, 2016
HELPFUL HINTS
Organization and Teaching Tips
Maneuvering the Middle® has been publishing teaching strategies, classroom tools and
organization tips since 2013. Click the buttons below to access some of our favorite tips and
tricks for the classroom.
MATH CONCEPTS
Getting Started with
Algebra Tiles
How to Model
Integer Operations
Ideas for Teaching
Proportional Relationships
How to Teach Slope
Across Grade Levels
TEACHER ORGANIZATION
Utilizing Google Drive and
Google Classroom Effectively
Teacher
Organization Hacks
Must Have
Teacher Supplies
Our Favorite
Technology Gadgets
INSTRUCTIONAL IDEAS
Ideas for Implementing
Math Small Groups
Test Corrections as a
Tool for Mastery
Turn Any Worksheet
into an Activity
Out of the Box Ideas for
Using Task Cards
©Maneuvering the Middle LLC, 2017
EIGHTH GRADE CURRICULUM
LINEAR
RELAT IONSHIPS
UNIT FOUR: 8.EE.5, 8.EE.6, 8.F.4
©MANEUVERING THE MIDDLE, 2016
EIGHTH GRADE CURRICULUM
UNIT FOUR: 8.EE.5, 8.EE.6, 8.F.4
EIGHTH GRADE CURRICULUM
UNIT FOUR: 8.EE.5, 8.EE.6, 8.F.4
LINEAR RELATIONSHIPS
EIGHTH GRADE CURRICULUM
UNIT FOUR: 8.EE.5, 8.EE.6, 8.F.4
LINEAR
RELATIONSHIPS
LINEAR
RELATIONSHIPS
EIGHTH GRADE CURRICULUM
LINEAR
RELAT IONSHIPS
UNIT FOUR: ANSWER KEY
©MANEUVERING THE MIDDLE, 2016
EIGHTH GRADE CURRICULUM
LINEAR
RELAT IONSHIPS
UNIT FOUR: ACTIVITIES
©MANEUVERING THE MIDDLE, 2016
EIGHTH GRADE CURRICULUM
LINEAR
RELAT IONSHIPS
UNIT FOUR: ASSESSMENTS
©MANEUVERING THE MIDDLE, 2016
Name _____________________________________
Unit: Linear Relationships
Student Handout 1
Date _____________________________Pd______
SLOPE AND RATE OF CHANGE
The side view of two ramps at a local skate park are shown.
A
c. Which ramp appears steeper? Justify your answer
using a and b.
12 ft
B
8 ft
b. What measurement is different between the ramps?
8 ft
a. What measurement do the ramps have in common?
8 ft
• When a linear relationship is graphed, the slope is a value used to describe the
______________ of the line.
SLOPE
• Slope is the ratio of the _____________ change compared to the ______________
RISE
change, or RUN . Slope is equal to the ________ of ______________ of the graph
and the linear relationship.
There are four types of slope as described and shown in the table below.
TYPES OF SLOPE
A _____________ slope
increases from left to
right.
y
A _____________ slope
decreases from left
to right.
A _____________ slope
is a horizontal line.
An _____________
slope is a vertical line.
y
y
y
x
x
x
x
Follow the steps described below to find the slope of the graphed line.
• Choose two __________ on the graphed line.
• Draw a right triangle to count the _______ and
the _______ between the points.
RISE
• Set up a _______ of
and simplify.
RUN
• Double check if the graph is _______________ or
_________________.
RISE:
RUN:
SLOPE:
©Maneuvering the Middle LLC, 2016
FINDING SLOPE FROM A GRAPH
In 1-6, find the slope of each graphed line.
1.
2.
Slope: ___________________
Slope: ___________________
5.
Slope: ___________________
Slope: ___________________
6.
Slope: ___________________
1
7. Luis thinks the slope is 2. Explain his mistake
and give the correct slope.
Slope: ___________________
8. The graph shows the linear relationship
between number of bagels a bagel shop has
remaining, and the number of customers
served so far that day. Find the rate of change
of bagels remaining with respect to the number
of customers served.
# BAGELS REMAINING
4.
3.
# CUSTOMERS
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Homework 1
Date _____________________________Pd______
SLOPE AND RATE OF CHANGE
In 1-6, complete each blank with the letter of the correct vocabulary term from the right.
1. The slope of a vertical line will always be _______________.
A. Positive
2. A graph with a _______________ slope decreases from left to right.
B. Undefined
3. The vertical change on a graph is described as the _______________.
C. Run
4. The horizontal change on a graph is described as the ______________.
D. Zero
5. The slope of a horizontal line will always be _______________.
E. Negative
6. A graph with a _______________ slope increases from left to right.
F. Rise
In 7-12, find the slope of each line. Use the bank of answer choices below to check your work.
Not all choices will be used.
-25
UNDEFINED
7.
Slope: ___________________
1
2
3
2
2
8.
Slope: ___________________
10.
-
1
4
0
-3
3
4
9.
Slope: ___________________
11.
Slope: ___________________
12.
Slope: ___________________
Slope: ___________________
©Maneuvering the Middle LLC, 2016
2
3
Name _____________________________________
Unit: Linear Relationships
Student Handout 2
Date _____________________________Pd______
THE SLOPE FORMULA
To find the slope of the graph, Aiden chose the two points shown
and counted the rise to be 2 and the run to be 6. He used these
2
1
values to set up the ratio 6 and simplified the slope to 3.
• Using the values in the ordered pairs, how else could Aiden have
found the rise to be 2?
(9, 7)
(3, 5)
●
●
• Using the values in the ordered pairs, how else could Aiden have
found the run to be 6?
THE SLOPE FORMULA
If a linear relationship contains two
ordered pairs (x1, y1) and (x2, y2), the slope
can be found using the formula below:
Ex. Use the formula
to find the slope of
the graphed line.
y2 − y1
x2 − x1
1. A graphed line passes through each of the following pairs of points. Use the slope formula to
find the slope of each line. Show all work.
POINTS
FORMULA AND WORK
SLOPE
A
(2, 4) and (1, 7)
B
(-1, 3) and (5, -5)
C
(8, 11) and (10, 22)
D
(-1, 9) and (-1, 6)
©Maneuvering the Middle LLC, 2016
In 2-3, each table represents a linear relationship. Choose two ordered pairs from the table and
use the slope formula to find the rate of change.
2.
3.
x
0
1
2
3
x
3
8
11
14
y
5
5.5
6
6.5
y
15
15
15
15
Formula: ________________ Slope: ___________
Formula: ________________ Slope: ___________
4. A web designer charges customers an initial consultation fee
plus an hourly rate. The table shows the linear relationship
between x, the number of hours and y, the total cost of hiring the
designer.
a. Find the rate of change.
b. What does the rate of change represent in the context
of the situation?
HOURS
TOTAL
COST
2
$205
5
$400
8
$595
10
$725
5. The post office calculates shipping costs based on the weight of the item in addition to a fee.
The cost to ship a 2-pound item is $6.09, while the cost to ship a 7-pound item is $8.84. Find the
rate of change of the cost with respect to the weight of the item.
___________________
6. Jude and Yin each chose two points on the graphed line below and drew right triangles to find
the slope of the line. Use their triangles to answer a-d.
a. Use Jude’s points to set up and simplify the slope
formula below.
b. Use Yin’s points to set up and simplify the slope
formula below.
c. How are the two triangles related to one another?
●
JUDE
●
●
YIN
●
d. What can we assume about the slope between
any two points on the same line? Explain.
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Homework 2
Date _____________________________Pd______
THE SLOPE FORMULA
In 1-3, find the slope of the graphed line that contains the given ordered pairs. Show all work.
1.
2.
(8, 7) and (2, 12)
3.
(12, -10) and (15, -8)
slope:___________
(1, 4) and (0, 11)
slope:___________
4. Madeline needs to find the slope of the line
that passes through the points (9, 12) and
(7, 4). She sets up the following work:
12 − 4
7−9
slope:___________
5
5. A graphed line has a slope of . Which of
3
the following points could the line contain?
A. (15, 13) and (0, 4)
Has she set up her work correctly? Why or
why not?
B. (3, 9) and (6, 14)
C. (0, 4) and (19, 9)
D. (5, 7) and (10, 10)
In 6-8, each table represents a linear relationship. Use the slope formula to find the slope or rate
of change shown in each table.
6.
7.
x
2
6
10
14
x
-6
4
9
20
y
8
28
48
68
y
5
10
12.5
18
__________
8. An interior designer charges customers
an initial consultation fee plus an hourly rate.
The table shows the linear relationship
between x, the number of hours, and y, the
total cost of hiring the designer.
__________
HOURS
0
2
8
15
TOTAL COST ($)
125
235
565
950
a. Find the rate of change.
b. What does the rate of change represent in the context of the situation?
©Maneuvering the Middle LLC, 2016
Use the graphed line and triangles ABC
and DEF below to answer 9-10.
9. Find the slope of AC.
A
●
10. Which is a true statement about the slope of AC
compared to the slope of DF?
D●
F●
C●
___________
a. The slope of AC is greater than the slope of DF.
E
b. The slope of AC is less than the slope of DF.
B
c. The slopes are equal because
5 −(−4) 2 −(−1)
= 2−0 .
4 − (−2)
d. The slopes are equal because
4 − (−2)
2−0
=
.
5 −(−4) 2 −(−1)
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Student Handout 3
Date _____________________________Pd______
SLOPE-INTERCEPT FORM: PART I
a. Find the slope of the graph. Where do you see this value in
Xander’s equation?
b. What value does the graph touch on the y-axis? Where do you
see this value in Xander’s equation?
y = 6x + 2
TOTAL MILES
Xander has biked 2 miles so far this week and plans to bike an
average of 6 miles each day over the next several days. Xander
wrote the equation and created the graph to represent x, the number
of days and y, the total number of miles traveled on his bike.
Xander’s equation is written in slope-intercept form which is described below.
SLOPEINTERCEPT
FORM
• Slope-intercept form, or y = ____x + ____, is
one way to write the equation of a
_____________ relationship.
DAYS
y = mx + b
• The y-intercept of a graph is the value of y
where the line ____________ the y-axis, or when
x = _____.
In 1-3, use the given information to write an equation of the line in slope-intercept form.
1. slope = -9, y-intercept = 2
2. m = 4.5, b = -10
y = ____ x + _____
3. A line has a slope of -5 and
passes through the origin.
___________________
___________________
4. Complete the table below by recording the slope, the y-intercept and a sketch of each linear
equation’s graph.
y=x−5
y = 3x
y = -5x + 7
SLOPE (m)
Y-INT (b)
y
GRAPH
y
x
y
x
x
©Maneuvering the Middle LLC, 2016
For each graph below, record the slope, y-intercept, and equation in slope-intercept form.
5.
6.
7.
m: _____ b: ______
m: _____ b: ______
m: _____ b: ______
equation: _______________
equation: _______________
equation: _______________
8.
9.
10.
m: _____ b: ______
m: _____ b: ______
m: _____ b: ______
equation: _______________
equation: _______________
equation: _______________
4
11. Matt is going to create a graph of the equation y = 5 x − 7. Mark each statement as true or
false and correct any false statements.
_____ b. Matt’s graph will increase from left to right.
12. Circle the name of any student who wrote an equation that could possibly
represent the graphed line shown at the right.
JAVIER
KARISSA
LIAM
y = -3x - 2
y = 2x - 3
y = 2x + 3
Summarize today’s lesson:
y
x
©Maneuvering the Middle LLC, 2016
_____ a. Matt’s graph will cross the y-axis at (-7, 0).
Name _____________________________________
Unit: Linear Relationships
Homework 3
Date _____________________________Pd______
SLOPE-INTERCEPT FORM: PART I
Apply your knowledge of slope-intercept form to answer the questions below.
1. Harper is going to create a graph of the
equation y = -0.5x + 12. Which of the following
will be true about the graph?
2. Khari graphed the line below. Which
equation could represent Khari’s graph?
a.
b.
c.
d.
a.
b.
c.
d.
The
The
The
The
graph
graph
graph
graph
will contain the origin.
will increase from left to right.
will cross the x-axis at (12, 0).
will have a slope of -0.5.
y
y
y
y
y
= -2x – 3
= 3x + 4
= -4x + 3
= -2x – 5
x
For each graph below, record the slope, y-intercept, and equation in slope-intercept form.
4.
5.
m: _____ b: ______
m: _____ b: ______
m: _____ b: ______
equation: _______________
equation: _______________
equation: _______________
6. Li wrote the equation below to represent the
graph shown. Explain her errors and correct
the equation.
1
y= x-3
2
7. For a and b, write an equation in
slope-intercept form that meets the given
criteria.
a. A negative slope and passes through the
origin
b. Slopes upward from left to right and has a
y-intercept below the x-axis.
8. Mr. Brown asked his students to write an equation that represents a line with a positive slope
and a negative y-intercept. Circle the name of any student who correctly completed the task.
EZRA
AALIYAH
JACOBY
y = -5x + 2.5
y = 4x - 7
y = -3x - 11
PENNY
4
y = x - 20
5
©Maneuvering the Middle LLC, 2016
3.
Name _____________________________________
Unit: Linear Relationships
Student Handout 4
Date _____________________________Pd______
SLOPE-INTERCEPT FORM: PART II
Manny needs to write an equation in slope-intercept
form to represent the linear relationship between x and y in
the table shown at the right.
x
0
1
4
11
18
y
6
2
-10
-38
-66
a. Describe how Manny can find m, the slope.
b. Describe how Manny can find b, the y-intercept.
c. Write an equation to represent the relationship.
In 1-4, write an equation in slope-intercept form to represent each linear relationship.
1.
2.
x
-5
0
5
10
x
2
4
6
8
y
-0.25
6
12.25
18.5
y
50
90
130
170
m: ________ b: _______
m: ________ b: _______
equation: __________________
equation: __________________
3.
4.
x
3
6
9
12
x
0
1
2
3
y
-6
-12
-18
-24
y
1
1
15
1
2
5
1
3
5
m: ________ b: _______
m: ________ b: _______
equation: __________________
equation: __________________
5. Luke’s family goes to the movies and purchases a large popcorn.
They are debating whether to purchase any drinks. The table shows
the total cost based on the number of drinks they decide to purchase.
a. Find the slope and explain what it represents.
b. Find the y-intercept and explain what it represents.
c. Write an equation in slope-intercept form: __________________
DRINKS
TOTAL
COST
0
$6.25
1
$10.00
2
$13.75
3
$17.50
4
$21.25
©Maneuvering the Middle LLC, 2016
In 6-9, write an equation in slope-intercept form to represent the given situation.
m: _______ b: _______ equation: _______________
7. The graph shows the linear relationship between the number
of gallons of water remaining in a storage tank and the number
of minutes it has been draining.
MINUTES
TEMPERATURE (°F)
1
425
2
400
3
375
4
350
5
325
WATER (GALLONS)
6. At the end of the day, a pizzeria turns off its pizza oven.
The table shows the linear relationship between the
temperature of the oven and the first five minutes after
it was turned off.
TIME (MINUTES)
m: _______ b: _______ equation: _______________
8. Carly wants to buy some fish to keep in her
room. At a local pet store, customers can pay
$12.50 for a fish tank and $0.20 for each fish
they purchase. Write an equation to represent
the relationship between t, the total cost and n,
the number of fish purchased.
9. Danny is diving for rings at the bottom of the
pool and is 8.7 feet below the surface of the
water. He grabs a ring and ascends 1.3 feet
per second. Write an equation to represent the
relationship between s, the number of seconds
and f, Danny’s depth in feet relative to the
surface of the water.
m: _______ b: ______
m: _______ b: ______
equation: _________________
equation: _________________
TOTAL COST
36
60
a. The cost of each class is $8.
b. The monthly membership fee is $36.
c. A student who attended 30 classes would pay $220.
78
114
150
©Maneuvering the Middle LLC, 2016
10. A karate academy charges a monthly membership fee plus an additional fee per karate class.
The table shows the linear relationship between the number of karate classes taken and the total
cost including the membership fee. Find the error in each statement and rewrite them to make
them true.
# OF CLASSES
1
5
8
14
20
Name _____________________________________
Unit: Linear Relationships
Homework 4
Date _____________________________Pd______
SLOPE-INTERCEPT FORM: PART II
In 1-2, write an equation in slope-intercept form to represent each linear relationship.
1.
2.
x
0
5
10
15
x
3
6
9
12
y
-2
40.5
83
125.5
y
5
-1
-7
-13
m: ________ b: _______
m: ________ b: _______
equation: __________________
equation: __________________
Apply your knowledge of slope-intercept form to answer each of the following questions.
3. Mia has $50 on a gift card to her favorite
coffee shop. Each time she visits the coffee
shop she spends $3.75 on her favorite drink.
Write an equation to represent the relationship
between n, the number of times she visits the
coffee shop, and b, the total balance on her
gift card.
4. A magician charges a $30 fee to
cover travel and expenses, plus
$19.99 per hour. Write an equation to
represent the relationship between h, the
number of hours, and t, the total charge for the
magician.
________________________
________________________
Robert pays for his family to go to the arcade. He pays an entrance fee for his group and an
additional amount per game that his family plays as shown in the graph. Use the graph to answer
5-7.
6. Find the y-intercept and interpret its meaning.
TOTAL COST ($)
5. Find the slope and interpret its meaning.
7. Write an equation to represent the relationship
between x, the number of games and y, the total cost.
NUMBER OF GAMES
©Maneuvering the Middle LLC, 2016
8. A hiker hikes at a steady rate throughout the day on a mountain. Which student wrote a correct
equation to represent the linear relationship shown on the table between x, the number of hours
hiked and y, the current altitude of the climber?
# HOURS HIKED
1
2
3
5
8
ALTITUDE (FEET)
5,650
5,525
5,400
5,150
4,775
MATEO
JULIE
OLIVER
y = 125x + 5,775
y = -125x + 5,775
y = -125x + 5,650
The table shows the linear relationship between the number of pages left to read in a novel and
the number of hours a student has already spent reading the novel. Mark each statement as true
or false. If false, rewrite the statement correctly.
______ 9. The student reads at a rate of 48 pages per hour.
______10. The number of pages in the novel is 644.
______11. The situation can be represented by the
equation y = -48x + 692.
HOURS
READ
PAGES
REMAINING
1
644
4
500
8
308
12
116
14
20
©Maneuvering the Middle LLC, 2016
Unit: Linear Relationships
Quiz 1
Name _____________________________________
Date _____________________________Pd______
QUIZ: SLOPE AND SLOPE-INTERCEPT FORM
Answers
1. Kayla thinks that the slope of a vertical line is undefined, while Joshua
argues that the slope of a vertical line is zero. Who is correct?
1. ______________
2. ______________
2. Find the rate of change shown
in the table.
3. Find the slope of the graph.
3. ______________
4. ______________
x
y
1
-3
2
-9
3
-15
4
-21
5. ______________
6. ______________
7. ______________
8. ______________
9. ______________
4. Which of the following triangles could lie on the line graphed in
question #3?
A.
B.
C.
44
21
D.
30
36
12
10. _____________
20
52
26
5. A line has a slope of zero. Which of the
following points could this line pass through?
6. Which of the following is true about the
graph of the equation y = -5x + 10?
A. (12, 9) and (12, 6)
A. The graph would increase from left to right.
B. (3, -6) and (7, -6)
B. The graph would pass through the origin.
C. (1, 4) and (2, 5)
C. The graph would have a positive y-intercept.
D. (-9, 7) and (9, -7)
D. The graph would have a slope of 10.
©Maneuvering the Middle LLC, 2016
7. Write an equation in slope-intercept form to
represent the linear relationship on the graph.
8. Write an equation in slope-intercept form to
represent the linear relationship shown in the
table.
x
-2
0
2
4
6
8
y
-8.2
-5
-1.8
1.4
4.6
7.8
9. Which of the following could be the graph of the equation y = -2x?
A.
B.
C.
D.
MILES REMAINING
10. Misty is driving on a scenic road trip, and the graph shows the number of hours traveled
compared to the number of miles remaining in the trip. Write an equation of the graph in
slope-intercept form.
HOURS TRAVELED
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Student Handout 5
Date _____________________________Pd______
GRAPHING LINEAR EQUATIONS
Sara plots points A and B to graph a linear relationship.
●
A
a. Draw a right triangle and determine the slope.
●B
b. Is the slope positive or negative and how do you know?
c. Use the slope to plot another point on the line.
1
Ex. y = 2x + 1
GRAPHING A LINE
• Identify the ____________ and ____________.
• Plot the ____________ on the graph.
• Determine the ____________ of the line using the slope.
• Find the next point by counting _______ over _______.
1. Use each linear equation to determine the type of slope and direction of the line from left to right.
a. y = 3x + 17
b. y = -6x + 8
c. y = -5
d. x = 4
type of
slope: _______________
type of
slope: _______________
type of
slope: _______________
type of
slope: _______________
direction: ___________
direction: ___________
direction: ___________
direction: ___________
1
2. y = x − 2
3
3. y = -2x + 4
m: _____
m: _____
b: ______
b: ______
direction: ____________
direction: ____________
©Maneuvering the Middle LLC, 2023
For each equation, record the slope, y-intercept, and the direction of the line from left to right. Then
create a graph of the equation.
Graph each linear equation in the space provided.
4. y = 3x – 4
5. y = -x
3
6. y = 3
1
7. y = -2x + 1
9. x = -2
8. y = 4x + 2
Apply your knowledge of graphing linear equations to answer the question below.
3
10. Ms. Thompson asked her students to graph the equation y = 5x – 2. The work of three students is
shown below. Circle the name of the student that correctly graphed the linear equation, and identify
the mistakes made by the other two students.
SHONDRA
FRANKIE
DANTE
●
●
●
●
●
●
©Maneuvering the Middle LLC, 2023
Name _____________________________________
Unit: Linear Relationships
Homework 5
Date _____________________________Pd______
GRAPHING LINEAR EQUATIONS
On questions 1-9, graph each linear equation in the space provided.
1. y = 3x + 1
2. y = -5x – 4
1
4. y = -3x – 2
5. y = -5x – 5
6. y = 4x + 3
7. x = 1
8. y = -4
9. y = 4x - 5
3
1
10. A line passes through point A and has the equation y = -2x – 6.
Using the slope, plot 5 more points the line contains on the coordinate
grid. Then, list the five points below.
●
A
_________ , _________ , _________ , _________ , _________
©Maneuvering the Middle LLC, 2023
2
3. y = -2x
Use your understanding of graphing linear equations to answer the question below.
11. Mrs. Schultz asked her students how they would graph the linear equation below. First, graph
the line on the provided coordinate grid. Then, read the discussion from the students and circle
any statements that are true.
MAXWELL
y = 2x + 1
First, I would plot the
y-intercept at (1, 0).
ELEANOR
From the y-intercept, I would
count up two and to the right
one and plot a point at (1, 3).
DYLAN
From the y-intercept, I would
count down two and to the left
one and plot a point at (-1, -1).
©Maneuvering the Middle LLC, 2023
Name _____________________________________
Unit: Linear Relationships
Student Handout 6
Date _____________________________Pd______
MULTIPLE REPRESENTATIONS
Practice representing linear relationships in multiple ways with the following examples. Use the
representation given to help you fill in the others.
[VERBAL DESCRIPTION]
[EQUATION]
A baby giraffe measures 6 feet tall when it is
1
born and grows an average of foot each
2
month. What is the relationship between x, the
number of months and y, the height of the
giraffe?
[TABLE]
MONTHS
[GRAPH]
PROCESS
HEIGHT (FT)
0
1
2
3
4
5
6
Use the representations in the example above to answer 1-5.
1. Explain how you found
ordered pairs to create your
graph.
2. What is the slope of the
graph, and what does it
represent?
4. What does the ordered pair (9, 10.5)
represent in the context of the situation?
3. What is the y-intercept of
the graph, and what does it
represent?
5. If the giraffe is 12 feet tall, how many
months old is it?
©Maneuvering the Middle LLC, 2016
Use the given information for each situation below to fill in the missing representations.
[VERBAL DESCRIPTION]
[EQUATION]
y = 12 + 1.5x
[TABLE]
[GRAPH]
JOEY’S PIZZA PARLOR
TOPPINGS
PROCESS
COST ($)
0
COST ($)
1
2
3
4
5
6
NUMBER OF TOPPINGS
[VERBAL DESCRIPTION]
[EQUATION]
[TABLE]
PROCESS
GASOLINE
REMAINING
(GALLONS)
10
8.5
20
8
30
7.5
40
7
50
6.5
[GRAPH]
GASOLINE REMAINING
(GALLONS)
MILES
DRIVEN
SMITH FAMILY ROAD TRIP
MILES DRIVEN
Summarize today’s lesson:
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Homework 6
Date _____________________________Pd______
MULTIPLE REPRESENTATIONS
Andy’s Appliance Repair charges a set fee for house calls and an additional fee for
each hour of labor. Use the graph shown below to fill in the missing representations.
[VERBAL DESCRIPTION]
[EQUATION]
[TABLE]
[GRAPH]
ANDY’S APPLIANCE REPAIR
HOURS
PROCESS
COST ($)
TOTAL COST ($)
0
1
2
3
4
5
6
HOURS
1. What is the slope of the
graph, and what does it
represent?
2. What is the y-intercept of
the graph, and what does it
represent?
4. How much would it cost for a 9-hour repair?
3. What does the ordered pair
(7, 440) represent in the
context of the situation?
5. If the cost of a repair was $740, how many
hours did it take?
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Student Handout 7
Date _____________________________Pd______
PROPORTIONAL AND NON-PROPORTIONAL RELATIONSHIPS
NON-PROPORT IONAL
• Can be written as _________ where k is
the slope or rate of change.
• Can be written as _____________ where m
is the slope and b does not equal 0
• Ex: ____________
• Ex: ____________
• The ratio of _____ is constant
• The ratio of ______ is not constant
TABLE
PROPORT IONAL
• Ex:
GRAPH
EQUAT ION
Linear relationships can be proportional or non-proportional. A proportional relationship means
that there is a constant _________ between the values of x and y. Complete the table below to
review the differences in proportional and non-proportional representations.
• Any graph that is
both ___________ and
contains the ___________
x
2
4
6
8
y
6
12
18
24
• Ex:
x
2
4
6
8
y
8
14
20
26
• Any graph that is
not ___________ or does
not contain the ___________
Complete each representation for the situation described below. Then, determine if the situation
is proportional based on each representation.
1. Hillary is looking for a gym to join. A local gym, Forever Fit, is offering a special deal where
new members pay $30 per month with no sign-up fee.
A. EQUAT ION
B. TABLE
MONTHS (X)
0
1
2
3
COST (Y)
D. PROPORT IONAL?
COST
Explain based on each representation:
• equation:
• table:
MONTHS
• graph:
©Maneuvering the Middle LLC, 2016
C. GRAPH
2. Javier is ordering custom sunglasses for an upcoming event. The website he is ordering from
will charge $2.50 per pair of sunglasses and $5 for shipping.
A. EQUAT ION
B. TABLE
PAIRS (X)
0
1
2
3
COST (Y)
C. GRAPH
D. PROPORT IONAL?
COST
Explain based on each representation:
• equation:
• table:
• graph:
PAIRS
In 3-8, label the representation as “proportional” or “non-proportional.” Justify your choice.
3.
4.
8
y = 7x
6. Denzel has $13.50 and
saves an additional $7.50 each
week.
5.
x
-9
-8
-7
-6
y
13.5
12
10.5
9
7.
8.
x
8
10
12
14
y
18
20
22
24
b. Which company represents a proportional
relationship? Explain.
●
●
●
●
●
# OF GUESTS
●
x
y
0
0
2
27
4
54
6
81
8
108
©Maneuvering the Middle LLC, 2016
a. Which company has the greater rate
of change? Explain.
TOTAL COST ($)
9. Kate is catering food for a luau-themed party and the representations below compare the cost
of two catering companies. Let x represent the number of guests and y represent the total cost
of the caterer.
COMPANY A
COMPANY B
Name _____________________________________
Unit: Linear Relationships
Homework 7
Date _____________________________Pd______
PROPORTIONAL AND NON-PROPORTIONAL RELATIONSHIPS
In A-D, mark each statement as true or false. If false, rewrite the statement correctly.
A
Two students created the graphs shown
below.
RAUL
BECKY
B
The table
represents the amount
of coffee in a coffee
pot based on the
number of minutes the
coffee has been
brewing.
TIME
(MIN)
COFFEE
(OZ)
2
4.8
3
7.2
4
9.6
5
12
y
is not constant.
x
____ 1. Both graphs represent linear
relationships.
____ 4. The ratio of
____ 2. Both graphs have a positive slope.
____ 5. The table represents a proportional
relationship between x and y.
____ 3. Both graphs represent proportional
relationships between x and y.
____ 6. The table can be represented by
y = x + 2.4.
The graph
represents the
balance in
Jimena’s
checking account
based on the
number of days
since her last
paycheck.
D
BALANCE ($)
C
Two students wrote the equations
shown below.
ERICA
ALIYAH
y = -0.5x
y = 2.5x − 8
____ 7. The relationship shown on the graph
is non-proportional.
____ 10. Graphs of both equations will pass
through the origin.
____ 8. The graph represents a linear
relationship with a negative slope.
____ 11. Only Erica’s equation is
proportional.
____ 9. The graph can be represented by
y = 450x − 25.
____ 12. Both equations have a negative
slope.
©Maneuvering the Middle LLC, 2016
DAYS
Name _____________________________________
Unit: Linear Relationships
Review
Date ___________________________ Pd ______
LINEAR RELATIONSHIPS STUDY GUIDE
Solve each of the problems below. Be sure to ask questions if you need more help with a topic.
I CAN DETERMINE RATE OF CHANGE.
1. Find the rate of
change from the table.
x
y
-3
10.5
-2
7
-1
3.5
0
0
3
-10.5
2. Find the slope of
the graph.
_________
3. The graph represents the cost per person at
a pottery painting
studio. Find the
●
rate of change.
4. Find the slope of the line that passes
through the following pairs of points.
TOTAL COST ($)
_________
a. (5, 4) and (-4, 3)
●
________
●
b. (10, 8) and (9, 13)
●
______________
# OF PEOPLE
5. Find the slope of
the graph.
________
6. A line has a slope of 7. One of the points on
the line is (3, 5). Which of the following could
be another point on the line?
A. (-1, -24)
B. (-2, -30)
C. (6, 24)
D. (10, 55)
______________
7. Find the rate of change from the table.
______________
x
0
2
4
6
y
-10
20
50
80
©Maneuvering the Middle LLC, 2016
I CAN USE SIMILAR TRIANGLES TO UNDERSTAND SLOPE.
Use the graph to answer 8-9.
A
8. Igor believes the slope of AC is greater than the slope of
DF, while Keenan believes the two slopes are equal.
Who do you agree with?
D
F
E
C
9. Justify your choice above.
B
I CAN DETERMINE RATE OF CHANGE AND INITIAL VALUE FROM MULTIPLE REPRESENTATIONS.
10. Use the graph
to fill in each blank.
11. Use the graph
to fill in each blank.
m: _____ b: _____
m: _____ b: _____
Equation:
Equation:
___________________
___________________
Proportional?_______
Proportional? ______
12. Use the equation to fill in each blank.
13. Use the equation to fill in each blank.
y=
y = -3.5x – 10
2
x
7
m: _______ b: _______ Proportional? _______
14. Use the table to fill in each blank.
x
2
4
6
8
y
20
50
80
110
m: _______ b: _______Proportional? _______
15. Kayla works at a coffee shop and earned
$6.25 an hour plus $8.50 in tips yesterday.
Write an equation to represent the relationship
between x, the number of hours worked and y,
the total amount Kayla earned.
m: _____ b: ______ Equation: ________________
Proportional? ________________
__________________
©Maneuvering the Middle LLC, 2016
16. Elyse has a gift card to a local movie theater. The graph
shows the amount of money remaining on her gift card
based on the number of movies she has seen.
a. Write an equation to represent the situation.
b. Interpret the slope and y-intercept in the context of
the situation.
GIFT CARD BALANCE ($)
I CAN WRITE AND INTERPRET LINEAR EQUATIONS.
●
●
●
●
●
●
# MOVIES
17. Trish is ordering travel mugs from a website
that charges a certain amount per mug plus a flat
rate for shipping as shown in the table.
MUGS
0
3
6
9
COST
$5.99
$32.24
$58.49
$84.74
a. Write an equation to represent the situation.
b. Interpret the slope and y-intercept in the context of the situation.
I CAN GRAPH PROPORTIONAL RELATIONSHIPS.
18. Andrew works at Grub Burger and earns
$8.00 an hour. Create a graph of the
relationship between x, the number of hours
and y, the total amount Andrew earns. Then
write an equation to represent the relationship.
19. Use the relationship in #18 to answer a-b:
a. If Andrew earned $52, how many hours did
he work?
EARNINGS ($)
b. If Andrew works 9 hours, how much money
will he make?
20. Reece works at Fries n More, and the
relationship between x, the number of hours
and y, Reece’s total earnings can be
represented by y = 8.25x. Who earns more per
hour?
HOURS
Equation:
©Maneuvering the Middle LLC, 2016
Name _____________________________________
Unit: Linear Relationships
Test
Date ___________________________ Pd ______
LINEAR RELATIONSHIPS UNIT TEST
2. Find the unit rate in the graph below.
DISTANCE (MILES)
1. Find the slope of the line graphed below.
Slope: _________
A.
B.
C.
D.
9 miles per hour
4.5 miles per hour
3 miles per hour
1.5 miles per hour
TIME (HOURS)
3. Which situation could be represented by the graph shown?
A. Garrett buys limes for $0.80 each.
B. Sophia buys 12-packs of soda for $1.75 each.
C. Jacob buys packs of gum for $1.50 each.
D. Allison purchases lemons for $0.75 each.
4. Which is a true statement about the slopes of MO and OQ?
Q
●
O
●
B. The slope of OQ is greater than the slope of MO.
6 – 0 10 – 6
C. The slopes are equal because 3 – 6 = 6 – 8 .
M
●
N
6–3 8–6
D. The slopes are equal because 6 – 0 = 10 – 6.
5. A line crosses through the points (0, 2) and (-10, -16). What is the slope of the line?
A.
5
9
9
B. -5
9
C. 5
1
D. 3
P
©Maneuvering the Middle LLC, 2016
A. The slope of MO is greater than the slope of OQ.
Solve the problems below. Be sure to show your thinking.
6. A car repair company charges a $15 fee for
an evaluation plus an hourly rate for any
services required.
HOURS
0
2
4
6
CHARGE ($)
15
165
315
465
7. Find the slope of the line that contains the
following points:
What is the hourly charge for services?
a. (17, -12) and (17, 8) ______________
___________
b. (6, -2) and (-3, 1) ______________
8. Which of the following equations represents
a line with a positive slope and a negative
y-intercept?
9. Write the equation of the graphed line.
A. y = 3.5x
B. y = 7.5x – 2
1
C. y = 4x + 7
D. y = -5x – 8
__________________
10. Which of the following situations best matches the data in the table?
A. Robbie has $8 in his account and spends $1.50 each day for the next
3 days.
B. Zach sells t-shirts for $9.50 each.
C. A newborn weighs 8 pounds at birth and gains 1.5 pounds each month
for the next 3 months.
D. Riley earns $8 an hour lifeguarding, plus $1.50 for any pool memberships
she sells.
x
y
0
8
1
9.5
2
11
3
12.5
11. Two students found the slope of the line shown. Tavion used the
points (-1, -3) and (0, -1) while Jess used the points (3, 5) and (0, -1).
Which of the following is a true statement?
●
A. The triangles drawn between each pair of points are similar.
y –y
B. The ratio of x2 – x1 will be the same for Tavion and Jess.
2
●
1
C. Both students should find a slope of 2.
●
D. All the above are true.
©Maneuvering the Middle LLC, 2016
Solve the problems below. Be sure to show your thinking.
12. Write an equation to represent the relationship shown in the table.
-1
0
1
2
3
y
-3
-7
-11
-15
-19
13. The graph below shows the relationship between the
number of hours Cody works and the amount of money he
earns at his job. Which of the following statements is NOT
true about the relationship?
A. The graph can be represented by y = 7.5x.
B. If Cody has earned $120, he has worked 16 hours.
MONEY EARNED
______________________
x
C. If Cody works 20 hours, he will earn $160.
D. The situation is a proportional relationship.
14. Which equation represents the linear
relationship in the table below?
HOURS WORKED
15. JJ purchases almonds in bulk at the
supermarket. Which best describes the slope
of the line graphed?
0
5
10
15
20
y
10
16.5
23
29.5
36
COST ($)
x
A. y = 1.3x + 10
B. y = x + 10
C. y = 3.3x + 10
D. y = 1.3x
POUNDS
A.
B.
C.
D.
Almonds
Almonds
Almonds
Almonds
cost $9 per pound.
cost $2 for 9 pounds.
cost $5 per pound.
cost $4.50 per pound.
9
16. A line has a slope of 5. If one of the points on the line is (-10, -16), which of the following
could be another point on the line?
A.
B.
C.
D.
(-5, -6)
(0, 2)
(4, 11)
(10, 22)
©Maneuvering the Middle LLC, 2016
Solve the problems below. Be sure to show your thinking.
17. A trampoline park charges guests a fee for
socks and an hourly rate as shown in the table.
Which is a correct interpretation of the
relationship?
A.
B.
C.
D.
18. Which of the following is a true statement
7
about the equation y = -8x + 10?
HOURS
CHARGE ($)
0
4.50
2
16.50
A. The slope is 10.
4
28.50
6
40.50
7
B. The slope is -8.
The cost for socks is $4.50.
The hourly rate charged is $12.50.
Both A and B.
Neither A nor B.
C. The graph of the equation would pass
through the origin.
D. The equation represents a proportional
relationship.
19. Which of the following situations could be modeled by the graph below?
A. Elaina has $100 in her bank account. Every 3 days,
she saves another $10 and adds it to her account.
B. Hunter can bench press 100 pounds, and he plans to
increase the weight by 10 pounds every 3 weeks.
C. Dawn’s pond is 100 meters deep, but her city hasn’t
received much rain. As a result, the pond level decreases
by 10 meters every 3 weeks.
D. All of the above.
A. Amanda started with 30 homework questions.
B. Amanda finishes 2 homework questions every
3 minutes.
3
C. The graph shows the equation y = -2x + 30.
PROBLEMS REMAINING
20. The graph shows the number of homework problems Amanda has remaining based on the
number of minutes she has been working. Which of the following statements is NOT true?
D. The graph represents a non-proportional relationship.
TIME (MINUTES)
©Maneuvering the Middle LLC, 2016