8th Grade Mathematics
UNIT 2: PROPORTIONAL AND LINEAR RELATIONSHIPS
Proportional and Linear Relationships (6 weeks)
This unit focuses on the concepts of proportional and linear relationships. In it,
we pay special attention to the proportionality and linearity of any two given
variables, and examine the relationship between them in multiple ways. In
addition to deriving and interpreting y = mx and y = mx + b, students will
model linearity between any two quantities through graphs, tables, equations,
and verbal descriptions. They will explore similar triangles and use them to
explain why the slope is constant between any two points on a given line.
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8th Grade Mathematics
Key Ideas:
Guiding/Focus Questions:
 Every linear relationship depends on
 How do we create a graph given a set of
one independent variable and one
data?
dependent variable.
 What conclusions can we draw from the
 We can express proportionality and
collection and interpretation of our data?
linearity in various ways, and each
 What is the difference between
representation has a use / purpose in
proportionality and linearity?
our investigations.
 What is the pattern of change in a linear
 *When two quantities vary in such a
relationship?
way that one of them is a constant
 How does the pattern of change for a linear
multiple of the other, the two
relationship appear in a table, a graph, or
quantities are “proportional”.
an equation?
Conversely, two quantities are
 How can we distinguish linear relationships
proportional when they vary in such a
from quadratic and other non-linear
way that one of them is a constant
relationships?
multiple of the other, i.e., they have a
 How can you determine if a linear
constant ratio.
relationship is increasing or decreasing?
 *When two quantities, x and y, vary in
 How are solutions of an equation of the
such a way that one of them is a
form y = mx + b related to the graph and
constant multiple of the other, a
the table for the equation?
model for that situation is y = kx
 How is the steepness of a set of stairs
where k is the constant of
related to a straight-line graph?
proportionality or the constant ratio
 How can you find the y-intercept and the
of y to x.
slope of a line from data in a table, graph,
 *The slope of the graph of a
or equation?
proportional relationship is the
 What information do you need to write an
constant of proportionality or the
equation for a linear relationship? Is the
constant ratio of y to x
expression for the dependent variable
 *The equation y = mx for a line
always the same?
through the origin and the equation y
= mx + b for a line intercepting the
vertical axis at b can be derived by
using 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.
 Slope and intercept can be
interpreted within the context of a
real-world problem.
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8th Grade Mathematics
NYS Common Core Standards for Mathematics
Focus Standards:
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. For example,
compare a distance-time graph to a distance-time equation to determine which of two moving
objects has greater speed.
8.EE.B.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.
Foundational Standards(Prerequisites):
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use
rate language in the context of a ratio relationship.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations.
Analyze proportional relationships and use them to solve real-world and mathematical
problems.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
Focus Standards for Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them. Students analyze given constraints
to make conjectures about the form and meaning of a solution to a given situation in one-variable
and two-variable linear equations, as well as in simultaneous linear equations. Students are
systematically guided to understand the meaning of a linear equation in one variable, the natural
occurrence of linear equations in two variables with respect to proportional relationships, and the
natural emergence of a system of two linear equations when looking at related, continuous
proportional relationships.
MP.2 Reason abstractly and quantitatively. Students decontextualize and contextualize
throughout the module as they represent situations symbolically and make sense of solutions
within a context. Students use facts learned about rational numbers in previous grade levels to
solve linear equations and systems of linear equations.
MP.3 Construct viable arguments and critique the reasoning of others. Students use assumptions,
definitions, and previously established facts throughout the module as they solve linear equations.
Students make conjectures about the graph of a linear equation being a line, then proceed to
prove this claim. While solving linear equations, they learn that they must first assume that a
solution exists, then proceed to solve the equation using properties of equality based on the
assumption. Once a solution is found, students justify that it is in fact a solution to the given
equation, thereby verifying their initial assumption. This process is repeated for systems of linear
equations.
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8th Grade Mathematics
MP.4 Model with mathematics. Throughout the module, students represent real-world situations
symbolically. Students identify important quantities from a context and represent their relationship
in the form of an equation, table, and graph. Students analyze the various representations and
draw conclusions and/or make predictions. Once a solution or prediction has been made, students
reflect on whether the solution makes sense in the context presented. One example of this is when
students determine how many buses are needed for a field trip. Students must interpret their
fractional solution and make sense of it as it applies to the real world.
MP.7 Look for and make use of structure. Students use the structure of an equation to make sense
of the information in the equation. For example, students write equations that represent the
constant rate of motion for a person walking. In doing so, they interpret an equation like 𝑦=35𝑥 as
the total distance a person walks, 𝑦, in 𝑥 amount of time at a rate of 35. Students look for patterns
or structure in tables and show that a rate is constant.
Content
 Linear Relationships represented as
tables of values, graphs, equations, and
verbal descriptions/real-world situations
 Slope-Intercept Form of Linear Equations
 Connections among representations of a
relationship as an equation, graph, or
table of values, and/or verbal
description/real-world situations
 Slope as a rate of change
 Proportionality and direct variation
Skills
 Calculate unit rate
 Graph / Plot points
 Recognize linear and non-linear
patterns
 Graph linear equations using slopeintercept form
 Interpret graphs of linear equations
 Represent linear relationships using
tables, graphs, equations, and verbal
descriptions/real-world situations
 Compare linear relationships based on
slope/rate of change in tables, graph,
equations, and verbal descriptions/realworld situations
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8th Grade Mathematics
Vocabulary/ Key Terms (with definitions and Spanish translations)
 Ratio
 Unit rate
 Proportionality
 Rate of change
 Linear
 Equation
 Direct variation
 Intercept
 Similar
 Similar triangles
 Constant of Proportionality
 Proportional Relationship
 Linear Relationship
 Slope
 Term
 Variable
 Coefficient
 Constant Term
ASSESSMENT EVIDENCE
Initial Assessment: Unit Readiness Test
Formative Assessments:
Quizzes
Exit Slips
Checks for Understanding
Short and Extended-Response questions used throughout the unit.
Mathematical Reflections
Formative Assessments Tasks
 Performance Task 1: Professional Typists
Students compare two different proportional relationships involving professional
typists represented in different ways.

Performance Task #2: Three Friends’ Savings
Students apply their understanding of linear relationships and simultaneous
equations to solve real-world problems involving savings accounts of three friends.
Summative Assessment/s: Unit Test
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8th Grade Mathematics
TEACHING PLAN
Teaching and Learning Activities:
I.
Administer the Unit Readiness Test to assess students’ performance against the
Foundational Standards.
II.
Use the guiding questions to focus each lesson.
[Suggested instructional approach: Launch-Explore-Summarize]
Lesson 1: Finding and Using Rates
Lesson 2: A Critical Look at Proportional Relationships
o Using Tables
o Using Graphs
o Using Equations (y = mx)
Lesson 3: Walking Rates and Linear Relationships
Lesson 4: Recognizing Linear Relationships: Tables, Graphs, and Equations
Lesson 5: Comparing Relationships
Lesson 6: Climbing Stairs: Using Rise and Run
Lesson 7: Finding the Slope of a Non-Vertical Line
o Slope as rate of change
o Use of similar triangles to prove constancy of slope for a given line
Lesson 8: Exploring Patterns with Lines
o The Line Joining Two Distinct Points of the Graph y = mx + b has Slope m
o There is Only One Line Passing through a Given Point with a Given Slope
o The Graph of a Linear Equation in Two Variables in a Line
o Every Line is a Graph of a Linear Equation
III.
Assess students on the unit.
 Final Performance Task
 Unit Test
Resources Needed:
 Connected Math Project 3 (CMP3): Moving Straight Ahead, Thinking with Mathematical
Models
 NYS Common Core Math Module 4: Linear Equations (Topics B & C)
https://www.engageny.org/resource/grade-8-mathematics-module-4
 Impact Mathematics Course 3
 Integrated Algebra
 Math Handbook
 MathXL (Pearson’s online homework, tutorial, and assessment system)
 Glencoe New York State Review Series
 http://www.livebinders.com/play/play?id=953710#anchor
 http://www.algebrahelp.com/
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8th Grade Mathematics
CALENDAR
Time
Spent
3 weeks
3 weeks
Standards
8.EE.5
8.EE.6
Topics To Cover
 Ratios, Rates, and
Unit Rates
 Proportional
Relationships
 Linear Relationships
 Comparing
Relationships
 Rate of Change
 Slope
 Slope-Intercept
Form of Linear
Equations
 Patterns with Lines
Resources Needed









Connected Math Project 3 (CMP3): Moving
Straight Ahead, Thinking with Mathematical
Models
NYS Common Core Math Module 4: Linear
Equations (Topics B & C)
https://www.engageny.org/resource/gra
de-8-mathematics-module-4
Impact Mathematics Course 3
Integrated Algebra
Math Handbook
MathXL (Pearson’s online homework,
tutorial, and assessment system)
Glencoe New York State Review Series
http://www.livebinders.com/play/play?id=9
53710#anchor
http://www.algebrahelp.com/
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