8th Grade Mathematics
Unit 5: Linear Relationships
UNIT 5: Linear Relationships
Unit/ Topic Length:
 This unit focuses on the subject of linear relationships. In it, we pay special attention
to the proportionality of any two given variables, and examine the relationship
between them in multiple ways. Students will be able to represent linearity between
any number of variables through graphs, tables, equations, and verbal expressions.
Essential Question:
How can we tell when two variables have a linear relationship?
Big Ideas/Enduring
Understandings
 Every linear relationship depends
on at least one independent
variable and one dependent
variable.
 We can express linearity in various
ways, and each representation has
a use / purpose in our
investigations.
 We can use linearity to estimate
relationships between any
bivariate data.
Guiding Questions:
1. What does “linear” mean?
2. Where do we see variables in real life?
3. What tools can we use to measure
linearity?
4. How can we represent the pattern
between x and y in a given situation?
5. How do we show ordered pairs in a
table?
6. How do we create a graph given a set of
data?
7. What conclusions can we draw from the
collection and interpretation of our
data?
8. What is the difference between
proportionality and linearity?
9. How can we distinguish linear
relationships from quadratic and other
non-linear relationships?
NYS Standards and Indicators Assessed:
8.A.15—
Students will understand that numerical information can be represented in multiple ways:
arithmetically, algebraically and graphically
8.A.16 –
Students will find a set of ordered pairs to satisfy a given linear numerical pattern
(expressed algebraically); then plot the ordered pairs and draw the line
8.A.3 –
Describe a situation involving relationships that matches a given graph
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8th Grade Mathematics
Unit 5: Linear Relationships
8.A.4 –
Create a graph given a description or an expression for a situation involving a linear or
nonlinear relationship
8.G.13 Determine the slope of a line from a graph and explain the meaning of slope as a constant
rate of change
8.G.17 Graph a line from an equation in slope-intercept form (y=mx+b)
8.G.14 Determine the y-intercept of a line from a graph and be able to explain the y-intercept
8.G.15 Graph a line using a table of values
8.G.16 Determine the equation of a line given the slope and y-intercept
NYS Common Core Standards for Mathematics Assessed:
8.EE
Understand the connections between proportional relationships, lines, and linear
equations.
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.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution,
infinitely many solutions, or no solutions. Show which of these possibilities is
the case by successively transforming the given equation into simpler forms,
until an equivalent equation of the form x = a, a = a, or a = b results (where a
and b are different numbers).
b. Solve linear equations with rational number coefficients, including
equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs, because points
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8th Grade Mathematics
Unit 5: Linear Relationships
of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by
inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because
3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear
equations in two variables. For example, given coordinates for two pairs of
points, determine whether the line through the first pair of points intersects
the line through the second pair.
8.SP
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as clustering,
outliers, positive or negative association, linear association, and nonlinear
association.
2. Know that straight lines are widely used to model relationships between two
quantitative variables. For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by judging the closeness of the
data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For example, in a linear
model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is associated with an additional 1.5 cm in
mature plant height.
See alignment of standards and indicators to authentic task.
Content











Graphs
Ordered Pairs
Lines
Variables
Tables
Equations
Slope
Proportional Relationships
Constant
Coefficient
Y-Intercept
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Skills
Graphing / Plotting points
Creating a table
Recognizing patterns
Writing an equation
Translating equations into verbal
expressions
 Identifying / finding the y-intercept
 Using the quotient of the differences
to find slope





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8th Grade Mathematics
Unit 5: Linear Relationships
Vocabulary/ Key Terms (with definitions and Spanish translations)
Coefficient
Constant Term
Direct Variation
Directly Proportional
Slope
Slope-Intercept Form
Standard Form
Bivariate
ASSESSMENT EVIDENCE
Authentic Performance Task(s):
Alignment to NYS Common Core
Standards for Mathematics:
Students will use data about pH
balance in water to make claims about
the relationship between amounts of
chemicals in water.
Math Standards for Content
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8.SP
Investigate patterns of association in bivariate
data.
1. Construct and interpret scatter plots for
bivariate measurement data to investigate
patterns of association between two
quantities. Describe patterns such as
clustering, outliers, positive or negative
association, linear association, and
nonlinear association.
2. Know that straight lines are widely used
to model relationships between two
quantitative variables. For scatter plots
that suggest a linear association,
informally fit a straight line, and
informally assess the model fit by judging
the closeness of the data points to the line.
3. Use the equation of a linear model to
solve problems in the context of bivariate
measurement data, interpreting the slope
and intercept. For example, in a linear
model for a biology experiment, interpret
a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is
associated with an additional 1.5 cm in
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8th Grade Mathematics
Unit 5: Linear Relationships
mature plant height.
Math Standards for Practice
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Diagnostic and Pre/Post Assessments:
1. Students will respond to the essential question at the start of the unit and at the end of
the unit. (pre/post)
Formative Assessments:
1. Discussions
2. Daily reflections at the end of class
3. Practice quizzes
4. Student mini-showcase
Summative Assessments:
1. quizzes (graded)
2. interim assessments
3. unit test
4. integrated projects
5. portfolio assignments
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8th Grade Mathematics
Unit 5: Linear Relationships
TEACHING PLAN
Teaching and Learning Activities:
1.
2.
3.
4.
5.
Administer diagnostic for 8th grade math (post occurs at the end of the unit).
Use the essential question as a pre-assessment. (individual journal entry)
Discuss the root of the word “linear” and where they might find linearity.
Introduce unit vocabulary and integrate vocabulary into unit to learn the words.
Use unit guiding questions to do lessons on how to represent linearity in
multiple ways.
6. Have students work in groups to complete the authentic task for the unit.
7. Read informational text together.
8. Use essential question as a post-assessment. (individual journal entry)
9. 11. Have students self-select pieces for the portfolio, reflect on selections and set
goals for improvement.
10. Administer the unit test.
Resources Needed:
a.
b.
c.
d.
e.
f.
IMPACT Curriculum
GLENCOE Math
Science Map: Unit 2
Rulers
Graph Paper
Chart Paper
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8th Grade Mathematics
Unit 5: Linear Relationships
CALENDAR
Time
Spent on
Standard
Standards
constant
8.A.15 - Understand
that numerical
information can be
represented in multiple
ways: arithmetically,
algebraically and
graphically
 Establish overview of the
unit
4 days
8.A.16 –
Find a set of ordered
pairs to satisfy a given
linear numerical pattern
(expressed
algebraically); then plot
the ordered pairs and
draw the line
2 days
8.A.3 - Describe a
situation involving
relationships that
matches a given graph
 Review ordered pairs,
coordinates
 Collecting data in a table
with more than one
dependent variable
 Represent given data in a
graph
 Reintroduce the word
“linear”
 Establish usefulness of
seeing patterns in table and
graph form
 Emphasize connection
between graphs shown and
data collected
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Topics To Cover
Main Curriculum
1.1
E, p. 4;
Investigation 1: D&U: A, pp. 67; D&U:B; p. 8; D&U:C, p. 9;
S&S, p. 9
Investigation 2: D&U: A, pp.
10-11; D&U:B, p. 11; S&S, p. 12
Investigation 3: T&D, p. 12;
D&U: A, p. 13; D&U:B, p. 14
(odd numbers); D&U:C, p. 14
(odd numbers); D&U: D, p.15,
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8th Grade Mathematics
Unit 5: Linear Relationships
2 days
8.A.4 –
Create a graph given a
description or an
expression for a
situation involving a
linear or nonlinear
relationship
 Use estimation techniques to
have a general sense if graph
will be linear
 Test for accuracy by finding
coordinates and placing
them as values in table
3 days
8.G.13 - Determine the
slope of a line from a
graph and explain the
meaning of slope as a
constant rate of change
 Establish the idea of
proportions
 Make connection between
proportions from percent
unit and into here
 Emphasize slope as a
constant rate
3 days
2 days
2 days
8.G.17 - Graph a line
from an equation in
slope-intercept form
(y=mx+b)
8.G.14 - Determine the
y-intercept of a line
from a graph and be
able to explain the yintercept
8.G.15 - Graph a line
using a table of values
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 Explain the difference
between equation and
expression
 Ensure all students know
that equations with two
variables usually have
infinite amounts of
solutions, but only one
output for every input
 Show the difference between
x as 0 and y as 0.
 Arrive at the conclusion that
the intersections for the line
and the axes create the
“intercepts”

#19,20;
S&S, p. 15
1.2
T&D, p. 24;
Investigation 1:D&U:A, pp.2526; D&U:B, p.26; D&U:C, pp.
27-28; S&S, p. 28
Investigation 2: E, p. 29;
D&U:A, pp. 29-30; D&U:B, p.
30; S&S, p. 30
1.3
T&D, p. 35;
Investigation 1: T&D, p. 35;
D&U:A, p. 36; D&U:B, pp. 3637;
D&U:C, p. 37; Ex, p. 38; S&S, p.
38
Investigation 2: D&U:A, pp. 3840; D&U:B, p. 40; S&S, p. 41
Investigation 3: T&D, p,42;
D&U:A, pp. 42; S&S, p. 43
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8th Grade Mathematics
Unit 5: Linear Relationships
2 days
8.G.16 - Determine the
equation of a line given
the slope and yintercept
 Use all the skills you’ve
developed in students for
the last couple of weeks to
take any given bivariate
relationship and create a
graph from any two bits of
information
Investigation 4: E, p. 44; Ex, p.
45; D&U:A, p. 45; D&U:B, pp.
45-46; D&U:C, p. 46; D&U:D,
pp. 46-47; S&S, p. 47
Investigation 5: T&D, p. 48;
D&U:A, pp. 48-49; T&D, p. 49;
D&U:B, p. 49; D&U:C, p. 50;
D&U:D, p. 50; S&S, p. 50
Inquiry Investigation 6: Linear
Designs, pp. 51-52
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