8th Grade Mathematics
UNIT 2: Linear Relationships and Simultaneous Equations
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 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.
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.
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8th Grade Mathematics
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
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.
Content
 Ratios, Rates, and Unit Rates
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Coordinate Plane
Coordinate and Ordered Pairs
Graphs
Proportional Relationships
Slope
Linear Relationships
Slope-Intercept Form of Linear
Equations
 Linear Equations in One Variable
 Systems of Linear Equations or
Simultaneous Equations
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Skills
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Determining the Unit rate
Graphing / Plotting points
Creating a table
Recognizing patterns
Writing an equation
Identifying / finding the y-intercept
Using the quotient of the differences
to find slope
 Solving Linear Equations in One
Variable
 Solving Simultaneous Equations
Graphically and Algebraically
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8th Grade Mathematics
Vocabulary/ Key Terms (with definitions and Spanish translations)
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Ratio
Unit rate
Constant of Proportionality
Proportional Relationship
Linear Relationship
Slope
Coefficient
Constant Term
Slope-Intercept Form
Standard Form
Simultaneous Equations/Systems of Linear Equations
ASSESSMENT EVIDENCE
Performance Tasks:
Alignment to NYS Common Core
Standards for Mathematics:
Performance Task #1: Professional
Typists
Students compare two different
proportional relationships
involving professional typists
represented in different ways.
Math Standards for Content
8.EE.5
8.EE.6
8.EE.7
8.EE.8
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.
Math Standards for Practice
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
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
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8th Grade Mathematics
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
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
CALENDAR
Time
Spent on
Standard
3 weeks
2 weeks
1 week
Standards
Topics To Cover
 Ratios, Rates, and Unit Rates
8.EE.5 - Graph
 Proportional Relationships
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 nonvertical 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.
 Slope
 Linear Relationships
 Slope-Intercept Form of
Linear Equations
8.EE.7 - Solve linear
equations in one variable.
 Solving Linear Equations in
One Variable
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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
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
7.1
T&D, p. 312;
Investigation 1:D&U:A, p. 314;
T & D, p. 314;
Example, p. 314;
D&U:B & C, p. 315
D&U:D, p. 316 ;
S&S, p. 316
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8th Grade Mathematics
2 weeks
8.EE.8 - Analyze and
solve pairs of
simultaneous linear
equations.
 Systems of Linear Equations
 Solving Simultaneous
Equations Graphically and
Algebraically
7.3
E, p.342, Investigation 1:
D&U:A, pp.343-344
Investigation 1: D&U:B, p.344345;
S&S, p.345
Investigation 2: E, p.346;
D&U:A, pp.346-347; D&U:B,
p.347-348;
S&S, p.348
Investigation 3: p.349; T&D,
p.350;
D&U:A, p.350; D&U:B, p.351;
S&S, p.351
Investigation 4: T&D, p.352;
Ex, p.353; T&D, p.353; D&U:A,
p.354
Investigation 4: D&U:B, p.355;
S&S, p.355
Inquiry Investigation 5: Use a
Spreadsheet, p.356-360
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