Download Math Grade 8 Analyzing and Solving Pairs of Simultaneous Linear

Analyzing and Solving Pairs of Simultaneous
Linear Equations
Mathematics Grade 8
In this unit, students manipulate and interpret equations in one variable, then progress to simultaneous equations. They look at the structure of
equations (SMP.7) to predict whether there is one solution, no solution, or infinite solutions. Students see the relationships between equations
and between algebraic and graphical representations. They extend prior knowledge of the properties of equality, representing quantities as
variables, and writing simple equations to construct and solve systems of two linear equations that model real-world and mathematical situations
(SMP.4). Students will be able to analyze the context of equations and problems in order to determine which method for solving simultaneous
equations is most efficient (SMP.3).
Before starting this unit, students need to complete the unit on Proportions, Lines, and Equations (8.EE.5, 8.EE.6)
These Model Curriculum Units are designed to exemplify the expectations outlined in the MA Curriculum Frameworks for English Language
Arts/Literacy and Mathematics incorporating the Common Core State Standards, as well as all other MA Curriculum Frameworks. These units
include lesson plans, Curriculum Embedded Performance Assessments, and resources. In using these units, it is important to consider the variability
of learners in your class and make adaptations as necessary.
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Table of Contents
Unit Plan ……………………………………………………………………………………………..………………………………………………….…...3
Lesson 1 …………………………………………………………………………………………………………………………..………….…………...….5
Lesson 2 ………………………………………………………………………………………………………………………………..…….……….….…26
Lesson 3 ………………………………………………………………………………………………………………………………………………….…46
Lesson 4 ………………………………………………………………………………………………….…………………………………………………70
Lesson 5 …………………………………………………………………………………………………………………………………………………….86
Lesson 6 …………………………………………………………………………………………………………………………………………………….98
Appendix: Optional Pre-Lesson……………………….……………………………………………………………………………………………107
Unit Resources …………………………………………………………………………………………………………………………..……………..128
CEPA …………………………………………………………………………………………………………………………………..……………………131
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Stage 1 Desired Results
ESTABLISHED GOALS
8.EE.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 the
given 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.EE.8
Analyze and solve linear equations and 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
Transfer
Students will be able to independently use their learning to…
Apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation
in order to make decisions, draw conclusions, and solve problems.
Meaning
UNDERSTANDINGS
ESSENTIAL QUESTIONS
Students will understand that…
1. How can equations be used to represent real-world and
U1 Linear equations in one variable are used
mathematical situations?
to model real-world situations. They may
result in one solution, infinitely many
2. Given a particular problem situation, how do we
solutions, or no solution.
determine which method of solving simultaneous equations
will be the most useful?
U2 Simultaneous equations are used to
model real-world situations. Solutions may
result in one common solution, an infinite
number of solutions or no common solution.
Acquisition
Students will know…
Students will be able to…
1. The properties of operations and the
1. Interpret a word problem, define the variable(s), and write
properties of equality can be used to
the equation(s).
write and solve equivalent equations with
rational number coefficients.
2. Use the properties of operations (associative, distributive,
commutative, and inverse) to transform equations.
2. Simultaneous linear equations can be
solved using multiple methods (tables,
3. Use the properties of equality to transform equations (see
graphing, algebraic methods).
2011 MA Curriculum Framework for Mathematics, p. 185)
3. The solution of simultaneous linear
equations solved algebraically
corresponds to the point(s) of intersection
of their graphs.
4. Solve problems involving linear equations in one variable.
4. Academic Vocabulary: model, coefficient,
6. Analyze when a linear equation in one variable has one
5. Analyze solutions to draw conclusions in the context of
the problem.
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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.
SMP.3
Construct viable arguments and critique the
reasoning of others.
SMP.4
Model with mathematics.
SMP.7
Look for and make use of structure.
elimination; graphical; linear; point of
intersection; simultaneous; slopeintercept form; substitution; system; one,
infinite, no solution; identity; infinite
solution (x=a); infinite solutions (a=a); or no solution (a =
b).
7. Produce examples of linear equations that represent one
solution, infinite solutions, and no solutions.
8. Solve problems involving simultaneous equations using
multiple methods (tabular, graphical, and algebraic).
9. Write systems of equations to represent mathematical
and real-world problems.
10. Estimate solutions to simple cases by inspection of
equations and/or graphs.
11. Prove that the solution of simultaneous equations solved
algebraically is the point(s) of intersection of the lines
graphed in a plane.
12. Justify the accuracy of solutions.
Evaluative Criteria
See CEPA Rubric.
Stage 2 - Evidence
Assessment Evidence
CURRICULUM EMBEDDED PERFOMANCE ASSESSMENT (PERFORMANCE TASKS)
PT
Powering Up Patriot School
Students act as technology consultants, using simultaneous equations to develop a recommendation for the
purchase of laptops and tablets for a school. They prepare a report (including a cost analysis and an itemized
budget) for the principal, superintendent, and members of the school committee.
OTHER EVIDENCE:
OE
Formative Assessments:
Lesson 1 – Writing Equations to Represent Situations
Lesson 3 – Ticket to Leave “Number of Solutions – One, None, or Infinitely Many”
Lesson 4 – Creating Systems of Equations
Lesson 5 – Substitution Cards
Stage 3 – Learning Plan
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Summary of Key Learning Events and Instruction
1.
2.
3.
4.
5.
6.
7.
Solving equations in one variable.
Categorizing linear equations as having one solution, no solutions, or infinite solutions.
Solving linear equations through graphing.
Solving simultaneous equations by substituting for one of the variables.
Solving simultaneous equations by eliminating one of the variables.
Connecting learning through a multi-step problem that requires knowledge of linear equations and systems of linear equations.
Curriculum Embedded Performance Assessment (CEPA).
Appendix: Optional Pre-lesson: Modeling real-world and mathematical situations with equations in one variable
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Lesson 1: Solving Equations in One Variable
Brief Overview of Lesson:
Students extend their understanding of writing algebraic equations by reasoning out verbal statements. (Example: “If you double a number and
add 8, the result is 20” can be written as 2n + 8 = 20). Attention to the nuances of language as well as to the relationships of unknowns and values
is essential. After reviewing the properties of operations and the properties of equality, students transform and solve equations, developing a
formal process for solving equations in one variable. Students check their solutions by calculator or by substituting the solution back into the
original equation. They connect equations to real-world situations in order to consider whether solutions make sense in the context of the story
problem. As you plan, consider the variability of learners in your class and make adaptations as necessary.
Prior Knowledge Required:
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.3 Apply the properties of operations to generate equivalent expressions.
6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problem; understand that a variable can represent
an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all
nonnegative rational numbers.
7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7. EE.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
7. EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
Estimated Time (minutes):
2 Days/ 50 minute lessons
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Resources for Lesson:
Classwork: Problem Set
Verbal Expressions
Equations Homework
Classwork: Solving Equations
Homework: Working Across the Equal Sign
Concrete models:
Algebra tiles (or template for algebra tile cut-outs)
Virtual manipulative algebra tiles are available at:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=216
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson 1: Solving Equations in One Variable
Time (minutes): Two 50-minute lessons
By the end of this lesson students will know and be able to:
 Write linear equations in one variable to model real-life and mathematical situations.
 Interpret a word problem, identify the variable, and write the equation.
 Use the properties of operations and the properties of equality to transform equations.
 Solve problems involving linear equations in one variable.
 Analyze solutions to draw conclusions in the context of the problem.
 Determine the accuracy of solutions.
Essential Question(s) addressed in this lesson:
 How can equations be used to represent real-world and mathematical situations?
Standard(s)/Unit Goal(s) to be addressed in this lesson:
8.EE.7 Solve linear equations in one variable.
8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
SMP.3 Construct viable arguments and critique the reasoning of others
SMP.4 Model with mathematics
SMP.7 Look for and make use of structure
Targeted Academic Language: model
Teacher Content and Pedagogy
In Grades 6 and 7, students have used the properties of operations and properties of equality to generate equivalent expressions and to solve linear equations in
one variable. They are able to add, subtract, factor, and expand linear expressions with rational coefficients. However, during the transition to the new
standards, some students may need additional instruction and practice in these concepts.
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In this lesson, students move to a mathematical translation of a more abstract verbal expression that does not have a story problem context.
They collect language/ terminology to parallel parts of an equation.
If students can write word problems that are modeled by given equations, they are demonstrating proficiency in moving between word problem contexts and
algebraic expressions and equations.
Prior knowledge needed for this lesson:
 Write, read, and evaluate expressions in which letters stand for numbers.
 Write expressions that record operations with numbers and with letters standing for numbers
 Use variables to represent quantities and write expressions when solving real-world or mathematical problem; understand that a variable can represent
an unknown number, or, depending on the purpose at hand, any number in a specified set.
 Solve real-world/ mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all
nonnegative rational numbers.
 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
 Construct simple equations and inequalities to solve problems by reasoning about quantities.
Anticipated Student Preconceptions/Misconceptions:
 Students may consider –n = 6 as a solution. They need to multiply or divide both sides of the equation by -1 to get n isolated.
 When using the distributive property, students are often confused about the signs of the multiplied addends.
Lesson Sequence:
Introduce the unit: You may recall that you have already studied how to read, write, and evaluate expressions and equations. You also used equations to solve
real world math problems. In order to assess students’ background knowledge, ask students: What is the difference between and expression and an equation?
Solicit student responses. Remind them they also studied Properties of Operations and ask the students what they remember about the properties.
Today we are beginning a new unit in which you will continue to analyze and solve linear equations. You will also learn how to solve pairs of linear equations.
Today’s lesson will focus on writing and solving linear equations to model real-life and mathematical situations (SMP.4 Model with Mathematics)

Opening: Entry ticket:
Write a word problem that can be modeled by this equation:
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x + (x – 4) = 40
Students may need to look back at previous problems. For students who find this very challenging, give a general context:
Someone has x of something, and someone else has 4 less of the same thing. Together, they have 40.
Share several problems. If possible, choose some problems that do not fit the equation. Ask the class to modify the word problem to reflect the equation.

Pose the following problem:
If a number is doubled, and 8 is added, the result is 20.
Students are used to reading from left to right, and in this example, they will be able to start on the left.
o
o
o

If a number is doubled
and 8 is added
the result is 20
(multiply 2 x n)
(2n + 8)
(2n + 8 = 20)
Sometimes a verbal expression needs to be interpreted in a different order:
The result is 27 when you subtract 12 from a number.
n – 12 = 27 or 27 = n – 12
Double a number which results in three plus five times the number.
2x = 3 + 5x
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
Create a wall chart where students can add language that indicates mathematical relationships or operations. Students should have a blank copy in their
folders or notebooks, and as the unit progresses, they can add to this page.
Sample Mathematical Language Chart
Multiplication
Division
twice as many …… x2
half as many …… ÷2
tripling a number …… x3
OO
OO OO OO
Addition
the sum of …..add
together
Subtraction
6 fewer than …… -6
OOOOOOOOO
OOOOOOOOO

Students practice translating verbal expressions into equations in one variable by completing a short problem set. Classwork: Problem Set

Students solve their equations. They then pick one or two problems to discuss with a partner, explaining their solution strategies. As students listen to
the solutions of their peers, they decide whether the solutions make sense and ask questions to better understand and evaluate the work (SMP.3
Construct viable arguments and critique the reasoning of others).
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The teacher may link some of these strategies to Properties of Operations.
Note: Students may memorize the names of these properties, but unless they have a strong conceptual understanding, they will not be able to apply them
when problem-solving. Although this is not the focus of this lesson, whenever possible, the properties should be referenced in the equation-solving process.
x + 3 = 17
Use the subtraction property of equality to subtract 3 from both sides of the equation.
x + 3 – 3 = 17 – 3
x = 14

Homework: Students translate six verbal expressions into equations; they then write three additional word problems to match given equations.
Verbal Expressions
Equations Homework
(***See Optional Enrichment)
DAY 2
Explicitly connect the lesson to the work from Day 1. Yesterday we worked on writing a word problem that could be modeled by a specific equation. You also
used the Properties of Operations to solve equations. Today you will work with writing equations to model problem solving situations.
 Pose the following problem:
In an election, Candidate A received half the votes of Candidate B. Candidate C received 1000 votes less than Candidate B. All three candidates
combined got 3,000 votes. How many votes did each candidate get?
Step 1: Students write the equation.
B + (B/2) + (B-1000) = 3000
Step 2: Students simplify.
B + B/2 + B – 1000 = 3000
2B + B/2 = 4000
2(2B + B/2) = 2(4000)
4B + B = 8000
5B = 8000
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B = 1600
Candidate A: 800 votes
Candidate B: 1600 votes
Candidate C: 600 votes
Step 3: Check – Does the solution work in the original equation?

Analyze the equation-solving process (SMP.3 Construct viable arguments and critique the reasoning of others).
o What steps did you take? Why?

Remind students that in order to solve a linear equation, the variable must be isolated on one side of the equation (it does not matter which side).

Like terms must be combined, and simplified if possible.

Students apply what they have learned about solving equations by completing a problem set of four problems (SMP.4 Model with mathematics).
Classwork: Solving Equations
They are required to:
o Write an equation to match a story problem.
o Use properties of equality and properties of operations to isolate the variable and solve the problem (SMP.7 Look for and make use of structure)
o Write down the steps that they took to solve the problem.

Teacher Note: Working across the equal sign:
Students have had ample opportunity to manipulate expressions, but they may not be used to working across the equal sign.
Take time to look at this problem:
12n = 144
Most students will easily be able to solve this by relying on a known math fact 12 x 12. However, they may not fully understand how the equation can
be manipulated to arrive at the solution. Students need to justify the solution as coming from the step before.
Ask students: What happened to the 12n? How does n = 12 relate to 12n = 144?
For many students, models (such as a balance scale) can help them to visualize concepts of equality.
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Some balance beam problems can be found here:
http://seeingmath.concord.org/case_studies/raw/9d501f6bc748cd26d5218510d234ddfc/lodocs/ma003/01b_activities/divingin/ma003_divingin.htm
http://nlvm.usu.edu/en/nav/frames_asid_324_g_4_t_2.html?open=instructions&from=category_g_4_t_2.html

Students generate a reference chart of solution strategies based on these and prior problems.
SITUATION
STRATEGY
Like terms on the same side of the equation
Combine like terms
Like terms on opposite sides of the equation
Collect all like terms on one side (doesn’t matter
which side)
Fractional coefficients
Multiply by reciprocal
Negative variable
Multiply/divide by -1
Distributive property
Multiply each addend in the sum
Like terms and constants on the same side of the
equation
Use the commutative and associative properties to
rearrange/combine like terms
Multi-step equation
Work backward to isolate the variable, but simplify
first

EXAMPLE
7n + 5n = 144
12n = 144
2n + 4 = 3n – 7
11 = n
2/5 n = 20
5 (2/5n) = 5(20)
-n = 10
n = -10
7(a + 6) = 112
7a + 42 = 112
6a – 8 – 3a + 7 = 11
(6a – 3a) + (-8 + 7) = 11
3a + -1 = 11
3a + 4 = -7a -4 + 6
3a + 4 = 7a +2
10a = -2
a = -1/5
Optional Enrichment Sheet: Working Across the Equal Sign: Students solve problems that demonstrate understanding of the properties of equality and
manipulation of equations.
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
Closing: On the chart, put a green sticker on the strategy that is easiest for you. Put a red sticker on the strategy that you find the most difficult.
Summarize the day’s lesson by asking students to reflect upon what was learned, and sharing either with the class or a partner.
Preview outcomes for the next lesson:
Students will be able to identify if a problem has no solution, one solution, or an infinite number of solutions.
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Lesson 1 Classwork: Problem Set: Verbal Expressions to Equations
Name: _________________________________ ____ Class: __________ Date:___________________
A. Write the following verbal expressions as equations:
1. The sum of a number n and 12 is 16.
2. The quotient of 15 divided by a number x is equal to 3.
3. The product of 5 and a number n is 25.
4. 10 times a number is greater than or equal to 50.
5. A distance of 55 meters that is 20 meters short than x meters.
6. A bill of $96 for n baseball caps, each costing d dollars.
7. A weight of 115 pounds that is 40 pounds heavier than p pounds.
8. Doubling d dollars is equal to $228.
B. Solve your equations:
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Lesson 1 Classwork: Problem Set: Verbal Expressions to Equations ANSWER KEY
A. Write the following verbal expressions as equations:
1. The sum of a number n and 12 is 16.
n + 12 = 16, n = 4
2. The quotient of 15 divided by a number x is equal to 3.
3. The product of 5 and a number n is 25.
15/x = 3, x = 5
5n = 25, n = 5
4. 10 times a number is greater than or equal to 50.
10x ≥ 50, x ≥ 5
5. A distance of 55 meters that is 20 meters short than x meters.
6. A bill of $96 for n baseball caps, each costing d dollars.
55 = x – 20, x = 75
nd = 96 or 96/n = d
7. A weight of 115 pounds that is 40 pounds heavier than p pounds. p + 40 = 115
8. Doubling d dollars is equal to $228.
B. Solve your equations.
2d = 228, d = $114
See above.
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Lesson 1 Homework: Verbal Expressions to Equations
Name: _________________________________ ____ Class: __________ Date:_________________
Write and solve equations for the following verbal expressions:
1. A number decreased by 75 is equal to -25. _______________________________________
2. A number minus one-half is equal to three-fourths. _________________________________
3. The sum of 32 and 88 and a number is equal to 156. ________________________________
4. A number increased by 22 is -93. _______________________________________________
5. Twice a number is less than 56. ________________________________________________
6. One and one-fourth times a number is 15. _______________________________________
Write a story to go with each equation. The first one is done for you.
1. 2(25) + n = 90
Sandy raked leaves for two days and made $25 each day. How much more does she need to make to earn $90?
2. 45 – 16 = y
_______________________________________________________________________
_______________________________________________________________________
3.
2
3
s + s = 50 _____________________________________________________________
_______________________________________________________________________
4. 18 + 67 + p = 104
_______________________________________________________________________
_______________________________________________________________________
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Lesson 1 Homework: Verbal Expressions to Equations ANSWER KEY
Write and solve equations for the following verbal expressions:
1.
A number decreased by 75 is equal to -25.
n – 75 = - 25
2.
A number minus one-half is equal to three-fourths.
3.
The sum of 32 and 88 and a number is equal to 156.
4.
A number increased by 22 is -93.
5.
Twice a number is less than 56.
6.
One and one-fourth times a number is 15.
n + 22 = - 93
2n < 56
(Solution: 50)
𝟏
𝟑
n-𝟐 =𝟒
(Solution:
𝟓
or
𝟒
𝟏
1𝟒 )
32 + 88 + n = 156 (Solution: 36)
(Solution: -115)
(Solution: n < 28)
𝟏
1𝟒 n = 15
or
𝟓
𝟒
n = 15
(Solution: 12)
Write a story to go with each equation. The first one is done for you.
1.
2(25) + n = 90
Sandy raked leaves for two days and made $25 each day. How much more does she need to make to earn $90?
2.
45 – 16 = y
___________________________________________________________________________
3.
2
3
s + s = 50
___________________________________________________________________________
4.
18 + 67 + p = 104
___________________________________________________________________________
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Lesson 1 Classwork Problem Set: Solving Equations
Name: _________________________________ ____ Class: __________ Date:___________________
1. We saved $50 to buy an anniversary gift for our parents. You saved y dollars. I saved $14.50.
How much money did you save?
Equation: _____________________________________________________________
Solution Steps
What I did in this step
2. A car gets 28 miles per gallon. If the car was driven for 250 miles, how many gallons of gas were
used?
Equation: _____________________________________________________________
Solution Steps
What I did in this step
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Lesson 1 Classwork Problem Set: Solving Equations
3. Mrs. O’Neill saved some money last summer. She spent half of it on some new clothes for her
children. She took the remainder, and spent one-third of that on a new tire for her car. She had
$200 left. What was the amount of money she saved last summer?
Equation: _____________________________________________________________
Solution Steps
What I did in this step
4. John owns three times as many video games as Mitch. Mitch owns twice as many video games as
Sam. If the three boys own 72 video games altogether, how many video games does John own?
Equation: _____________________________________________________________
Solution Steps
What I did in this step
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Lesson 1 Homework: Working Across the Equal Sign
Name: _________________________________ ____ Class: __________ Date:___________________
Solve each equation.
Put an X to show which Property (or Properties) of Equality you used to solve the equation.
Equation
1.
2.
3.
p
3
Addition
Property of
Equality
Subtraction
Property of
Equality
Multiplication
Property of
Equality
Division
Property of
Equality
=1
6=r–3
k
2
=3
4.
2q = 6
5.
m+2=6
6.
3 = -3u
7.
4=
8.
Solution
j
4
v
3
=4
9.
s + 5 = 17
10.
8 = v-2
11.
2g – 1 = 9
12.
8 = 2 + 2d
13.
4 = 8 – 2k
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14.
10 = 2h + 2
Equation
15.
10 = 8x + 2
16.
-3 = -m + 3
17.
-3 + 2h = 5
18.
w
3
Solution
Addition
Property of
Equality
Subtraction
Property of
Equality
Multiplication
Property of
Equality
Division
Property of
Equality
+ 8 = 10
d
19.
1= –1
20.
4r + 12 = 20
3
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Lesson 1 Homework: Working Across the Equal Sign ANSWER KEY
Solve each equation.
Put an X to show which Property (or Properties) of Equality you used to solve the equation.
Equation
1.
2.
p
3
=1
6=r–3
Solution
r=9
k=6
4.
2q = 6
q=3
5.
m+2=6
m=4
6.
3 = -3u
u = -1
7.
4=
8.
k
2
j
4
v
Subtraction
Property of
Equality
p=3
=3
3.
Addition
Property of
Equality
Multiplication
Property of
Equality
Division
Property of
Equality
X
X
X
X
X
X
v = 12
X
j = 16
X
3
=4
9.
s + 5 = 17
s = 12
X
10.
8 = v-2
v = 10
X
11.
2g – 1 = 9
g=5
X
12.
8 = 2 + 2d
d=3
X
X
13.
4 = 8 – 2k
k=2
X
X
X
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Lesson 2 Homework: Working Across the Equal Sign ANSWER KEY
14.
10 = 2h + 2
Equation
h=4
Solution
X
Addition
Property of
Equality
Subtraction
Property of
Equality
X
Multiplication
Property of
Equality
Division
Property of
Equality
15.
10 = 8x + 2
x=1
X
X
16.
-3 = -m + 3
m=6
X
X
17.
-3 + 2h = 5
h=4
18.
w
3
+ 8 = 10
w=6
19.
1= –1
d=6
20.
4r + 12 = 20
r=2
d
X
X
X
X
X
X
3
X
X
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Lesson 2: How Many Solutions?
Brief Overview of Lesson:
Students will categorize linear equations in one variable into one of the following three categories:
1. One solution: the equation is sometimes true; it is true for one specific value of x.
2. No solution: the equation can never be true; there are no values of x that will make the equation true.
3. Infinite solutions: the equation is always true; no matter what value is assigned to x, the equation will be true.
As you plan, consider the variability of learners in your class and make adaptations as necessary.
Prior Knowledge Required:
8.EE.7 Solve linear equations in one variable.
8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
Estimated Time (minutes):
Two 50 minute lessons
Resources for Lesson
Formative Assessment: Shopping Scenarios
Directions for Always, Sometimes, Never Activity
Solution for Categorizing Equations as Always True, Sometimes True, Never True
Ticket to Leave
Homework: Working with Linear Equations
Chart paper, sticky notes, internet access and projector with speakers
www.teachingchannel.org “Sorting and Classifying Equations Overview” video. (9:50)
www.teachingchannel.org “Sorting and Classifying Equations: Class Discussion” video. (8:24)
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson # 2: How Many Solutions?
Time (minutes): Two 50 minute lessons
By the end of this lesson students will know and be able to:



Identify the different forms of solutions and connect the form to the number of solutions (one solution: x=a; infinite solution: a=a; no solution: a = b)
Produce examples of linear equations that represent one solution, infinite solutions, and no solutions.
Estimate solutions to simple cases by inspection of equations (and/or graphs).
Essential Question(s) addressed in this lesson:
 How can equations be used to represent real-world and mathematical situations?
Targeted Academic Language: solution, identity, infinite
Standard(s)/Unit Goal(s) to be addressed in this lesson:
8.EE.7a 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 the given results (where a and b
are different numbers).
SMP.3 Construct viable arguments and critique the reasoning of others.
SMP.7 Look for and make use of structure.
Teacher Content and Pedagogy
After working in pairs or small groups to make decisions regarding assigned equations, students will be asked to critique the work of others (SMP.3 Construct
viable arguments and critique the reasoning of others).
Collaboration and opportunities to share work and explanations are an important part of this lesson.
Students should try to justify their decisions mathematically rather than “I tried some other numbers and they didn’t work”.
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Use this “Math Talk” protocol during problem-solving activities:
 Think about the problem – what strategies could you use?
 How do the problems and strategies compare to work you have done previously?
 Share your ideas with a partner.
 Work on your own to rethink the problem and arrive at an answer.
 Share your solution with a partner.
 Share with entire class.
Notes on One, Infinite, or No Solutions
Linear equations in one variable can have one solution, infinite solutions, or no solutions.
When the equation has one solution, the variable has one value that makes the equation true, and the equation is sometimes true.
Example: 10 – 3x = 4.
The only value for x that makes this equation true is 2.
When the equation has infinite solutions, the equation is true for all real numbers, and the equation is always true.
Example : 3x + 6 = 3 (x + 2).
When the distributive property is applied correctly, the expressions on each side of the equal sign are equivalent; therefore, the value for the two sides of the
equation will be the same regardless of which value is chosen for x.
Example: 2x + 3x + 9 = 5x + 9
When like terms are combined, the expressions on each side of the equal sign will be equivalent.
When an equation has no solution, there is no value that could possibly make the equation true. The equation is never true.
Example: 2x + 5 = 2 (x + 5)
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The two expressions are not equivalent. The equation simplifies to 5 = 10.Because all of these equations are linear, there will only be ONE solution when the
equation is sometimes true. It is important for students to explore this. As equations become more complex (exponents), equations that are sometimes true will
sometimes have more than one solution.
Key mathematical terms can be reinforced during this discussion:
solution
identity
infinite
Notes for Always, Sometimes, or Never True Activity
The use of different color markers on the poster (a different color for each member of the group) will allow the teacher to monitor who is completing the work.
As students are creating their posters in groups, the teacher observes and encourages student thinking via effective questioning:
 How do you know for sure?
 What are you thinking?
 Sam put the equation in this column. Nick, do you think it should go there? Why/why not? (SMP 3)
 Can you convince me that it would be true for EVERY number? (SMP 3)
 How could you change this equation, so it would belong in a different column?
 Can you create a new equation for each column?
 What do you notice about the equations that are in this column?
Anticipated Student Preconceptions/Misconceptions
 Errors with application of properties. Example: assuming that subtraction is commutative (5 – x = x - 5)
 Incorrect use of the equal sign. Example: student writes 5 – x = 6 = x = –1.
 Finding one solution and missing other solutions; not continuing to find other values or not finding a value that does not work.
 Belief that an equation is not true because the expression on each side of the equal signs looks different.
 Failure to fully simplify expressions.
Lesson Sequence
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DAY 1

Opening: Explicitly connect the lesson to the previous lesson. Remind students that equations can be solved in a variety of ways. Reference the chart that
the class made the day before.

Activator: Present students with the three shopping scenarios in the Shopping Scenarios handout.
Students work with a partner to interpret the equations provided to determine how many boxes of cookies each person bought.
Do a quick share-out following this introductory activity. What do students notice? (The Shopping Scenarios problems have one solution, no solution, or
infinite solutions. Students may be confused by equations that are not one-solution equations.) Students begin to look at the structure of equations to draw
conclusions about the number of solutions. (SMP.7)

Display the following equation: 3x + 2 = 11
Ask: Can you think of a value for x that makes this equation FALSE?

Display the equation again: 3x + 2 = 11
Ask: Can you think of a value for x that makes this equation TRUE?”
When a value is found, challenge students to try to find other values for x that will make the
equation true.
Ask: Would we describe this equation as always true, never true, or sometimes true?
When is it true? (x = 3)
Are there any other values for x that make it true? How do you know?

Repeat the same process with the following equations:
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x + 7 = 7 + x (always true for all values of x … )
5 – x = 2 x + 20 (sometimes true; x = -5)
x – 3 = x + 2 (never true)
Students are encouraged to look at the structure of the equations in order to make generalizations about the solutions (SMP.7).
Include the correct mathematical “label”:
o sometimes true = one solution
o never true = no solution
o always true = infinite solutions
Post each equation with its solution.
Encourage multiple students to contribute. Elicit a variety of answers, encouraging students to “think outside the box”. Discuss the reasons for their
choices.
Record exemplary responses. Post these, so students can reference them at a later date. IF necessary, have students support their answers with
calculations.

Always, Sometimes, or Never True small group activity:
o
Watch video: “Sorting and Classifying Equations Overview” video (9:50) at www.teachingchannel.org. (Enter the title in the search field.)
Explain to students that they will be completing a similar activity.
o
Divide students into groups of three.
o
Distribute the set of equations, chart paper, markers (a different color for each student in the group), and directions.
o
Review the three possible outcomes.
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ALWAYS TRUE or
INFINITE SOLUTIONS
NEVER TRUE or
NO SOLUTION
SOMETIMES TRUE or
ONE SOLUTION
means
The equation is true for any value of x.
means
There are no values of x that make the equation true.
means
There is at least one value of x that makes the equation true.
 To prove that an equation is SOMETIMES TRUE, you need two examples: one value for x that makes the equation true, and one that makes it false.
 Review directions: Directions for Always, Sometimes, Never True.
 Students spend the remainder of the class period completing the poster.
 Lesson Closing- have students reflect upon the lesson and share out either with a partner or in a whole class discussion.
DAY 2
Introduce the lesson and explicitly connect it to the previous lesson.
 Share posters … Conduct a gallery walk, so students can critique each others’ work.

Display charts in different areas of the room (hang on walls or lay on tables/desks).

Before beginning, distribute sticky notes to each student.

Provide the following directions:
If you disagree with where an equation has been placed, write three things on a sticky note:
1) Why you disagree.
2) Which column the equation belongs in.
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3) Why you think the equation belongs there. Place the sticky note on the chart.
This task requires students to critique the reasoning of others (SMP.3).

If you see a comment that has already been written and you agree, put a check mark on the sticky note.

If you see an explanation that you particularly like or something you didn’t think of, leave a note telling the group what you liked about it. (SMP.3)

Students walk silently around the room and review the work of others. Set a limit regarding the number of students looking at each poster. Or,
students can be divided equally among the charts and given a time limit before moving on to the next poster. It is not necessary to have all students
critique all posters; two are sufficient.
 Closing/Summarizer
o
o
o
o
o
o
o
o
Conduct a whole class discussion.
Highlight exemplary explanations.
Give me an equation that is… (ask for one of each type)
Why did you put that equation in that column?
Can anyone add to this explanation?
What did you learn by looking at other students’ work?
Which equations were the most difficult to categorize? Why?
Is there an easy way to tell which column an equation belongs in just by looking at it?
 Watch “Sorting and Classifying Equations: Class Discussion” www.teachingchannel.org (8:24)
 Ticket to Leave: Students use their understanding of the structure of equations to classify equations as always, sometimes, or never true, and create an
example of each type of equation on their own. (SMP.7)
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 Homework: Working with Linear Equations
Formative Assessment:
Three Shopping Scenarios check student understanding of the meaning of the terms and variables in linear equations.
Preview outcomes for the next lesson:
Students will be able to solve simultaneous linear equations through the use of graphing.
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Lesson 2 Shopping Scenarios Formative Assessment: What Does the Equation Tell Me?
Name: _________________________________ ____ Class: __________ Date:___________________
Read each shopping scenario.
Review the equations that have been written to represent them.
What can you say about the number of packages of cookies each person bought?
Description
A.
Noelle went shopping and
spent $19. She bought one
container of ice cream and
some packages of cookies.
Equation
What can you say about the number of
packages of cookies each person bought?
5c + 4 = 19
Last week, on Monday, Anna
bought 3 packages of cookies
and spent $5 more on other
items
On Tuesday, she bought the
exact same thing.
B.
This week, Anna bought 6
packages of cookies. She
also spent $10 more on
other items.
2(3c+ 5) = 6c + 10
She said she spent the same
amount of money last week
as this week.
C.
Last week, Jared bought 8
packages of cookies and
spent $8 more on some
cheese spread.
This week, on Monday,
Tuesday, Wednesday, and
Thursday, Jared bought two
packages of cookies each
day.
8c + 8 = 4(2c)
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He said he spent the same
amount of money last week
as this week.
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Name: _________________________________ ____ Class: __________ Date:___________________
Lesson 2: Directions for Always, Sometimes, or Never True Group Activity
1. Each person in your group selects a marker of a different color. This is the only
color that you may write with.
2. Divide your chart paper into three equal columns.
3. Label the columns:
Always True – Infinite Solutions
Sometimes True – One Solution
Never True – No Solution
4. Select an equation. Try out different values for x. Discuss with your group.
Record the equation in the column where it belongs.
5. In writing, next to each equation, explain why you chose to place the equation
where you did.
 If you think the equation is sometimes true, give values of x for which it is
true and for which it is false.
 If you think the equation is always true or never true, explain how you can
be sure this is the case.
Guidelines:
1. Take turns recording the equation and writing the explanation.
2. Be sure you agree with your partners. If you are not sure, challenge the
explanation.
3. Describe your thinking in your own words.
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Lesson 2: Equations for Always, Sometimes, Never Activity
Name: _________________________________ ____ Class: __________ Date:___________________
Decide whether each equation is always true, sometimes true, or never true. Record each equation in your
ALWAYS TRUE, SOMETIMES TRUE, NEVER TRUE chart. Remember to record your reasoning and some
examples next to each equation on the chart.
1.
2–x=x–2
2.
3+x=x+3
3.
x+5=x–3
4.
3x – 5 = 2x
5.
x
2
=x
6.
2(x + 1) = 2x + 1
7.
6x = x
8.
7x + 14 = 7(x + 2)
9.
10.
11.
10
2x
=5
2x+4
2
=x+2
5x – 5 = 5(x + 1)
12. 4x = 4
13.
14.
1
2
3
4
3
x–5+ x=x+x–5
2
1
x + 8 = (3x + 16)
4
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Lesson 2: Equations for Always, Sometimes, Never Activity ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
ANSWER KEY
Always True –
Sometimes True –
Never True –
Infinite Solutions
One Solution
No Solution
2–x=x–2
3+x=x+3
x+5=x–3
(Solution: x = 2)
3x – 5 = 2x
7x + 14 = 7(x + 2)
2(x + 1) = 2x + 1
(Solution: x = 5)
2x+4
=
2
1
2
3
2
x+2
x
2
=x
5x – 5 = 5(x + 1)
(Solution: x = 0)
6x = x
x–5+ x=x+x–5
(Solution: x = 0)
10
2x
3
4
1
4
x + 8 = (3x + 16)
=5
(Solution: x = 1)
4x = 4
(Solution: x = 1)
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Lesson 2: Ticket to Leave
Name: _________________________________ ____ Class: __________ Date:___________________
For which values are the following equations true?
Number of Solutions
Equation
For what values of x is the equation true?
(one, none, infinite)
Ex.
6x + 3 =15
1.
Only true when x = 2.
one
none
infinite
12 - x = 15
one
none
infinite
2.
x-3=3-x
one
none
infinite
3.
3 (x + 4) = 3x + 4
one
none
infinite
4.
x
=6
2
one
none
infinite
5.
2 (x + 3) = 2x + 6
one
none
infinite
Create three examples of your own:
6.
One solution
7.
None
8.
Infinite solutions
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Lesson 2: Ticket to Leave ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
For which values are the following equations true?
Number of Solutions
Equation
For what values of x is the equation true?
(one, none, infinite)
Ex.
6x + 3 =15
Only true when x = 2.
one
none
infinite
1.
12 - x = 15
Only true when x = -3.
one
none
infinite
2.
x-3=3-x
Only true when x = 0.
one
none
infinite
3.
3 (x + 4) = 3x + 4
Never true 12 ≠ 4.
one
none
infinite
4.
x
=6
2
Only true when x = 12.
one
none
infinite
5.
2 (x + 3) = 2x + 6
True for all real numbers.
one
none
infinite
Create three examples of your own:
6.
One solution
7.
None
8.
Infinite solutions
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Lesson 2 Homework: Working with Linear Equations
Name: _________________________________ ____ Class: __________ Date:___________________
x
-3
2
3
x
0
2
4
x
-1
0
2
x
-1
0
2
y
-3
7
9
y
5
7
9
y
5
1
7
y
1
3
7
A
B
C
D
1. Which of these tables of values satisfy the equation y = 2x + 3?
_______________________________________________________
2. Explain how you know.
___________________________________________________________________________
___________________________________________________________________________
3. Complete the tables below for the lines y = 2x + 3 and x = 1 - 2y.
Then use the values to graph the lines.
y = 2x + 3
x
-2
x = 1 - 2y
0
y
x
5
y
0
5
0
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4. Do the equations y = 2x + 3 and x = 1 - 2y have one common solution, no common solutions, or
infinitely many common solutions? Explain how you know.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________ over 
Lesson 2 Homework: Working with Linear Equations
5. Draw a straight line on the graph that has no common solutions with the line y = 2x + 3.
What is the equation of your new line? Explain your answer.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
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Lesson 2 Homework: Working with Linear Equations ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
x
-3
2
3
x
0
2
4
x
-1
0
2
x
-1
0
2
y
-3
7
9
y
5
7
9
y
5
1
7
y
1
3
7
A
B
C
D
1. Which of these tables of values satisfy the equation y = 2x + 3?
TABLE D
2. Explain how you know.
Answers may vary. Ex: The y-intercept in the table is (0,3) and the slope of the table is 2/1
because the x-values add one each time and the y-values add two each time. I added the point
(1,5) to my table to clearly see the constant rate of change.
3. Complete the tables below for the lines y = 2x + 3 and x = 1 - 2y.
Then use the values to graph the lines.
y = 2x + 3
x = 1 - 2y
x
-2
0
1
x
0
1
5
y
-1
3
5
y
-0.5
0
-2
Check students’ graphs.
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4. Do the equations y = 2x + 3 and x = 1 - 2y have one common solution, no common solutions, or
infinitely many common solutions? Explain how you know.
One Solution (-1,1) Students explanations will vary. Ex: Point of Intersection.
over 
5. Draw a straight line on the graph that has no common solutions with the line y = 2x + 3.
What is the equation of your new line? Explain your answer.
Student answers will vary. Ex: y = 2x + 5 will have no common solutions with the line
y = 2x + 3.
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Lesson 3: Strategies for Solving Simultaneous Equations (Graphing)
Brief Overview of Lesson:
Students graph pairs of equations to verify that systems of equations may have one solution in common, no solutions in common, or an infinite number of
solutions in common. They learn to categorize systems by inspection of the equations or graphs. They solve real-world and mathematical problems using
simultaneous equations. As you plan, consider the variability of learners in your class and make adaptations as necessary.
Prior Knowledge Required:
This lesson is written under the assumption that there has been no formal study of linear functions in Grade 8.
It will be helpful if they know how to put equations into slope-intercept form.
6.NS.8 Solve real-world and mathematical problems by graphing in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to
find distances between points with the same first coordinate or the same second coordinate.
7.RP.2c Represent proportional relationships by equations.7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the
situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.7.EE.4 Use variables to represent quantities in a real-world or mathematical
problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Estimated Time (minutes):
2 days (50 minutes each)
Resources for Lesson:
Lesson 2 Homework: Working with Linear Equations
Equations, Tables, and Graphs Cards
Homework: Simultaneous Equations Problem Set
Formative Assessment: Creating Systems of Equations
Graphing calculators
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson # 3: Strategies for Solving Simultaneous Equations (Graphing)
Time (minutes): Two 50 minute lessons
By the end of this lesson students will know and be able to:
 Solve problems involving simultaneous equations using multiple methods (tabular, graphical, and algebraic). (S8)
 Write systems of equations to represent mathematical and real-world problems. (S9)
 Estimate solutions to simple cases by inspection of equations and/or graphs. (S10)
Essential Question(s) addressed in this lesson:
How can equations be used to represent real-world and mathematical situations?
Standard(s)/Unit Goal(s) to be addressed in this lesson:
8.EE.8 Analyze and solve linear equations and 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.
SMP.3 Construct viable arguments and critique the reasoning of others.
SMP.7 Look for and make use of structure.
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Teacher Content and Pedagogy
Teachers can use a variety of Talk Moves to encourage mathematical discourse during the course of the unit. Talk Moves prevent students from “checking out.”
Talk Moves (https://www.teachingchannel.org/videos/student-participation-strategy)
1. Repeating…Can you explain what he just said (or did)?
2. Adding on… Can you add to what she just said?
3. Revising our Thinking…..So, you are starting to wonder…Does this change your thinking?....How have you revised your understanding?
(Gives students permission to revise their thinking and change their minds.)
Background Information on Simultaneous Equations / Systems of Equations
Simultaneous equations are also known as systems of equations. Given two equations, such as x + y = 10 and x – y = 2, the goal is to find values of x and y that
will make both equations true. In this example, one must find two numbers whose sum is ten AND whose difference is two.
Students at this level discover the three types of solutions (one solution, no solutions, or infinite solutions) as they graph simultaneous equations and then
learn to solve them algebraically.
It may be easier for students to make connections to prior knowledge by working with problems where two linear functions are compared.
Example:
Henry and Jose are gaining weight for football. Henry weighs 205 pounds and is gaining 2 pounds per week. Jose weighs 195 pounds and is gaining 3 pounds per
week. When will they weigh the same?
Then students can start to solve problems that lead to simultaneous equations that involve two variables with the context imposing two different constraints.
Example:
Tickets for the class show are $3 for students and $10 for adults. The auditorium holds 450 people. The show was sold out and the class raised $2750 in ticket
sales. How many students bought tickets?
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This problem involves two equations … one representing the number of student and adult tickets sold, and one representing the value of the tickets sold.
Simultaneous equations have one solution when the graphs of the two lines meet at one point (intersecting lines). The solution is the ordered pair that
represents the point of intersection.
In the example below, the solution is x = 0, y = 2.
Example:
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Simultaneous equations have no solution when the graphs of the two lines are parallel; the slopes are identical. The lines will never intersect.
Example:
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Simultaneous equations have infinite solutions when the graphs of the two lines are the same (coincident). The set of ordered pairs representing all the points
on the line are identical.
Example:
Anticipated Student Preconceptions/Misconceptions
Students may have difficulty understanding how a problem can be solved using a systems of equations. They may not see the relationship between the
equations or the variables.
Lesson Sequence:
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DAY 1 and 2 (Teacher determines how to break the lesson into two days.)
 Opening: Review Homework from Lesson 2: Working with Linear Equations.
o
o
o
In groups of four, have students share and discuss responses.
Reach consensus for Question 4.
Whole class discussion: “How do you know?” Use Talk Moves to encourage increased student participation.
 Make the connection between linear equations and simultaneous linear equations: During previous lessons, students learned that linear equations can have
one solution, infinite solutions, or no solutions. The same is true for simultaneous equations.
 Students graph the growth of two plants in Tiffany’s Plants. They answer the question: When will the two plants be the same height? There are three
scenarios: one with no solution, one with one solution, and one with infinite solutions.
What do students notice about the graphs of the plants? What conclusions can they make?
 Present three pairs of linear equations:
y = 3x + 4 and y +5 = 3x
4x + y = 3 and y = 2x
y = -5x + 2 and 10x = 4 – 2y


Have students use graphing calculators or Geogebra, Grapher app to sketch graphs of each pair of equations.
(Note: Students will need to rewrite equations in slope-intercept form before entering equations into graphing calculator.)
Determine the type of solution:
o one (intersecting)
o infinite (lines are the same – coincident)
o none (parallel).
Share and discuss results. Look at table of values to help justify results.
Classifying Solutions to Simultaneous Equations Activity
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 Distribute Equations & Graphs cards to pairs of students.
 Direct students to choose any two of the Equations/Tables/Graphs cards.
 Students graph both equations on one graph.
 The pair of students decides whether the two equations have one common solution, no common solutions, or infinitely many common solutions.
 Have students include a one or two sentence explanation justifying the solutions for the pairs of linear functions (SMP.3).
 They then choose a different pair of equations.

Conduct a whole class discussion with each group, sharing one of their answers and the strategies used to make their decision. Focus the discussion on the
link between the graphical representations of the equations and their common solutions.

Guide students to see that the solution(s) to a system of two linear equations in two variables corresponds to the intersection points of their graphs.
Also, reinforce how a graph represents no solution or an infinite number of solutions.

Classwork and Homework: Simultaneous Equations Problem Set. Teacher selects problems from these examples for classroom practice and homework.
Students write equations and graph them. They identify attributes of the lines in order to determine if there is one solution, no solution, or an infinite
number of solutions (SMP.7).

Closing: Ask students: What do you observe? (Students notice that any pair of equations are either parallel, intersect at one point, or intersect at all
points.)

Formative Assessment: Creating Systems of Linear Equations
Students work independently. They graph 4x – 2y = 3.
Then they use the graph to create systems of equations that have:




Exactly 1 solution.
Exactly 2 solutions.
No solutions.
Infinitely many solutions.
Preview outcomes for the next lesson: Students will be able to use at least two methods for solving simultaneous equations (graphing and substitution).
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Lesson 3: Tiffany’s Plants
Name: _________________________________ ____ Class: __________ Date:___________________
1. Tiffany received two plants for her birthday. She decided to record their growth every two weeks.
Plant A
Plant B
Week
Height (in.)
Week
Height (in.)
0
4
0
1
2
5
2
3
4
6
4
5
6
7
6
7
A. Graph the growth of each plant on the graph below. Label Plant A and Plant B on the graph.
B. Write an equation for each plant’s growth.
C. When will Plant A and Plant B be the same height?
How is this represented on the graph?
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Lesson 3: Tiffany’s Plants
2. Tiffany decided to do her science fair project about comparing the growth rate of plants. She planted a
third plant and compared it to Plants A and B. To keep the variables in her project similar to Plants A and B,
she recorded Plant C’s growth every two weeks. Below is the table of the growth for Plant C:
Plant C
Week
Height (in.)
0
6
2
7
4
8
6
9
A. Graph the growth of Plant C on the same graph.
B. Write an equation for Plant C’s growth.
C. When will the plant C be the same height as plant A?
D. What should Tiffany report to her Science teacher about the growth rate of Plants A, B, and C?
Write a few sentences describing the growth patterns of these plants. Be sure to use appropriate
mathematical vocabulary in your descriptions.
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Lesson 3: Tiffany’s Plants
3. At the Science Fair, one of the judges told Tiffany that she had grown and recorded the growth of a fourth
plant. She was interested to find out how it grew in comparison to Tiffany’s Plant A. Below is the data that
the science fair judge provided:
Plant D
Measured at:
Height (in.)
½ week
4.25
1 ½ weeks
4.75
2 ½ weeks
5.25
3 ½ weeks
5.75
A. Graph the growth of Plant D on the same graph with Plants A, B, and C.
B. Write an equation for Plant D’s growth.
C. When will Plants A and D be the same height?
D. Compare Plant D’s rate of growth to Plant A’s rate of growth.
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Lesson 3: Tiffany’s Plants ANSWER KEY
1. Tiffany received two plants for her birthday. She decided to record their growth every two weeks.
Plant A
Plant B
Week
Height (in.)
Week
Height (in.)
0
4
0
1
2
5
2
3
4
6
4
5
D. Graph the growth of each plant on the graph below. Label Plant A and Plant B on the graph.
E. Write an equation for each plant’s growth.
𝟏
𝟐
Plant A: y = x + 4
Plant B: y = x + 1
F. At what week will Plant A and Plant B be the same height?
How is this represented on the graph?
Week 6
At the point of intersection of the two lines, (6, 7)
Height (Inches)
Week
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Lesson 3: Tiffany’s Plants ANSWER KEY
2. Tiffany decided to do her science fair project about comparing the growth rate of plants. She planted a
third plant and compared it to Plants A and B. To keep the variables in her project similar to Plants A and B,
she recorded Plant C’s growth every two weeks. Below is the table of the growth for Plant C:
Plant C
Week
Height (in.)
0
6
2
7
4
8
6
9
a. Graph the growth of Plant C on the same graph.
b. Write an equation for Plant C’s growth.
𝟏
Plant C: y = 𝟐 x + 6
c. At what week will the Plant C be the same height as Plant A?
They will never be the same height at the same time.
d. What should Tiffany report to her Science teacher about the growth rate of Plants A, B, and C?
Write a few sentences describing the growth patterns of these plants. Be sure to use
appropriate mathematical vocabulary in your descriptions.
Plant B has the fastest growth rate. It has the steepest slope. Plants A and C have the same rate
of growth. The equations for Plants A and C are the same except for the y intercept, which tells
how tall the plants were when Tiffany first got them. The slope, or rate of change, is the same
for both.
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Lesson 3: Tiffany’s Plants ANSWER KEY
3. At the Science Fair, one of the judges told Tiffany that she had grown and recorded the growth of a fourth
plant. She was interested to find out how it grew in comparison to Tiffany’s Plant A. Below is the data that
the science fair judge provided:
Plant D
Measured at:
Height (in.)
½ week
4.25
1 ½ weeks
4.75
2 ½ weeks
5.25
3 ½ weeks
5.75
A. Graph the growth of Plant D on the same graph with Plants A, B, and C.
B. Write an equation for Plant D’s growth.
𝟏
Plant D: y = 𝟐 x + 4
C. When will Plants A and D be the same height?
They are always the same height.
D. Compare Plant D’s rate of growth to Plant A’s rate of growth.
Plant A and Plant D grow at the same rate. Since they started at the same height, they will
continue to be the same height in future weeks.
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Lesson 3: Equations and Graph Cards
Name: _________________________________ ____ Class: __________ Date:___________________
Choose any two equation cards and graph both lines on one graph.
What patterns and similarities do you see?
Choose two new cards and graph their lines. What do you notice?
y = 2x + 4
x + 2y = 8
y = 2x - 1
y = 2(x + 2)
1
y = − 2x + 4
1
x = − 2 – 2y
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set
Name: _________________________________ ____ Class: __________ Date:___________________
Write two equations for each problem. Then graph each equation on the same coordinate plane to
determine the solution. Use extra graph paper if you need it.
1. To get to work, Meg has to take the bus from her house into town and back every day. She has two
options. Which is the best option for her?
Option 1: She can pay $3 per ride.
Option 2: She can buy a monthly pass for $40 and then pay $2 per ride.
2. There are some rabbits and chickens in a field. Together they have 55 heads and 162 feet. How many
rabbits? How many chickens?
3. Last week Andrea worked a total of 14 hours at her two jobs and earned a total of $96.
Her babysitting job pays $6 per hour. Her cashier’s job pays $8 per hour.
How many hours did she work at each job?
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set
4. Your family is planning a 7-day trip to Florida. You estimate that it will cost $350 per day in Miami and
$500 per day in Orlando. Your budget for the 7 days is $2900. How many days should you spend in
each city?
5. The price of refrigerator A is $800. The electric bill for refrigerator A is $60 per year.
The price of refrigerator B is $1200. The electric bill for refrigerator B is $50 per year.
After how many years will the cost of owning each refrigerator be the same?
Which refrigerator would you recommend buying? Why?
6. A rectangle has a length that is 3 units greater than its width.
Another rectangle has a length that is two times the width.
The perimeters of both rectangles are the same.
What are the dimensions of each rectangle?
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set
7. Chloe bought 8 CDs and 6 DVDs which altogether cost $78. On another occasion she bought 4 CDs and
18 DVDs which cost $114. On both shopping trips, each CD cost the same amount and each DVD cost
the same amount. How much does one CD cost? How much does one DVD cost?
8. In one day a movie theater collected $3600 from 600 people. The prices of tickets were $8 for adults
and $5 for children. How many adult tickets and how many children’s tickets were purchased?
Challenge:
9. Find the equation of the line that will produce a right triangle.
Side 1:
y = -2x + 8
Side 2:
y = 1/3 x
Side 3:
_____________
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
SIMULTANEOUS EQUATIONS – Problem Set
Write two equations for each problem, and then solve. Use extra graph paper if you need it.
1. To get to work, Meg has to take the bus from her house into town and back every day. She has two
options. Which is the best option for her?
Option 1: She can pay $3 per ride.
Option 2: She can buy a monthly pass for $40 and then pay $2 per ride.
y = total cost, x = number of rides
y = 3x
y = 2x + 40
If Meg is going to ride the bus $40 both options cost the same. If she plans to ride the bus less than 40
times option 1 is best. If she plans to ride the bus more than 40 times option 2 is best.
2. There are some rabbits and chickens in a field. Together they have 55 heads and 162 feet. How many
rabbits? How many chickens?
r = number of rabbits, c = number of chickens
r + c = 55
4r + 2c = 162
There are 29 chickens and 26 rabbits.
3. Last week Andrea worked a total of 14 hours at her two jobs and earned a total of $96.
Her babysitting job pays $6 per hour. Her cashier’s job pays $8 per hour.
How many hours did she work at each job?
b = babysitting hours, c = cashier hours
b + c = 14
6b + 8c = 96
She worked 8 hours babysitting and 6 hours at her cashier’s job
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set ANSWER KEY
4. Your family is planning a 7-day trip to Florida. You estimate that it will cost $350 per day in Miami and
$500 per day in Orlando. Your budget for the 7 days is $2900. How many days should you spend in
each city?
m = number of days in Miami, r = number of days in Orlando
m+r=7
350m + 500r = 2900
You should spend 4 days in Miami and 3 days in Orlando.
5. The price of refrigerator A is $800. The electric bill for refrigerator A is $60 per year.
The price of refrigerator B is $1200. The electric bill for refrigerator B is $50 per year.
A = cost of owning refrigerator A, B = cost of owning refrigerator B, y = number of years
A = 60y + 800
B = 50y + 1200
After how many years will the cost of owning each refrigerator be the same?
60y + 800 = 50y + 1200
After 40 years the cost of owning each refrigerator will be the same.
Which refrigerator would you recommend buying? Why?
If you plan on owning the refrigerator for less than 40 years then Refrigerator A is the best option
because it will be cheaper. If you plan on owning the refrigerator for more than 40 years then
Refrigerator B is the cheaper option.
6. A rectangle has a length that is 3 units greater than its width.
Another rectangle has a length that is two times the width.
The perimeters of both rectangles are the same.
W = width, L = length R1 = rectangle number 1, R2 = rectangle number 2, P = perimeter
PR1 = 2W + 2(W+3)
PR2 = 2W + 2(2W)
What are the dimensions of each rectangle?
Rectangle 1 has a width of 3 units and a length of 6 units.
Rectangle 2 has a width of 3 units and a length of 6 units.
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Lesson 3: Classwork & Homework: Simultaneous Equations Problem Set ANSWER KEY
7. Chloe bought 8 CDs and 6 DVDs which altogether cost $78. On another occasion she bought 4 CDs and
18 DVDs which cost $114. On both shopping trips, each CD cost the same amount and each DVD cost the
same amount. How much does one CD cost? How much does one DVD cost?
c = cost per CD, d = cost per DVD
8c + 6d = 78
4c + 18d = 114
One CD costs $6.00 and one DVD costs $5.00.
8. In one day a movie theater collected $3600 from 600 people. The prices of tickets were $8 for adults and
$5 for children. How many adult tickets and how many children’s tickets were purchased?
a = number of adults, c = number of children
a + c = 600
8a + 5c = 3600
There were 200 adult tickets purchased and 400 children tickets purchased.
Challenge:
9. Find the equation of the line that will produce a right triangle.
Side 1:
y = -2x + 8
Side 2:
y = 1/3 x
Side 3:
y = -3x + 8
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Lesson 3: Creating Systems of Equations
Name: _________________________________ ____ Class: __________ Date:__________________
1. Graph this equation: 4x – 2y = 3
2. Use the graph you made to answer
questions A – D. First, draw each line,
and then, write the equation. If you
cannot find a second equation, write
“Not Possible”.
A. Find another linear equation to create a system of linear equations that has exactly one
solution.
Equation 1: 4x – 2y = 3
Equation 2:_____________________________
B. Find another linear equation to create a system of linear equations that has exactly two
solutions.
Equation 1: 4x – 2y = 3
Equation 2:_____________________________
C. Find another linear equation to create a system of linear equations that has no solutions.
Equation 1: 4x – 2y = 3
Equation 2:_____________________________
D. Find another linear equation to create a system of linear equations that has infinitely
many solutions.
Equation 1: 4x – 2y = 3
Equation 2:_____________________________
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Lesson 4: Strategies for Solving Simultaneous Equations (Substitution)
Brief Overview of Lesson:
In this lesson students explore an alternate strategy for solving simultaneous linear equations known as substitution. When the solution of a system of linear
equations is an ordered pair that is not located near the origin, or that does not contain whole numbers, the graphic method will not be useful. The substitution
method is just one of many algebraic methods, used to solve a system of equations without using its graphs. As you plan, consider the variability of learners in
your class and make adaptations as necessary.
Prior Knowledge Required:
6.EE.3 Apply the properties of operations to generate equivalent expressions
6.EE.5 Understand solving and equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or
inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x +p = q and px = q for cases in which p, q, and x are all known
negative rational numbers.
6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the
reasonableness of answers using mental computation and estimation strategies.
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
Estimated Time (minutes):
Resources for Lesson:
Two 50 minute lessons
Sample Card Set, Math Teachers’ Café Handout
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson 4: Strategies for Solving Simultaneous Equations (Substitution)
Time (minutes): Two 50 minute lessons
By the end of this lesson students will know and be able to:

Solve simultaneous equations by substitution. (S8)
Essential Question(s) addressed in this lesson:


How can equations be used to represent real-world and mathematical situations?
Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful?
Standard(s)/Unit Goal(s) to be addressed in this lesson (type each standard/goal exactly as written in the framework):
8.EE.8 Analyze and solve linear equations and 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.
SMP.7 Look for and make use of structure.
Teacher Content and Pedagogy
There are several methods for solving simultaneous equations, or systems. Students explored graphical models in the previous lesson. When an exact answer is
needed, these graphical models have some limitations. One algebraic method is substitution.
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This lesson is intended to lead students to uncover the limitations of graphical solutions in some situations and to explore other ways to solve systems of
equations.
Do not teach the substitution method as a handy trick or procedure. In fact, it is recommended that the method remain unnamed until the end of the lesson.
Students need to see the connection between this algebraic method and the point of intersection on the graphs of the two lines.
First, students identify if the system has one solution. The point of intersection makes both equations true. If so, students can use the Substitution Property of
Equality to create an equation in one variable.
The Substitution Property of Equality: If a=b, then a can be substituted for b in any expression containing a.
Substitution
Students have previously solved equations using substitution when given an exact value for one of the values. This would be a good reference point from which
to start.
Example 1: Solve the equation y = 5x – 7 when x = -2.
Example 2: Solve the equation y = 5x – 7 when y = 3.
In addition to the Activator Activity, algebra balance scales can also be used to provide a visual model of the concept of substitution.
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Anticipated Student Preconceptions/Misconceptions
Students may not realize that solutions to simultaneous equations are written as a coordinate pair (e.g., x = ___ does not sufficiently describe the point(s) of
intersection.)
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Lesson Sequence
DAY 1
 Opening: Revisit the previous day’s formative assessment: 4x – 2y =3.
Ask students: Were you able to create equations that showed at least one solution, no solutions, or infinite solutions?
Strategically select student work to display on a document reader or on chart paper. Ask students to turn and talk to discuss the strategies that they see.
o What were some strategies for writing the equations?
o What difficulties did they have in finding an equation with one solution, no solution, or infinite solutions?
o What misconceptions do students have?

Consider this system: y = 5x and y = 2x + 2. Students graph this system and determine whether there is one solution, no solutions, or many solutions.
Ask students to state the solution. Students should notice the difficulty of finding an exact solution. Using the graphical method, students can only
approximate a solution.
Have students evaluate both equations for the point (2/3, 10/3).
Have students Think, Pair, Share with a partner about the question, “Is there another way to solve this problem other than graphing the system on a
coordinate grid?” “When might it be important to find the exact solution?” Allow for students to talk about their ideas with each other. Have students
share their ideas at the end of this time.

If we know that y = 5x and y = 2x + 2, and we know that there is one solution for this system of equations, then there must be one point on the line y =
5x that intersects with one point on the line y = 2x + 2. At that point of intersection, for that same x value, the equations have the same y value. We saw
that when x = 2/3, y was 10/3 in both equations. So, (2/3, 10/3) is the point of intersection. Does that mean that 5x and 2x + 2 have the same value at that
point?
Y = 5x
y = 2x + 2
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If y = y at that point of intersection, is it true that 5x = 2x + 2?
Can you solve this equation? (Algebraic approach to solving systems.) Students will find the x value. They have to realize that in a system of equations,
the solution is a point, with both an x and y value.

Post the next two problems on the board and have students try them on their own. Students pair up. One person solves it graphically and finds the point of
intersection. The other person solves it algebraically. Compare solutions with each other, and then with another pair.
Problem 1:
y=x–2
y = -x + 12
(Solution: (7,5))
Problem 2:
y = -3x
x + y = -2
(Solution: (1, -3))
The word substitute or substitution may come up in the discussions. Mention that this method is often called the substitution method. Ask students
why.
Connect algebraic representations to visual representations. Problems can be described and solved in multiple ways.
As a Ticket to Leave, ask: When might you use the graphing method, and when might you use the substitution method?
Students need to analyze the structure of the given equations in order to determine which method to use (SMP.7)
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DAY 2

Equation Cards Activity
1. Students work in pairs, using the Sample Cards to practice solving simultaneous equations algebraically. Each pair is given a card set in zip-lock bags. Does
the pair of equations have one, no, or infinite solutions?
Each zip-lock bag contains two cards. Cards are folded so that there is a front and a back to each card. Cards #1-5 have y isolated in both equations so
students can set the equations equal to each other. Cards # 6 – 10 have either x or y isolated so that one equation can be substituted into the other.
2. Students will then solve the simultaneous equations algebraically using substitution. They put their cards back in their zip-lock bag, and swap with another
team. Repeat.
3. Have a few groups go to the board, post, and explain their solutions.

Students revisit two examples from the previous day:
Example 1: y = x – 2
y = -x + 12

Example 2: y = 3x
x + y = 20
They graph each pair of equations. How do the solutions compare to those derived from using substitution method on the previous day?
Closing: Ticket to Leave: Students write a response: What are the advantages of solving simultaneous equations with graphing? What are the advantages
of solving them with the substitution method? Which one makes more sense to you?
Assign Homework: Math Teachers’ Café (practice solving simultaneous equations with substitution)
Preview outcomes for the next lesson: Students will be able to use a variety of strategies to solve systems of equations, including graphing, substitution, and
elimination.
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Lesson 4: Sample Card Set
Card Set #1:
y
5x–8
y
5x+3
y
7 x – 21
Card Set #2:
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y
5x+9
y
-3 x + ¼
Card Set #3
y
-5 x – ¾
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Card Set #4
y
2/3 x + 7
y
1/3 x + 9
y
-2 x + 7
y
–x –x + 3 + 4
Card Set #5
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Card Set #6
4x–1
y
2x -y=5
2x – (
)=5
Card Set #7
2y+4
x
2x – y = 5
2(
)–y =5
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Card Set #8
2/5 x – 3
y
-2x = -3 – y
5
-2x = –3 – (
5
)
Card Set #9
-1/4 y + 3
x
-4x + y = 20
-4 (
) + y = 20
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Card Set #10
1/3x – 7
y
–x + y = 1
3
2
–x + (
3
)= 1
2
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Answer Key
Card Set #1
No Solution -8 ≠ 3
Card Set #2
One Solution (15, 84)
Card Set #3
One Solution (-1/2, 7/4)
Card Set #4
One Solution (6, 11)
Card Set #5
Infinitely Many Solutions -2x + 7 = -x –x +3 + 4
Card Set #6
One Solution (-2, -9)
Card Set #7
One Solution (-1, 2)
Card Set #8
Infinitely Many Solution 2/5x – 3 = 2/5x – 3
Card Set #9
One Solution (7/8, 17/2)
Card Set #10
No Solution -7 ≠ ½
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Lesson 4: Math Teachers’ Café
Name: _________________________________ ____ Class: __________ Date:___________________
Solving Systems Using Substitution – Jokingly
Directions – Show all work on a separate piece of paper. Solve each system of equations by the substitution
method. Then, write the word next to the correct answer in the box containing the selected solutions.
What’s the best dessert in the Math Teachers’ Café?
Answer to #4
Answer to #1
Answer to #6
Answer to #7
Answer to #9
!
1. y = 5x
x + y = 30
2. 2x + 5y = 26
y = x/4
4. 4p + 2q = 10
p – 5q = 8
5. 5c – 2d = 0
6d = 1 + c
3. B – 3a = 1
b = 4a
6. 3s – t = 1
3s = 9
7. The sum of two numbers is 36 and their difference is 6. What are the numbers?
8. There are 312 students at the school dance. There are 48 more girls than boys. How many girls
are at the dance? How many boys are at the dance?
9. The school’s Drama Club charges $10 for an adult ticket and $5 for a child’s ticket. One night 410
tickets are sold for a total of $3625. How many adults attended the show? How many children
attended the show?
Answer Options:
( 180, 132)
(3, -1)
(5, 25)
vanilla
A
slice
(3,8)
(1, 4)
( 315, 95)
of
piece
pi
( 24, 6 )
One
( 21, 15)
chocolate
(1/14, 5/28)
cake
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Math Teachers’ Café ANSWER KEY
Solving Systems Using Substitution – Jokingly
Directions – Show all work on a separate piece of paper. Solve each system of equations by the substitution
method. Then, write the word next to the correct answer in the box containing the selected solutions.
What’s the best dessert in the Math Teachers’ Café?
Answer to #4
A
9. y = 5x
x + y = 30
(5, 25)
4. 4p + 2q = 10
p – 5q = 8
(3, -1)
Answer to #1
slice
Answer to #6
Answer to #7
Answer to #9
of
chocolate
pi !
2. 2x + 5y = 26
y = x/4
3. B – 3a = 1
b = 4a
(24, 6)
(1, 4)
5. 5c – 2d = 0
6d = 1 + c
(1/14, 5/28)
6. 3s – t = 1
3s = 9
(3, 8)
7. The sum of two numbers is 36 and their difference is 6. What are the numbers?
(21, 15)
8. There are 312 students at the school dance. There are 48 more girls than boys. How many girls
are at the dance? How many boys are at the dance?
(180, 132)
10. The school’s Drama Club charges $10 for an adult ticket and $5 for a child’s ticket. One night 410
tickets are sold for a total of $3625. How many adults attended the show? How many children
attended the show?
(315, 95)
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Lesson 5: Strategies for Solving Simultaneous Equations (Elimination)
Brief Overview of Lesson:
Students continue to practice different strategies for solving simultaneous linear equations. They learn another method known as elimination. They add
equations together to eliminate variables, often by multiplying each equation by a constant so that the resulting coefficient of one of the variables is 0.
Previously, students have mastered the method of substitution (solving an equation for one variable, and then substituting the solution into the other equation
to solve). As you plan, consider the variability of learners in your class and make adaptations as necessary.
Prior Knowledge Required:
6.EE.3 Apply the properties of operations to generate equivalent expressions.
6.EE.5 Understand solving and equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or
inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x +p = q and px = q for cases in which p, q, and x are all known
negative rational numbers.
6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the
reasonableness of answers using mental computation and estimation strategies.
7.E.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
Estimated Time (minutes):
Resources for Lesson:
One 50 minute lesson
Classwork: Using the Elimination Strategy
Homework: Solving Systems of Equations Using the Elimination Method
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson 5: Strategies for Solving Simultaneous Equations (Elimination)
Time (minutes): 1 day (50 minutes)
By the end of this lesson students will know and be able to (write out clear and concise objectives for this lesson):
 Solve simultaneous linear equations by elimination. (S8)
 Write systems of equations to represent mathematical and real-world problems. (S6)
Essential Question(s) addressed in this lesson:


How can equations be used to represent real-world and mathematical situations?
Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful?
Standard(s)/Unit Goal(s) to be addressed in this lesson (type each standard/goal exactly as written in the framework):
8.EE.8 Analyze and solve linear equations and 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.
SMP.3 Construct viable arguments and critique the reasoning of others.
Teacher Content and Pedagogy
Another method for solving systems of linear equations is the elimination method. The elimination method uses addition or subtraction to cancel one variable.
This method may also be referred to as the Addition-or-Subtraction method or the Combination method.
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Remind students of the Properties of Equality: Adding, subtracting, multiplying, or dividing by the same number on each side of an equation does not change
the solution.
Before teaching students how to use the elimination strategy, it is important for them to understand the concepts behind this method. They need to be able to
explain mathematically why it works.
The balance scale model will help students to visualize why adding (or subtracting) two equations is possible and how it can help to eliminate one variable.
The addition/subtraction property of equality states that if a = b and c = d, then a + c = b + d, or a – c = b - d
This property can be modeled for students using the scales, and reinforces the concept of the equal sign.
Anticipated Student Preconceptions/Misconceptions:
 Students may be able to carry out the procedure of eliminating a variable in a system of equations, but they may have no conceptual understanding of why
this is possible or why it results in a correct solution.

Adding two equations together (or subtracting one equation from another) is a new idea. Teachers need to spend time on conceptual understanding,
building on students’ knowledge of solving systems of linear equations by other means.
Lesson Sequence
DAY 1

Opening: Explicitly connect this lesson to the previous lesson. Introduce this lesson by articulating to the students what they will be learning today.
Students are given a system of equations to solve. It is anticipated that students will use prior knowledge of substitution or graphing to solve this system.
Show students this system of equations:
2x + 2y = 9
5x – 2y = 12
Have students discuss with a partner: How would you solve this system of equations?
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Allow students to work on a solution for a few minutes. (All students may not reach a solution, but all students should have a chance to grapple with how
“messy” this system is to solve when they use substitution or graphing. Students will probably struggle with this solution. Students may prefer a graphical
approach; however the solution is a fraction, and may have to be an estimate.)

Using Turn and Talk and give credit to answer: What was easy about solving this system? What was difficult?
Students pair up to discuss the response. A whole-class share-out is done, where one partner shares the other partner’s ideas.

Build conceptual understanding for adding the two linear equations together by using a balance scale model.
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Refer back to the last example: 2x + 2y = 9
5x – 2y = 12
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Since
We can add the two equations together by replacing the shapes in the balance pan.
If students add each side of the equation, the result is 7x = 21, or x = 3. They can then solve for y.
Teacher Note: Emphasize that students need to find a value for both the x and y variable.

Students use the Math Talk Protocol (see Lesson 3) to review:
o
o
o
o
Why can you add 2x + 2y and 5x – 2y?
Why is it equal to 9 + 12?
How does this help you to solve the system of equations? (It eliminates one of the variables.)
Will this work for all systems of equations? Why or why not?
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o

Is there a way to use this strategy without the picture of a balance scale? Try it on paper.
Students solve two systems of equations where one variable can easily be eliminated by addition.
2x + y = 18
4x – y = 12

x + y = 32
x – y = 16
Ask: What if you had this system of equations?
5s + 2t = 6
9s + 2t = 22
How could you eliminate one of the variables using the method you just learned? (Subtract instead of add.)
Classwork: Elimination Strategy Practice
Students complete a practice sheet with simple elimination problems.

The last problem will not be eliminated easily. Students would have to multiply one equation in order to solve. They will not be familiar with this idea yet
but the problem should be posed as a challenge.
Final problem:
-x + 4y = 3
3x + 2y = 5
If they are stuck, go on to the next part of the lesson.

Students solve these equations:
4n = 12
2n = 6
3n = 9
2x
3x
4x
n =3
n =3
n =3
2n = 6
3n = 9
4n = 12
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Ask: Since all of these equations have the same solution (n = 3), are they equivalent equations?
Why or why not? How would you prove it? Give students some time to discuss this in small groups to be able to answer these questions. They should be
ready to justify their responses (SMP.3).
Some students may try to graph the three equations. This would show that they are equivalent equations. Others may notice that each equation is a
multiple of x = 3.

Go back to the last problem on the problem set.
Final problem:
-x + 4y = 3
3x + 2y = 5
Ask students to find a multiple of one equation so that the coefficient of one of the variables can easily be used to eliminate the variable when the two
equations are added or subtracted. (Example: 2(3x + 2y) = 2(5), or 6x + 4y = 10.)

Closing:
Summarize that the elimination strategy is another method for solving simultaneous linear equations like the examples from previous lessons.
Ask: When would you use the elimination method vs. the substitution method?
What are the advantages and disadvantages of each?

Homework: Solving Systems of Equations Using the Elimination Method
Preview outcomes for the next lesson:
Students make connections between various strategies for solving systems of equations, including justifying which method is the most efficient for the problem
presented (SMP.3)
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Lesson 5 Classwork: Elimination Strategy Practice
Name: _________________________________ ____ Class: __________ Date:___________________
Solve each system using the elimination method.
1.
x – y = 13
7x + y = 11
2.
2r – 2s = -1
-2r + 5s = 4
3. 4x – 2y = 17
7x + 2y = -6
4. c – 3d = 9
c + 3d = 21
Create two equations from the following word problem and solve using elimination.
5. There are 735 students at the Rogers Middle School. There are 41 more boys than girls.
How many boys are there?
CHALLENGE:
6. –x + 4y = 3
3x + 2y = 5
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Lesson 5 Classwork: Elimination Strategy Practice ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
Solve each system using the elimination method.
1.
x – y = 13
7x + y = 11
(3, -10)
2.
2r – 2s = -1
-2r + 5s = 4
(1, ½)
3. 4x – 2y = 17
7x + 2y = -6
(1, -6.5)
4. c – 3d = 9
c + 3d = 21
(15, 2)
Create two equations from the following word problem and solve using elimination.
5. There are 735 students at the Rogers Middle School. There are 41 more boys than girls. How many boys
are there?
b + g = 735 b = number of boys, g = number of girls
g + 41 = b
Solution: Number of boys = 388 and Number of girls = 347
CHALLENGE:
6. –x + 4y = 3
3x + 2y = 5
Solution: (1, 1)
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Lesson 5 Homework: Elimination Strategy Practice
Name: _________________________________ ____ Class: __________ Date:___________________
Solve each system using the elimination method.
1.
x – 5y = -1
3x + 5y = 17
2.
3s + 4t = 18
-3s – 7t = 18
4.
2x + 3y = 6
4x + 6y = 12
5. 12x – 13y = 2
-6x + 6.5y = -2
3. 1/2p + 2q = -10
-1/2p – 5q = -8
Create two equations from the following word problems and solve using elimination.
6. The sum of two numbers is 28 and their difference is 8. What are the numbers?
7. The width of a rectangle is 15 cm less than its length. The perimeter is 450 cm. What is the area?
8. At Halloween, a local farmer creates a giant maze using his corn fields. He charges $4 for an adult
ticket and $2 for a child’s ticket. One night 221 tickets were sold for a total of $634. How many
children walked through the maze that night?
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Lesson 5 Homework: Elimination Strategy Practice ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
Solve each system using the elimination method.
1.
x – 5y = -1
3x + 5y = 17
(4, 1)
2.
3s + 4t = 18
-3s – 7t = 18
(14, -6)
4.
2x + 3y = 6
4x + 6y = 12
0 = 0 infinite solutions
5. 12x – 13y = 2
-6x + 6.5y = -2
0 = -2 no solution
3. 1/2p + 2q = -10
-1/2p – 5q = -8
(-44, 6)
Create two equations from the following word problems and solve using elimination.
6. The sum of two numbers is 28 and their difference is 8. What are the numbers?
x + y = 28
x = first number, y = second number
x–y=8
Solution: (38, -10)
7. The width of a rectangle is 15 cm less than its length. The perimeter is 450 cm. What is the area?
2W + 2L = 450
W = width, L = length
W = L – 15
Solution: Length = 120 cm and Width = 105 cm
8. At Halloween, a local farmer creates a giant maze using his corn fields. He charges $4 for an adult
ticket and $2 for a child’s ticket. One night 221 tickets were sold for a total of $634. How many
children walked through the maze that night?
4a + 2c = 634 a = number of adult tickets, c = number of children tickets
a + c = 221
Solution: Number of children tickets = 125 and Number of adult tickets = 96
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Lesson 6: Connections
Brief Overview of Lesson:
This is a culminating lesson in which students use their understanding of simultaneous equations to solve a problem that requires multiple steps. They should be
fluent in solving systems of equations graphically, using substitution, and using elimination strategies. Given different real-world or mathematical problems,
students will analyze which method of solving is most efficient, discussing their choices with classmates (SMP.3). As you plan, consider the variability of learners
in your class and make adaptations as necessary.
Prior Knowledge Required:
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the
reasonableness of answers using mental computation and estimation strategies.
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
8.EE.7 Solve linear equations in one variable.
8.EE.8 Analyze and solve pairs of simultaneous linear equations.
Estimated Time (minutes):
Two 50 minutes lessons
Resources for Lesson:
Problem Set: Which Way is the Most Efficient?
Homework: Thinking About Systems of Equations
Summative Assessment: Sam’s and Katherine’s Mistakes
Answer Key for Sam’s and Katherine’s Mistakes
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson 6: Connections
Time (minutes): Two 50-minute lessons
By the end of this lesson students will know and be able to



Identify the different forms of solutions and connect the form to the number or solutions (one solution: x = a, infinite solution: a = a, no solution: a = b) (S10)
Produce examples of linear equations that represent one solution, infinite solutions, and no solutions. (S7)
Solve problems involving simultaneous equations using multiple methods. (S8)
Essential Question(s) addressed in this lesson:
Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful?
Standard(s)/Unit Goal(s) to be addressed in this lesson:
8.EE.8 Analyze and solve linear equations and 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.
SMP.3
Construct viable arguments and critique the reasoning of others.
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Teacher Content and Pedagogy
What does “efficient” mean?
It is essential that the teacher connect the graphical, elimination, and substitution models at this time if this does not occur naturally during the discussion.
Ask students to define what “most efficient” means in terms of mathematics. Among other ideas, they may suggest:





The quickest way
The easiest way
The way that fits best with the type of problem
The most accurate way
The way that makes the most sense to me
List all ideas on the board and come to a consensus about what an “efficient way” represents.
Lesson Sequence
DAY 1

Opening: Say to the students: So far we have learned how to solve equations using one of three methods: Graphing, Substitution, Elimination. Do a Fist of
Five survey about each method. (Student puts fist on chest, showing 0-5 fingers. 5 = expert and could teach someone else; 0 = I have no idea)
Assign students one problem and ask them to solve it using the method of their choice.
y - 3x = 12
2y + 3x = 6
Conduct a Gallery Walk: Students place their solutions on their desks and walk silently around the room to view the work. Debrief: What did you notice?
What kinds of strategies did people use? Is one strategy more efficient than another?
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
Problem Set: Which Way is the Most Efficient?
Students work on a problem set to find the solutions to systems of simultaneous linear equations. For each problem, the student must decide which
method is the most efficient way to solve the problem and why.
Students share their solutions in a small group and try to convince the group why their method for solving is the most efficient.

Ticket to Leave:
o
o
o
Tell which method you prefer and why.
Ask one question about any of the three solving methods.
Create a system of simultaneous equations, telling which method would be the most efficient to solve this system and why.
DAY 2
Optional Enrichment Assignment
Students are given the following Summative Assessment:
Sam’s and Katherine’s Mistakes
Two students solved a system of simultaneous equations. In each equation, a stands for a number, and b stands for a number.
ax + y = 5
2x – by = 13
7
Katherine misread a and got x = 2 and y = -2.
Sam misread b and got x = 3 and y = -7.
Other than these mistakes, they did all calculations correctly.
Can you find the correct solution for (x,y)? If so, what is it?
Problem courtesy of Andrew Chen
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Teacher Note on Sam’s and Katherine’s Mistake
This is a relatively difficult problem.
First, students must be able to transform the equations into slope-intercept form.
Then, they must reason about the correct and incorrect coefficients, and how to find the correct values using multiple solution strategies.
Step 1: Students work independently for 10-15 minutes on their own while the teacher circulates.
Step 2: Students are then allowed to work at a table group to share ideas and continue the problem-solving activity.
Step 3: The teacher selects a few students to share work via a document camera. The student explains his/her reasoning. The teacher should scaffold the order
of the students. E.g., the first student might explain how he/she decided to put the equations in y = mx + b form, and why. The second student might reason
ideas for working with the incorrect coefficients. The third student might identify a, etc. If necessary, students can be sent back to their small groups to
incorporate new ideas learned from the sharing.
Step 4: When the problem has been solved and various solutions shared, students then individually complete a written explanation of how the problem was
solved. This piece of work serves as a summative assessment.
Preview outcomes for the next lesson:
End of Unit; CEPA to follow
Summative Assessment:
Katherine and Sam’s Mistakes
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Lesson 6 Classwork: Which is Most Efficient?
Name: _________________________________ ____ Class: __________ Date:___________________
1. For each system below, decide which strategy to use. Which method would be the most efficient,
convenient and accurate? Do not solve the systems yet! When you are done, compare your choices with
others in your group. Be prepared to justify your reasons for choosing one strategy over another.
Equations
Strategy
Solution
Graphing, Substitution, Elimination
y=6
x+3=y
3x + y = 1
4x + y = 2
2x – 4y = 8
X = 2y + 5
-6x + 2y = 120
-3x + y = 60
x–5=y
2(x – 5) – y = 7
5x + 3y = -6
2x – 9y = 18
2. Erica works in a soda-bottling factory. As bottles roll past her on a conveyor belt, she puts caps on
them. Unfortunately, Erica sometimes breaks a bottle before she can cap it. She gets paid 4 cents
for each bottle she successfully caps, but her boss deducts 2 cents from her pay for each bottle
she breaks.
Erica is having a bad morning. Fifteen bottles have come her way, but she has broken some, and
she has only earned 6 cents so far today. How many bottles has Erica capped, and how many has
she broken?
A. Write a system of equations to represent this situation.
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B. Solve the system of equations using two different methods.
Lesson 6 Classwork: Which is Most Efficient? ANSWER KEY
Name: _________________________________ ____ Class: __________ Date:___________________
1. For each system below, decide which strategy to use. Which method would be the most efficient,
convenient and accurate? Do not solve the systems yet! When you are done, compare your choices with
others in your group. Be prepared to justify your reasons for choosing one strategy over another.
Equations
Strategy
Graphing, Substitution, Elimination
Solution
y=6
x+3=y
Answers will vary for this
column. Sample Answers:
SUBSTITUTION
(3,6)
3x + y = 1
4x + y = 2
ELIMINATION
(1,-2)
2x – 4y = 8
X = 2y + 5
SUBSTITUTION
no solution
-6x + 2y = 120
-3x + y = 60
SUBSTITUTION
infinite solutions
x–5=y
2(x – 5) – y = 7
SUBSTITUTION
(12,7)
5x + 3y = -6
2x – 9y = 18
GRAPHICALLY
(0, -2)
2. Erica works in a soda-bottling factory. As bottles roll past her on a conveyor belt, she puts caps on
them. Unfortunately, Erica sometimes breaks a bottle before she can cap it. She gets paid 4 cents
for each bottle she successfully caps, but her boss deducts 2 cents from her pay for each bottle
she breaks.
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Erica is having a bad morning. Fifteen bottles have come her way, but she has broken some, and
she has only earned 6 cents so far today. How many bottles has Erica capped, and how many has
she broken?
A. Write a system of equations to represent this situation.
Erica has capped 6 bottles and broken 9 bottle.
B. Solve the system of equations using two different methods. See student work.
Lesson 6: Enrichment: Sam and Katherine’s Mistakes
Name: _________________________________ ____ Class: __________ Date:___________________
Two students solved a system of simultaneous equations.
In each equation, a stands for a number, and b stands for a number.
ax + y = 5
2x – by = 13
Katherine misread a and got x =
7
2
and y = -2.
Sam misread b and got x = 3 and y = -7.
Other than these mistakes, they did all calculations correctly.
Can you find the correct solution? If so, what is it?
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Lesson 6: Enrichment: Sam and Katherine’s Mistakes ANSWER KEY
Solution:
Two students solved a system of simultaneous equations.
In each equation, a stands for a number, and b stands for a number.
ax + y = 5
2x – by = 13
7
Katherine misread a and got x = 2 and y = -2.
Sam misread b and got x = 3 and y = -7.
Other than these mistakes, they did all calculations correctly.
Can you find the correct solution (x,y)? If so, what is it?
By substituting in Katherine’s values, we get:
ax + y = 5
7
(2 )a + (-2) = 5
Solving for a, a = 2.
By substituting in Sam’s values, we get:
ax + y = 5
(3 )a + (-7) = 5
Solving for a, a = 4.
2x – by = 13
2(3) – b (-7) = 13
2x – by = 13
7
2(2) – b (-2) = 13
Solving for b, b = 3
Solving for b, b = 1
Katherine believes that a = 2 and b = 3.
Sam believes that a = 4 and b = 1.
Since we know that Katherine’s b value is correct and Sam’s a value is correct, the correct
equations are:
4x + y = 5
2x – 3y = 13
Students may solve this pair of simultaneous equations using any strategy.
The solution is: x = 2, y = -3
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Name: _________________________________ ____ Class: __________ Date:___________________
GRAPHING
SUBSTITUTION
ELIMINATIO
N
1. Which method of solving systems of equations do you prefer? Why?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
2. Write one question that you have about one of these strategies.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
3. Create a system of simultaneous equations, and tell which method would be the most efficient to solve this
system and why.
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Appendix: Optional Pre-Lesson
Modeling Real-World and Mathematical Situations with Equations in One Variable
Brief Overview of Lesson:
Students translate story problems into single variable equations. They may use manipulatives or representational drawings to develop equations. Students
develop an abstract understanding of how to write one variable equations based on real-world or mathematical situations.
Prior Knowledge Required:
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2a Write expressions that record operations with numbers and with letters standing for numbers.
6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problem; understand that a variable can represent
an unknown number, or, depending on the purpose at hand, any number in a specified set.
7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
Estimated Time (minutes):
One 50-minute lesson
Resources for Lesson:
Formative Assessment
Classwork: Matching Problems, Models, and Equations
Answer Key for Classwork: Matching Problems, Models, and Equations
Problem Set: Writing Equations from Word Problems
Answer Key for Problem Set: Writing Equations from Word Problems
Manipulatives such as blocks and two-color counters
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Optional Pre-Lesson: Modeling Real-World and Mathematical Situations with Equations in One Variable
Time (minutes): One 50-minute lesson
By the end of this lesson students will know and be able to:

Interpret a word problem, identify the variable, and write the equation.
Essential Question(s) addressed in this lesson:

How can equations be used to represent real-world and mathematical situations?
Standard(s)/Unit Goal(s) to be addressed in this lesson (type each standard/goal exactly as written in the framework):
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities
SMP.4 Model with mathematics
SMP.7 Look for and make use of structure
Teacher Content and Pedagogy
For many students, it is difficult to translate word problems or verbal descriptions into equations in one variable. Students who are ELL and students with
language-based disabilities may find this especially challenging. A short formative assessment is provided to gauge student skill before beginning the lesson.
Begin with examples that are mathematically simple so that the task focuses on the translation from words to equations rather than on the mathematics. Many
students will be able to solve the problems easily; however, they will not always be able to formulate an equation that describes the situation.
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Similar concepts were addressed in Grades 6 and 7, although they related to expressions rather than equations. Students should know how to read, write, and
evaluate expressions in which letters stand for numbers. They will be familiar with the idea that variables represent unknown quantities in a real-world or
mathematical problems. They can use the properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational
coefficients.
Tape diagrams (also known as “bar modeling”) are an effective visual model for helping students to frame the thinking that is required to develop an equation
leading to the solution. Students need to be explicitly taught this strategy.
A tape diagram is a drawing that looks like a segment of tape, used to illustrate number relationships. They are also known as strip diagrams, bar models or
graphs, fraction strips or length models.
A fruit punch recipe has a 3 to 2 ratio of apple juice to grape juice.
Apple Juice
Grape Juice
This diagram represents any mixture of apple juice and grape juice with a ratio of 3 to 2. The total amount of juice is partitioned into 5 parts of equal size,
represented by the 5 shaded boxes.
For example, if the diagram represents 5 cups of juice, then each box represents 1 cup.
If the diagram represents 15 cups of juice, then each box represents 3 cups.
If the total amount of juice is 1 gallon, than each part represents 1/5 gallon and there are 3/5 gallon of apple juice and 2/5 gallon of grape juice.
The emphasis of this lesson is on describing mathematical and real-world situations algebraically.
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These standards are covered in former grades, but with the CCS’ emphasis on multiple representations and reasoning, it is important for some students to have
additional opportunities to activate prior knowledge to build conceptual understanding.
Based on the Formative Assessment, teachers can choose which models and questions to highlight with their students.
Future lessons will focus on problem-solving strategies.
See: Classwork: Matching Problems, Models, and Equations for additional examples.
Formative assessment: Two problems gauge student experience with translating word problems into pictorial and algebraic representations.
Lesson Sequence:
DAY 1
 Prior to this lesson, you may want to administer the two-problem Formative Assessment to gauge students’ knowledge of translating word problems
into pictorial and algebraic representations.


Opening: Trigger students’ prior knowledge of linear equations by conducting a Popcorn Activity: What do you remember about linear equations?
(Popcorn Activity to encourage mathematical discourse: Give the prompt to students. One student contributes an idea. The student (or the teacher)
immediately calls on another student, who contributes one idea. This continues until as many ideas as possible are heard.)
Pose this problem to students:
Tom made $7 more than Sarah. Together they made $25. How much did Tom make?
How much did Sarah make?
Have students solve the problem individually. Students share solution methods (quick think-pair-share). Chart the different strategies. If there are no
concrete representations, present a model (algebra tiles, cubes, etc.)

Using the problem above, show students how to connect a visual model to an equation.
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
Present the following concrete model to students and ask:
o How can we compare the amount of money that Sarah and Tom have?
o What does the “smiley face” represent? What do the tiles represent?
o How can we represent the “smiley face” algebraically?
o How can we represent Sarah’s money algebraically?
o How can we represent Tom’s money algebraically?
o How do we use these algebraic expressions to write an equation that matches the problem?
x
x+7
x + (x + 7) = 25
Students may need multiple opportunities to practice connecting specific models to a story problem. Have students work alone or in pairs to create models
and equations. Have groups share solutions with the class so that students can begin to build a bank of strategies.

Additional practice problems:
Teachers can use some or all of the problems below to solidify student understanding. The representations shown are merely suggestions; students can
develop their own. All reasonable, accurate models should be accepted.
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1. Mrs. Smith is three times as old as her daughter Hannah. If you add their ages together, the sum is 60. How old is Hannah?
Students may write an equation like this:
H + S = 60 (H = Hannah’s age: S = Mrs. Smith’s age)
However, Hannah’s and Mrs. Smith’s ages cannot be determined from this equation.
Encourage students to think of one variable in terms of the other, as there is a relationship between the two variables that is described in the problem.
Students should think of Mrs. Smith’s age as 3 x Hannah’s age. Some students might also express Hannah’s age as 1/3 of Mrs. Smith’s age.
2. Draw a rectangle where the longer side is twice the length of the
side.
Write a formula to find the perimeter.
If the width is 7 cm, what is the perimeter?
shorter
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3. There are 2/5 as many boys as girls. If there are 50 boys, how many girls are there?
4. Ardu has $500 in his savings account. He wants to save a total of $750. If he can save $25 a week, how many weeks will it take him to reach his goal?
500 + 25(W) = 750
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5. Evan ate 3 more pieces of pizza than Jason, who ate one more piece than Adam. Together, they ate 13 pieces. How many pieces of pizza did Jason eat?
3A + 4 = 13 (Adam ate 3 pieces, so Jason ate 4 pieces)

Classwork: Students complete a practice sheet: Matching Problems, Models, and Equations. Teacher models the first problem to share thinking about how
to move between the problem, visual models, and equations.
DAY 2
 Students independently solve this problem in as many ways as they can:
Laura’s father is four times as old as Laura. Their combined ages are forty. How old is Laura? How old is her father?
Students share with a partner how they approached the problem, including what questions they asked themselves and the steps that they took. They then
work with another pair of partners in a group of four to come to consensus about one strategy that would be helpful to a student trying to solve this kind of
problem. Post each group’s selected strategy for reference throughout the lesson.

Through class discussions, students informally think about a more generalized problem-solving strategy. As students offer strategies and ideas, ask, “Can
you build on what she just said?”
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Ask students guiding questions while engaging in authentic problem-solving situations.
1.
2.
3.
4.
5.
What are we solving for?
What is known? (quantities, values)? What is unknown?
How are the unknown or quantities related to each other or to a value?
Can we write one unknown in terms of another unknown or a value?
How can this situation be represented as an equation?
1. What are we solving for?
2. What is known?
3. How are the unknowns/
values related?
4. Can we write one
unknown in terms of
another?
5. How can this situation be
represented as an
equation?

Laura’s age
Laura’s father’s age
Laura’s father is 4x as old as
she is.
Their combined ages = 40
4 x Laura = father
Laura + father = 40
Laura = L
Laura’s father = 4L
4L + L = 40
5L = 40
How do we choose letter variables?
o Any letter or symbol can stand for an unknown quantity.
o In algebraic thinking, it is conventional to use a letter.
o n and x are frequently used, but it can be helpful to use a letter that helps us to remember the unknown. (Examples: L for Laura’s age, w for
width, m for minutes, d for dollars, etc.)
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
Students work on a problem set: Writing Equations from Word Problems. Encourage students to use manipulatives, pictures, tape diagrams, etc. to help
them understand the meaning of the problem. Circulate among the students, addressing any particularly challenging problem types.

Closing: Quick Write: How do equations help us to interpret word problems?

HW: Assign any unfinished problems from Writing Equations from Word Problems for homework.
Preview outcomes for the next lesson:
Students will be able to write and solve linear equations in one variable.
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Pre-Lesson: Formative Assessment
Name: _________________________________ ____ Class: __________ Date:___________________
Draw a visual representation of each problem below. Then, write the equation.
Example:
Alex bought oranges and grapes. The oranges cost twice as much as the grapes. The total cost was
$6.00. How much did Alex spend on oranges?
Grapes $
Oranges $ $
$ + $ $ = $6
3$ =6
$ =2
Grapes cost $2 and oranges cost $4.
1. In Harristown, 1 out of every 3 people lives in an apartment. The population of Harristown is 480. How
many people do not live in apartments?
2. Rachel is 5 years older than Mike. The sum of Mike’s age and twice Rachel’s age equals 45. How old is
Mike?
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Pre-Lesson: Formative Assessment ANSWER KEY
Draw a visual representation of each problem below. Then, write the equation.
Example:
Alex bought oranges and grapes. The oranges cost twice as much as the grapes. The total cost was
$6.00. How much did Alex spend on oranges?
Grapes $
Oranges $ $
$ + $ $ = $6
3$ =6
$ =2
1.
Grapes cost $2 and oranges cost $4.
In Harristown, 1 out of every 3 people lives in an apartment. The population of Harristown is 480. How
many people do not live in apartments?
Live in an apartment
Do not live in an apartment
+
=
480 therefore the number living in an apartment is 160, while the number of
people not living in an apartment is 320.
2.
Rachel is 5 years older than Mike. The sum of Mike’s age and twice Rachel’s age equals 45. How old is
Mike?
PreMatching
Models,
Lesson
Homework:
Problems,
and Equations
Name: _________________________________ ____ Class: __________ Date:___________________
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WORD PROBLEMS
Pete’s dog weighs 12 pounds more than Jill’s
dog. The dogs weigh 36 pounds together. How
much does Jill’s dog weigh?
Five envelopes each contain the same amount
of money. After $14 is removed, $36 is left.
How much was in each envelope?
Five statues are in a box that weighs one
pound. The total weight of the box and the
statues is 36 pounds. How much does each
statue weigh?
A family of 5 gets a $4 discount on their dinner
bill. The total cost is $36. What would be the
cost for each person with no discount?
Lan has some baseball cards, and his brother
has twice as many as Lan. Together, they have
24 baseball cards. How many does Lan have?
Niko has some baseball cards, and his brother
has 1/3 as many as Niko. Niko has 36 more
than his brother. How many does Niko have?
If Moishe had twice as much money as he has
now, he would have $36. How much does he
have now?
Mike has $36. The amount he has is $2 more
than half the amount his brother has. How
much does his brother have?
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MODELS (PICTURES)
-$14
+2
Niko
Niko’s brother
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EQUATIONS
x + ( x+12) = 36
5x – 14 = 36
5 x + 1 = 36
5x - 4 = 36
x + 2x = 24
2x = 36
𝑥−
1
𝑥 = 36
3
1
𝑥 + 2 = 36
2
Pre-Lesson Homework: Matching Problems, Models, and Equations ANSWER KEY
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Teacher Note: Accept any reasonable visual model for the problem; accept variations in equations as long as
they accurately represent the problem.
1
2
3
4
5
6
7
8
Word Problem
Pete’s dog weighs 12 pounds more than
Jill’s dog. The dogs weigh 36 pounds
together. How much does Jill’s dog
weigh?
Model (Picture)
x + ( x+12) = 36
Five statues are in a box that weighs one
pound. The total weight of the box and
the statues is 36 pounds. How much does
each statue weigh?
5 x + 1 = 36
Lan has some baseball cards, and his
brother has twice as many as Lan.
Together, they have 24 baseball cards.
How many does Lan have?
If Moishe had twice as much money as he
has now, he would have $36. How much
does he have now?
Equation
x + 2x = 24
M
M
=36
Five envelopes each contain the same
amount of money. After $14 is removed,
$36 is left. How much was in each
envelope?
A family of 5 gets a $4 discount on their
dinner bill. The total cost is $36. What
$ $ $ $
would be the cost for each person with no
discount?
Niko has some baseball cards, and his
brother has 1/3 as many as Niko. Niko has Niko
36 more than his brother. How many
Niko’s brother
does Niko have?
Mike has $36. The amount he has is $2
+2
more than half the amount his brother
has. How much does his brother have?
2x = 36
-$14
5x – 14 = 36
$
- $4
5x - 4 = 36
𝑥−
1
𝑥 = 36
3
1
𝑥 + 2 = 36
2
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Pre-Lesson Problem Set: Writing Equations from Word Problems
Name: _________________________________ ____ Class: __________ Date:___________________
Word Problem
1
Shari’s dog weighs 18 pounds more
than Kayla’s dog. The total weight of
the dogs is 93 pounds. How much
does Shari’s dog weigh?
2
When Justin is 14 years older, he will
be 31 years old. How old is he now?
3
Six identical gifts are placed in a box
that weighs one pound. The total
weight is 49 pounds. How much does
each gift weigh?
4
Joe bought a box of cookies. He had
one dozen more at home. When he
divides all the cookies among six
friends, each person gets 7 cookies.
How many in a box?
5
Michael has some baseball cards. His
brother has 1/4 as many as Michael.
Michael has 24 more than his brother.
How many does Michael have?
Visual Model
Equation
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6
After Tom reads 36 pages of his
magazine, he still has 12 pages to
read. How many pages are in the
magazine?
7
Last week, Connie had three times as
much money as she has now. Last
week she had $63. How much money
does he have now?
8
The SHS football team was divided
into two groups, Each group had 26
boys. How many boys are on the
team?
9
Meghan had twice as many math
problems to do as Amy. Together,
they had a total of 33 problems to do.
How many problems did each girl do?
10
Four envelopes each contain the same
amount of money. Twenty-four dollars
is removed, 76 is left. How much
money was in each envelope?
11
Mary has $80. After she buys 6 books,
she has $8 left. What was the average
price of each book?
12
Janice and Patty made $45 selling
lemonade. Because Janice worked
longer than Patty, she kept 1½ times
as much money as Patty. How much
money did Janice make?
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Pre-Lesson Problem Set: Writing Equations from Word Problems
Name: _________________________________ ____ Class: __________ Date:___________________
Word Problem
1
2
Shari’s dog weighs 18 pounds
more than Kayla’s dog. The total
weight of the dogs is 93 pounds.
How much does Shari’s dog
weigh?
When Justin is 14 years older,
he will be 31 years old. How old
is he now?
Visual Model
Equation
(Accept any reasonable visual
model that accurately
represents the problem.)
k + k + 18 = 94
j + 14 = 31
3
4
5
Six identical gifts are placed in a
box that weighs one pound. The
total weight is 49 pounds. How
much does each gift weigh?
Joe bought a box of cookies. He
had one dozen more at home.
When he divides all the cookies
among six friends, each person
gets 7 cookies. How many in a
box?
Michael has some baseball
cards. His brother has 1/4 as
many as Michael. Michael has
24 more than his brother. How
many does Michael have?
6g + 1 = 49
(c + 12)/6 = 7
m = 24 + m/4
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6
7
8
9
10
11
12
After Tom reads 36 pages of his
magazine, he still has 12 pages
to read. How many pages are in
the magazine?
p – 12 = 36
Last week, Connie had three
times as much money as she has
now. Last week she had $63.
How much money does he have
now?
3a = 63
The SHS football team was
divided into two groups, Each
group had 26 boys. How many
boys are on the team?
t/2 = 26
Meghan had twice as many
math problems to do as Amy.
Together, they had a total of 33
problems to do. How many
problems did each girl do?
a + 2a = 33
Four envelopes each contain the
same amount of money.
Twenty-four dollars is removed,
76 is left. How much money was
in each envelope?
4e – 24 = 76
Mary has $80. After she buys 6
books, she has $8 left. What
was the average price of each
book?
80 – 6b = 8
Janice and Patty made $45
selling lemonade. Because
Janice worked longer than
Patty, she kept 1½ times as
much money as Patty. How
much money did Janice make?
m + 3/2 m = 45
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List of Unit Resources
Lesson 1:
Classwork: Problem Set
Verbal Expressions
Equations Homework
Classwork: Solving Equations
Homework: Working Across the Equal Sign
Concrete models:
Algebra tiles (or template for algebra tile cut-outs)
Virtual manipulative algebra tiles are available at:
www.glencoe.com/sites/common_assets/.../VMF-Interface.html
http://illuminations.nctm.org/ActivityDetail.aspx?ID=216
Lesson 2:
Formative Assessment: Shopping Scenarios
Directions for Always, Sometimes, Never Activity
Solution for Categorizing Equations as Always True, Sometimes True, Never True
Ticket to Leave
Homework: Working with Linear Equations
Chart paper
Sticky notes
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Internet access and projector with speakers
www.teachingchannel.org “Sorting and Classifying Equations Overview” video. (9:50) and “Sorting and Classifying Equations: Class Discussion” video. (8:24)
Lesson 3:
Lesson 2 Homework: Working with Linear Equations
Equations, Tables, and Graphs Cards
Homework: Simultaneous Equations Problem Set
Formative Assessment: Creating Systems of Equations
Graphing calculators
Lesson 4:
Sample Card Set
Math Teachers’ Café
Lesson 5:
Classwork: Using the Elimination Strategy
Homework: Solving Systems of Equations Using the Elimination Method
Lesson 6:
Problem Set: Which Way is the Most Efficient?
Homework: Thinking About Systems of Equations
Summative Assessment: Sam’s and Katherine’s Mistakes
Answer Key for Sam’s and Katherine’s Mistakes
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Pre-Lesson:
Formative Assessment
Classwork: Matching Problems, Models, and Equations
Answer Key for Classwork: Matching Problems, Models, and Equations
Problem Set: Writing Equations from Word Problems
Answer Key for Problem Set: Writing Equations from Word Problems
Manipulatives such as blocks and two-color counters
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(CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/
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Curriculum Embedded Performance Assessment (CEPA)
Powering Up Patriot School
The goal is to develop a proposal to guide purchase decisions to upgrade classroom technology at Patriot Middle School.
You are an educational technology consultant hired by the school to determine which plan best meets the needs of the school.
You will present your findings to the District Technology Committee which includes the principal, the superintendent, and members of the school committee.
Two classrooms at Patriot Middle School tested out tablets and laptops in the classroom. One classroom bought 2 tablets and 3 laptops for $1,700 from
EdCompute. The other classroom bought 5 tablets and 2 laptops for $2,600 from EdCompute. The cost of each tablet was the same. The cost of each laptop
was the same.
The new technology was successful. Now the school wants to purchase more technology devices. The goal is to purchase 25 tablets and 10 laptops for each
classroom. The school can continue to purchase technology from EdCompute, or they can purchase technology from SmartTech. SmartTech sells laptops for
the same price as EdCompute. However, they offer tablets for $250 each. At SmartTech, there is also a one-time school contract fee of $10,000.
The district has a budget for each classroom of $13,500. You need to consider the needs of the school, prepare a plan for purchasing equipment, and provide a
budget that makes best use of the funds.
You will develop a proposal (either a written report or a multimedia presentation) for the Technology Committee to help them make their decision.
Your plan should include:
• a cost analysis
• an itemized budget for recommended purchases
• a description of when it is better to buy from one company rather than the other
All parts of the plan must be supported with mathematical evidence (equations, tables, graphs, etc.)
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CEPA Teacher Instructions:
To clarify expectations for the written (or oral) proposal, review the rubric with the students prior to beginning the task.
Be sure to have graphing calculators, graphing software, and graph paper available for student use.
The information given below (#1-4) is a resource for teachers. Students need to be able to figure out the steps for themselves. This includes realizing what
information they will need to find first, how to use a system of equations to find that information, and how to compare the total costs graphically and/or
mathematically.
1. In Part A, students solve a system of equations using any method that they chose. They need to use this information to find the cost of purchasing 25
tablets and 10 laptops for each classroom from each company.
2. In Part B, students need to calculate the cost of buying 25 tablets and 10 laptops from each company. The cost per classroom is then used as the x
coefficient in an equation which can be used to find the total cost for each company.
𝑦 = 𝑚𝑥 + 𝑏 (where y = total cost (C); m = the cost per classroom; x = the number of classrooms (n); and b = the contract fee)
EdCompute:
SmartTech:
C1 = 13,000n
C2 = 9,250n + 10,000
3. In Part C, the students are asked to graph the equations. They may use graphing calculators or graphing software. There is also a blank graph set up as a
scaffolding option. Students may struggle with the scale of the y-axis since the total cost is quite large in comparison to the number of classrooms
represented on the x-axis. Students may have a difficult time distinguishing the break-even point from a hand-drawn graph. Graphing calculators or
software with a trace feature may be helpful.
4. Part D requires students to prepare a proposal. Buying from SmartTech is more cost effective if you are buying for more than 2 classrooms. Students may
also prove this mathematically by setting C1= C2 and solving.
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CEPA Student Instructions:
Powering Up Patriot School
The goal of this task is to develop a proposal for buying new classroom technology at Patriot Middle School. The school can purchase equipment
from either EdCompute or SmartTech.
You are an educational technology consultant hired by the school to determine which company’s plan best meets the needs of the school.
You will present your findings to the District Technology Committee which includes the principal, the superintendent, and members of the school
committee.
Two classrooms at Patriot Middle School tested out tablets and laptops from EdCompute.
The cost of each tablet was the same. The cost of each laptop was the same.
Classroom A bought 2 tablets and 3 laptops for $1,700.
Classroom B bought 5 tablets and 2 laptops for $2,600.
The new technology was successful. Now the school wants to purchase more technology devices for additional classrooms. The goal is to purchase
25 tablets and 10 laptops for each classroom. The school can continue to purchase technology from EdCompute, or they can purchase technology
from SmartTech.
SmartTech sells laptops for the same price as EdCompute. SmartTech’s tablets cost $250 each.
At SmartTech, there is also a one-time school contract fee of $10,000.
The district has a budget of $13,500 per classroom. What would be your advice to the District Technology Committee for the purchase of
classroom technology devices? You should consider the needs of the school, prepare a plan for purchasing equipment, and provide a budget.
Technology devices may be purchased for one classroom, some classrooms, or many classrooms. Your job is to decide which company offers the
best deal.
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Part A: Find the cost of a tablet and a laptop at each company. (Do not include SmartTech’s contract fee.)
Part B: To compare the two companies: Consider the cost per classroom at each company and write a system of equations that models the cost
(C) of the new equipment based on the number of classrooms (n) ordering new equipment.
Part C: Graph both equations on the same graph, comparing the number of classrooms to the total cost - you may use a graphing calculator or
graphing software.
Part D: You will develop a proposal (either a written report or a multimedia presentation) for the Technology Committee to help them make their
decision.
Your plan should include:
• An itemized budget that breaks down the cost of each individual item as well as the total cost for the recommended purchases.
(The itemized budget is usually presented in table form.)
• A cost analysis which describes when it is better to buy from one company rather than the other.
All parts of the plan must be supported with mathematical evidence (equations, tables, graphs, etc.)
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CEPA Rubric
Score
Mathematical Work (CEPA Parts A,B,C)
4
Response demonstrates thorough understanding of how to solve simultaneous equations that represent real world contexts.
 Accurately computes the cost per laptop and per tablet from each company.
 Equations for each company show the total cost for n classrooms.
 The graph models a comparison of the number of classrooms (x axis) to total cost (y axis) for each company, using an appropriate
scale. A point of intersection is specifically identified and can be interpreted so that it is clear which company has the best deal under
various conditions.
3
Response demonstrates understanding of how to solve simultaneous equations that represent real world contexts.
 Accurately computes the cost per laptop and per tablet from each company.
 Equations for each company show the total cost for n classrooms.
 The graph models a comparison of the number of classrooms (x axis) to total cost (y axis) for each company. A point of intersection is
evident and can be interpreted.
2
Response demonstrates some understanding of how to solve simultaneous equations that represent real world contexts.
 Computes the cost per laptop and per tablet from each company but may have minor computational errors that do not detract from
the student’s conceptual understanding.
 Equations for each company show the total cost for n classrooms.
 The graph may be somewhat unclear or may contain an error that affects the solution.
 Without additional work, it is difficult to prepare a convincing proposal for the purchase technology equipment.
1
Response demonstrates limited understanding of how to solve simultaneous equations that represent real world contexts.
 Computes the cost per laptop and per tablet from each company but may have significant computational errors.
 Demonstrates inability to write equations to represent the total cost of n classrooms for both companies.
 The graph is unclear.
 The student requires significant reteaching and support to develop the mathematical understanding to support a viable argument.
0
Response is unrelated to the CEPA task, illegible, not of sufficient length to score, or shows no understanding of the standards addressed.
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(CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/
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Score
4
3
2
1
0
Written Explanation (CEPA Part D)
Proposal makes a highly convincing case for choosing a company based on a cost analysis. The explanation identifies when it is better to buy
from one company rather than the other based on the number of classrooms.
 Cost analysis identifies the point where it becomes more cost efficient to buy from a different company, and connects this
information to the mathematics of the unit (such as the point of intersection and/or the solution of the system).
 The budget is itemized and all computations are correct.
 Writing is exceptionally focused and clear.
 Mathematical language is precise, sophisticated, and present throughout.
Proposal makes a convincing case for choosing a company based on a cost analysis. The explanation identifies when it is better to buy from
one company rather than the other based on the number of classrooms.
 Cost analysis identifies the point where it becomes more cost efficient to buy from a different company.
 The budget is itemized and all computations are correct.
 Writing is focused and clear.
 Mathematical language is present and is used correctly.
Proposal chooses a company based on a cost analysis. The explanation is not backed up by mathematical evidence as to why the selected
company has been chosen.
 A comparison of the two companies is attempted, but is not fully supported or has mathematical inaccuracies.
 The budget is itemized and most computations are correct.
 Written explanation may be somewhat difficult to follow.
 Use of mathematical language is inconsistent and/or infrequent.
 With some support, the proposal could be transformed into a convincing proposal.
Proposal chooses a company based on a cost analysis. Mathematical evidence to support the choice is either missing or inaccurate.
 Cost analysis lacks a comparison between the two companies.
 The budget contains many incorrect computations.
 Written explanation is difficult to follow.
 Use of mathematical language is missing.
 The student requires significant reteaching and support to develop a viable argument.
Response is unrelated to the CEPA task, illegible, not of sufficient length to score, or shows no understanding of the standards addressed.
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(CC BY-NC-SA 3.0). Educators may use, adapt, and/or share. Not for commercial use. To view a copy of the license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/
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Optional Graph for CEPA
Number of Classrooms (n)
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