LESSON
2.2
Name
Creating and Solving
Equations
2.2
Creating and Solving Equations
Resource
Locker
Explore
The student is expected to:
A-CED.A.1
Creating Equations from Verbal
Descriptions
You can use what you know about writing algebraic expressions to write an equation that represents a real-world
situation.
Create equations… in one variable and use them to solve problems. Also
A-CED.A.3, A-REI.B.3
Suppose Cory and his friend Walter go to a movie. Each of their tickets costs the same amount, and they share a
frozen yogurt that costs $5.50. The total amount they spend is $19.90. How can you write an equation that describes
the situation?
Mathematical Practices
COMMON
CORE
Date
Essential Question: How do you use an equation to model and solve a real-world problem?
Common Core Math Standards
COMMON
CORE
Class
MP.5 Using Tools
A
Identify the important information.
is
Language Objective
The word
Explain to a partner how to solve an equation with the variable on both
sides.
The word total tells you that the operation involved in the relationship is
tells you that the relationship describes an equation.
addition
.
What numerical information do you have? The frozen yogurt costs $5.50,
and the total amount spent is $19.90.
ENGAGE
Use a verbal description or a table to write an
equation, solve the equation, then check that your
answer makes sense.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the various
ways savings can be invested to make more money.
Then preview the Lesson Performance Task.
B
Write a verbal description.
cost
Choose a name for the variable. In this case, use c for
a ticket
The verbal description is: Twice the cost of
© Houghton Mifflin Harcourt Publishing Company
Essential Question: How do you use
an equation to model and solve a
real-world problem?
the cost of each ticket
What is the unknown quantity?
.
plus the cost
of the frozen yogurt equals the total amount .
C
algebraic
To write an equation, write a numerical or
expression for each quantity
and insert an equal sign in the appropriate place. An equation is:
2c + 5.50 = 19.90
.
Reflect
1.
How can you use a verbal model to write an equation for the situation described?
Use the information to write the model
Twice the cost
of a ticket (dollars)
Module 2
+
Cost of frozen
(dollars)
Lesson 2
55
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made throu
be
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EDIT--Chan
DO NOT Key=NL-B;CA-B
Correction
Total cost
=
yogurt (dollars)
Date
Class
Name
Creating
2.2
ion: How
Quest
Essential
A-CED.A.1
A-REI.B.3
COMMON
CORE
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an equat
ons… in one
variable and
use them
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problem?
.A.3,
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bal
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Equatio
Creating
s
Description
Explore
A1_MNLESE368170_U1M02L2.indd 55
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Resource
Locker
HARDCOVER PAGES 4554
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Turn to these pages to
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© Houghto
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
Lesson 2
55
Module 2
L2.indd
0_U1M02
SE36817
A1_MNLE
55
Lesson 2.2
55
14/05/14
11:00 PM
5/14/14 2:20 PM
2.
Could you write a different equation to describe the situation? Explain your reasoning.
Yes; you could write a different but equivalent equation, such as 19.90 - 2c = 5.50, or
EXPLORE
c + c + 5.50 = 19.90.
Explain 1
Creating Equations from Verbal
Descriptions
Creating and Solving Equations Involving
the Distributive Property
When you create an equation to model a real-world problem, your equation may involve the Distributive Property.
When you solve a real-world problem, you should always check that your answer makes sense.

INTEGRATE TECHNOLOGY
Write and solve an equation to solve each problem.
Example 1
Students may complete the Explore activity either in
the book or online.
Aaron and Alice are bowling. Alice’s score is twice the
difference of Aaron’s score and 5. The sum of their scores
is 320. Find each student’s bowling score.
Write a verbal description of the basic situation.
CONNECT VOCABULARY
The sum of Aaron’s score and Alice’s score is 320.
The word equation begins with the root equa-. Have
students think of some other words that begin with
equa-. Discuss what all these words have in common.
Choose a variable for the unknown quantity and write an
equation to model the detailed situation.
Let a represent Aaron’s score. Then 2(a - 5) represents
Alice’s score.
a + 2(a - 5) = 320
QUESTIONING STRATEGIES
Solve the equation for a.
a + 2a - 10 = 320
Distributive Property
3a - 10 = 320
3a - 10 + 10 = 320 + 10
Addition Property of Equality
3a = 330
3a _
330
_
=
3
3
a = 110
Division Property of Equality
So, Aaron’s score is 110 and Alice’s score is 2(a - 5) = 2(110 - 5) = 2(105) = 210.
Check that the answer makes sense.
110 + 210 = 320, so the answer makes sense.

What words in a verbal description could be
represented by the equal sign in an
equation? equals, is, is equal to, is the same as
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Digital
Vision/Getty Images
a + 2(a - 5) = 320
EXPLAIN 1
Creating and Solving Equations
Involving the Distributive Property
QUESTIONING STRATEGIES
Mari, Carlos, and Amanda collect stamps. Carlos has five more stamps than Mari, and
Amanda has three times as many stamps as Carlos. Altogether, they have 100 stamps. Find
the number of stamps each person has.
What is the first step in solving an equation
that contains parentheses? Remove the
parentheses by distributing any factor in front of
the parentheses to all terms within the parentheses.
Write a verbal description of the basic situation.
The total of the numbers of stamps Mari, Carlos, and Amanda have is 100.
Module 2
56
Lesson 2
AVOID COMMON ERRORS
PROFESSIONAL DEVELOPMENT
A1_MNLESE368170_U1M02L2 56
Math Background
In this lesson, students will use the Distributive Property along with the properties
of equality (addition, subtraction, multiplication, and division) to solve equations.
Solving equations involves using inverse operations to get the variable alone on
one side of the equation and the solution on the other side of the equation.
At each step of the solving process, students use properties to derive equivalent
equations, or equations with the same solution set.
3/19/14 3:16 PM
Students may try to multiply a coefficient in front of
parentheses by only one of the enclosed terms.
Remind students to multiply the coefficient by every
term within the parentheses.
Creating and Solving Equations 56
Choose a variable for the unknown quantity and write an equation to model the detailed
situation.
s+5
Let s represent the number of stamps Mari has. Then Carlos has
3(s + 5)
and Amanda has
s+
s+5
(
stamps.
s+5
+3
Solve the equation for s.
s+
s+5
(
s+5
+3
)=
100
)=
100
s + s + 5 + 3s + 15 = 100
Distributive Property
5s + 20 = 100
5
So, Mari has
63
has
16
stamps,
Combine like terms
s=
80
Subtraction Property of Equality
s=
16
Division Property of Equality
21
stamps, Carlos has
stamps, and Amanda
stamps.
Check that the answer makes sense.
16
+ 21
+ 63
= 100
; the answer makes sense.
© Houghton Mifflin Harcourt Publishing Company
100 = 100
Reflect
3.
Would a fractional answer make sense in this situation?
No; a fractional number of stamps does not make sense.
4.
Discussion What might it mean if a check revealed that the answer to a real-world problem did not
make sense?
Possible answers: The equation may have been written incorrectly; the equation may have
been correct, but been solved incorrectly; there may not be any solution to the problem.
Module 2
57
Lesson 2
COLLABORATIVE LEARNING
A1_MNLESE368170_U1M02L2 57
Whole Class Activity
Give each student a card with an expression such as 2x + 4 or 3x – 7 written on it.
Have students pair up and solve the equation that is formed by setting their
expressions equal to each other. Students should pair up with as many students as
time allows.
Peer-to-Peer Activity
Have students work in pairs. Give each pair an equation to solve. Have each
student solve the equation and write an explanation next to each step of the
solution, then have students compare their steps.
57
Lesson 2.2
3/19/14 3:14 PM
Your Turn
5.
A rectangular garden is fenced on all sides with 256 feet of fencing.
The garden is 8 feet longer than it is wide. Find the length and width
of the garden.
2w + 2(w + 8) = 256
2w + 2w + 16 = 256
4w + 16 = 256
4w = 240
w = 60
EXPLAIN 2
(w + 8) feet
Write and solve an equation to solve the problem.
Creating and Solving Equations with
Variables on Both Sides
w feet
Distributive Property
QUESTIONING STRATEGIES
Combining like terms
Subtraction Property of Equality
What words or phrases in a word problem tell
you what the variable in your equation should
represent? how many, what number of
Division property of Equality
Width = 60 ft, length = 60 + 8 = 68 ft
Explain 2
Why should you use the original form of the
equation to check your solution? It is possible
that you made a mistake while transforming the
equation. If you use the original equation, you won’t
repeat the same error.
Creating and Solving Equations with Variables
on Both Sides
In some equations, variables appear on both sides. You can use the properties of equality to collect the variable terms
so that they all appear on one side of the equation.
Example 2

Write and solve an equation to solve each problem.
Janine has job offers at two companies. One company offers a starting salary of $28,000
with a raise of $3000 each year. The other company offers a starting salary of $36,000 with
a raise of $2000 each year. In how many years would Janine’s salary be the same with both
companies? What will the salary be?
AVOID COMMON ERRORS
When students have a variable on both sides of an
equation, they often forget to pay attention to the
sign of the coefficient of the variable. To avoid this
error, encourage students to add and subtract variable
terms so that the result is a positive coefficient. Often
this means to subtract the variable term with the
lesser coefficient on both sides. For example, for the
first step in solving 3x + 2 = x + 10, subtracting x,
rather than 3x, from both sides will result in an
equation with a positive coefficient for x.
Write a verbal description of the basic situation.
Let n represent the number of years it takes for the salaries to be equal.
28,000 + 3000n = 36,000 + 2,000n
28,000 + 3000n - 2000n = 36,000 + 2,000n - 2000n
28,000 + 1000n = 36,000
28,000 + 1000n - 28,000 = 36,000 - 28,000
Subtraction Property of Equality
Combine like terms.
Subtraction Property of Equality
1000n = 8000
1000n
8000
_
= _
Division Property of Equality
1000
1000
n=8
© Houghton Mifflin Harcourt Publishing Company
Base Salary A plus $3000 per year raise = Base Salary B + $2000 per year raise
INTEGRATE TECHNOLOGY
28,000 + 3,000(8) = 36,000 + 2,000(8)
52,000 = 52,000
In 8 years, the salaries offered by both companies will be $52,000.
Module 2
58
Lesson 2
DIFFERENTIATE INSTRUCTION
A1_MNLESE368170_U1M02L2 58
Manipulatives
Put 7 pencils in a closed container. Have one student hold the container in one
hand and 5 pencils in the other hand. Have another student hold 12 pencils. Tell
the class that both students have the same number of pencils. Ask how they can
figure out the number of pencils in the container without opening it. Give or take
pencils away from each student as suggestions are made. When the class has
arrived at an answer, check by counting the number of pencils in the container.
Relate this activity to solving the equation x + 5 = 12.
3/19/14 3:14 PM
A graphing calculator can be used to check the
solution to an equation with variables on both
sides. For example, to solve 3x + 2 = x + 10, enter
Y1 as 3x + 2 and Y2 as x + 10. Graph both lines,
adjusting the window if necessary. Then select the
intersect feature to determine the intersection point
of the two lines. Point out that the x-coordinate of the
intersection point, which is 4, is the solution.
Creating and Solving Equations 58
B
One moving company charges $800 plus $16 per hour. Another moving company charges
$720 plus $21 per hour. At what number of hours will the charge by both companies be the
same? What is the charge?
Write a verbal description of the basic situation. Let t represent the number of hours that the move takes.
Moving Charge A plus $16 per hour = Moving Charge B plus $21 per hour
800 + 16 t = 720 +
800 + 16 t -
21 t
16 t = 720 + 21 t -
16 t
800 = 720 +
5
t
800 - 720 = 720 +
5
t - 720
80 =
Subtraction Property of Equality
t
5
80
5
_ = _t
5
Subtraction Property of Equality
Division Property of Equality
5
t = 16
16
The charges are the same for a job that takes
hours.
Substitute the value 16 in the original equation.
800 + 16t = 720 + 21t
(
800 + 16
16
) = 720 + 21(
16
)
© Houghton Mifflin Harcourt Publishing Company
800 + 256 = 720 + 336
1056
After
16
=
1056
hours, the moving charge for both companies will be
$1056
.
Reflect
6.
Suppose you collected the variable terms on the other side of the equal sign to solve the equation. Would
that affect the solution?
No; the solution of the equation is the value that makes it true. The solution method does
not change the solution.
Module 2
59
Lesson 2
LANGUAGE SUPPORT
A1_MNLESE368170_U1M02L2.indd 59
Visual Cues
Encourage students to break down a written description of a real-world problem
by identifying and highlighting key parts of the description. For example, they
might highlight the unknown quantity in one color and highlight each of the
given quantities in a different color. Then they can look for a phrase meaning is
equal to and identify the two quantities that are equal to one another.
59
Lesson 2.2
5/14/14 2:20 PM
Your Turn
Write and solve an equation to solve each problem.
7. Claire bought just enough fencing to enclose either a
8.
rectangular garden or a triangular garden, as shown. The
two gardens have the same perimeter. How many feet of
fencing did she buy?
(2x - 1)
(3x - 3)
(x - 3)
(2x - 1)
2x
2(3x - 3) + 2(x - 3) = (2x - 1) + (2x - 1) + 2x
6x - 6 + 2x - 6 = 2x - 1 + 2x - 1 + 2x
EXPLAIN 3
A veterinarian is changing the diets of
two animals, Simba and Cuddles. Simba
currently consumes 1200 Calories per day.
That number will increase by 100 Calories
each day. Cuddles currently consumes
3230 Calories a day. That number will
decrease by 190 Calories each day. The
patterns will continue until both animals
are consuming the same number of
Calories each day. In how many days will
that be? How many Calories will each
animal be consuming each day then?
Constructing Equations from an
Organized Table
QUESTIONING STRATEGIES
When you create a table to represent the
information in a real-world problem, what do
the rows and columns represent? Possible answer:
Each row might represent a different person in the
problem, and each column represents the situation
at different times.
1200 + 100n = 3230 - 190n
8x - 12 = 6x - 2
n=7
2x = 10
1200 + 100(7) = 3230 - 190(7)
x=5
1900 = 1900
Substitute the value 5 in the original equation.
2(3(5) - 3) + 2(5 - 3) = (2(5) - 1) + (2(5) - 1) + 2(5)
28 = 28
In 7 days, each animal will be
consuming 1900 Calories per day.
Claire bought 28 feet of fencing.
Explain 3
AVOID COMMON ERRORS
Constructing Equations from an Organized Table
When a problem asks for several related quantities,
such as the ages of three people, students may
become confused and forget which quantity the
variable represents. Constructing a table is especially
helpful in solving this type of problem, because it
forces students to write out how each quantity is
related to the variable.
You can use a table to organize information and see relationships.
Example 3
Construct and solve an equation to solve the problem.
Kim works 4 hours more each day than Jill does, and Jack works 2 hours less each day than Jill does. Over 2
days, the number of hours Kim works is equal to the difference of 4 times the number of hours Jack works
and the number of hours Jill works. How many hours does each person work each day?
Analyze Information
© Houghton Mifflin Harcourt Publishing Company
Identify the important information.
• Kim works
4
hours more per day than Jill does.
• Jack works
2
hours less per day than Jill does.
Formulate a Plan
Make a table using the information given. Let x be the number of hours Jill works in one day.
Hours Worked Per Day
Hours Worked Over 2 Days
2(x + 4)
Kim
x+4
Jill
x
2x
x-2
2(x - 2)
Jack
Over 2 days, the number of hours Kim works is equal to the difference of 4 times the number of
hours Jack works and the number of hours Jill works.
2(x + 4) = 4 · 2(x - 2) - 2x
Module 2
A1_MNLESE368170_U1M02L2 60
60
Lesson 2
14/10/14 7:53 PM
Creating and Solving Equations 60
Solve
ELABORATE
2(x + 4) = 4 · 2(x - 2) - 2x
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Discuss with students how to use inverse
2(x + 4) =
2x +
8 (x - 2) - 2x
= 8x - 16 - 2x
8
2x + 8 =
Distributive Property
6 x - 16
2x + 8 + 16 = 6x - 16 +
operations to solve an equation.
Simplify.
16
Addition Property of Equality
x
Subtraction Property of Equality
2x + 24 = 6x
2x + 24 -
AVOID COMMON ERRORS
x = 6x -
2
24 =
Make sure students understand that they must keep
an equation balanced by adding/subtracting or
multiplying/dividing on both sides of the equation,
not just one side.
4
Jill works
QUESTIONING STRATEGIES
How do you know if a solution to a real-world
problem is correct? You verify that
the original equation was written correctly by
re-reading the problem, and then substitute your
answer for the variable in the original equation.
6
Division Property of Equality
4
=x
hours per day, Kim works 10 hours per day, and Jack works
4
hours per day.
Justify and Evaluate
Substitute x = 6 into the original equation.
2(6 + 4) = 4 · 2(6 - 2) - 2x
( 10 ) = 8(
2
© Houghton Mifflin Harcourt Publishing Company
SUMMARIZE THE LESSON
x
4
x
24 = _
_
6
How is the process of solving an equation with
a leading coefficient of 1 different from
solving an equation with a leading coefficient not
equal to 1? After the variable term is isolated, you
must divide by the leading coefficient if it is not 1.
4
2
4
20 = 20
) - 2(
6
)
Your Turn
Write and solve an equation to solve the problem.
9.
Lisa is 10 centimeters taller than her friend Ian. Ian is 14 centimeters taller than Jim. Every month, their
heights increase by 2 centimeters. In 7 months, the sum of Ian’s and Jim’s heights will be 170 centimeters
more than Lisa’s height. How tall is Ian now?
Lisa
lan
Jim
Height now
Height after 7 months
h + 10
h
h - 14
(h + 10) + 14
h + 14
(h - 14) + 14
(h + 14) + (h - 14) + 14 = (h + 10) + 14 + 170; h = 180; Ian is 180 cm tall now.
Module 2
A1_MNLESE368170_U1M02L2.indd 61
61
Lesson 2.2
61
Lesson 2
5/14/14 2:20 PM
Elaborate
EVALUATE
10. How can you use properties to solve equations with variables on both sides?
Use the Properties of Equality and the Distributive Property to isolate the variable on one
side of the equation using inverse operations.
11. How is a table helpful when constructing equations?
A table can help organize the information and let you recognize relationships and, possibly,
how they change over time, which makes writing an equation easier.
12. When solving a real-world problem to find a person’s age, would a negative solution make sense? Explain.
No, because a person cannot have a negative age.
ASSIGNMENT GUIDE
13. Essential Question Check-In How do you write an equation to represent a real-world situation?
Read the information carefully so that you can write a verbal description, build a verbal model,
or use a table to represent the situation. Use the numerical information you have along with
appropriate algebraic expressions for the unknown quantities to write an equation.
Evaluate: Homework and Practice
Write an equation for each description. Possible answers given.
1.
The sum of 14 and a number is equal to 17.
2.
A number increased by
10 is 114.
14 + x = 17
The difference between a number and 12 is 20.
4.
Ten times the sum of half a number and 6 is 8.
(
Two-thirds a number plus 4 is 7.
6.
_2x + 4 = 7
3
7.
Hector is visiting a cousin who lives 350 miles away. 8.
He has driven 90 miles. How many more miles does
he need to drive to reach his cousin’s home?
d + 90 = 350
Module 2
)
1
x+6 = 8
10 _
2
n - 12 = 20
5.
Tanmayi wants to raise $175 for a school
fundraiser. She has raised $120 so far. How
much more does she need to reach her goal?
m + 120 = 175
Exercise
Explore
Creating Equations from Verbal
Descriptions
Exercises 1–8
Example 1
Creating and Solving Equations
Involving the Distributive Property
Exercises 9–12,
21, 23
Example 2
Creating and Solving Equations with
Variables on Both Sides
Exercises 13–16,
22, 24
Example 3
Constructing Equations from an
Organized Table
Exercises 17–20,
25
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Students can check their solutions for
correctness by substituting the value into the original
equation and verifying that the solution makes the
equation true.
The length of a rectangle is twice its width.
The perimeter of the rectangle is 126 feet.
2(2w) + 2w = 126
Lesson 2
62
A1_MNLESE368170_U1M02L2.indd 62
Practice
n + 10 = 114
© Houghton Mifflin Harcourt Publishing Company
3.
• Online Homework
• Hints and Help
• Extra Practice
Concepts and Skills
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
1–5
1 Recall of Information
MP.2 Reasoning
6–9
2 Skills/Concepts
MP.4 Modeling
10
2 Skills/Concepts
MP.2 Reasoning
11–16
2 Skills/Concepts
MP.4 Modeling
17–19
3 Strategic Thinking
MP.1 Problem Solving
20
3 Strategic Thinking
MP.4 Modeling
5/14/14 2:20 PM
Creating and Solving Equations 62
Write and solve an equation for each situation.
COGNITIVE STRATEGIES
9.
Remind students that an equation is like a balanced
scale. To keep the balance, whatever you do on one
side of the equation, you must also do on the other
side. Tell students that in order to solve the equation,
they have to undo the operations on each side of the
scale. Have students name each operation in an
equation and then identify the operation that
undoes it.
h + 2(h - 6) = 18; h = 10
So, Alice hit 10 home runs and Peter hit
2(10 - 6) = 8 home runs.
10. The perimeter of a parallelogram is 72 meters. The width of
the parallelogram is 4 meters less than its length. Find the length and the
width of the parallelogram.
2ℓ + 2 (ℓ - 4) = 72; ℓ = 20
ℓ meters
(ℓ - 4) meters
The length of the parallelogram is 20 meters and the width is
20 - 4 = 16 meters.
11. One month, Ruby worked 6 hours more than Isaac, and Svetlana worked 4 times as many hours as Ruby.
Together they worked 126 hours. Find the number of hours each person worked.
AVOID COMMON ERRORS
h + (h + 6) + 4(h + 6) = 126; h = 16
Isaac worked 16 hours, Ruby worked 22 hours, and Svetlana worked 88 hours.
12. In one day, Annie traveled 5 times the sum of the number of hours Brian traveled and 2. Together they
traveled 20 hours. Find the number of hours each person traveled.
h + 5(h + 2) = 20; h = 1__23
(
)
Brian traveled for 1__23 hours and Annie traveled for 5 1__23 + 2 = 18__13 hours.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t)©Peter
Kim/Shutterstock
When solving problems using division, some students
may automatically divide the larger number by the
smaller number every time. For example, a student
may try to solve 6x = 3 by dividing both sides by 3.
Remind students that they must isolate the variable,
and that the variable is being multiplied by 6,
not by 3.
In one baseball season, Peter hit twice the difference of the
number of home runs Alice hit and 6. Altogether, they hit
18 home runs. How many home runs did each player hit
that season?
13. Xian and his cousin Kai both collect stamps. Xian has 56 stamps, and
Kai has 80 stamps. The boys recently joined different stamp-collecting
clubs. Xian’s club will send him 12 new stamps per month. Kai’s club
will send him 8 new stamps per month. After how many months will
Xian and Kai have the same number of stamps? How many stamps
will each have?
56 + 12m = 80 + 8m; m = 6
56 + 12(6) = 128 and 80 + 8(6) = 128
After 6 months, they will each have 128 stamps.
14. Kenya plans to make a down payment plus monthly payments in order to buy a motorcycle. At one dealer
she would pay $2,500 down and $150 each month. At another dealer, she would pay $3,000 down and $125
each month. After how many months would the total amount paid be the same for both dealers? What
would that amount be?
2500 + 150m = 3000 + 125m; m = 20
2500 + 150(20) = 5500 and 3000 + 125(20) = 5500
After 20 months, the total amount paid to both dealers will be $5500.
Module 2
Exercise
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Lesson 2.2
Lesson 2
63
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
21
2 Skills/Concepts
MP.4 Modeling
22
3 Strategic Thinking
MP.2 Reasoning
23
2 Skills/Concepts
MP.4 Modeling
24
3 Strategic Thinking
MP.2 Reasoning
25
3 Strategic Thinking
MP.3 Logic
5/14/14 2:20 PM
15. Community Gym charges a $50 membership fee and a $55 monthly fee. Workout Gym charges a
$200 membership fee and a $45 monthly fee. After how many months will the total amount of money
paid to both gyms be the same? What will the amount be?
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Before students solve problems involving
50 + 55m = 200 + 45m; m = 15
50 + 55(15) = 875 and 200 + 45(15) = 875
After 15 months, the total amount paid to both gyms is $875.
perimeter, it may be helpful to review that the
perimeter of a figure is the total distance around its
edges. Ask students how they can calculate the
perimeter of a rectangle. The perimeter of a
rectangle is 2ℓ + 2w, where ℓ is the length and
w is the width.
16. Tina is saving to buy a notebook computer. She has two options. The first option is to put $200 away
initially and save $10 every month. The second option is to put $100 away initially and save $30 every
month. After how many months would Tina save the same amount using either option? How much would
she save with either option?
200 + 10n = 100 + 30n; n = 5 200 + 10(5) = 250 and 100 + 30(5) = 250
After 5 months Tina will save $250 using either option.
Use the table to answer each question.
Starting Salary
Yearly Salary Increase
Company A
$24,000
$3000
Company B
$30,000
$2400
Company C
$36,000
$2000
COGNITIVE STRATEGIES
Point out that a good pattern to use when solving
equations is to clear parentheses, if there are any, by
using the Distributive Property, and then finish
solving by using the properties of equality.
17. After how many years are the salaries offered by Company A and Company B the same?
24,000 + 3,000n = 30,000 + 2,400n; n = 10 years
18. After how many years are the salaries offered by Company B and Company C the same?
30,000 + 2,400n = 36,000 + 2,000n; n = 15 years
19. Paul started work at Company B ten years ago at the salary shown in the table. At the same time, Sharla
started at Company C at the salary shown in the table. Who earned more during the last year? How much
more?
20. George’s page contains twice as many typed words as Bill’s page and Bill’s page contains 50 fewer words
than Charlie’s page. If each person can type 60 words per minute, after one minute, the difference between
twice the number of words on Bill’s page and the number of words on Charlie’s page is 210. How many
words did Bill’s page contain initially? Use a table to organize the information.
George
Charlie
Bill
Initial number
of words
Total number of words
after one minute
2x
x + 50
x
2x + 60
(x + 50) + 60
x + 60
© Houghton Mifflin Harcourt Publishing Company
In the last year, Sharla earned $2000 more than Paul. Paul earned $30,000 + 10($2400) =
$54,000. Sharla earned $36,000 + 10($2000) = $56,000.
2(x + 60) - ((x + 50) + 60) = 210; x = 200
Bill’s page contained 200 words initially.
Module 2
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Lesson 2
3/19/14 3:13 PM
Creating and Solving Equations 64
21. Geometry Sammie bought just enough fencing to border
either a rectangular plot or a square plot, as shown. The
perimeters of the plots are the same. How many meters of
fencing did she buy?
INTEGRATE TECHNOLOGY
Have students use the equation-solving tools
available in graphing calculators or online.
Students should consider how technology can be
used to verify solutions.
(x + 2) meters
(x + 2) meters
2( 3x + 2 ) + 2( x - 1 ) = 4( x + 2 ); x = 1.5
2(3( 1.5 ) + 2) + 2( 1.5 – 1 ) = 14 and 4( 1.5 + 2 ) = 14.
Sammie bought 14 meters of fencing.
(3x + 2) meters
H.O.T. Focus on Higher Order Thinking
(x - 1) meters
22. Justify Reasoning Suppose you want to solve the equation 2a + b = 2a, where a and b are nonzero real
numbers. Describe the solution to this equation. Justify your description.
JOURNAL
There is no solution; Sample answer: If you subtract 2a from each side, you get the
statement b = 0. Since a and b are nonzero real numbers, this statement is false. Therefore,
there is no solution.
23. Multi-Step A patio in the shape of a rectangle, is fenced on all sides with
ℓ feet
134 feet of fencing. The patio is 5 feet less wide than it is long.
Have students write a journal entry in which they
describe how to find the solution of an equation that
contains parentheses and has the variable on both
sides.
a. What information can be used to solve the problem? How can you find
the information?
(ℓ - 5) feet
Possible answer: You need the length and width of the patio. You can write algebraic
expressions for the length and width. Use the expressions in the formula for the perimeter
of a rectangle, and then solve to find the length and the width.
b. Describe how to find the area of the patio. What is the area of the patio?
Find the length and width using the formula for the perimeter of a rectangle. Let ℓ be the
length and let ℓ - 5 be the width. 2ℓ + 2( ℓ - 5 ) = 134; ℓ = 36. The length of the patio is
36 ft and the width is 36 - 5 = 31 ft. The area is 36 × 31 = 1,116 ft 2.
© Houghton Mifflin Harcourt Publishing Company
24. Explain the Error Kevin and Brittany write an equation to represent the following relationship, and
both students solve their equation. Who found the correct equation and solution? Why is the other person
incorrect?
5 times the difference of a number and 20 is the same as half the sum of 4 more than 4
times a number.
Kevin:
Brittany:
1
5(x - 20) = _(4x + 4)
2
1
5(20 - x) = _(4x + 4)
2
5x - 100 = 2x + 2
100 - 5x = 2x + 2
3x - 100 = 2
100 - 7x = 2
3x = 102
-7x = -98
x = 34
x = 14
Kevin is correct. Brittany expressed the difference of a number and 20 incorrectly. Kevin’s
expression x - 20 was correct.
Module 2
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Lesson 2.2
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Lesson 2
5/14/14 2:20 PM
25. What If? Alexa and Zack are solving the following problem.
AVOID COMMON ERRORS
The number of miles on Car A is 50 miles more than the number of miles on Car B, and the number of
miles on Car B is 30 miles more than the number of miles on
Car C. All the cars travel 50 miles in 1 hour. After 1 hour, twice the number of miles on Car A is 70 miles
less than 3 times the number of miles on Car C. How many miles were there on Car B initially?
Because of the need to use parentheses within
parentheses, some students may write the equation
incorrectly. Suggest that students use brackets in
place of the outer parentheses, making the equation.
Alexa assumes there are m miles on Car B. Zack assumes there are m miles on Car C. Will Zack’s answer be
the same as Alexa’s answer? Explain.
Alexa’s
Table
Initial Miles
Miles after
One Hour
Car A
m + 50
( m + 50 ) + 50
Car B
m
Car C
m - 30
Zack’s
Table
Initial Miles
Miles after
One Hour
Car A
m + 80
( m + 80 ) + 50
m + 50
Car B
m + 30
( m + 30 ) + 50
( m - 30 ) + 50
Car C
m
m + 50
2(( m + 50 ) + 50) = 3(( m - 30 ) + 50) - 70
m = 210 miles
1.06(a + 550) = 2[212 + 1.06(2a)].
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Simple interest is interest paid on the original
2(( m + 80 ) + 50) = 3( m + 50 ) - 70; m = 180
principal only. Simple interest can be calculated using
the formula l = Prt, where l is the interest, P is the
principal, r is the interest rate per year, and t is the
time in years. Note that interest at a bank is more
likely to be calculated using compound interest,
which is earned not only on the original principal,
but also on all interest earned previously. Accounts
earning compound interest make more money than
accounts earning simple interest. The formula for
compound interest involves an exponential function,
which students will learn later in this course.
Initially: A: 260 mi, B: 210 mi, C: 180 mi; Zack’s answer is the same as Alexa’s.
Lesson Performance Task
Stacy, Oliver, and Jivesh each plan to put a certain amount of money into their savings accounts that earn simple
interest of 6% per year. Stacy puts $550 more than Jivesh, and Oliver puts in 2 times as much as Jivesh. After a year,
the amount in Stacy’s account is 2 times the sum of $212 and the amount in Oliver’s account. How much does each
person initially put into his or her account? Who had the most money in his or her account after a year? Who had the
least? Explain.
Amount in Account Initially
Amount in Account after One Year
a + 550
1.06(a + 550)
2a
1.06(2a)
Jivesh
a
1.06a
© Houghton Mifflin Harcourt Publishing Company
Stacy
Oliver
1.06(a + 550) = 2(212 + 1.06 (2a))
1.06a + 583 = 424 + 4.24a
583 = 424 + 3.18a
159 = 3.18a
50 = a
Initially: Jivesh $50, Stacy $600, Oliver $100
Stacy: 1.06(50 + 550) = 1.06(600) = 636
Oliver: 1.06(2 · 50) = 1.06(100) = 106
Jivesh: 1.06 · 50 = 53
So Stacy had the most in her account after one year and Jivesh had the least.
Module 2
66
Lesson 2
EXTENSION ACTIVITY
A1_MNLESE368170_U1M02L2 66
Have students research the Rule of 72 and use it to estimate how long it will take
to double an investment of $100 in an account earning 8% interest compounded
once a year. Then ask them how this compares to the value of $100 invested in an
account earning 8% simple interest for the same amount of time. Students should
find that the time needed for an investment to double can be estimated by
dividing 72 by the compound interest rate, so $100 earning 8% compound interest
will double to $200 in 9 years. However, $100 earning 8% simple interest for
9 years is worth only 100 + 100(0.08)(9) = $172.
14/10/14 7:56 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Creating and Solving Equations 66