Algebraic Expressions - glencoe

Transcription

CHAPTE R
5
Algebraic Expressions
The
cconnectED.mcgraw-hill.com
onn
BIG Idea
Investigate
How does the order
of operations help
you evaluate
expressions?
Animations
Vocabulary
Multilingual
eGlossary
Learn
Personal Tutor
sions
Words
Graphing Calculator
Expressions
ic Expres
Algebra
Variables
Virtual
Manipulatives
Make this Foldable
to help you organize
your notes.
Audio
Foldables
Practice
Self-Check Practice
Review Vocabulary
thematical process such as
operation operación a ma
lication, and division
addition, subtraction, multip
operation
Worksheets
+( -
Assessment
)
Key Vocabulary
English
algebra
algebraic expression
evaluate
variable
266
Español
álgebra
expresión numérica
evaluar
variable
When Will I Use This?
Me too!
I heard there’s
a cool movie!
I’m in!
This ad says
the movie is
NOT included in
the family night
admission.
Let me
see
that!
Hey!
So, everyone
wants to go to
the Science
Center?
It costs an extra
$7.50 for a movie
pass on Family
Night!!
I’d like to pass on
the movie pass then...
No way! If we don’t
see the movie, it’s
not worth going!
Hmm... What’s the
BEST deal for us to
go see the movie?
Your Tthuiis rn!
You will sollve
ter.
problem in the chap
267
Are You Ready
for the Chapter?
Text Option
Take the Quick Check below. Refer to the Quick Review for help.
Write the following using exponents.
1. 6 × 6 × 6 × 6 × 6
2. 25 × 25 × 25 × 25 × 25 × 25 × 25
3. 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Evaluate.
4. 73
5. 122
6. 94
7. 27
8. 95
9. 106
EXAMPLE 1
Write 10 × 10 × 10 × 10 × 10 using
exponents.
The base is 10. Since 10 is used as a
factor five times, the exponent is 5.
10 × 10 × 10 × 10 × 10 = 105
EXAMPLE 2
10. GEOGRAPHY This distance from
Columbia, Missouri, to Memphis,
Tennessee, is 192 miles. Find this
distance.
Evaluate 54.
Add or subtract. Write in simplest form.
EXAMPLE 3
5 _
11. _
+5
6
Find 3 7 - 1 1 .
6
4
1
_
13. - _
5
2
3
1
15. 2_ + _
7
4
17. FUNDRAISING
What fraction
more of the
coupon books
did Jabar sell
than Guto?
7 _
12. _
-3
8
8
8 _
14. _
+2
3
9
5
1
16. 3_
- 2_
6
10
Coupon Book Sales
Student
Guto
Billy
Domanick
Jabar
Online Option
268
Y have two options for checking
You
pprerequisite skills for this chapter.
Fraction of
Total Sales
1
_
12
3
_
40
_1
3
2
_
15
The base is 5. The exponent is 4. So, 5 is
used as a factor four times.
54 = 5 × 5 × 5 × 5 or 625
_ _
8
2
7
7
3_
= 3_
8
8
1
4
= 1_
- 1_
Rename using the LCD, 8.
2
8
___________
_
2 3 Subtract.
8
Take the Online Readiness Quiz.
Meaning of Division
Look for these other meanings when
you are solving a word problem.
One meaning of division is to
put objects into equal groups.
But there are other
meanings too.
• To share
Zach and his friend are going to share
3 apples equally. How many apples will
each boy have?
• To take away equal amounts
Isabel is making bookmarks from a piece of
ribbon. Each bookmark is 6.5 centimeters long.
How many bookmarks can she make from a
piece of ribbon that is 26 centimeters long?
26 cm
6.5 cm
6.5 cm
6.5 cm
6.5 cm
• To find how many times greater
The Nile River, the longest river on Earth,
is 4,160 miles long. The Rio Grande River
is 1,900 miles long. About how many times
as long is the Nile than the Rio Grande?
Nile River 4,160 mi
Rio Grande 1,900 mi Rio Grande 1,900 mi
Practice
Identify the meaning of division shown in each problem. Then solve
the problem.
1. The Jackson family wants to buy a flat screen television that costs
$1,200. They plan to pay in six equal payments. What will be the
amount of each payment?
2. A full-grown blue whale can weigh 150 tons. An adult African elephant
weighs about 5 tons. How many times as great does a blue whale
weigh than an African elephant?
3. Each story in an office building is about 4 meters tall. The Eiffel Tower
in Paris, France, is 300 meters tall. How many stories tall is the Eiffel
Tower?
GLE 0606.1.6 Read and interpret the language of mathematics and use written/oral communication to
express mathematical ideas precisely.
Reading Math
269
Multi-Part
Lesson
1
Write and Evaluate Expressions
PART
A
Main Idea
Find the value of
expressions using the
order of operations.
Vocabulary
V
n
numerical
expression
order of operations
Get ConnectED
GLE 0606.3.4
Use expressions, equations
and formulas to solve
problems. SPI 0606.3.2 Use
order of operations and
parentheses to simplify
expressions and solve
problems. Also addresses
GLE 0606.3.3, SPI 0606.3.4.
B
C
D
E
Numerical Expressions
SNACKS The table shows
the cost of different snacks
at a concession stand.
1. How much would
3 boxes of popcorn
cost? 4 sandwiches?
2. Find the total cost of
buying 3 boxes of popcorn
and 4 sandwiches.
Item
Price ($)
Box of Popcorn
2
Juice or Pop
1
Sandwich
4
3. What two operations did you use in Exercises 1 and 2?
Explain how to find the answer to Exercise 2 using these
operations.
A numerical expression like 3 × 2 + 4 × 4 is a combination of
numbers and operations. The order of operations tells you which
operation to perform first so that everyone finds the same value
for an expression.
Order of Operations
1. Simplify the expressions inside grouping symbols, like parentheses.
2. Find the value of all powers.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Use Order of Operations
Find the value of each expression.
4+3×5
4+3×5
= 4 + 15
= 19
10 - 2 + 8
Multiply 3 and 5.
Add 4 and 15.
a. 10 + 2 × 15
270
10 - 2 + 8
= 8 + 8 Subtract 2 from 10 first.
= 16
Add 8 and 8.
b. 16 ÷ 2 × 4
Parentheses and Exponents
20 ÷ 4 + 17 × (9 - 6)
20 ÷ 4 + 17 × (9 - 6) = 20 ÷ 4 + 17 × 3
Subtract 6 from 9.
= 5 + 17 × 3
Divide 20 by 4.
= 5 + 51
Multiply 17 by 3.
= 56
Add 5 and 51.
3 × 62 + 4
Exponents
To ﬁnd the value of
numbers written with
exponents, use the base as
a factor the number of
times indicated by the
exponent.
62 = 6 × 6
= 36
3 × 62 + 4 = 3 × 36 + 4
Find 62.
= 108 + 4
Multiply 3 and 36.
= 112
Add 108 and 4.
d. 24 ÷ 23 + 6
c. 25 × (5 - 2) ÷ 5 - 12
SHOPPING Write an expression
for the total cost of 3 lotions,
2 candles, and 4 lip balms.
Find the total cost.
Words
Expression
Cost of Items
Item
Cost ($)
Lotion
Candle
Lip balm
5
7
2
cost of 3 lotions plus cost of 2 candles plus cost of 4 lip balms
3 × $5
+
2 × $7
+
4 × $2
3 × $5 + 2 × $7 + 4 × $2
= $15 + 2 × $7 + 4 × $2 Multiply 3 and 5.
= $15 + $14 + 4 × $2
Multiply 2 and 7.
= $15 + $14 + $8
Multiply 4 and 2.
= $37
Real-World Link
R
TThe largest hot pretzel
ever baked weighed
40 pounds and was
5 feet across.
The total cost of the items is $37.
e. SNACKS Alexis and 3 friends decide to have a snack at the mall.
Each person buys a hot pretzel for $3, a dipping sauce for $1,
and a drink for $2. Write an expression for the total
cost of the snacks. Then find the total cost.
Lesson 1A Write and Evaluate Expressions
271
Examples 1–4
Example 5
1. 9 + 3 - 5
2. 10 - 3 + 9
3. (26 + 5) × 2 - 15
4. 18 ÷ (2 + 7) × 2 + 1
5. 52 + 8 ÷ 2
6. 19 - (32 + 4) + 6
7. THEATER Tickets to a play cost $10 for members and $24 for nonmembers.
Write an expression to find the total cost of 4 nonmember tickets and
2 member tickets. Then find the total cost.
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1–4
8. 8 + 4 - 3
Example 5
9. 9 + 12 - 15
10. 38 - 19 + 12
11. 22 - 17 + 8
12. 7 + 9 × (3 + 8)
13. (9 + 2) × 6 - 5
14. 63 ÷ (10 - 3) × 3
15. 66 × (6 ÷ 2) + 1
16. 27 ÷ (3 + 6) × 5 - 12
17 55 ÷ 11 + 7 × (2 + 14)
18. 53 - 12 ÷ 3
19. 26 + 62 ÷ 4
20. 15 - 23 ÷ 4
21. 22 ÷ 2 × 32
22. TICKETS Admission to a circus is $16 for adults and $10 for children.
Write an expression to find the total cost of 3 adult tickets and
4 children’s tickets. Then find the total cost.
23. MOVIES Tyree and four friends go to the movies. Each person buys a
movie ticket for $7, a snack for $3, and a drink for $2. Write an
expression for the total cost of the trip to the movies. Then find the
total cost.
B
24. 8 × (24 - 3) + 8
25. 12 ÷ 4 + (52 - 6)
26. 9 + 43 × (20 - 8) ÷ 2 + 6
27. 96 ÷ 42 + (25 × 2) - 15 - 3
1
28. APPLES Addison is making caramel apples. She has 2_
bags of apples.
2
One full bag has 8 apples, and each apple weighs 5 ounces. Write an
expression that could be used to find the total number of ounces of
apples Addison has. Then find the total number of ounces.
29 SNACKS A wholesaler sells rolls of fruit snacks
in two sizes of bags. The table shows the number
of rolls that come in each bag. Write an expression
that could be used to determine the number of
rolls in 3 large bags and 2 small bags. Then find
the number of rolls.
272
Bag
Number
of Rolls
Large
10
Small
5
C 30. FIND THE ERROR Luis is finding 9 - 6 + 2.
Find his mistake and correct it.
9–6+2=9–8
=1
31. REASONING Use the expression 34 - 12 ÷ 2 + 7.
a. Place parentheses in the expression so that the value of the
expression is 18.
b. Place parentheses in the expression to find a value other than 18.
Then find the value of the new expression.
32. CHALLENGE Write an expression with a value of 10. It should contain four
numbers and two different operations.
33. E
WRITE MATH Why is it important to have the order of operations?
TTest Practice
34. Arleta is 2 years younger than Josh,
and Josh is 5 years older than Monica,
who is 9 years old. Which expression
could you use to find Arleta’s age?
36. Kailey wants to buy 4 pencils and
3 notebooks. Which expression shows
how to find the total cost of 4 pencils
and 3 notebooks?
A. 9 + 5 + 2
B. 2 + 9 - 5
Pencils
$0.50
Notebooks
$2.25
C. 9 - 5 + 2
A. 3($0.50) + 4($2.25)
D. 9 + 5 - 2
B. 4($0.50) - 3($2.25)
2
35. Denzel had 3_
boxes of party favors.
5
One full box contained 15 bags of
favors, and each bag had 3 items in it.
Which expression could be used to find
the total number of items Denzel had?
2
F. 3_
+ 15 × 3
5
_
G. 3 2 × 15 × 3
5
_
H. 3 2 × 15 + 3
5
2
(15 + 3)
I. 3_
5
C. 4($0.50) + 3($2.25)
D. 4($0.50) × 3($2.25)
37.
SHORT RESPONSE Pierre evaluates
52 ÷ 5 × 3 + 9 as shown.
52 ÷ 5 × 3 + 9 = 25 ÷ 5 × 3 + 9
=5×3+9
= 5 × 12
= 60
What should Pierre have done
differently to evaluate the expression
correctly? What is the correct answer?
Lesson 1A Write and Evaluate Expressions
273
Multi-Part
Lesson
1
PART
A
Main Idea
Evaluate algebraic
expressions.
Vocabulary
V
a
algebra
variable
evaluate
term
like terms
constant
Get ConnectED
B
C
D
E
Algebra: Variables and
Expressions
ART SUPPLIES A box
contains an unknown
number of crayons.
There are 2 crayons
outside of the box.
The total number of
crayons is represented
by the bar diagram
below.
unknown number of crayons 2 crayons
GLE 0606.3.2
Interpret and represent
algebraic relationships with
variables in expressions,
simple equations and
inequalities. GLE 0606.3.4
SPI 0606.3.2, SPI 0606.3.4,
SPI 0606.3.5.
1. Suppose there are 14 crayons in the box. Find the total number
of crayons.
2. Assume you have two boxes of crayons each with the same
unknown number of crayons inside. Draw a bar diagram that
represents the total number of crayons in both boxes.
Algebra is a language of symbols including variables. A variable
is a symbol, usually a letter, used to represent a number.
Algebraic expressions contain at least one variable and at least
one operation. For example, the expression n + 2 represents the
sum of an unknown number and two.
Any letter can be
used as a variable.
+
The letter x is often used as a variable. To avoid confusion with the
symbol ×, there are other ways to show multiplication.
2·3
5t
st
2 times 3
5 times t
s times t
The variables in an expression can be replaced with any number.
Once the variables have been replaced, you can evaluate, or find
the value of, the algebraic expression.
274
Evaluate Algebraic Expressions
Evaluate 16 + b if b = 25.
16 + b = 16 + 25 Replace b with 25.
= 41
Add 16 and 25.
Evaluate x - y if x = 64 and y = 27.
x - y = 64 - 27
Replace x with 64 and y with 27.
= 37
Subtract 27 from 64.
Evaluate 5t + 4 if t = 3.
Multiplication In algebra,
the symbol · is often used
to represent multiplication,
as the symbol × may
be confused with the
variable x.
5t + 4 = 5 · 3 + 4 Replace t with 3.
= 15 + 4
Multiply 5 and 3.
= 19
Add 15 and 4.
Evaluate each expression if a = 6 and b = 4.
a. a + 8
b. a - b
c. a · b
d. 2a - 5
GEOMETRY In Isabela’s woodworking
class, she learned that the expression
180(n - 2), where n represents the
number of sides of a figure, can be used
to find the sum of the degree measures
of all of the angles. What is the total
number of degrees of the 8-sided picture
frame she is making for a project?
180(n - 2) = 180(8 - 2) Replace n, the number of sides, with 8.
= 180(6)
Subtract.
= 1,080
Multiply.
So, the total number of degrees is 1,080°.
Everyday Use
Variable able to change or
vary, as in variable winds
Math Use
Variable a symbol used to
represent a number
e. FAIRS Admission to a fair is $7 per person, and each ride ticket
costs $2. The total cost for admission and t ride tickets is 7 + 2t.
Find the total cost for admission and 5 ride tickets.
Lesson 1B Write and Evaluate Expressions
275
Examples 1–3
Example 4
Evaluate each expression if m = 4 and z = 9.
1. 3 + m
2. z + 5
3. z - m
4. m - 2
5. 4m - 2
6. 2z + 3
7. PARTY FAVORS The amount of money that remains from a 20-dollar bill
after Malina buys 4 party favors for p dollars each is 20 - 4p. Find the
amount remaining if each favor cost $3.
Examples 1–3
Evaluate each expression if m = 2 and n = 16.
8. m + 10
9. n + 8
10. 9 - m
11. 22 - n
12. n ÷ 4
13. 12 ÷ m
14. n · 3
15. 6 · m
16. m + n
17. n - m
18. n - 6
19. m - 1
Evaluate each expression if a = 4, b = 7, and c = 11.
Example 4
20. b - a
21. c - b
22. 5c + 6
23 2b + 7
24. 3a - 4
25. 4b - 10
26. RACING To find the average speed of a race car, use the expression d ÷ t,
where d represents distance and t represents time. Find the speed s of
a race car that travels 508 miles in 4 hours.
27. FINANCIAL LITERACY A pizza deliverer earns $7 per hour plus $1.50 for
each delivery made. The expression 7h + 1.50d can be used to find the
total earnings after h hours and d deliveries have been made. How
much money will the deliverer earn after working 15 hours and
making 8 deliveries?
B 28. c2 + a
31. 4b ÷ 5
29. b2 - 5c
30. 3a ÷ 4
32. 5b · 2
33. 2ac
34. If y = 4, what is the value of 5y - 3?
35. What is the value of st ÷ (6r) if r = 5, s = 32, and t = 45?
36. MUSIC As a member of a music club, you can order CDs for $14.99
each. The music club also charges $4.99 for each shipment. The
expression 14.99n + 4.99 represents the cost of n CDs. Find the total
cost for ordering 3 CDs.
276
37. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a–b.
I wonder
w
how
mu
much it will
co
cost us.
a. What is the total cost for one individual admission and one
individual movie pass on Family Night?
b. The expression 14.50x can be used to find the total cost for x tickets
on Family Night for admission and the movie. What is the cost for
3 tickets?
Evaluate each expression if x = 3, y = 12, and z = 8.
38. 4z + 8 - 6
39. 6x - 9 ÷ 3
40. 15 + 9x ÷ 3
41. 7z ÷ 4 + 5x
42. y2 ÷ (3z)
43. z2 - (5x)
44. GEOMETRY To find the area of a trapezoid, use
1
the expression _
h(b1 + b2), where h represents
2
the height, b1 represents the length of the top
base, and b2 represents the length of the bottom
base. What is the area of the trapezoidal table?
4 ft
2 ft
3 ft
45 TAXES Julian earns $13.50 per hour. His company
deducts 23% of his pay each week for taxes. Julian uses the expression
0.77(13.50h) to compute his earnings after taxes for the hours h he
works. What will be his earnings after taxes, if he works 40 hours?
C 46. OPEN ENDED Create two algebraic expressions involving
multiplication that have the same meaning.
47. CHALLENGE Isandro and Yvette each have a calculator. Yvette starts at
100 and subtracts 7 each time. Isandro starts at zero and adds 3 each
time. If they press the keys at the same time, will their displays ever
show the same number? If so, what is the number?
48. E WRITE MATH Compare and contrast numerical expressions and
algebraic expressions. Use examples in your explanation.
Lesson 1B Write and Evaluate Expressions
277
TTest Practice
49. The height of the triangle below can
be found using the expression 48 ÷ b
where b is the base of the triangle. Find
the height of the triangle.
51. The table shows the total medal counts
for different countries from the 2008
Summer Olympic games.
Total Medal Count
Country
height
8 ft
base
Germany
41
United States
110
Canada
x
A. 4 ft
France
40
B. 6 ft
Russia
72
Japan
25
C. 8 ft
D. 10 ft
50.
Number of
Medals
GRIDDED RESPONSE The expression
4s can be used to find the perimeter of
a square where s represents the length
of a side. What is the perimeter in
inches of a square with a side length
of 26.2 inches?
Which expression represents the total
number of medals earned by all the
countries listed in the table?
F. 288 - x
G. 2x + 288
H. x - 288
I. 288 + x
More
About Variables and Expressions
When addition or subtraction signs separate an algebraic expression into
parts, each part is called a term. Like terms contain the same variables.
For example, 3x and 6x are like terms. But 4x and 7y are not like terms.
A term without a variable is called a constant. In the expression below,
10 is a constant.
three terms
2x + 10 + 8x
like terms
Identify the like terms in each expression.
52. 5m + 2m + 7
53. 9d + 5 + 2d + 4h
54. 6p + 3 + 2p
55. 9k - 3k + 5
56. x - 7 + 3x + 5x
57. 4c + 8f + 3c + 6
58. 12v - v + 12b + 3b
59. 14x - 7x + 3x + 4
60. 3t + 2h - 8h - 4t + 5
278
Multi-Part
Lesson
1
PART
A
B
C
D
E
Write Expressions
Main Idea
Use models to write
expressons.
Get ConnectED
GLE 0606.3.2
SPI 0606.3.4, SPI 0606.3.5.
BOOKS Lula read 10 books.
Kelly read 4 more books
than Lula.
Lula
10 books
Kelly
10 books
4 books
The expression 10 + 4 represents the number of books Kelly read.
You can use a similar method to write algebraic expressions.
COLLECTIONS Kevin has 6 more baseball cards than Elian. Write
an algebraic expression to represent the number of baseball
cards Kevin has.
Elian has an unknown number of baseball cards c.
Elian
c cards
Kevin has 6 more baseball cards than Elian.
Elian
c cards
Kevin
c cards
6 cards
So, Kevin has c + 6 baseball cards.
TEXT MESSAGES Sam sent 10 fewer messages in July than in
August. Write an algebraic expression to represent the number
of text messages Sam sent in July.
Sam sent an unknown
number of messages
m in August.
Sam sent 10 fewer
messages in July.
So, Sam sent m - 10 messages
in July.
August
m messages
August
m messages
July
m messages
10 fewer
Lesson 1C Write and Evaluate Expressions
279
TRAVEL Ilana’s family drove three times as many miles on Saturday as
they did on Friday. Write an algebraic expression to represent the
number of miles driven on Saturday.
Ilana’s family drove an unknown number of miles m
on Friday.
Friday
m miles
They drove three times as many miles on Saturday.
Friday
m miles
Saturday
m miles
m miles
m miles
So, they drove 3m miles on Saturday.
DOLPHINS A bottlenose dolphin can swim d miles per hour. Humans swim
one third as fast as dolphins. Write an expression that could be used to
find how fast humans can swim.
Dolphins can swim an unknown number of miles per hour d.
d miles per hour
Dolphins
Humans swim one third as fast as dolphins.
d miles per hour
Dolphins
Humans
So, humans can swim d ÷ 3 miles per hour.
the Results
1. Suppose a speedboat can travel twice as fast as a dolphin. Write an
expression that could be used to find how fast the boat can travel.
2. What is another way of writing the speed humans can travel, d ÷ 3?
3. MAKE A CONJECTURE Write an expression that is represented by the
bar diagram.
w
280
w
w
3
and Apply
Write a real-world problem and algebraic expression for each situation
modeled below.
4.
6.
Year 1
p people
Year 2
p people
Dasan
b baseball caps
Dion
b baseball caps
5.
43 people
7.
Sixth
Grade
h inches
Seventh
Grade
h inches
Kent
m square miles
Ames
m square miles
12
2 caps
8.
Harry
m minutes
2 inches
9.
Bag of
Apples
p pounds
Bag of
Oranges
Janice
10. DONATIONS Marcela collected an unknown number of dollars for the
children’s fund. Delmar collected $57 more than Marcela. Write an
algebraic expression that can be used to determine how much Delmar
collected.
11. PLUMBING While installing water pipes, Marco had an unknown length of
piping. If he cut the piping into 5 pieces, write an algebraic expression
that can be used to determine the length of each pipe.
12. HAIR Tamera grew her hair an unknown length to be
cut and donated to charity. Mala has grown her hair
3 inches less than Tamera’s. Write an algebraic
expression that can be used to determine the length
of Mala’s hair.
13. TUTORING Pablo earns an unknown dollar amount
each hour he tutors. If he tutors 3 hours, write an
algebraic expression that can be used to determine
how much money he makes.
Write each algebraic expression as a word phrase.
14. t + 8
15. r - 4
16. 5w
17.
18. 7 + m
19. 9 - p
20. 11 · x
21. y ÷ 10
_c
3
Lesson 1C Write and Evaluate Expressions
281
Multi-Part
Lesson
1
PART
A
Main Idea
Write verbal phrases
as simple algebraic
expressions.
Vocabulary
V
defining the variable
d
Get ConnectED
GLE 0606.3.2
GLE 0606.1.4, SPI 0606.3.4,
SPI 0606.3.5.
B
D
C
E
Algebra: Write Expressions
AIRPORTS Missouri has 8 major
commercial airports. California
has 24 major commercial airports.
1. Alabama has 4 fewer airports
than Missouri. What operation
would you use to determine
how many airports are located
in Alabama? Explain.
2. California has three times as many airports as Georgia. What
operation would you use to find how many airports Georgia
has? Explain.
You can write verbal phrases as algebraic expressions. The first
step is to choose a variable and decide what the variable
represents. This is called defining the variable. To write an
algebraic expression, follow these steps.
1
2
WORDS
3
VARIABLE
Describe the
situation. Use
only the most
important words.
EXPRESSION
Define a variable by
choosing a variable
to represent the
unknown quantity.
Translate your
verbal model
into an algebraic
expression.
Write Verbal Phrases as
Write the phrase eight dollars more than Ryan earned as an
algebraic expression.
Words
eight dollars more than Ryan earned
Variable
Let d represent the number of dollars Ryan earned.
Model
Ryan’s earnings
d
T
d
Expression
282
The expression is d + 8.
8
Write the phrase ten dollars less than the original price as an
Less Than You can write
ten more than a number as
either 10 + p or p + 10.
But ten less than a number
can only be written as
p - 10.
Words
ten dollars less than the original price
Variable
Let p represent the original price.
the original price
Model
p
p
$10
Expression
The expression is p - 10.
Write the phrase as an algebraic expression.
a. 4 points fewer than the Bulls scored
b. the number of inches in an unknown number of feet
c. the total cost of a shirt and an $8 pair of socks
Write Two-Step Expressions
SHOPPING Terri bought 2 bottles of nail polish. Each one cost
the same amount. She also bought a magazine that cost $5.
Write an expression to represent the total amount she spent.
Step 1 The nail polish costs an unknown amount. Define a
variable to represent the cost of the nail polish.
nail polish
d dollars
Step 2 She bought 2 bottles of polish plus a magazine.
Real-World Link
R
It is estimated that
$40 to $50 billion
each year is spent on
cosmetics in the
United States.
nail polish
d dollars
Total amount
d dollars
d dollars
$5
So, the expression is 2 × d + 5 or 2d + 5.
d. FOOTBALL Shawn rushed for a certain number of yards
during a recent football game. Greg rushed for 8 more than
twice Shawn’s yardage. Define a variable and write an
expression to represent the number of yards Greg ran.
Lesson 1D Write and Evaluate Expressions
283
Examples 1 and 2
Define a variable and write each phrase as an algebraic expression.
1. Darren saved four times more money than Elliot
2. Jada read half as many pages as George
3. the width of a box is 4 inches less than the length
4. the cost of a CD and a $12 DVD
Example 3
5. FOOD Shoko and her friends bought three
medium drinks and a box of popcorn at the
movies. Define a variable and write an
expression to represent the total amount of
money they spent.
MovieLand
.FEJVN%SJOL
1PQDPSO
Examples 1 and 2
Define a variable and write each phrase as an algebraic expression.
6. six feet less than the width
7. four times as many apples
8. ten more shoes than Ruben
9
six years less than Tracey’s age
10. Leon studies 6 hours more per week than Theodore
11. the Steelers scored one third of the points that the Panthers scored
12. Sarita has twice as many ringtones as Mary
13. Kellie spent $5 dollars less on dinner than James
Example 3
14. BOWLING Melinda goes bowling on Saturday
afternoons. She bowls three games and pays
for shoe rental. Define a variable and write an
expression to represent the total cost Melinda pays.
Bowl-A-Rama
One Game
Shoe Rental
$2
15 GOVERNMENT The United States House of Representatives has 35 more
members than four times the number of members in the United States
Senate. Define a variable and write an expression to represent the
number of members in the House of Representatives.
B 16.
MULTIPLE REPRESENTATIONS Dani
uses the table to help her convert
measurements when she is sewing.
Number of feet
3
6
9
12
Number of yards
1
2
3
4
a. WORDS Write an expression that describes the relationship between
the number of feet and the number of yards.
b. SYMBOLS Write an expression for the number of yards in f feet.
c. NUMBERS Find the number of yards in 63 feet.
284
C 17. FIND THE ERROR Elisa is writing
an algebraic expression for the
phrase 5 less than a number. Find
her mistake and correct it.
5–n
18. CHALLENGE Wendy earns $2 for every table she serves plus 20% of the
total customer order. Define a variable and write an expression to
represent the amount of money she earns for one table.
19. E WRITE MATH If n represents the amount of songs stored on an
MP3 player, analyze the meaning of the expressions n + 7, n - 2,
4n, and n ÷ 2.
TTest Practice
20. The rates for renting a car are shown.
21.
Glades Car Rental
Cost per day
$19.00
Cost per mile
$0.15
EXTENDED RESPONSE Carmen
bought a plant that is 4 centimeters
tall. Each week, the plant grows an
additional 2 centimeters.
Part A Make a table that shows the
height of the plant after 5 weeks.
Which expression could be used to
find the cost of renting a car for 3 days
and driving m miles?
Part B Write an expression to show the
height of the plant after w weeks.
A. 19 + 0.15m
Part C What is the height of the plant
after 10 weeks?
B. 19m + 3 × 0.15
C. 3 × 19 + 0.15m
D. 3m + 19 × 0.15
22. 6c - 6 + 10
23. 3 + 3b ÷ 2
24.
a3
(Lesson 1B)
÷ (5a)
25. GOLF One package of golf balls costs $28 and a package of
golf tees costs $4. Write an expression to find the total
cost of 2 packages of golf balls and 3 packages of golf tees.
Then find the total cost. (Lesson 1A)
Lesson 1D Write and Evaluate Expressions
285
Multi-Part
Lesson
1
PART
A
B
C
D
E
Problem-Solving Investigation
Main Idea Solve problems by acting them out.
Act It Out
ARIANA: I am organizing cafeteria tables for our
volleyball banquet. The tables are rectangular
and can seat 6 people. If we line up the tables,
we could put more tables in the cafeteria and
seat more people. I wonder how many people we can
seat with four tables.
YOUR MISSION: Act it out to determine the number
of people that can be seated with four tables.
Understand
You know that the rectangular tables can seat 6 people, the tables will be lined
up, and Ariana has four tables.
Plan
Draw a rectangle to represent one table.
Use counters to represent people seated.
Solve
Find the number of people that can be seated at four tables.
Four tables can seat 18 people.
Check
Use the expression 4x + 2, where x represents the number of tables.
So, 4 × 4 + 2 = 18.
1. Explain how the act it out strategy could help you check the
reasonableness of answers.
GLE 0606.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation,
and reasonableness of the solution.
286
•
•
•
•
Act it out.
Make a table.
Make an organized list.
Choose an operation.
8. SCHOOL The birth months of the students
in Miss Desimio’s geography class are
shown. How many more students were
born in June than in August?
Use the act it out strategy to solve
Exercises 2–4.
2. GEOMETRY Assume the pattern continues
in the figures below. Find the number of
squares in Figure 5.
Figure 1
Figure 2
Figure 3
3 TEAMS Twenty-four students will be
divided into four equal-size teams. Each
student will count off, beginning with
the number 1 as the first team. If Nate
is the eleventh student to count off,
to which team number will he be
assigned?
4. RUNNING Mike, Juliana, Dane, and
Dakota are entered in a 4-person relay
race. In how many orders can they run
the relay, if Mike must run last? List
them.
Use any strategy to solve Exercises 5–12.
5. SEATING Six students are sitting at a lunch
table. Two more students arrive, and at
the same time three students leave. How
many students are at the table now?
6. GEOMETRY If the pattern below
continues, how many toothpicks are
used to create Figure 6?
'JHVSF
'JHVSF
Birth Months
June
July
April
March
July
June
October
May
August
June
April
October
May
October
April
September
December
January
9. ANIMALS Chantal has one cat and one
hamster. The cat weighs 9.75 pounds,
and the hamster weighs 1.8 pounds.
About how many times more does the
cat weigh than the hamster?
10. DVDs The table shows the cost of DVD
rentals at a video store. Russell purchases
a 3-night rental and gets a new release
rental for half price. If he has $20 to spend,
how much money will he have left?
DVD Rentals
Type of Rental
Cost
New Release
$4.50
1-Night
$3.75
2-Night
$4.25
3-Night
$5.25
11. SCHOOL Every 8 minutes, Daniela
can study twelve Spanish vocabulary
words. How many Spanish vocabulary
words can she study in 1 hour and
20 minutes?
'JHVSF
7. INTERNET Corbin needs to visit three Web
sites for a homework assignment. In how
many orders can he visit the Web sites?
12. E WRITE MATH Write a real-world
problem that could be solved by using
the act it out strategy. Then explain how
you would act it out.
Lesson 1E Write and Evaluate Expressions
287
Mid-Chapter
Check
(Lesson 1A)
1. 10 - 6 + 20
2. 25 ÷ (15 - 10) × 2
3. 32 + 32 ÷ 2
15. CLIMBING A rock climber is at an altitude
of a feet before she climbs another
80 feet. Write an expression for her new
altitude. (Lesson 1D)
4. 12 - (43 ÷ 8) + 1
5. MULTIPLE CHOICE The cost of a shirt and
pair of pants is shown.
$23
$16
16. MULTIPLE CHOICE Cameron has $5 more
than Necie. If Necie has d dollars, which
expression represents the number of
dollars Cameron has? (Lesson 1D)
F. d - 5
H. 5 - d
G. d + 5
I. 5d
17. Write an expression to describe the
relationship of the data in the table.
(Lesson 1D)
Which of the following expressions can
be used to find the cost of 4 shirts and
3 pairs of pants? (Lesson 1A)
n
40
5
56
7
A. 4 × $16 × 3 × $23
80
10
B. 4 × $16 + 3 × $23
C. 4 + $16 × 3 + $23
D. 4 + $16 + 3 + $23
Evaluate each expression if x = 6.
6. x + 11
7. 54 ÷ x
8. 3x - 7
9. 4(x - 5)
10. 10 + 7x
(Lesson 1B)
11. 2x ÷ 6
Define a variable and then write each
phrase as an algebraic expression. (Lesson 1D)
12. the cost of two DVDs
13. 10 feet less than the height of a tree
14. the number of photos on Lori’s phone
is two fewer than 4 times the photos on
Brad’s phone
288
18. AGE Tia is 8 years younger than her sister
Annette. Annette is y years old. Write an
algebraic expression that describes Tia’s
age. (Lesson 1D)
19. FOOD Each container in an 8-pack of
yogurt contains y ounces. Write an
algebraic expression that describes how
many ounces an 8-pack of yogurt
contains. (Lesson 1D)
20. TRACK The track coach needs to choose a
runner to run 1st, 2nd, and 3rd from 4 of
his top runners. How many different
3-person combinations can he choose?
Use the act it out strategy. (Lesson 1E)
Multi-Part
Lesson
2
PART
Properties
A
Main Idea
Use the Commutative,
Associative, and
Identity Properties to
simplify expressions.
Vocabulary
V
p
properties
equivalent expressions
Get ConnectED
SPI 0606.3.4
Rewrite expressions to
represent quantities in
different ways. SPI 0606.3.7
Use algebraic expressions
and properties to analyze
numeric and geometric
patterns. Also addresses
SPI 0606.1.4.
B
C
Algebra: Properties
BAKING Angelica and Nari are baking
cook
cookies
ki for a bake sale fundraiser.
Angelica baked 6 sheets with 10 cookies
each and Nari baked 10 sheets with
6 cookies each.
1. How many total cookies did
Angelica bake?
2. How many total cookies did
Nari bake?
3. What do you notice about your
answers for Exercises 1 and 2?
4. MAKE A CONJECTURE What can you conclude about the order in
which the factors were multiplied?
Properties are statements that are true for any number. The
expressions 6 × 10 and 10 × 6 are called equivalent expressions
because they have the same value. This illustrates the
Commutative Property.
Properties
Commutative
Properties
Associative
Properties
Identity
Properties
The order in which two numbers are added or
multiplied does not change their sum or product.
7+9=9+7
4·6=6·4
a+b=b+a
a·b=b·a
The way in which three numbers are grouped when
they are added or multiplied does not change their sum
or product.
3 + (9 + 4) = (3 + 9) + 4
8 · (5 · 7) = (8 · 5) · 7
a + (b + c) = (a + b) + c
a · (b · c) = (a · b) · c
The sum of an addend and 0 is the addend. The
product of a factor and 1 is the factor.
13 + 0 = 13
7·1=7
a+0=a
a·1=a
Lesson 2A Properties
289
Use Properties
Determine whether the two expressions are equivalent. If so,
tell what property is applied. If not, explain why.
15 + (5 + 8) and (15 + 5) + 8
The numbers are grouped differently. They are equivalent by
the Associative Property.
So, 15 + (5 + 8) = (15 + 5) + 8.
(20 - 12) - 3 and 20 - (12 - 3)
The expressions are not equivalent because the Associative
Property is not true for subtraction.
So, (20 - 12) - 3 ≠ 20 - (12 - 3).
34 + 0 and 34
The expressions are equivalent by the Identity Property.
So, 34 + 0 = 34.
20 ÷ 5 and 5 ÷ 20
The expressions are not equivalent because the Commutative
Property does not hold for division.
So, 20 ÷ 5 ≠ 5 ÷ 20.
a. 5 × (6 × 3) and (5 × 6) × 3
b. 27 ÷ 3 and 3 ÷ 27
BASKETBALL In a recent season, the Kansas Jayhawks had
15 guards, 4 forwards, and 3 centers on their roster. Write two
equivalent expressions using the Associative Property that
can be used to find the total number of players on their roster.
Real-World Link
R
TThe Kansas Jayhawks
basketball team
has won three
NCAA National
Championships. In
2008 they beat
Memphis 75-68.
290
The Associative Property states that the grouping of numbers
when they are added does not change the sum, so 15 + (4 + 3)
is the same as (15 + 4) + 3.
c. INCOME Brandi earned $7 babysitting and $12 cleaning out
the garage. Write two equivalent expressions using the
Commutative Property that can be used to find the total
amount she earned.
AREA The area of a triangle can
be found using the expression
_1 bh, where b is the base and
12 ft
2
h is the height. Find the area
of the triangle shown.
_1 bh = _1 (15)(12)
2
Mental Math It is easier
to mentally multiply 6 x 15
than 7.5 x 12.
2
1
=_
(12)(15)
2
15 ft
Replace b with 15 and h with 12.
Commutative Property
= 6(15)
Multiply from left to right.
= 90 square feet
Multiply.
The area of the triangle is 90 square feet.
Check
1
You can also use the Associative Property. Find _
(15 · 12).
2
1
Since 15 × 12 is 180, _
of 180 is 90. 2
d. FINANCIAL LITERACY Vickie earned $6 an hour while working
1
11 hours over the weekend. She put _
of what she earned in
3
a savings account. Find how much she has put into the
account.
Examples 1–4
Determine whether the two expressions are equivalent. If so, tell what
property is applied. If not, explain why.
1. (35 + 17) + 43 and 35 + (17 + 43)
2. 0 + 124 and 0
72 - (63 - 8) and (72 - 63) - 8
4. 59 × 1 and 59
3
5. 80 ÷ (10 ÷ 2) and (80 ÷ 10) ÷ 2
Example 5
6. (7 · 25) · 8 and 7 · (25 · 8)
7. FOOD Anita’s mother hosted a party for
Anita’s birthday. The table shows the costs.
Use the Associative Property to write two
equivalent expressions that could be used
to find the total amount spent.
Party Costs
Item
Cost ($)
Cake
12
Hot dogs and
hamburgers
24
Drinks
Example 6
6
8. VACATION Nadia bought suntan lotion for
$12, sunglasses for $15, and a towel for $18.
Use the Associative Property to mentally
find the total of her purchases. Explain the
steps you used.
291
Examples 1–4
Determine whether the two expressions are equivalent. If so, tell what
property is applied. If not, explain why.
9. (8 + 27) + 52 and 8 + (27 + 52)
Example 5
10. 64 + 0 and 64
11. (25 - 9) - 5 and 25 - (9 - 5)
12. 23 · 1 and 23
13. (3 · 6) · 9 and 3 · (6 · 9)
14. 8 ÷ 2 and 2 ÷ 8
15. 36 ÷ (12 ÷ 3) and (36 ÷ 12) ÷ 3
16. 46 + 15 and 15 + 46
17. 0 + 32 and 0
18. 13 · 1 and 1
19. GEOMETRY Find the perimeter of the
triangle shown.
1
Example 6
1
4 2 in.
20. GYMNASTICS At a gymnastics meet, a
gymnast scored an 8.95 on the vault and
a 9.2 on the uneven bars. Write two
equivalent expressions that could be used
to find her total score.
5 2 in.
6 in.
21 TRAVEL Each day, about 75,000 people visit Paris, France. Use the
Commutative Property to write two equivalent expressions that could
be used to find the number of people that visit over a 5-day period.
22. CHARITY Ella sold 37 necklaces for $20 each at the craft fair. She is
going to donate half of the money she earned to charity. Use the
Commutative Property to mentally find how much money she will
donate. Explain the steps you used.
B ALGEBRA Use one or more properties to rewrite each expression as an
expression that does not use parentheses.
23. (y + 1) + 4
24. 2 + (x + 4)
25. 4 + (b + 0)
26. (6 + m) + 9
27. 1 · (n · 8)
28. (6 · r) · 7
29. (w · 12) · 3
30. 20 · (6 · y)
31. (k · 1) · 36
32. MILEAGE The graphic shows
the driving distance between
certain cities in Florida.
a. Write a sentence that
compares the mileage
from Miami to Jacksonville
to Tampa, and the mileage
from Tampa to Jacksonville
to Miami.
b. Refer to part a. Name the
property that is illustrated
by this sentence.
10
Jacksonville
i
Miam
From onville
ks
to Jac miles
8
33
e
onvill
Jacks
m
o
r
F
pa
to Tam es
l
i
m
188
Orlando
75
Tampa
95
4
FLO RID A
ALGEBRA Find the value of x that makes a true statement.
33 17 + x = 3 + 17
292
34. 32 · (2 · 5) = (32 · x) · 5 35. 24 + x = 24
C 36. OPEN ENDED Write two equivalent expressions that illustrate the
Associative Property of Addition.
37. NUMBER SENSE Determine whether (18 + 35) × 4 = 18 + 35 × 4 is
true or false. Explain.
38. CHALLENGE A counterexample is an example showing that a statement
is not true. Provide a counterexample to the following statement.
Division of whole numbers is commutative.
39. E WRITE MATH How are the identity properties for multiplication
and addition the same? different?
TTest Practice
40. Which of the following expressions
could you use to find the total number
of desks at both Medina and Monroe
Middle Schools?
Middle
School
Number of
Classrooms
Number of Desks
per Classroom
Medina
12
25
Monroe
12
25
Yorktown
15
20
41.
EXTENDED RESPONSE Jared deposits
$2 into his savings account every
day for 6 weeks.
Part A Using the Associative Property,
write two equivalent expressions that
could be used to find how much money
he saved after 6 weeks.
Part B How much money will Jared
have deposited in his bank account after
6 weeks? Explain how you solved.
A. (2 × 12) × 25
C. 25(12 + 2)
B. 2 + (12 + 25)
D. 2(12 + 25)
More
About Properties
Another property is known as the Multiplicative Property of Zero.
Multiplicative Property of Zero
Words
The product of a number and zero is zero.
Examples a × 0 = 0
8×0=0
0×a=0
0×8=0
Do not confuse this property with the Identity Property of Addition, a + 0 = a.
Name the property shown by each statement.
42. 6 × 0 = 0
43. 7 + 0 = 7
44. 3 × 1 = 3
45. 8 + 0 = 8
46. 0 × 16 = 0
47. 1 × 20 = 20
293
Multi-Part
Lesson
2
PART
Properties
A
B
C
The Distributive Property
Main Idea
Model the Distributive
Property.
To find the area of a rectangle, multiply the length and
wi
idth
h T
o find the area of a rectangle formed by two smaller
width.
To
rectangles, you can use either one of two methods.
Get ConnectED
Find the area of the blue and yellow rectangles.
SPI 0606.1.5
Model algebraic expressions
using algebra tiles.
SPI 0606.3.4 Rewrite
expressions to represent
quantities in different ways.
Also addresses SPI 0606.1.4,
SPI 0606.3.7.
Method 1
8
Add the lengths. Then multiply.
4
5(8 + 4) = 5(12) Add.
5
= 60
Simplify.
Method 2 Find each area. Then add.
8
4
5 · 8 + 5 · 4 = 40 + 20 Multiply.
5
5
= 60
Simplify.
In Method 1, you found that 5(8 + 4) = 60. In Method 2, you
found that 5 · 8 + 5 · 4 = 60.
So, 5(8 + 4) = 5 · 8 + 5 · 4.
and Apply
Draw a model showing that each equation is true.
1. 2(4 + 6) = (2 · 4) + (2 · 6)
2. 4(3 + 2) = (4 · 3) + (4 · 2)
3. 7(10 + 8) = (7 · 10) + (7 · 8)
4. 6(20 + 3) = (6 · 20) + (6 · 3)
the Results
5. Refer to Exercise 3 above. How could you evaluate 7(18)
mentally?
6. MAKE A CONJECTURE Evaluate 9(33) mentally. Explain the steps
you used.
294
You can also use algebra tiles to model expressions with variables. Refer
to the set of algebra tiles below.
The green rectangular tile
represents the variable x.
x
The yellow square tile
represents the integer 1.
1
Just like 2(3) means 2 groups of 3, 2(x + 1) means 2 groups of x + 1.
x
x
2
1
1
= 2
x
x
+ 2
1
1
Use algebra tiles to tell whether the expressions 2(2x + 1) and 4x + 2
are equivalent.
Model the expression 2(2x + 1).
Y
Y
Y
Y
1
1
There are two groups with
2x + 1 in each group.
Group like tiles together.
1
Y
Y
Y
Y
1
No tiles were added or taken away from the expression 2(2x + 1).
The expressions 2(2x + 1) and 4x + 2 are equivalent.
the Results
Use algebra tiles to tell whether the pairs of expressions are equivalent.
7. 3(x + 1) and 3x + 3
8. 4(x + 1) and 4x + 1
9. 3(2x + 1) and 6x + 2
10. 2(3x + 2) and 6x + 4
11. MAKE A CONJECTURE Use what you learned in this activity to make a
conjecture about the expressions 5(2x + 3) and 10x + 15.
12. REASONING Use what you learned in this activity to rewrite the
expressions below without parentheses.
a. 2(x + 1)
b. 6(x + 4)
c. 3(5x + 6)
13. E WRITE MATH A friend decides that 4(x + 3) = 4x + 3. How would
you explain to your friend that 4(x + 3) = 4x + 12? Include drawings
in your explanation.
Lesson 2B Properties
295
Multi-Part
Lesson
2
PART
Properties
A
Main Idea
Use the Distributive
Property to compute
multiplication
problems mentally and
to rewrite algebraic
expressions.
Get ConnectED
SPI 0606.1.5
Model algebraic expressions
using algebra tiles.
SPI 0606.3.4 Rewrite
expressions to represent
quantities in different ways.
Also addresses SPI 0606.1.4,
SPI 0606.3.7.
B
C
BASEBALL Three friends
went to a baseball game. Each
ticket cost $20 and all three
friends bought a baseball hat
for $15 each.
1. What does the expression
3(20 + 15) represent?
2. Evaluate the expression in
Exercise 1.
3. Evaluate the expression
3 × 20 + 3 × 15. What do
you notice?
The expressions 3(20 + 15) and 3 × 20 + 3 × 15 show how the
Distributive Property combines addition and multiplication.
Distributive Property
Words
To multiply a sum by a number, multiply each addend by
the number outside the parentheses.
Example
Numbers
Algebra
2(7 + 4) = 2 × 7 + 2 × 4
a(b + c) = ab + ac
Compute Products Mentally
_
Find 9 × 4 1 mentally using the Distributive Property.
3
1
1
=9 4+_
9 × 4_
3
3
)
1
= 9(4) + 9(_
3)
Write 4_ as 4 + _.
= 36 + 3
Multiply.
= 39
Add.
(
1
3
1
3
Find each product mentally. Show the steps you used.
a. 5 × 2_
3
5
296
1
b. 12 × 2_
4
c. 2 × 3.6
JEWELRY Francesca is making earrings and bracelets for four
friends. Each pair of earrings uses 4.5 centimeters of wire and
each bracelet uses 13 centimeters. Write two equivalent
expressions using the Distributive Property. Then find how
much total wire is needed.
Multiplication There are
different ways of showing
multiplication.
2×3
2·3
2(3)
Using the Distributive Property, 4(4.5) + 4(13) and 4(4.5 + 13)
are two equivalent expressions.
Method 1
Multiply. Then add.
4(4.5) + 4(13) = 18 + 52 or 70 centimeters
amount of wire
for 4 bracelets
amount of wire for
4 pairs of earrings
Method 2 Add. Then multiply.
4(4.5 + 13) = 4(17.5) or 70 centimeters
amount of wire for one pair
of earrings and one bracelet
Using either method, Francesca needs 70 centimeters of wire.
d. EXERCISE Each day, Martin lifts weights for 10 minutes and
runs on the treadmill for 25 minutes. Use the Distributive
Property to find the total minutes that Martin exercises for
7 days.
Everyday Use
The Distributive Property can also be used to rewrite algebraic
expressions containing variables.
Distribute to divide and
give out in shares
Rewrite Algebraic Expressions
Math Use
Distributive a property in
which multiplication is
applied to addition of two
or more numbers
Use the Distributive Property to rewrite 2(x + 3).
2(x + 3) = 2(x) + 2(3) Distributive Property
= 2x + 6
Multiply.
1
Y
1
1
Y
1
1
1
Use the Distributive Property to rewrite each expression.
e. 8(x + 3)
f. 5(x + 9)
g. 2(x + 3)
Lesson 2C Properties
297
Example 1
1. 4 × 38
Example 2
Example 3
2
2. 9 × 8_
3. 11 × 27
3
4. STATE FAIR Six friends are going to the state fair. The cost of one
admission is $9.50, and the cost for one ride on the Ferris wheel is
$1.50. Write two equivalent expressions and then find the total cost.
Use the Distributive Property to rewrite each algebraic expression.
5. 3(x + 1)
6. 5(x + 8)
7. 4(x + 6)
Example 1
8. 9 × 44
Example 2
1
9. 4 × 5_
8
10. 3 × 3.9
12. DIGITAL CAMERAS The table shows prices for
digital prints. Use the Distributive Property
to find the total cost to print 12 one-hour and
12 next day photos.
11 7 × 3.8
Digital Prints
Cost
Next Day
$0.08
One-Hour
$0.17
13 ANIMALS A coyote can run up to 43 miles per hour while a rabbit can
run up to 35 miles per hour. How many more miles will a coyote run
in six hours than a rabbit at these rates?
Example 3
Use the Distributive Property to rewrite each algebraic expression.
14. 8(x + 7)
15. 6(x + 11)
16. 8(x + 1)
17. 4(x + 2)
B 18. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a–b.
I wonder
wo
what’s the
wh
cheapest.
che
a. Write two equivalent expressions that demonstrate the Distributive
Property for the cost of x tickets for admission and movie passes on
Family Night.
b. Is it less expensive for a youth to pay regular admission with a
movie pass or go on Family Night? Explain.
298
C 19. CHALLENGE Evaluate the expression 0.1(3.7) mentally. Justify your
response using the Distributive Property.
20. OPEN ENDED Write two equivalent expressions involving decimals
that illustrate the Distributive Property.
21. E WRITE MATH A friend rewrote the expression 5(x + 2) as 5x + 2.
Write a few sentences to your friend explaining the error. Then,
rewrite the expression 5(x + 2) correctly.
TTest Practice
22. Which of the following statements
represents the Distributive Property?
A. 7x + 1 = 7(x + 1)
23. Four friends ate lunch together at a
deli. Each friend ordered the items in
the table.
B. 7(x + 1) = 7x + 7
Item
Cost ($)
C. 7x + 7 = 7(x + 7)
Sandwich
2.75
D. 7(x + 1) = x + 7
Drink
1.25
Which expression represents the total
cost of the four meals?
F. 4($4)
H. 4($2.50)
G. 4($3)
I. 4($1.50)
24. LUNCH Maya spends $3 for her lunch and $2 for breakfast each day
Monday through Friday. Use the Associative Property to find how
much money she spends on lunch and breakfast for 4 weeks. (Lesson 2A)
25. SNAILS A snail at the bottom of a 10-foot hole crawls up 3 feet each day,
but slips back 2 feet each night. How many days will it take the snail to
reach the top of the hole and escape? (Lesson 1E)
Define a variable. Then write each phrase as an algebraic expression.
(Lesson 1D)
26. Lee-Ann has volunteered 8 more hours than Tricia
27. the cost of a pair of jeans is 3 times the cost of a book
Evaluate each expression if x = 2, y = 10, and z = 4.
28. 4z - 10
29. y ÷ 5 + x
(Lesson 1B)
30. x2 + (y ÷ x)
Lesson 2C Properties
299
in Engineering
It’s a
p
e
p
i
r
l
y
S
Ride!
Do you love riding the twisting, turning, plunging slides at water parks? Do you
have ideas that would make them more fun and exciting? If so, you should think
about a career designing water slides! Water slide engineers apply engineering
principles, the newest technology, and their creativity to design state-of-the-art
water slides that are both innovative and safe. These engineers are responsible
for designing not only the winding flumes that riders slide down, but also the
pumping systems that allow the slides to have the appropriate flow of water.
Are you interested in a career as a water slide
engineer? Take some of the following courses
in high school.
• Algebra
• Computer-Aided Drafting
• Engineering Calculus
Get ConnectED
300 Algebraic Expressions
• Engineering
Technology
• Physics
GLE 0606.1.7
Use the information in the table to solve
each problem.
Water Slides
1. The table shows the relationship
between the number of minutes
and the gallons of water pumped
out on The Black Hole. Write an
expression to determine the
number of gallons pumped out
for any number of minutes.
Number of
Minutes
(m)
3
6
9
Water Slide, Park
Big Thunder,
Rapids Water Park
The Black Hole,
Wet ‘n Wild
Crush ‘n’ Gusher,
Typhoon Lagoon
Gulf Scream,
Adventure Island
Water
Pumped
Out (g)
3,000
6,000
9,000
Fact
At the steepest drop, riders travel about 30 feet
per second.
Riders plummet 500 feet as water is pumped out
at 1,000 gallons per minute.
The water jet nozzle on each slide pumps out
about 23 gallons of water per second.
Riders hurl down a 210-foot slide at 25 miles
per hour.
2. Refer to the fact about Big Thunder. Define a
variable. Then write an expression that could be
used to find the number of feet that riders travel
in any number of seconds.
3. Write two equivalent expressions that could be
used to find the number of gallons of water
pumped out of the Crush ‘n’ Gusher after
90 seconds. Then determine the number of
gallons pumped in 90 seconds.
4. Explain how you could use the Distributive
Property to find how many gallons of water are
1
pumped out of The Black Hole in 2_ minutes.
2
Problem Solving in Engineering
301
Chapter Study
Guide and Review
Be sure the following
Key Concepts are noted
in your Foldable.
Key Vocabulary
algebra
defining the variable
equivalent expressions
evaluate
numerical expression
order of operations
xpre
ssio
ns
properties
ns
les
ab
Va
ri
Wo
rds
ic E
ssio
ebra
Ex
pre
Alg
variable
Key Concepts
(Lesson 1)
• Simplify the expression inside grouping symbols,
like parentheses.
• Find the value of all powers.
• Multiply and divide in order from left to right.
• Add and subtract in order from left to right.
(Lesson 1)
• Algebraic expressions are combinations of
variables, numbers, and at least one operation.
Properties (Lesson 2)
• Commutative Properties The order in which
numbers are added or multiplied does not change
the sum or product.
5+4=4+5
3·8=8·3
• Associative Properties The way in which
numbers are grouped does not change the sum
or product.
2 + (8 + 4) = (2 + 8) + 4
6 · (3 · 9) = (6 · 3) · 9
• Distributive Property To multiply a sum by a
number, multiply each addend by the number
outside the parentheses.
3(4 + 5) = 3(4) + 3(5)
302
Vocabulary Check
State whether each sentence is true or false.
If false, replace the underlined word or
number to make a true sentence.
1. The expressions 2(x + 7) and 2x + 14 are
equivalent.
2. The Commutative Property can be
2
used to rewrite the expression 3 × 5_
2
as 3(5) + 3 _
.
(3)
3
3. Numerical expressions have the same
value.
4. The expression x + 8 is an example of an
5. The expression 7x means seven more
than x.
6. When following the order of operations,
adding and subtracting are done last.
Multi-Part Lesson Review
Lesson 1
MA.6.A.1.2,
MA.6.A.1.2
(Lesson 1A)
EXAMPLE 1
Find the value of
28 ÷ 2 - 1 × 5.
7. 4 × 6 + 2 × 3
8. 8 + 33 × 4
Use the order of operations.
9. 10 + 15 ÷ 5 - 6
28 ÷ 2 - 1 × 5 = 14 - 1 × 5 Divide 28 by 2.
10. 112 - 6 + 3 × 15
= 14 - 5
Multiply 1 and 5.
=9
Subtract 5 from 14.
11. TRAVEL On a family trip, Natalie
counted 3 groups of 5 motorcycles and
an additional 7 solo motorcycles. Write
an expression for the number of
motorcycles Natalie saw. Then find the
number of motorcycles.
Algebra: Variables and Expressions
MA.6.A.1.2
(Lesson 1B)
Evaluate each expression if a = 18 and
b = 6.
Evaluate 9 - k3 if k = 2.
EXAMPLE 2
9 - k3 = 9 - 23
Replace k with 2.
12. ab
=9-8
23 = 8
13. a2 ÷ b
=1
Subtract 8 from 9.
14. 3b2 + a
15. 2a - 10
Evaluate each expression if x = 6, y = 8,
and z = 12.
16. 2x + 4y
EXAMPLE 3
Evaluate 10 + mn if m = 3
and n = 5.
10 + mn = 10 + (3)(5)
m = 3 and n = 5
= 10 + 15
Multiply 3 and 5. MA.6.A.1.2
= 25
Add 10 and 15.
17. 3z2 + 4x
18. z ÷ 3 + xy
19. HOME REPAIR Lloyd will tile a square
kitchen floor with square ceramic tile.
The number of tiles needed is equal to
a2 ÷ b2, where a is the floor length in
inches and b is the length of the tile in
inches. If a = 96 and b = 8, how many
tiles are needed?
Chapter Study Guide and Review
303
Lesson 1
(continued)
Algebra: Write Expressions
(Lesson 1D)
Define a variable. Then write each phrase
as an algebraic expression.
20. Chad ran 3 fewer miles than Rey
21. the cost of renting a video game and a
$3 late fee
22. the length of a painting is 4 times the
width
23. MEDIA STORAGE
Memory
Type
(GB)
The table shows the
amount of memory
Music
used on Caleb’s MP3
Video
0.5
player. Define a
variable and write an expression to
represent the total memory used on his
MP3 player.
EXAMPLE 4
Bret hit three more home
runs than Elias. Define a variable. Then
write an expression to represent the
number of home runs Bret hit.
Elias
h home runs
Let h represent the number of home runs
Elias hit.
Elias
h home runs
Bret
h home runs
3 home runs
The expression is h + 3.
24. RECYCLING The Environmental Club
collected bags of aluminum cans to
recycle. Each bag weighed 13 pounds.
Define a variable and write an
expression to represent the total weight
of cans collected.
PSI: Act It Out
(Lesson 1E)
Solve. Use the act it out strategy.
25. WALKING Germaine walks at the rate of
10 feet every five seconds while Nolan
walks at the rate of 15 feet every five
seconds. If Germaine has a head start of
25 feet, after how many seconds will
they be at the same spot?
26. GAMES Eight friends are seated in a
circle. Juana walks around the circle
and lightly taps every third person on
the shoulder. How many times does
she need to walk around the circle in
order to have tapped each person on
the shoulder at least once?
304
EXAMPLE 5
Drew wants to center the
word manatee on a square sheet of paper
that is 10 inches long. If the dimensions of
the word are 7 1 by 2 inches, what should
2
be the top and side margins of the word?
_
1
Cut a 7_
-by-2-inch rectangular piece of
2
paper with the word manatee written on it.
Place it roughly in the center of a square
sheet of paper 10 inches long. Measure the
top and side margins to see if they are
equal and adjust if necessary. The top
margin should be 4 inches.
1
inches.
The side margin should be 1_
4
Lesson 2
Properties
Algebra: Properties
(Lesson 2A)
Determine whether the two expressions
are equivalent. If so, tell which property
is applied. If not, explain why.
27. 48 ÷ 6 and 6 ÷ 48
28. 8 + (5 + 9) and (8 + 5) + 9
30. 0 + 14 and 14
31. MAGAZINES Charlie sold 14 magazine
subscriptions, Mike sold 16, and Patty
sold 11. Write two equivalent
expressions that could be used to
find the total subscriptions sold.
32. INCOME Horacio earns $7.50 for each
hour he works at a bookstore. Write
two equivalent expressions that could
be used to find the amount he earns
after working 5 hours. Then find the
amount he earns.
EXAMPLE 7
Ming ran 6 laps each day
for 4 days. Each lap is 1 of a mile. Find
4
the number of miles she ran.
_
The Associative Property states that the
grouping of the factors does not change
the product.
_1 (6)(4) = _1 (4)(6)
4
4
Associative Property
= (1)(6)
Multiply.
=6
Multiply.
Ming ran 6 miles.
(Lesson 2C)
Find each product mentally. Show the
steps you used.
2
33. 7 × 4_
1
34. 9 × 2_
35. 6 × 5.4
36. 8 × 4.6
7
Determine whether the
expressions 23 + 82 and 82 + 23 are
equivalent. If so, tell what property is
applied. If not, explain why.
The order is different. The expressions are
equivalent by the Commutative Property.
So, 23 + 82 = 82 + 23.
29. 17 · 1 and 1
EXAMPLE 6
3
Use the Distributive Property to rewrite
each expression.
37. 4(x + 3)
38. 8(x + 1)
39. 3(3x + 6)
40. 7(x + 5)
41. CAR SHOW Admission to a car show
costs $9.50. Lunch at the Snack Shop
costs $5.50. Find the total cost for four
admissions to the car show and four
lunches at the Snack Shop.
EXAMPLE 8
Find 5 × 6.4 mentally.
5 × 6.4 = 5(6 + 0.4)
Write 6.4 as 6 + 0.4.
= 5(6) + 5(0.4)
= 30 + 2
Multiply.
= 32
Add.
EXAMPLE 9
Use the Distributive
Property to rewrite the expression
6(x + 7).
6(x + 7) = 6(x) + 6(7) Distributive Property
= 6x + 42
Multiply.
305
Practice
Chapter Test
1. MULTIPLE CHOICE Justin raked leaves for
18 hours and mowed lawns for 25 hours.
Justin’s Landscaping
Raking leaves $6/h
Mowing lawns $8/h
Which of the following can be used to
find the total amount Justin earned?
6. PHONES As part of a family cell phone
plan, Monique pays her parents the
monthly fee and the charge for any text
messages she sends. Define a variable.
Then write an expression to represent the
total monthly cost she pays her parents.
Cell Phone Charges
Monthly fee
$9.00
Each text message
$0.02
Evaluate each expression if a = 4 and b = 3.
A. (6 + 18) + (8 + 25)
7. a + 12
8. 27 ÷ b
B. (6 · 18) + (8 · 25)
9. a3 - 2b
10. b2 + 5a
C. (6 + 18) · (8 + 25)
D. (6 · 18) · (8 · 25)
2. 12 - 3 × 2 + 15
11. ART Tia is making a sign with her name
to hang in her bedroom. She wants each
letter of her name to be a different color.
How many different ways can she write
her name using red, green, and yellow
markers?
3. 72 ÷ 23 - 4 × 2
Find each product mentally. Show the steps
you used.
4. (14 + 3) × (6 ÷ 2)
12. 5 × 37
13. 9 × 4.2
14. 6 × 1.6
15. 8 × 43
5. MULTIPLE CHOICE Stella and Raquel
ordered two beverages, two dinners,
and a dessert. Which of the following
expressions can be used to find the amount
each should pay, not including tax?
Item
Cost ($)
Beverage
1.50
Dinner
12.99
Dessert
3.50
F. 1.50 + 2 × 12.99 + 3.50 ÷ 2
G. (2 × 1.50 + 2 × 12.99 + 3.50) ÷ 2
H. 2 × (1.50 + 12.99 + 3.50)
I. (2 × 1.50 + 12.99) + 3.50 ÷ 2
306
Use the Distributive Property to rewrite
each algebraic expression.
16. 3(x + 4)
17. 12(x + 1)
18. 8(x + 6)
19. 5(x + 2)
20.
EXTENDED RESPONSE A class of
30 students is visiting Washington,
D.C. The hotel costs $115 per student, and
the transportation costs $45 per student.
Part A Use the Distributive Property
to write two equivalent expressions that
could be used to find the total cost.
Part B Find the total hotel and
transportation cost for the 30 students.
Explain how you solved.
Preparing for
Standardized Tests
Multiple Choice: Use the Blank Space
in Test Book
You are not required to show your work for multiple-choice questions.
However, you may use the blank space in the test book to do your calculations.
Shaunda bought 3 bags of pretzels at $2.69 per bag and 3 boxes of
crackers at $3.15 per box. Which expression represents the total cost
of Shaunda’s purchase?
A. 3(2.69 × 3.15)
C. 3 + (2.69 × 3.15)
B. 3(2.69 + 3.15)
D. (3 + 2.69) × (3 + 3.15)
cost of 3 bags
of pretzels
3(2.69)
plus
cost of 3 boxes
of crackers
equals
cost of 3 bags
and 3 boxes
+
3(3.15)
=
3(2.69 + 3.15)
So, the cost of 3 bags of pretzels and 3 boxes of crackers is 3(2.69) + 3(3.15)
or 3(2.69 + 3.15).
The correct answer is B.
Mrs. Quinn’s homeroom class
ordered pizzas. They bought 2 of
each type of pizza in the table.
Which expression could be used
to find the total amount of money
spent on pizzas?
Large Pizza Prices
Number of
Toppings
Price
($)
1
8.99
2
10.49
3
11.99
F. 2 + (8.99 + 10.49 + 11.99)
G. 2(8.99 + 10.49 + 11.99)
Some questions
m
seem hard, bu ay
t itt mayy
help you solve
them if you
read them care
ffuulllylyl a
second time.
H. (2 × 8.99) × (2 × 10.49) × (2 × 11.99)
I.
(2 + 8.99) + (2 + 10.49) + (2 + 11.99)
Preparing for Standardized Tests
307
Test Practice
Read each question. Then fill in the correct
answer on the answer sheet provided by
your teacher or on a sheet of paper.
4.
1. The perimeter of a rectangle can be
found using the expression 2ℓ + 2w,
where ℓ represents length and w
represents width. Find the perimeter of
the front of a new building whose design
is shown below.
GRIDDED RESPONSE The Manny
family is installing a large patio in their
backyard. Find the area of the patio in
square yards.
3.7 yd
6.2 yd
90 ft
5. Which of the following illustrates the
Distributive Property?
120 ft
F. 3(2x + 4) = 5x + 4
G. 3(2x + 4) = 5x + 7
A. 180 feet
C. 240 feet
H. 3(2x + 4) = 6x + 4
B. 210 feet
D. 420 feet
I. 3(2x + 4) = 6x + 12
2. Ora bought the variety
pack of granola bars
shown. The box
contains 24 granola
bars. How much
does one granola
bar cost?
F. $0.29
6.
SHORT RESPONSE What is the value
of 45 ÷ (7 + 2) - 1?
7. The table shows the constant speed that
each driver is driving.
G. $0.30
H. $1.35
I. $3.33
3. The cost of renting roller blades is $4
plus $3.50 for each hour that the roller
blades are rented. Which expression can
be used to find the cost in dollars of
renting roller blades for h hours?
Speed (mph)
Using the Distributive Property, how
many more miles will Ms. Santiago drive
in 3 hours than Mr. Reynolds?
A. 4h + 3.5
A. 5 miles
B. 3.5 - 4h
B. 15 miles
C. 3.5(h + 4)
C. 180 miles
D. 3.5h + 4
D. 195 miles
308
8. Each figure below is divided into
sections of equal size. Which figure has
87.5% of its total area shaded?
F.
12. Which expression is equivalent to
5 + 42 × 2?
H.
13.
G.
I.
A. 21 × 2
C. 92 × 2
B. 5 + 32
D. 5 + 82
GRIDDED RESPONSE A garden has
five light poles in the positions shown.
9. Which expression is NOT an example of
the Commutative Property?
A. b - t = t - b
C. f + a = a + f
B. rt = tr
D. 5d = d5
A party planner wants to connect each
light pole directly to each of the other
poles with a string of lights for an
10. At a state fair, each person pays $8 for
admission plus $2 for each ride. While at
the fair, Elizabeth goes on 6 rides. Which
expression can be used to find the total
amount Elizabeth spends?
11.
F. $8 + 6 × $2
H. ($8 + $2) + 6
G. ($8 + $2) × 6
I. $8 × 6 × $2
n(n - 1)
2
outdoor party. The expression _,
where n represents the number of poles,
can be used to determine how many
strings are needed. How many strings
are needed to connect the five poles?
14.
GRIDDED RESPONSE Norene is
buying a coat that is on sale. Write 30%
as a decimal.
)&
EXTENDED RESPONSE Cary earns
$5 per hour raking leaves for her
next-door neighbor. She owes her
mom $6.
Part A Suppose Cary raked leaves
for h hours and paid her mother the
$6 she owed her. Write an expression
to represent the amount of money Cary
will have.
E<<
Part B Use your expression from
Part A to find how much money she
will have if she rakes leaves for 3 hours.
Explain how you solved.
NEED EXTRA HELP?
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3-1D
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1-1E
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GLE 3.4
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GLE 3.4
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Test Practice
309

Algebraic Expressions - glencoe

Transcription

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