Table Of ContentWrite a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
The word less suggests subtraction. So, the verbal
1-1 Variables and Expressions expression 6 less a number t can be represented by
the algebraic expression 6 – t.
Write a verbal expression for each algebraic 6. 7 more than 11 times a number
expression.
SOLUTION:
1. 2m
Let n represent a number. The words more than
SOLUTION:
suggest addition and the word times suggests
Because the 2 and the m are written next to each multiplication. So, the verbal expression 7 more than
other, they are being multiplied. So, the verbal 11 times a number can be represented by the
expression the product of 2 and m can be used to algebraic expression 11n + 7.
describe the algebraic expression 2m.
7. 1 minus the quotient of r and 7
2. SOLUTION:
The word minus suggests subtraction and the word
SOLUTION: quotient suggests division. So, the verbal expression
1 minus the quotient of r and 7 can be represented
The expression shows the product of the factors
by the algebraic expression .
4 4
and r . The factor r represents a number raised
8. two fifths of a number j squared
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be SOLUTION:
used to describe the algebraic expression . The words two–fifths of a number suggest
multiplication. The squared means to raise to the
3. a2 – 18b second power. So, the verbal expression two–fifths
of a number j squared can be represented by the
SOLUTION: algebraic expression .
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b 9. n cubed increased by 5
represents 18 times b. So, the verbal expression a
SOLUTION:
squared minus 18 times b can be used to describe
2 The word cubed means to raise to the third power.
the algebraic expression a – 18b.
The words increased by suggest addition. So, the
verbal expression n cubed increased by 5 can be
Write an algebraic expression for each verbal
3
expression. represented by the algebraic expression n + 5.
4. the sum of a number and 14
10. GROCERIES Mr. Bailey purchased some
SOLUTION:
groceries that cost d dollars. He paid with a $50 bill.
Let n represent a number. The word sum suggests Write an expression for the amount of change he will
addition. So, the verbal expression the sum of a receive.
number and 14 can be represented by the algebraic
SOLUTION:
expression n + 14.
To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
5. 6 less a number t $50. So, Mr. Bailey will receive 50 – d in change.
SOLUTION:
Write a verbal expression for each algebraic
The word less suggests subtraction. So, the verbal
expression.
expression 6 less a number t can be represented by
11. 4q
the algebraic expression 6 – t.
SOLUTION:
6. 7 more than 11 times a number
Because 4 and q are written next to each other, they
are being multiplied. So, the verbal expression four
SOLUTION:
times a number q can be used to describe the
Let n represent a number. The words more than
algebraic expression 4q.
suggest addition and the word times suggests
eSolutmionusltMipalnicuaalti-oPno.w Seore,d thbey Cvoegrnbearlo expression 7 more than Page1
11 times a number can be represented by the 12.
algebraic expression 11n + 7.
SOLUTION:
7. 1 minus the quotient of r and 7
Because and y are written next to each other,
SOLUTION:
The word minus suggests subtraction and the word they are being multiplied. So, the verbal expression
quotient suggests division. So, the verbal expression one eighth of a number y can be used to describe
1 minus the quotient of r and 7 can be represented the algebraic expression .
by the algebraic expression .
13. 15 + r
8. two fifths of a number j squared
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. So, the
The words two–fifths of a number suggest
verbal expression 15 plus r can be used to describe
multiplication. The squared means to raise to the
the algebraic expression 15 + r.
second power. So, the verbal expression two–fifths
of a number j squared can be represented by the
14. w – 24
algebraic expression .
SOLUTION:
9. n cubed increased by 5 The expression shows the difference of two terms.
So, the verbal expression w minus 24 can be used to
SOLUTION: describe the algebraic expression w – 24.
The word cubed means to raise to the third power.
The words increased by suggest addition. So, the 15. 3x2
verbal expression n cubed increased by 5 can be
3 SOLUTION:
represented by the algebraic expression n + 5.
The expression shows the product of the factors 3
10. GROCERIES Mr. Bailey purchased some and x2. The factor x2 represents a number raised to
groceries that cost d dollars. He paid with a $50 bill. the second power. So, the verbal expression 3 times
Write an expression for the amount of change he will x squared can be used to describe the algebraic
receive. 2
expression 3x .
SOLUTION:
To find the amount of change Mr. Bailey will 16.
receive, subtract the cost of the groceries, d, from
$50. So, Mr. Bailey will receive 50 – d in change.
SOLUTION:
The expression shows the quotient of two terms. The
Write a verbal expression for each algebraic
4
expression. term r represents a number raised to the fourth
11. 4q power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the
SOLUTION:
Because 4 and q are written next to each other, they algebraic expression .
are being multiplied. So, the verbal expression four
times a number q can be used to describe the
17. 2a + 6
algebraic expression 4q.
SOLUTION:
The expression shows the sum of two terms. The
12.
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times
SOLUTION:
a can be used to describe the algebraic expression
2a + 6.
Because and y are written next to each other,
they are being multiplied. So, the verbal expression 18. r4 ∙ t3
one eighth of a number y can be used to describe
SOLUTION:
the algebraic expression .
The expression shows the product of two factors.
4
The factor r represents a number raised to the
13. 15 + r
3
fourth power. The factor t represents a number
SOLUTION:
raised to the third power. So, the verbal expression
The expression shows the sum of two terms. So, the
the product of a number r raised to the fourth
verbal expression 15 plus r can be used to describe power and a number t cubed can be used to
the algebraic expression 15 + r. describe the algebraic expression r4 ∙ t3.
14. w – 24
Write an algebraic expression for each verbal
expression.
SOLUTION:
19. x more than 7
The expression shows the difference of two terms.
So, the verbal expression w minus 24 can be used to
SOLUTION:
describe the algebraic expression w – 24.
The words more than suggest addition. So, the
verbal expression x more than 7 can be represented
2
15. 3x by the algebraic expression 7 + x.
SOLUTION:
The expression shows the product of the factors 3 20. a number less 35
2 2
and x . The factor x represents a number raised to
SOLUTION:
the second power. So, the verbal expression 3 times
Let n represent a number. The word less suggests
x squared can be used to describe the algebraic
subtraction. So, the verbal expression a number less
2
expression 3x . 35 can be represented by the algebraic expression n
– 35.
16.
21. 5 times a number
SOLUTION: SOLUTION:
The expression shows the quotient of two terms. The Let n represent a number. The word times suggests
4 multiplication. So, the verbal expression 5 times a
term r represents a number raised to the fourth
number can be represented by the algebraic
power. So, the verbal expression r to the fourth
expression 5n.
power divided by 9 can be used to describe the
22. one third of a number
algebraic expression .
SOLUTION:
17. 2a + 6 Let n represent a number. The words one third of a
number suggest multiplication. So, the verbal
SOLUTION:
expression one third of a number can be
The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the represented by the algebraic expression .
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
23. f divided by 10
2a + 6.
SOLUTION:
18. r4 ∙ t3 The words divided by suggest division. So, the
verbal expression f divided by 10 can be
SOLUTION:
The expression shows the product of two factors. represented by the algebraic expression .
4
The factor r represents a number raised to the
3
fourth power. The factor t represents a number 24. the quotient of 45 and r
raised to the third power. So, the verbal expression
SOLUTION:
the product of a number r raised to the fourth
The word quotient suggests division. So, the verbal
power and a number t cubed can be used to
expression the quotient of 45 and r can be
describe the algebraic expression r4 ∙ t3.
represented by the algebraic expression .
Write an algebraic expression for each verbal
expression.
25. three times a number plus 16
19. x more than 7
SOLUTION:
SOLUTION:
Let n represent a number. The word times suggests
The words more than suggest addition. So, the
multiplication, and the word plus suggests addition.
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x. So, the verbal expression three times a number plus
16 can be represented by the algebraic expression
3n + 16.
20. a number less 35
26. 18 decreased by 3 times d
SOLUTION:
Let n represent a number. The word less suggests SOLUTION:
subtraction. So, the verbal expression a number less The word decreased suggests subtraction, and the
35 can be represented by the algebraic expression n word times suggests multiplication. So, the verbal
– 35. expression 18 decreased by 3 times d can be
represented by the algebraic expression 18 – 3d.
21. 5 times a number
27. k squared minus 11
SOLUTION:
Let n represent a number. The word times suggests SOLUTION:
multiplication. So, the verbal expression 5 times a The word squared means a number raised to the
number can be represented by the algebraic second power. The word minus suggests subtraction.
expression 5n. So, the verbal expression k squared minus 11 can
2
be represented by the algebraic expression k – 11.
22. one third of a number
SOLUTION: 28. 20 divided by t to the fifth power
Let n represent a number. The words one third of a SOLUTION:
number suggest multiplication. So, the verbal The words divided by suggest division. So, the
expression one third of a number can be verbal expression 20 divided by t to the fifth power
represented by the algebraic expression . can be represented by the algebraic expression .
23. f divided by 10 29. GEOMETRY The volume of a cylinder is π times
the radius r squared multiplied by the height. Write
SOLUTION:
an expression for the volume.
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
represented by the algebraic expression .
24. the quotient of 45 and r
SOLUTION:
SOLUTION:
The word quotient suggests division. So, the verbal
The words times and multiplied by suggest
expression the quotient of 45 and r can be
multiplication. So, the volume of a cylinder can be
represented by the algebraic expression . written as the algebraic expression πr2h.
30. FINANCIAL LITERACY Jocelyn makes x dollars
25. three times a number plus 16
per hour working at the grocery store and n dollars
SOLUTION: per hour babysitting. Write an expression that
Let n represent a number. The word times suggests describes her earnings if she babysat for 25 hours
multiplication, and the word plus suggests addition. and worked at the grocery store for 15 hours.
So, the verbal expression three times a number plus
SOLUTION:
16 can be represented by the algebraic expression
To write an expression for how much Jocelyn made
3n + 16.
babysitting, multiply the number of hours she babysat,
26. 18 decreased by 3 times d 25, by her hourly rate, n. This can be represented by
the algebraic expression 25n.
SOLUTION:
The word decreased suggests subtraction, and the To write an expression for how much Jocelyn made
word times suggests multiplication. So, the verbal working at the grocery store, multiply the number of
expression 18 decreased by 3 times d can be hours she worked, 15, by her hourly rate, x. This can
represented by the algebraic expression 18 – 3d. be represented by the algebraic expression 15x.
27. k squared minus 11 To write an expression for her total earnings, find the
sum of the amount she earned babysitting and the
SOLUTION:
amount she earned working at the grocery store.
The word squared means a number raised to the
This can be written as the expression 25n + 15x.
second power. The word minus suggests subtraction.
So, the verbal expression k squared minus 11 can Write a verbal expression for each algebraic
be represented by the algebraic expression k2 – 11. expression.
2
31. 25 + 6x
28. 20 divided by t to the fifth power
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. The
The words divided by suggest division. So, the
2
term 6x means six times the square of a number.
verbal expression 20 divided by t to the fifth power
2
So, the algebraic expression 25 + 6x can be
can be represented by the algebraic expression . described by the verbal expression twenty–five plus
six times a number squared.
29. GEOMETRY The volume of a cylinder is π times
the radius r squared multiplied by the height. Write 2
32. 6f + 5f
an expression for the volume.
SOLUTION:
The expression shows the sum of two terms. The
2
term 6f means six times the square of a number.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
SOLUTION: five times the number.
The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be
33.
2
written as the algebraic expression πr h.
SOLUTION:
30. FINANCIAL LITERACY Jocelyn makes x dollars
The expression shows the quotient of two terms. The
per hour working at the grocery store and n dollars
5
per hour babysitting. Write an expression that term 3a means three times a number that has been
describes her earnings if she babysat for 25 hours raised to the fifth power. So, the algebraic expression
and worked at the grocery store for 15 hours. can be described by the verbal expression three
times a number raised to the fifth power divided
SOLUTION: by two.
To write an expression for how much Jocelyn made
babysitting, multiply the number of hours she babysat, 34. CCSS SENSE-MAKING A certain smartphone
25, by her hourly rate, n. This can be represented by family plan costs $55 per month plus additional usage
the algebraic expression 25n. costs. If x is the number of cell phone minutes used
above the plan amount and y is the number of
To write an expression for how much Jocelyn made megabytes of data used above the plan amount,
working at the grocery store, multiply the number of interpret the following expressions.
hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x. a. 0.25x
To write an expression for her total earnings, find the b. 2y
sum of the amount she earned babysitting and the
amount she earned working at the grocery store. c. 0.25x + 2y + 55
This can be written as the expression 25n + 15x.
Write a verbal expression for each algebraic SOLUTION:
expression. a. Since x is the number of cell phone minutes, then
31. 25 + 6x2 0.25x would be the cost of extra minutes at $0.25 per
minute.
SOLUTION:
The expression shows the sum of two terms. The b. Since y is the number of megabytes of data used
term 6x2 means six times the square of a number. above the plan amount, then 2y would be the cost of
2 extra data used at $2 per megabyte.
So, the algebraic expression 25 + 6x can be
described by the verbal expression twenty–five plus
c. The expression 0.25x + 2y + 55 represents the
six times a number squared.
extra minute charges and the extra data usage
charge plus the monthly family plan cost of $55. The
32. 6f2 + 5f expression represents the total monthly cost for the
family.
SOLUTION:
The expression shows the sum of two terms. The
2
term 6f means six times the square of a number. 35. DREAMS It is believed that about of our dreams
The term 5f means five times a number. So, the
2 involve people that we know.
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number. a. Write an expression to describe the number of
dreams that feature people you know if you have d
dreams.
33.
b. Use the expression you wrote to predict the
SOLUTION: number of dreams that include people you know out
The expression shows the quotient of two terms. The of 28 dreams.
5
term 3a means three times a number that has been
raised to the fifth power. So, the algebraic expression
SOLUTION:
can be described by the verbal expression three
a. To write an expression to describe the number of
times a number raised to the fifth power divided
by two. dreams that feature people you know if you have d
dreams, multiply by d or .
34. CCSS SENSE-MAKING A certain smartphone
family plan costs $55 per month plus additional usage
costs. If x is the number of cell phone minutes used
b. To predict the number of dreams that include
above the plan amount and y is the number of
people you know out of 28 dreams, replace d with 28
megabytes of data used above the plan amount,
in the expression .
interpret the following expressions.
a. 0.25x
b. 2y
c. 0.25x + 2y + 55
So, you would predict having 21 dreams that include
people you know.
SOLUTION:
36. SPORTS In football, a touchdown is awarded 6
a. Since x is the number of cell phone minutes, then
points and the team can then try for a point after a
0.25x would be the cost of extra minutes at $0.25 per
touchdown.
minute.
b. Since y is the number of megabytes of data used a. Write an expression that describes the number of
points scored on touchdowns and points after
above the plan amount, then 2y would be the cost of
touchdowns by one team in a game.
extra data used at $2 per megabyte.
c. The expression 0.25x + 2y + 55 represents the b. If a team wins a football game 27–0, write an
extra minute charges and the extra data usage equation to represent the possible number of
charge plus the monthly family plan cost of $55. The touchdowns and points after touchdowns by the
expression represents the total monthly cost for the winning team.
family.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
35. DREAMS It is believed that about of our dreams
teams?
involve people that we know.
SOLUTION:
a. Write an expression to describe the number of a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
dreams that feature people you know if you have d
dreams. expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
b. Use the expression you wrote to predict the
number of dreams that include people you know out
of 28 dreams. b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
SOLUTION:
a. To write an expression to describe the number of c. If a team wins a football game 21–7, then 6T + p
dreams that feature people you know if you have d = 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
dreams, multiply by d or . the game by both teams.
Let T = 4 and p = 4.
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
in the expression .
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
teams.
So, you would predict having 21 dreams that include
people you know. 37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
36. SPORTS In football, a touchdown is awarded 6 with like bases.
points and the team can then try for a point after a
touchdown. a. TABULAR Copy and complete the table.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of b. ALGEBRAIC Write an equation for the pattern
touchdowns and points after touchdowns by the in the table.
winning team.
c. VERBAL Make a conjecture about the exponent
c. If a team wins a football game 21–7, how many of the product of two powers with like bases.
possible number of touchdowns and points after
SOLUTION:
touchdowns were scored during the game by both
a.
teams?
SOLUTION:
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns b. The exponent of the product is the sum of the
by one team in a game. exponents of the factors. So, the algebraic equation
102 × 10x = 10(2 + x) represents the pattern.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns c. The exponent of the product of two powers is the
and points after touchdowns by the winning team. sum of the exponents of the powers with the same
bases.
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns 38. REASONING Explain the differences between an
and points after touchdowns that were scored during algebraic expression and a verbal expression.
the game by both teams.
SOLUTION:
Algebraic expressions include variables, numbers,
Let T = 4 and p = 4.
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
So, it is possible that 4 touchdowns and 4 points after
real-life quantity, such as time in minutes or distance
touchdowns were scored during the game by both
in feet. Then use the variable to write an algebraic
teams.
expression to represent one of your daily activities.
Describe in words what your expression represents,
37. MULTIPLE REPRESENTATIONS In this
and explain your reasoning.
problem, you will explore the multiplication of powers
with like bases.
SOLUTION:
Sample answer: x is the number of minutes it takes to
a. TABULAR Copy and complete the table. walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
b. ALGEBRAIC Write an equation for the pattern three times the sum of n squared and 3. Is either
in the table. of them correct? Explain your reasoning.
c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a.
b. The exponent of the product is the sum of the
SOLUTION:
exponents of the factors. So, the algebraic equation
Consuelo is correct. The verbal expression says that
2 x (2 + x)
10 × 10 = 10 represents the pattern.
the sum of n squared and 3 is multiplied by 3. So,
parentheses are necessary. James left out the
c. The exponent of the product of two powers is the
2
parentheses around n + 3.
sum of the exponents of the powers with the same
bases.
41. CHALLENGE For the cube, x represents a positive
whole number. Find the value of x such that the
38. REASONING Explain the differences between an
volume of the cube and 6 times the area of one of its
algebraic expression and a verbal expression.
faces have the same value.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”. SOLUTION:
The volume of a cube can be found by multiplying
39. OPEN ENDED Define a variable to represent a the length times the width times the height. Because
real-life quantity, such as time in minutes or distance the sides of a cube all have the same length, V = x •
in feet. Then use the variable to write an algebraic 3
x • x, or x . The area of one of the faces of the cube
expression to represent one of your daily activities.
can be found by multiplying the length times the
Describe in words what your expression represents,
2
width. So, A = x • x, or x .
and explain your reasoning.
SOLUTION: To find the value of x such that the volume of the
Sample answer: x is the number of minutes it takes to cube and 6 times the area of one of its faces have
walk between my house and school. 2x + 15 the same value, find a value for x such that x3 = 6x2.
represents the amount of time in minutes I spend
walking each day since I walk to and from school
x x3 = 6x2 Yes/No
and I take my dog on a 15 minute walk.
4 No
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
6 Yes
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
SOLUTION:
own words.
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So, SOLUTION:
parentheses are necessary. James left out the Sample answer: An algebraic expression is a math
2 phrase that contains one or more numbers or
parentheses around n + 3.
variables. To write an algebraic expression from real
world situation, first assign variables. Then determine
41. CHALLENGE For the cube, x represents a positive
the arithmetic operations done on the variables.
whole number. Find the value of x such that the
Finally, put the terms in order.
volume of the cube and 6 times the area of one of its
faces have the same value.
43. Which expression best represents the volume of the
cube?
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
A the product of three and five
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
B three to the fifth power
2
width. So, A = x • x, or x .
C three squared
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
D three cubed
the same value, find a value for x such that x3 = 6x2.
SOLUTION:
x x3 = 6x2 Yes/No The volume of a cube can be found by multiplying
the length times the width times the height. Because
4 No
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
6 Yes
So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation. F 2lw
Include a definition of algebraic expression in your
own words.
G l + w
SOLUTION:
Sample answer: An algebraic expression is a math H 2l + 2w
phrase that contains one or more numbers or
variables. To write an algebraic expression from real J 4(l + w)
world situation, first assign variables. Then determine
SOLUTION:
the arithmetic operations done on the variables.
Finally, put the terms in order. To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression
43. Which expression best represents the volume of the 2l + 2w best represents the perimeter of the
cube? rectangle.
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the
length of the window l.
A the product of three and five
SOLUTION:
The word times suggests multiplication and the words
B three to the fifth power
divided by suggest division. So, represents the
C three squared
yards of fabric needed in terms of the length of the
window l.
D three cubed
46. GEOMETRY Find the area of the rectangle.
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
A 14 square meters
So, Choice D is the correct answer.
B 16 square meters
44. Which expression best represents the perimeter of
the rectangle?
C 50 square meters
D 60 square meters
SOLUTION:
F 2lw
G l + w
So, the area of the rectangle is 16 square meters.
H 2l + 2w
Choice B is the correct answer.
J 4(l + w)
47. AMUSEMENT PARKS A roller coaster
SOLUTION: enthusiast club took a poll to see what each
To find the perimeter of a rectangle, find the sum of member’s favorite ride was.. Make a bar graph of
twice the length and twice the width. The expression the results.
2l + 2w best represents the perimeter of the
rectangle. Our Favorite Rides
Number
Ride
Choice H is the correct answer. of Votes
Big Plunge 5
45. SHORT RESPONSE The yards of fabric needed
Twisting
to make curtains is 3 times the length of a window in 22
Time
inches, divided by 36. Write an expression that
The Shiner 16
represents the yards of fabric needed in terms of the
length of the window l. Raging Bull 9
The
SOLUTION: 25
Bat
The word times suggests multiplication and the words
Teaser 6
divided by suggest division. So, represents the The
12
Adventure
yards of fabric needed in terms of the length of the
window l.
SOLUTION:
46. GEOMETRY Find the area of the rectangle.
Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
the roller coaster chosen.
A 14 square meters
B 16 square meters
C 50 square meters
D 60 square meters
SOLUTION: 48. SPORTS The results for an annual 5K race are
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
race.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
Our Favorite Rides
SOLUTION:
Number
Ride Order the data from least to greatest. The times in
of Votes
order from least to greatest are 14:48, 14:58, 15:06,
Big Plunge 5 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
Twisting 20:47, 20:49, 21:35.
22
Time The times are given in minutes and seconds. Rewrite
The Shiner 16 the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
Raging Bull 9
seconds.
The
25
Bat
Time in Time in
Teaser 6
Min:Sec Seconds
The
14:48 888
12
Adventure
14:58 898
15:06 906
SOLUTION: 15:48 948
15:54 954
Draw a bar to represent each roller coaster. The
16:10 970
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies 16:30 990
the roller coaster chosen. 19:27 1167
19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
48. SPORTS The results for an annual 5K race are 2
shown below. Make a box-and-whisker plot for the Q = 1239
3
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the There are no outliers.
race.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35. So, the mean is 5.6.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying Order the data from least to greatest.
the number of minutes by 60 and then adding the {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
seconds. Because there is an even number of data, the median
is the mean of 6 and 7.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898
15:06 906
15:48 948 So, the median is 6.5.
15:54 954
16:10 970 The number 7 appears most often, so the mode is 7.
16:30 990
19:27 1167
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
19:58 1198
20:21 1221 SOLUTION:
20:39 1239
20:47 1247
20:49 1249
21:35 1295
So, the mean is 0.4.
Then determine the quartiles.
Order the data from least to greatest.
Q = 948 {–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
1
Q = 1078.5
2 Because there is an even number of data, the median
Q = 1239 is the mean of 0 and 0.
3
There are no outliers.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
Find the mean, median, and mode for each set modes are 0 and –1.
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8} 51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION: SOLUTION:
So, the mean is 5.6. So, the mean is 15.25.
Order the data from least to greatest. Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}. {3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 6 and 7. Because there is an even number of data, the median
is the mean of 15 and 16.
So, the median is 6.5.
So, the median is 15.5.
The number 7 appears most often, so the mode is 7.
The number 24 appears most often, so the mode is
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0} 24.
SOLUTION: 52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
The area of Lisa’s trampoline is 72 square feet.
Because there is an even number of data, the median
is the mean of 0 and 0. Find each product or quotient.
53.
SOLUTION:
Multiply the numerators and denominators.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION: 54.
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
55.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
Evaluate each expression.
trampoline in square feet?
56.
SOLUTION:
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
SOLUTION:
Multiply the numerators and denominators.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
54.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
SOLUTION: with a common denominator of 6.
59. 10.34 + 14.27
55. SOLUTION:
10.34 + 14.27 = 24.61
SOLUTION:
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
Evaluate each expression.
56.
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
61. 37.02 – 15.86
with denominators of 45 then add the numerators.
SOLUTION:
37.02 – 15.86 = 21.16
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
Write a verbal expression for each algebraic
The word less suggests subtraction. So, the verbal
expression.
expression 6 less a number t can be represented by
1. 2m
the algebraic expression 6 – t.
SOLUTION:
6. 7 more than 11 times a number
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal SOLUTION:
expression the product of 2 and m can be used to
Let n represent a number. The words more than
describe the algebraic expression 2m.
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than
2. 11 times a number can be represented by the
algebraic expression 11n + 7.
SOLUTION:
7. 1 minus the quotient of r and 7
The expression shows the product of the factors
SOLUTION:
4 4
and r . The factor r represents a number raised
The word minus suggests subtraction and the word
to the fourth power. So, the verbal expression two quotient suggests division. So, the verbal expression
thirds times r raised to the fourth power can be 1 minus the quotient of r and 7 can be represented
by the algebraic expression .
used to describe the algebraic expression .
8. two fifths of a number j squared
3. a2 – 18b
SOLUTION:
SOLUTION:
The words two–fifths of a number suggest
The expression shows the difference of two terms.
multiplication. The squared means to raise to the
The term a2 represents a squared. The term 18b second power. So, the verbal expression two–fifths
represents 18 times b. So, the verbal expression a of a number j squared can be represented by the
squared minus 18 times b can be used to describe algebraic expression .
2
the algebraic expression a – 18b.
9. n cubed increased by 5
Write an algebraic expression for each verbal
expression. SOLUTION:
4. the sum of a number and 14 The word cubed means to raise to the third power.
The words increased by suggest addition. So, the
SOLUTION:
verbal expression n cubed increased by 5 can be
Let n represent a number. The word sum suggests 3
represented by the algebraic expression n + 5.
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
10. GROCERIES Mr. Bailey purchased some
expression n + 14.
groceries that cost d dollars. He paid with a $50 bill.
Write an expression for the amount of change he will
5. 6 less a number t receive.
SOLUTION: SOLUTION:
The word less suggests subtraction. So, the verbal To find the amount of change Mr. Bailey will
expression 6 less a number t can be represented by receive, subtract the cost of the groceries, d, from
the algebraic expression 6 – t. $50. So, Mr. Bailey will receive 50 – d in change.
6. 7 more than 11 times a number Write a verbal expression for each algebraic
expression.
SOLUTION:
11. 4q
Let n represent a number. The words more than
suggest addition and the word times suggests SOLUTION:
multiplication. So, the verbal expression 7 more than Because 4 and q are written next to each other, they
11 times a number can be represented by the are being multiplied. So, the verbal expression four
algebraic expression 11n + 7. times a number q can be used to describe the
algebraic expression 4q.
7. 1 minus the quotient of r and 7
SOLUTION: 12.
The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression SOLUTION:
1 minus the quotient of r and 7 can be represented
by the algebraic expression . Because and y are written next to each other,
they are being multiplied. So, the verbal expression
8. two fifths of a number j squared
one eighth of a number y can be used to describe
SOLUTION: the algebraic expression .
The words two–fifths of a number suggest
multiplication. The squared means to raise to the 13. 15 + r
second power. So, the verbal expression two–fifths
SOLUTION:
of a number j squared can be represented by the
algebraic expression . The expression shows the sum of two terms. So, the
verbal expression 15 plus r can be used to describe
the algebraic expression 15 + r.
9. n cubed increased by 5
SOLUTION: 14. w – 24
The word cubed means to raise to the third power.
SOLUTION:
The words increased by suggest addition. So, the
The expression shows the difference of two terms.
verbal expression n cubed increased by 5 can be
So, the verbal expression w minus 24 can be used to
3
represented by the algebraic expression n + 5. describe the algebraic expression w – 24.
10. GROCERIES Mr. Bailey purchased some 2
15. 3x
groceries that cost d dollars. He paid with a $50 bill.
Write an expression for the amount of change he will SOLUTION:
receive. The expression shows the product of the factors 3
2 2
SOLUTION: and x . The factor x represents a number raised to
the second power. So, the verbal expression 3 times
To find the amount of change Mr. Bailey will
x squared can be used to describe the algebraic
1-1 Vreacreiiavbel, essu abtnrdac Et txhper ceosssti oonf st he groceries, d, from
$50. So, Mr. Bailey will receive 50 – d in change. expression 3x2.
Write a verbal expression for each algebraic
expression. 16.
11. 4q
SOLUTION:
SOLUTION:
The expression shows the quotient of two terms. The
Because 4 and q are written next to each other, they
4
term r represents a number raised to the fourth
are being multiplied. So, the verbal expression four
power. So, the verbal expression r to the fourth
times a number q can be used to describe the
algebraic expression 4q. power divided by 9 can be used to describe the
algebraic expression .
12.
17. 2a + 6
SOLUTION:
SOLUTION:
Because and y are written next to each other, The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
they are being multiplied. So, the verbal expression
verbal expression 6 more than the product 2 times
one eighth of a number y can be used to describe
a can be used to describe the algebraic expression
the algebraic expression .
2a + 6.
13. 15 + r 18. r4 ∙ t3
SOLUTION: SOLUTION:
The expression shows the sum of two terms. So, the The expression shows the product of two factors.
verbal expression 15 plus r can be used to describe 4
The factor r represents a number raised to the
the algebraic expression 15 + r.
3
fourth power. The factor t represents a number
14. w – 24 raised to the third power. So, the verbal expression
the product of a number r raised to the fourth
SOLUTION: power and a number t cubed can be used to
The expression shows the difference of two terms. describe the algebraic expression r4 ∙ t3.
So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24. Write an algebraic expression for each verbal
expression.
15. 3x2 19. x more than 7
SOLUTION: SOLUTION:
The expression shows the product of the factors 3 The words more than suggest addition. So, the
2 2 verbal expression x more than 7 can be represented
and x . The factor x represents a number raised to
by the algebraic expression 7 + x.
the second power. So, the verbal expression 3 times
x squared can be used to describe the algebraic
expression 3x2. 20. a number less 35
SOLUTION:
16. Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less
SOLUTION: 35 can be represented by the algebraic expression n
– 35.
The expression shows the quotient of two terms. The
4
term r represents a number raised to the fourth 21. 5 times a number
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the SOLUTION:
Let n represent a number. The word times suggests
algebraic expression .
eSolutionsManual-PoweredbyCognero multiplication. So, the verbal expression 5 times aP age2
number can be represented by the algebraic
17. 2a + 6 expression 5n.
SOLUTION: 22. one third of a number
The expression shows the sum of two terms. The
SOLUTION:
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times Let n represent a number. The words one third of a
a can be used to describe the algebraic expression number suggest multiplication. So, the verbal
2a + 6. expression one third of a number can be
18. r4 ∙ t3 represented by the algebraic expression .
SOLUTION:
23. f divided by 10
The expression shows the product of two factors.
4 SOLUTION:
The factor r represents a number raised to the
The words divided by suggest division. So, the
3
fourth power. The factor t represents a number
verbal expression f divided by 10 can be
raised to the third power. So, the verbal expression
the product of a number r raised to the fourth represented by the algebraic expression .
power and a number t cubed can be used to
describe the algebraic expression r4 ∙ t3. 24. the quotient of 45 and r
Write an algebraic expression for each verbal SOLUTION:
expression. The word quotient suggests division. So, the verbal
19. x more than 7 expression the quotient of 45 and r can be
SOLUTION: represented by the algebraic expression .
The words more than suggest addition. So, the
verbal expression x more than 7 can be represented
25. three times a number plus 16
by the algebraic expression 7 + x.
SOLUTION:
20. a number less 35 Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
SOLUTION: So, the verbal expression three times a number plus
Let n represent a number. The word less suggests 16 can be represented by the algebraic expression
subtraction. So, the verbal expression a number less 3n + 16.
35 can be represented by the algebraic expression n
– 35. 26. 18 decreased by 3 times d
SOLUTION:
21. 5 times a number
The word decreased suggests subtraction, and the
SOLUTION: word times suggests multiplication. So, the verbal
Let n represent a number. The word times suggests expression 18 decreased by 3 times d can be
multiplication. So, the verbal expression 5 times a represented by the algebraic expression 18 – 3d.
number can be represented by the algebraic
expression 5n. 27. k squared minus 11
SOLUTION:
22. one third of a number
The word squared means a number raised to the
SOLUTION: second power. The word minus suggests subtraction.
Let n represent a number. The words one third of a So, the verbal expression k squared minus 11 can
number suggest multiplication. So, the verbal be represented by the algebraic expression k2 – 11.
expression one third of a number can be
28. 20 divided by t to the fifth power
represented by the algebraic expression .
SOLUTION:
The words divided by suggest division. So, the
23. f divided by 10
verbal expression 20 divided by t to the fifth power
SOLUTION:
can be represented by the algebraic expression .
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
29. GEOMETRY The volume of a cylinder is π times
represented by the algebraic expression . the radius r squared multiplied by the height. Write
an expression for the volume.
24. the quotient of 45 and r
SOLUTION:
The word quotient suggests division. So, the verbal
expression the quotient of 45 and r can be
represented by the algebraic expression .
SOLUTION:
25. three times a number plus 16 The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be
SOLUTION:
2
written as the algebraic expression πr h.
Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition. 30. FINANCIAL LITERACY Jocelyn makes x dollars
So, the verbal expression three times a number plus
per hour working at the grocery store and n dollars
16 can be represented by the algebraic expression
per hour babysitting. Write an expression that
3n + 16.
describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours.
26. 18 decreased by 3 times d
SOLUTION:
SOLUTION:
To write an expression for how much Jocelyn made
The word decreased suggests subtraction, and the
babysitting, multiply the number of hours she babysat,
word times suggests multiplication. So, the verbal
25, by her hourly rate, n. This can be represented by
expression 18 decreased by 3 times d can be
the algebraic expression 25n.
represented by the algebraic expression 18 – 3d.
To write an expression for how much Jocelyn made
27. k squared minus 11
working at the grocery store, multiply the number of
SOLUTION: hours she worked, 15, by her hourly rate, x. This can
The word squared means a number raised to the be represented by the algebraic expression 15x.
second power. The word minus suggests subtraction.
So, the verbal expression k squared minus 11 can To write an expression for her total earnings, find the
2 sum of the amount she earned babysitting and the
be represented by the algebraic expression k – 11.
amount she earned working at the grocery store.
This can be written as the expression 25n + 15x.
28. 20 divided by t to the fifth power
SOLUTION: Write a verbal expression for each algebraic
expression.
The words divided by suggest division. So, the
2
verbal expression 20 divided by t to the fifth power 31. 25 + 6x
can be represented by the algebraic expression . SOLUTION:
The expression shows the sum of two terms. The
2
29. GEOMETRY The volume of a cylinder is π times term 6x means six times the square of a number.
the radius r squared multiplied by the height. Write So, the algebraic expression 25 + 6x2 can be
an expression for the volume. described by the verbal expression twenty–five plus
six times a number squared.
2
32. 6f + 5f
SOLUTION:
The expression shows the sum of two terms. The
SOLUTION: term 6f2 means six times the square of a number.
The words times and multiplied by suggest The term 5f means five times a number. So, the
multiplication. So, the volume of a cylinder can be 2
algebraic expression 6f + 5f can be described by the
2
written as the algebraic expression πr h. verbal expression six times a number squared plus
five times the number.
30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
33.
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours. SOLUTION:
The expression shows the quotient of two terms. The
SOLUTION:
5
term 3a means three times a number that has been
To write an expression for how much Jocelyn made
raised to the fifth power. So, the algebraic expression
babysitting, multiply the number of hours she babysat,
can be described by the verbal expression three
25, by her hourly rate, n. This can be represented by
times a number raised to the fifth power divided
the algebraic expression 25n.
by two.
To write an expression for how much Jocelyn made
34. CCSS SENSE-MAKING A certain smartphone
working at the grocery store, multiply the number of
family plan costs $55 per month plus additional usage
hours she worked, 15, by her hourly rate, x. This can
costs. If x is the number of cell phone minutes used
be represented by the algebraic expression 15x.
above the plan amount and y is the number of
megabytes of data used above the plan amount,
To write an expression for her total earnings, find the
interpret the following expressions.
sum of the amount she earned babysitting and the
amount she earned working at the grocery store.
a. 0.25x
This can be written as the expression 25n + 15x.
Write a verbal expression for each algebraic b. 2y
expression.
2 c. 0.25x + 2y + 55
31. 25 + 6x
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. The
a. Since x is the number of cell phone minutes, then
2
term 6x means six times the square of a number. 0.25x would be the cost of extra minutes at $0.25 per
2
So, the algebraic expression 25 + 6x can be minute.
described by the verbal expression twenty–five plus
six times a number squared. b. Since y is the number of megabytes of data used
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
2
32. 6f + 5f
c. The expression 0.25x + 2y + 55 represents the
SOLUTION:
extra minute charges and the extra data usage
The expression shows the sum of two terms. The
charge plus the monthly family plan cost of $55. The
2
term 6f means six times the square of a number. expression represents the total monthly cost for the
The term 5f means five times a number. So, the family.
algebraic expression 6f2 + 5f can be described by the
verbal expression six times a number squared plus
five times the number.
35. DREAMS It is believed that about of our dreams
involve people that we know.
33.
a. Write an expression to describe the number of
SOLUTION:
dreams that feature people you know if you have d
The expression shows the quotient of two terms. The dreams.
5
term 3a means three times a number that has been
raised to the fifth power. So, the algebraic expression b. Use the expression you wrote to predict the
can be described by the verbal expression three number of dreams that include people you know out
times a number raised to the fifth power divided of 28 dreams.
by two.
34. CCSS SENSE-MAKING A certain smartphone SOLUTION:
family plan costs $55 per month plus additional usage a. To write an expression to describe the number of
costs. If x is the number of cell phone minutes used
dreams that feature people you know if you have d
above the plan amount and y is the number of
megabytes of data used above the plan amount, dreams, multiply by d or .
interpret the following expressions.
a. 0.25x b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
b. 2y in the expression .
c. 0.25x + 2y + 55
SOLUTION:
a. Since x is the number of cell phone minutes, then
0.25x would be the cost of extra minutes at $0.25 per So, you would predict having 21 dreams that include
minute. people you know.
b. Since y is the number of megabytes of data used 36. SPORTS In football, a touchdown is awarded 6
above the plan amount, then 2y would be the cost of points and the team can then try for a point after a
extra data used at $2 per megabyte. touchdown.
c. The expression 0.25x + 2y + 55 represents the a. Write an expression that describes the number of
extra minute charges and the extra data usage points scored on touchdowns and points after
charge plus the monthly family plan cost of $55. The touchdowns by one team in a game.
expression represents the total monthly cost for the
family.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of
touchdowns and points after touchdowns by the
35. DREAMS It is believed that about of our dreams winning team.
involve people that we know.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
a. Write an expression to describe the number of
touchdowns were scored during the game by both
dreams that feature people you know if you have d teams?
dreams.
SOLUTION:
b. Use the expression you wrote to predict the a. Let T be the number of touchdowns and p be the
number of dreams that include people you know out number of points scored after touchdowns. So, the
of 28 dreams. expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
SOLUTION:
a. To write an expression to describe the number of b. If a team scores 27 points in a game, then 6T + p
dreams that feature people you know if you have d = 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
dreams, multiply by d or .
c. If a team wins a football game 21–7, then 6T + p
b. To predict the number of dreams that include = 28 represents the possible number of touchdowns
people you know out of 28 dreams, replace d with 28 and points after touchdowns that were scored during
the game by both teams.
in the expression .
Let T = 4 and p = 4.
So, you would predict having 21 dreams that include
people you know.
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
36. SPORTS In football, a touchdown is awarded 6
teams.
points and the team can then try for a point after a
touchdown.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
a. Write an expression that describes the number of with like bases.
points scored on touchdowns and points after
touchdowns by one team in a game.
a. TABULAR Copy and complete the table.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of
touchdowns and points after touchdowns by the
winning team.
c. If a team wins a football game 21–7, how many b. ALGEBRAIC Write an equation for the pattern
possible number of touchdowns and points after in the table.
touchdowns were scored during the game by both
teams? c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a. Let T be the number of touchdowns and p be the SOLUTION:
number of points scored after touchdowns. So, the a.
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns b. The exponent of the product is the sum of the
and points after touchdowns by the winning team. exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns
c. The exponent of the product of two powers is the
and points after touchdowns that were scored during
sum of the exponents of the powers with the same
the game by both teams.
bases.
Let T = 4 and p = 4. 38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
So, it is possible that 4 touchdowns and 4 points after verbal expression. The expression 2x + 3 is the
touchdowns were scored during the game by both algebraic expression that represents the verbal
teams. expression “three more than a double a number”.
37. MULTIPLE REPRESENTATIONS In this 39. OPEN ENDED Define a variable to represent a
problem, you will explore the multiplication of powers real-life quantity, such as time in minutes or distance
with like bases. in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
a. TABULAR Copy and complete the table. Describe in words what your expression represents,
and explain your reasoning.
SOLUTION:
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
b. ALGEBRAIC Write an equation for the pattern walking each day since I walk to and from school
in the table. and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing
c. VERBAL Make a conjecture about the exponent
an algebraic expression for the verbal expression
of the product of two powers with like bases.
three times the sum of n squared and 3. Is either
SOLUTION: of them correct? Explain your reasoning.
a.
b. The exponent of the product is the sum of the
exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
c. The exponent of the product of two powers is the
SOLUTION:
sum of the exponents of the powers with the same
Consuelo is correct. The verbal expression says that
bases.
the sum of n squared and 3 is multiplied by 3. So,
38. REASONING Explain the differences between an parentheses are necessary. James left out the
algebraic expression and a verbal expression. parentheses around n2 + 3.
SOLUTION:
41. CHALLENGE For the cube, x represents a positive
Algebraic expressions include variables, numbers,
whole number. Find the value of x such that the
and symbols. Verbal expressions contain words. For
volume of the cube and 6 times the area of one of its
example, “three more than a double a number” is a
faces have the same value.
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
SOLUTION:
expression to represent one of your daily activities.
The volume of a cube can be found by multiplying
Describe in words what your expression represents,
the length times the width times the height. Because
and explain your reasoning.
the sides of a cube all have the same length, V = x •
SOLUTION: x • x, or x3. The area of one of the faces of the cube
Sample answer: x is the number of minutes it takes to can be found by multiplying the length times the
walk between my house and school. 2x + 15 2
width. So, A = x • x, or x .
represents the amount of time in minutes I spend
walking each day since I walk to and from school
To find the value of x such that the volume of the
and I take my dog on a 15 minute walk.
cube and 6 times the area of one of its faces have
40. CCSS CRITIQUE Consuelo and James are writing the same value, find a value for x such that x3 = 6x2.
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either x x3 = 6x2 Yes/No
of them correct? Explain your reasoning.
4 No
6 Yes
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
SOLUTION: to have the same value.
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So, 42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
parentheses are necessary. James left out the
Include a definition of algebraic expression in your
2
parentheses around n + 3. own words.
41. CHALLENGE For the cube, x represents a positive SOLUTION:
whole number. Find the value of x such that the Sample answer: An algebraic expression is a math
volume of the cube and 6 times the area of one of its phrase that contains one or more numbers or
faces have the same value. variables. To write an algebraic expression from real
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
Finally, put the terms in order.
43. Which expression best represents the volume of the
cube?
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
A the product of three and five
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
B three to the fifth power
the same value, find a value for x such that x3 = 6x2.
C three squared
x x3 = 6x2 Yes/No
4 No D three cubed
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
6 Yes
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
So, the sides must have a length of 6 for the volume So, Choice D is the correct answer.
of the cube and 6 times the area of one of its faces
to have the same value. 44. Which expression best represents the perimeter of
the rectangle?
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
Sample answer: An algebraic expression is a math F 2lw
phrase that contains one or more numbers or
variables. To write an algebraic expression from real G l + w
world situation, first assign variables. Then determine
the arithmetic operations done on the variables. H 2l + 2w
Finally, put the terms in order.
J 4(l + w)
43. Which expression best represents the volume of the
cube?
SOLUTION:
To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
rectangle.
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
A the product of three and five
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
B three to the fifth power represents the yards of fabric needed in terms of the
length of the window l.
C three squared
SOLUTION:
The word times suggests multiplication and the words
D three cubed
divided by suggest division. So, represents the
SOLUTION:
The volume of a cube can be found by multiplying yards of fabric needed in terms of the length of the
the length times the width times the height. Because window l.
the sides of a cube all have the same length, V = x •
3 46. GEOMETRY Find the area of the rectangle.
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
A 14 square meters
B 16 square meters
C 50 square meters
F 2lw
D 60 square meters
G l + w SOLUTION:
H 2l + 2w
J 4(l + w)
So, the area of the rectangle is 16 square meters.
SOLUTION:
To find the perimeter of a rectangle, find the sum of
Choice B is the correct answer.
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the 47. AMUSEMENT PARKS A roller coaster
rectangle.
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
Choice H is the correct answer. the results.
45. SHORT RESPONSE The yards of fabric needed
Our Favorite Rides
to make curtains is 3 times the length of a window in
Number
inches, divided by 36. Write an expression that Ride
of Votes
represents the yards of fabric needed in terms of the
length of the window l. Big Plunge 5
Twisting
SOLUTION: 22
Time
The word times suggests multiplication and the words
The Shiner 16
divided by suggest division. So, represents the Raging Bull 9
The
yards of fabric needed in terms of the length of the 25
Bat
window l.
Teaser 6
46. GEOMETRY Find the area of the rectangle. The
12
Adventure
SOLUTION:
Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
A 14 square meters the roller coaster chosen.
B 16 square meters
C 50 square meters
D 60 square meters
SOLUTION:
48. SPORTS The results for an annual 5K race are
So, the area of the rectangle is 16 square meters. shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
Choice B is the correct answer. the box-and-whisker plot tells about the times for the
race.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
Our Favorite Rides
Number
Ride
of Votes
Big Plunge 5
Twisting
SOLUTION:
22
Time
Order the data from least to greatest. The times in
The Shiner 16 order from least to greatest are 14:48, 14:58, 15:06,
Raging Bull 9 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
The 20:47, 20:49, 21:35.
Bat 25 The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
Teaser 6
the number of minutes by 60 and then adding the
The
12 seconds.
Adventure
Time in Time in
SOLUTION: Min:Sec Seconds
Draw a bar to represent each roller coaster. The 14:48 888
vertical scale is the number of members who voted 14:58 898
for each rollercoaster. The horizontal scale identifies 15:06 906
the roller coaster chosen. 15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167
19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
48. SPORTS The results for an annual 5K race are Then determine the quartiles.
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
Q = 948
the box-and-whisker plot tells about the times for the 1
race. Q = 1078.5
2
Q = 1239
3
There are no outliers.
Find the mean, median, and mode for each set
SOLUTION:
of data.
Order the data from least to greatest. The times in
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, SOLUTION:
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
So, the mean is 5.6.
Time in Time in
Order the data from least to greatest.
Min:Sec Seconds
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
14:48 888
Because there is an even number of data, the median
14:58 898 is the mean of 6 and 7.
15:06 906
15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167
So, the median is 6.5.
19:58 1198
20:21 1221
The number 7 appears most often, so the mode is 7.
20:39 1239
20:47 1247
20:49 1249 50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
21:35 1295
SOLUTION:
Then determine the quartiles.
Q = 948
1
So, the mean is 0.4.
Q = 1078.5
2
Q3 = 1239 Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
There are no outliers.
Because there is an even number of data, the median
is the mean of 0 and 0.
Find the mean, median, and mode for each set
of data.
So, the median is 0.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION: The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
So, the mean is 5.6.
Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median
So, the mean is 15.25.
is the mean of 6 and 7.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
So, the median is 6.5.
The number 7 appears most often, so the mode is 7.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
So, the median is 15.5.
SOLUTION:
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
So, the mean is 0.4.
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5} SOLUTION:
Because there is an even number of data, the median
is the mean of 0 and 0.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the SOLUTION:
modes are 0 and –1. Multiply the numerators and denominators.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
54.
So, the mean is 15.25.
Order the data from least to greatest.
SOLUTION:
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
55.
So, the median is 15.5.
The number 24 appears most often, so the mode is SOLUTION:
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
Evaluate each expression.
56.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient. SOLUTION:
53. The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
SOLUTION:
Multiply the numerators and denominators.
57. 5.67 – 4.21
SOLUTION:
54. 5.67 – 4.21 = 1.46
58.
SOLUTION:
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
55.
59. 10.34 + 14.27
SOLUTION:
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
Evaluate each expression.
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
56.
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
The word less suggests subtraction. So, the verbal
expression 6 less a number t can be represented by
the algebraic expression 6 – t.
6. 7 more than 11 times a number
Write a verbal expression for each algebraic
SOLUTION:
expression.
1. 2m Let n represent a number. The words more than
suggest addition and the word times suggests
SOLUTION:
multiplication. So, the verbal expression 7 more than
Because the 2 and the m are written next to each 11 times a number can be represented by the
other, they are being multiplied. So, the verbal algebraic expression 11n + 7.
expression the product of 2 and m can be used to
describe the algebraic expression 2m. 7. 1 minus the quotient of r and 7
SOLUTION:
2. The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression
SOLUTION: 1 minus the quotient of r and 7 can be represented
The expression shows the product of the factors by the algebraic expression .
4 4
and r . The factor r represents a number raised 8. two fifths of a number j squared
to the fourth power. So, the verbal expression two SOLUTION:
thirds times r raised to the fourth power can be
The words two–fifths of a number suggest
used to describe the algebraic expression . multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
3. a2 – 18b of a number j squared can be represented by the
algebraic expression .
SOLUTION:
The expression shows the difference of two terms.
9. n cubed increased by 5
The term a2 represents a squared. The term 18b
SOLUTION:
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe The word cubed means to raise to the third power.
2 The words increased by suggest addition. So, the
the algebraic expression a – 18b.
verbal expression n cubed increased by 5 can be
3
Write an algebraic expression for each verbal represented by the algebraic expression n + 5.
expression.
4. the sum of a number and 14 10. GROCERIES Mr. Bailey purchased some
groceries that cost d dollars. He paid with a $50 bill.
SOLUTION:
Write an expression for the amount of change he will
Let n represent a number. The word sum suggests receive.
addition. So, the verbal expression the sum of a
SOLUTION:
number and 14 can be represented by the algebraic
expression n + 14. To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
$50. So, Mr. Bailey will receive 50 – d in change.
5. 6 less a number t
Write a verbal expression for each algebraic
SOLUTION:
expression.
The word less suggests subtraction. So, the verbal
11. 4q
expression 6 less a number t can be represented by
the algebraic expression 6 – t. SOLUTION:
Because 4 and q are written next to each other, they
6. 7 more than 11 times a number
are being multiplied. So, the verbal expression four
times a number q can be used to describe the
SOLUTION:
algebraic expression 4q.
Let n represent a number. The words more than
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than 12.
11 times a number can be represented by the
algebraic expression 11n + 7. SOLUTION:
7. 1 minus the quotient of r and 7 Because and y are written next to each other,
SOLUTION: they are being multiplied. So, the verbal expression
The word minus suggests subtraction and the word one eighth of a number y can be used to describe
quotient suggests division. So, the verbal expression the algebraic expression .
1 minus the quotient of r and 7 can be represented
by the algebraic expression .
13. 15 + r
8. two fifths of a number j squared SOLUTION:
The expression shows the sum of two terms. So, the
SOLUTION:
verbal expression 15 plus r can be used to describe
The words two–fifths of a number suggest
the algebraic expression 15 + r.
multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
14. w – 24
of a number j squared can be represented by the
algebraic expression . SOLUTION:
The expression shows the difference of two terms.
9. n cubed increased by 5 So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24.
SOLUTION:
The word cubed means to raise to the third power. 15. 3x2
The words increased by suggest addition. So, the
verbal expression n cubed increased by 5 can be SOLUTION:
3 The expression shows the product of the factors 3
represented by the algebraic expression n + 5.
2 2
and x . The factor x represents a number raised to
10. GROCERIES Mr. Bailey purchased some the second power. So, the verbal expression 3 times
groceries that cost d dollars. He paid with a $50 bill. x squared can be used to describe the algebraic
Write an expression for the amount of change he will 2
expression 3x .
receive.
SOLUTION:
16.
To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
SOLUTION:
$50. So, Mr. Bailey will receive 50 – d in change.
The expression shows the quotient of two terms. The
4
Write a verbal expression for each algebraic term r represents a number raised to the fourth
expression. power. So, the verbal expression r to the fourth
11. 4q power divided by 9 can be used to describe the
SOLUTION: algebraic expression .
Because 4 and q are written next to each other, they
are being multiplied. So, the verbal expression four
17. 2a + 6
times a number q can be used to describe the
algebraic expression 4q. SOLUTION:
The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
12.
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
SOLUTION:
2a + 6.
Because and y are written next to each other,
18. r4 ∙ t3
they are being multiplied. So, the verbal expression
one eighth of a number y can be used to describe SOLUTION:
the algebraic expression . The expression shows the product of two factors.
4
The factor r represents a number raised to the
3
13. 15 + r fourth power. The factor t represents a number
raised to the third power. So, the verbal expression
SOLUTION:
the product of a number r raised to the fourth
The expression shows the sum of two terms. So, the
power and a number t cubed can be used to
verbal expression 15 plus r can be used to describe describe the algebraic expression r4 ∙ t3.
the algebraic expression 15 + r.
Write an algebraic expression for each verbal
14. w – 24
expression.
SOLUTION: 19. x more than 7
The expression shows the difference of two terms.
SOLUTION:
So, the verbal expression w minus 24 can be used to
The words more than suggest addition. So, the
describe the algebraic expression w – 24.
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x.
2
15. 3x
SOLUTION: 20. a number less 35
The expression shows the product of the factors 3
SOLUTION:
2 2
and x . The factor x represents a number raised to
Let n represent a number. The word less suggests
the second power. So, the verbal expression 3 times
subtraction. So, the verbal expression a number less
x squared can be used to describe the algebraic
35 can be represented by the algebraic expression n
2
expression 3x . – 35.
21. 5 times a number
16.
SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
The expression shows the quotient of two terms. The multiplication. So, the verbal expression 5 times a
4 number can be represented by the algebraic
term r represents a number raised to the fourth
expression 5n.
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the
22. one third of a number
algebraic expression .
SOLUTION:
Let n represent a number. The words one third of a
17. 2a + 6 number suggest multiplication. So, the verbal
expression one third of a number can be
SOLUTION:
The expression shows the sum of two terms. The represented by the algebraic expression .
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times
23. f divided by 10
a can be used to describe the algebraic expression
2a + 6.
SOLUTION:
The words divided by suggest division. So, the
18. r4 ∙ t3 verbal expression f divided by 10 can be
SOLUTION:
represented by the algebraic expression .
The expression shows the product of two factors.
4
The factor r represents a number raised to the
24. the quotient of 45 and r
3
fourth power. The factor t represents a number
SOLUTION:
raised to the third power. So, the verbal expression
The word quotient suggests division. So, the verbal
the product of a number r raised to the fourth
expression the quotient of 45 and r can be
power and a number t cubed can be used to
describe the algebraic expression r4 ∙ t3. represented by the algebraic expression .
Write an algebraic expression for each verbal
expression. 25. three times a number plus 16
19. x more than 7 SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
The words more than suggest addition. So, the
So, the verbal expression three times a number plus
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x. 16 can be represented by the algebraic expression
3n + 16.
20. a number less 35 26. 18 decreased by 3 times d
SOLUTION: SOLUTION:
Let n represent a number. The word less suggests The word decreased suggests subtraction, and the
subtraction. So, the verbal expression a number less word times suggests multiplication. So, the verbal
1-1 V3a5r ciaanb lbees raenpdre Esexnpterde sbsyio tnhes algebraic expression n expression 18 decreased by 3 times d can be
– 35. represented by the algebraic expression 18 – 3d.
21. 5 times a number 27. k squared minus 11
SOLUTION: SOLUTION:
Let n represent a number. The word times suggests The word squared means a number raised to the
multiplication. So, the verbal expression 5 times a second power. The word minus suggests subtraction.
number can be represented by the algebraic So, the verbal expression k squared minus 11 can
expression 5n. 2
be represented by the algebraic expression k – 11.
22. one third of a number
28. 20 divided by t to the fifth power
SOLUTION:
SOLUTION:
Let n represent a number. The words one third of a The words divided by suggest division. So, the
number suggest multiplication. So, the verbal verbal expression 20 divided by t to the fifth power
expression one third of a number can be
can be represented by the algebraic expression .
represented by the algebraic expression .
29. GEOMETRY The volume of a cylinder is π times
23. f divided by 10 the radius r squared multiplied by the height. Write
an expression for the volume.
SOLUTION:
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
represented by the algebraic expression .
24. the quotient of 45 and r
SOLUTION:
SOLUTION:
The words times and multiplied by suggest
The word quotient suggests division. So, the verbal
multiplication. So, the volume of a cylinder can be
expression the quotient of 45 and r can be
2
written as the algebraic expression πr h.
represented by the algebraic expression .
30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
25. three times a number plus 16
per hour babysitting. Write an expression that
SOLUTION: describes her earnings if she babysat for 25 hours
Let n represent a number. The word times suggests and worked at the grocery store for 15 hours.
multiplication, and the word plus suggests addition.
SOLUTION:
So, the verbal expression three times a number plus
To write an expression for how much Jocelyn made
16 can be represented by the algebraic expression
babysitting, multiply the number of hours she babysat,
3n + 16.
25, by her hourly rate, n. This can be represented by
26. 18 decreased by 3 times d the algebraic expression 25n.
SOLUTION: To write an expression for how much Jocelyn made
The word decreased suggests subtraction, and the working at the grocery store, multiply the number of
word times suggests multiplication. So, the verbal hours she worked, 15, by her hourly rate, x. This can
expression 18 decreased by 3 times d can be be represented by the algebraic expression 15x.
represented by the algebraic expression 18 – 3d.
To write an expression for her total earnings, find the
27. k squared minus 11 sum of the amount she earned babysitting and the
amount she earned working at the grocery store.
SOLUTION:
This can be written as the expression 25n + 15x.
The word squared means a number raised to the
eSolutsioencsoMnadn puaolw-Peorw. Terhede bwyoCrodg nmerionus suggests subtraction. Write a verbal expression for each algebraicP age3
So, the verbal expression k squared minus 11 can expression.
be represented by the algebraic expression k2 – 11. 31. 25 + 6x2
28. 20 divided by t to the fifth power SOLUTION:
The expression shows the sum of two terms. The
SOLUTION:
2
term 6x means six times the square of a number.
The words divided by suggest division. So, the
2
verbal expression 20 divided by t to the fifth power So, the algebraic expression 25 + 6x can be
described by the verbal expression twenty–five plus
can be represented by the algebraic expression . six times a number squared.
29. GEOMETRY The volume of a cylinder is π times 2
32. 6f + 5f
the radius r squared multiplied by the height. Write
an expression for the volume. SOLUTION:
The expression shows the sum of two terms. The
2
term 6f means six times the square of a number.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number.
SOLUTION:
The words times and multiplied by suggest
33.
multiplication. So, the volume of a cylinder can be
2
written as the algebraic expression πr h.
SOLUTION:
The expression shows the quotient of two terms. The
30. FINANCIAL LITERACY Jocelyn makes x dollars
5
per hour working at the grocery store and n dollars term 3a means three times a number that has been
per hour babysitting. Write an expression that raised to the fifth power. So, the algebraic expression
describes her earnings if she babysat for 25 hours can be described by the verbal expression three
and worked at the grocery store for 15 hours. times a number raised to the fifth power divided
by two.
SOLUTION:
To write an expression for how much Jocelyn made 34. CCSS SENSE-MAKING A certain smartphone
babysitting, multiply the number of hours she babysat, family plan costs $55 per month plus additional usage
25, by her hourly rate, n. This can be represented by costs. If x is the number of cell phone minutes used
the algebraic expression 25n. above the plan amount and y is the number of
megabytes of data used above the plan amount,
To write an expression for how much Jocelyn made interpret the following expressions.
working at the grocery store, multiply the number of
hours she worked, 15, by her hourly rate, x. This can a. 0.25x
be represented by the algebraic expression 15x.
b. 2y
To write an expression for her total earnings, find the
sum of the amount she earned babysitting and the c. 0.25x + 2y + 55
amount she earned working at the grocery store.
This can be written as the expression 25n + 15x.
SOLUTION:
Write a verbal expression for each algebraic a. Since x is the number of cell phone minutes, then
expression. 0.25x would be the cost of extra minutes at $0.25 per
31. 25 + 6x2 minute.
SOLUTION: b. Since y is the number of megabytes of data used
The expression shows the sum of two terms. The above the plan amount, then 2y would be the cost of
term 6x2 means six times the square of a number. extra data used at $2 per megabyte.
2
So, the algebraic expression 25 + 6x can be
c. The expression 0.25x + 2y + 55 represents the
described by the verbal expression twenty–five plus
extra minute charges and the extra data usage
six times a number squared.
charge plus the monthly family plan cost of $55. The
expression represents the total monthly cost for the
32. 6f2 + 5f family.
SOLUTION:
The expression shows the sum of two terms. The
35. DREAMS It is believed that about of our dreams
2
term 6f means six times the square of a number.
The term 5f means five times a number. So, the involve people that we know.
2
algebraic expression 6f + 5f can be described by the
a. Write an expression to describe the number of
verbal expression six times a number squared plus
five times the number. dreams that feature people you know if you have d
dreams.
33.
b. Use the expression you wrote to predict the
number of dreams that include people you know out
SOLUTION: of 28 dreams.
The expression shows the quotient of two terms. The
5
term 3a means three times a number that has been
SOLUTION:
raised to the fifth power. So, the algebraic expression
a. To write an expression to describe the number of
can be described by the verbal expression three
dreams that feature people you know if you have d
times a number raised to the fifth power divided
by two.
dreams, multiply by d or .
34. CCSS SENSE-MAKING A certain smartphone
family plan costs $55 per month plus additional usage
b. To predict the number of dreams that include
costs. If x is the number of cell phone minutes used
people you know out of 28 dreams, replace d with 28
above the plan amount and y is the number of
in the expression .
megabytes of data used above the plan amount,
interpret the following expressions.
a. 0.25x
b. 2y
So, you would predict having 21 dreams that include
c. 0.25x + 2y + 55
people you know.
SOLUTION: 36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
a. Since x is the number of cell phone minutes, then
touchdown.
0.25x would be the cost of extra minutes at $0.25 per
minute.
a. Write an expression that describes the number of
b. Since y is the number of megabytes of data used points scored on touchdowns and points after
touchdowns by one team in a game.
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
b. If a team wins a football game 27–0, write an
c. The expression 0.25x + 2y + 55 represents the equation to represent the possible number of
extra minute charges and the extra data usage touchdowns and points after touchdowns by the
charge plus the monthly family plan cost of $55. The winning team.
expression represents the total monthly cost for the
family. c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
teams?
35. DREAMS It is believed that about of our dreams
SOLUTION:
involve people that we know.
a. Let T be the number of touchdowns and p be the
a. Write an expression to describe the number of number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points
dreams that feature people you know if you have d
dreams. scored on touchdowns and points after touchdowns
by one team in a game.
b. Use the expression you wrote to predict the
b. If a team scores 27 points in a game, then 6T + p
number of dreams that include people you know out
of 28 dreams. = 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
SOLUTION: c. If a team wins a football game 21–7, then 6T + p
a. To write an expression to describe the number of = 28 represents the possible number of touchdowns
dreams that feature people you know if you have d and points after touchdowns that were scored during
the game by both teams.
dreams, multiply by d or .
Let T = 4 and p = 4.
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
in the expression .
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
teams.
So, you would predict having 21 dreams that include 37. MULTIPLE REPRESENTATIONS In this
people you know. problem, you will explore the multiplication of powers
with like bases.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a a. TABULAR Copy and complete the table.
touchdown.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
b. If a team wins a football game 27–0, write an b. ALGEBRAIC Write an equation for the pattern
equation to represent the possible number of in the table.
touchdowns and points after touchdowns by the
winning team. c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
c. If a team wins a football game 21–7, how many
SOLUTION:
possible number of touchdowns and points after
a.
touchdowns were scored during the game by both
teams?
SOLUTION:
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points b. The exponent of the product is the sum of the
scored on touchdowns and points after touchdowns exponents of the factors. So, the algebraic equation
by one team in a game. 102 × 10x = 10(2 + x) represents the pattern.
b. If a team scores 27 points in a game, then 6T + p c. The exponent of the product of two powers is the
= 27 represents the possible number of touchdowns sum of the exponents of the powers with the same
and points after touchdowns by the winning team. bases.
c. If a team wins a football game 21–7, then 6T + p 38. REASONING Explain the differences between an
= 28 represents the possible number of touchdowns algebraic expression and a verbal expression.
and points after touchdowns that were scored during
SOLUTION:
the game by both teams.
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
Let T = 4 and p = 4.
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
So, it is possible that 4 touchdowns and 4 points after
in feet. Then use the variable to write an algebraic
touchdowns were scored during the game by both
expression to represent one of your daily activities.
teams.
Describe in words what your expression represents,
and explain your reasoning.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
SOLUTION:
with like bases.
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
a. TABULAR Copy and complete the table. represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
b. ALGEBRAIC Write an equation for the pattern of them correct? Explain your reasoning.
in the table.
c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a.
SOLUTION:
b. The exponent of the product is the sum of the
Consuelo is correct. The verbal expression says that
exponents of the factors. So, the algebraic equation
the sum of n squared and 3 is multiplied by 3. So,
2 x (2 + x)
10 × 10 = 10 represents the pattern.
parentheses are necessary. James left out the
2
parentheses around n + 3.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
41. CHALLENGE For the cube, x represents a positive
bases.
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
38. REASONING Explain the differences between an
faces have the same value.
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal SOLUTION:
expression “three more than a double a number”. The volume of a cube can be found by multiplying
the length times the width times the height. Because
39. OPEN ENDED Define a variable to represent a the sides of a cube all have the same length, V = x •
real-life quantity, such as time in minutes or distance 3
x • x, or x . The area of one of the faces of the cube
in feet. Then use the variable to write an algebraic
can be found by multiplying the length times the
expression to represent one of your daily activities.
2
width. So, A = x • x, or x .
Describe in words what your expression represents,
and explain your reasoning.
To find the value of x such that the volume of the
SOLUTION: cube and 6 times the area of one of its faces have
Sample answer: x is the number of minutes it takes to the same value, find a value for x such that x3 = 6x2.
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
x x3 = 6x2 Yes/No
walking each day since I walk to and from school
and I take my dog on a 15 minute walk. 4 No
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
6 Yes
of them correct? Explain your reasoning.
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
Consuelo is correct. The verbal expression says that SOLUTION:
the sum of n squared and 3 is multiplied by 3. So, Sample answer: An algebraic expression is a math
parentheses are necessary. James left out the phrase that contains one or more numbers or
2 variables. To write an algebraic expression from real
parentheses around n + 3.
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
41. CHALLENGE For the cube, x represents a positive
Finally, put the terms in order.
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
43. Which expression best represents the volume of the
faces have the same value.
cube?
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
A the product of three and five
the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube
B three to the fifth power
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
C three squared
To find the value of x such that the volume of the
D three cubed
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2. SOLUTION:
The volume of a cube can be found by multiplying
x x3 = 6x2 Yes/No the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
4 No
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
6 Yes So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an F 2lw
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
G l + w
own words.
SOLUTION: H 2l + 2w
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or J 4(l + w)
variables. To write an algebraic expression from real
SOLUTION:
world situation, first assign variables. Then determine
the arithmetic operations done on the variables. To find the perimeter of a rectangle, find the sum of
Finally, put the terms in order. twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
43. Which expression best represents the volume of the rectangle.
cube?
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the
length of the window l.
SOLUTION:
A the product of three and five
The word times suggests multiplication and the words
B three to the fifth power divided by suggest division. So, represents the
yards of fabric needed in terms of the length of the
C three squared
window l.
D three cubed 46. GEOMETRY Find the area of the rectangle.
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
x • x, or x3. Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube. A 14 square meters
So, Choice D is the correct answer. B 16 square meters
44. Which expression best represents the perimeter of
C 50 square meters
the rectangle?
D 60 square meters
SOLUTION:
F 2lw
G l + w So, the area of the rectangle is 16 square meters.
H 2l + 2w Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
J 4(l + w)
enthusiast club took a poll to see what each
SOLUTION: member’s favorite ride was.. Make a bar graph of
To find the perimeter of a rectangle, find the sum of the results.
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the Our Favorite Rides
rectangle.
Number
Ride
of Votes
Choice H is the correct answer.
Big Plunge 5
Twisting
45. SHORT RESPONSE The yards of fabric needed 22
Time
to make curtains is 3 times the length of a window in
The Shiner 16
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the Raging Bull 9
length of the window l. The
25
Bat
SOLUTION:
Teaser 6
The word times suggests multiplication and the words
The
12
divided by suggest division. So, represents the Adventure
yards of fabric needed in terms of the length of the
window l. SOLUTION:
Draw a bar to represent each roller coaster. The
46. GEOMETRY Find the area of the rectangle.
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
the roller coaster chosen.
A 14 square meters
B 16 square meters
C 50 square meters
D 60 square meters
48. SPORTS The results for an annual 5K race are
SOLUTION: shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
race.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
SOLUTION:
Our Favorite Rides
Order the data from least to greatest. The times in
Number
Ride order from least to greatest are 14:48, 14:58, 15:06,
of Votes
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
Big Plunge 5 20:47, 20:49, 21:35.
Twisting The times are given in minutes and seconds. Rewrite
22
Time the times so that they are in seconds by multiplying
The Shiner 16 the number of minutes by 60 and then adding the
seconds.
Raging Bull 9
The
25 Time in Time in
Bat
Min:Sec Seconds
Teaser 6
14:48 888
The
12 14:58 898
Adventure
15:06 906
15:48 948
SOLUTION: 15:54 954
16:10 970
Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted 16:30 990
for each rollercoaster. The horizontal scale identifies 19:27 1167
the roller coaster chosen. 19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
48. SPORTS The results for an annual 5K race are Q = 1239
3
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of There are no outliers.
the box-and-whisker plot tells about the times for the
race.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, So, the mean is 5.6.
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite Order the data from least to greatest.
the times so that they are in seconds by multiplying {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
the number of minutes by 60 and then adding the Because there is an even number of data, the median
seconds. is the mean of 6 and 7.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898
15:06 906
So, the median is 6.5.
15:48 948
15:54 954
The number 7 appears most often, so the mode is 7.
16:10 970
16:30 990
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
19:27 1167
19:58 1198 SOLUTION:
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295 So, the mean is 0.4.
Then determine the quartiles. Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Q = 948
1
Because there is an even number of data, the median
Q = 1078.5
2 is the mean of 0 and 0.
Q = 1239
3
There are no outliers.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
Find the mean, median, and mode for each set
of data. 51. {17, 24, 16, 3, 12, 11, 24, 15}
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
So, the mean is 15.25.
So, the mean is 5.6.
Order the data from least to greatest.
Order the data from least to greatest. {3, 11, 12, 15, 16, 17, 24, 24}.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median Because there is an even number of data, the median
is the mean of 6 and 7. is the mean of 15 and 16.
So, the median is 6.5. So, the median is 15.5.
The number 7 appears most often, so the mode is 7. The number 24 appears most often, so the mode is
24.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
52. SPORTS Lisa has a rectangular trampoline that is 6
SOLUTION: feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5} The area of Lisa’s trampoline is 72 square feet.
Because there is an even number of data, the median Find each product or quotient.
is the mean of 0 and 0. 53.
SOLUTION:
Multiply the numerators and denominators.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
54.
SOLUTION:
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16. 55.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
Evaluate each expression.
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
56.
SOLUTION:
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
SOLUTION:
57. 5.67 – 4.21
Multiply the numerators and denominators.
SOLUTION:
5.67 – 4.21 = 1.46
58.
54. SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
SOLUTION:
59. 10.34 + 14.27
SOLUTION:
55. 10.34 + 14.27 = 24.61
60.
SOLUTION:
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
Evaluate each expression.
56.
SOLUTION:
61. 37.02 – 15.86
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators. SOLUTION:
37.02 – 15.86 = 21.16
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
Write a verbal expression for each algebraic
SOLUTION:
expression.
1. 2m Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
SOLUTION:
number and 14 can be represented by the algebraic
Because the 2 and the m are written next to each expression n + 14.
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
5. 6 less a number t
describe the algebraic expression 2m.
SOLUTION:
The word less suggests subtraction. So, the verbal
2.
expression 6 less a number t can be represented by
the algebraic expression 6 – t.
SOLUTION:
The expression shows the product of the factors 6. 7 more than 11 times a number
4 4
and r . The factor r represents a number raised SOLUTION:
Let n represent a number. The words more than
to the fourth power. So, the verbal expression two
suggest addition and the word times suggests
thirds times r raised to the fourth power can be
multiplication. So, the verbal expression 7 more than
used to describe the algebraic expression .
11 times a number can be represented by the
algebraic expression 11n + 7.
3. a2 – 18b
7. 1 minus the quotient of r and 7
SOLUTION:
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression
represents 18 times b. So, the verbal expression a
1 minus the quotient of r and 7 can be represented
squared minus 18 times b can be used to describe
by the algebraic expression .
2
the algebraic expression a – 18b.
8. two fifths of a number j squared
Write an algebraic expression for each verbal
expression. SOLUTION:
4. the sum of a number and 14
The words two–fifths of a number suggest
multiplication. The squared means to raise to the
SOLUTION:
second power. So, the verbal expression two–fifths
Let n represent a number. The word sum suggests
of a number j squared can be represented by the
addition. So, the verbal expression the sum of a
algebraic expression .
number and 14 can be represented by the algebraic
expression n + 14.
9. n cubed increased by 5
5. 6 less a number t SOLUTION:
The word cubed means to raise to the third power.
SOLUTION:
The words increased by suggest addition. So, the
The word less suggests subtraction. So, the verbal
verbal expression n cubed increased by 5 can be
expression 6 less a number t can be represented by
3
represented by the algebraic expression n + 5.
the algebraic expression 6 – t.
6. 7 more than 11 times a number 10. GROCERIES Mr. Bailey purchased some
groceries that cost d dollars. He paid with a $50 bill.
SOLUTION:
Write an expression for the amount of change he will
Let n represent a number. The words more than receive.
suggest addition and the word times suggests
SOLUTION:
multiplication. So, the verbal expression 7 more than
To find the amount of change Mr. Bailey will
11 times a number can be represented by the
algebraic expression 11n + 7. receive, subtract the cost of the groceries, d, from
$50. So, Mr. Bailey will receive 50 – d in change.
7. 1 minus the quotient of r and 7
Write a verbal expression for each algebraic
SOLUTION: expression.
The word minus suggests subtraction and the word 11. 4q
quotient suggests division. So, the verbal expression
SOLUTION:
1 minus the quotient of r and 7 can be represented
Because 4 and q are written next to each other, they
by the algebraic expression .
are being multiplied. So, the verbal expression four
times a number q can be used to describe the
8. two fifths of a number j squared
algebraic expression 4q.
SOLUTION:
The words two–fifths of a number suggest 12.
multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
SOLUTION:
of a number j squared can be represented by the
algebraic expression . Because and y are written next to each other,
they are being multiplied. So, the verbal expression
9. n cubed increased by 5
one eighth of a number y can be used to describe
SOLUTION: the algebraic expression .
The word cubed means to raise to the third power.
The words increased by suggest addition. So, the
13. 15 + r
verbal expression n cubed increased by 5 can be
3 SOLUTION:
represented by the algebraic expression n + 5.
The expression shows the sum of two terms. So, the
10. GROCERIES Mr. Bailey purchased some verbal expression 15 plus r can be used to describe
groceries that cost d dollars. He paid with a $50 bill. the algebraic expression 15 + r.
Write an expression for the amount of change he will
receive. 14. w – 24
SOLUTION: SOLUTION:
To find the amount of change Mr. Bailey will The expression shows the difference of two terms.
receive, subtract the cost of the groceries, d, from So, the verbal expression w minus 24 can be used to
$50. So, Mr. Bailey will receive 50 – d in change. describe the algebraic expression w – 24.
Write a verbal expression for each algebraic 15. 3x2
expression.
11. 4q SOLUTION:
The expression shows the product of the factors 3
SOLUTION:
2 2
and x . The factor x represents a number raised to
Because 4 and q are written next to each other, they
the second power. So, the verbal expression 3 times
are being multiplied. So, the verbal expression four
x squared can be used to describe the algebraic
times a number q can be used to describe the
2
algebraic expression 4q. expression 3x .
12. 16.
SOLUTION: SOLUTION:
The expression shows the quotient of two terms. The
Because and y are written next to each other, 4
term r represents a number raised to the fourth
they are being multiplied. So, the verbal expression power. So, the verbal expression r to the fourth
one eighth of a number y can be used to describe power divided by 9 can be used to describe the
the algebraic expression .
algebraic expression .
13. 15 + r
17. 2a + 6
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. So, the
The expression shows the sum of two terms. The
verbal expression 15 plus r can be used to describe
term 2a represents the product of 2 and a. So, the
the algebraic expression 15 + r.
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
14. w – 24
2a + 6.
SOLUTION:
The expression shows the difference of two terms. 18. r4 ∙ t3
So, the verbal expression w minus 24 can be used to
SOLUTION:
describe the algebraic expression w – 24.
The expression shows the product of two factors.
15. 3x2 The factor r4 represents a number raised to the
3
fourth power. The factor t represents a number
SOLUTION:
raised to the third power. So, the verbal expression
The expression shows the product of the factors 3
the product of a number r raised to the fourth
2 2
and x . The factor x represents a number raised to power and a number t cubed can be used to
the second power. So, the verbal expression 3 times describe the algebraic expression r4 ∙ t3.
x squared can be used to describe the algebraic
expression 3x2. Write an algebraic expression for each verbal
expression.
19. x more than 7
16.
SOLUTION:
SOLUTION: The words more than suggest addition. So, the
The expression shows the quotient of two terms. The verbal expression x more than 7 can be represented
4 by the algebraic expression 7 + x.
term r represents a number raised to the fourth
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the 20. a number less 35
SOLUTION:
algebraic expression .
Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less
17. 2a + 6
35 can be represented by the algebraic expression n
SOLUTION: – 35.
The expression shows the sum of two terms. The
21. 5 times a number
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times
SOLUTION:
a can be used to describe the algebraic expression
Let n represent a number. The word times suggests
2a + 6.
multiplication. So, the verbal expression 5 times a
number can be represented by the algebraic
18. r4 ∙ t3
expression 5n.
SOLUTION:
22. one third of a number
The expression shows the product of two factors.
4 SOLUTION:
The factor r represents a number raised to the
3 Let n represent a number. The words one third of a
fourth power. The factor t represents a number
number suggest multiplication. So, the verbal
raised to the third power. So, the verbal expression
expression one third of a number can be
the product of a number r raised to the fourth
power and a number t cubed can be used to
represented by the algebraic expression .
describe the algebraic expression r4 ∙ t3.
Write an algebraic expression for each verbal 23. f divided by 10
expression.
SOLUTION:
19. x more than 7
The words divided by suggest division. So, the
SOLUTION: verbal expression f divided by 10 can be
The words more than suggest addition. So, the
represented by the algebraic expression .
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x.
24. the quotient of 45 and r
20. a number less 35 SOLUTION:
The word quotient suggests division. So, the verbal
SOLUTION:
expression the quotient of 45 and r can be
Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less represented by the algebraic expression .
35 can be represented by the algebraic expression n
– 35.
25. three times a number plus 16
21. 5 times a number SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
Let n represent a number. The word times suggests
So, the verbal expression three times a number plus
multiplication. So, the verbal expression 5 times a
16 can be represented by the algebraic expression
number can be represented by the algebraic
3n + 16.
expression 5n.
26. 18 decreased by 3 times d
22. one third of a number
SOLUTION:
SOLUTION:
The word decreased suggests subtraction, and the
Let n represent a number. The words one third of a
word times suggests multiplication. So, the verbal
number suggest multiplication. So, the verbal
expression 18 decreased by 3 times d can be
expression one third of a number can be
represented by the algebraic expression 18 – 3d.
represented by the algebraic expression .
27. k squared minus 11
23. f divided by 10 SOLUTION:
The word squared means a number raised to the
SOLUTION:
second power. The word minus suggests subtraction.
The words divided by suggest division. So, the So, the verbal expression k squared minus 11 can
verbal expression f divided by 10 can be 2
be represented by the algebraic expression k – 11.
represented by the algebraic expression .
28. 20 divided by t to the fifth power
24. the quotient of 45 and r SOLUTION:
The words divided by suggest division. So, the
SOLUTION:
verbal expression 20 divided by t to the fifth power
The word quotient suggests division. So, the verbal
expression the quotient of 45 and r can be can be represented by the algebraic expression .
represented by the algebraic expression .
29. GEOMETRY The volume of a cylinder is π times
the radius r squared multiplied by the height. Write
25. three times a number plus 16 an expression for the volume.
SOLUTION:
Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
So, the verbal expression three times a number plus
16 can be represented by the algebraic expression
3n + 16.
SOLUTION:
26. 18 decreased by 3 times d
The words times and multiplied by suggest
SOLUTION: multiplication. So, the volume of a cylinder can be
The word decreased suggests subtraction, and the written as the algebraic expression πr2h.
word times suggests multiplication. So, the verbal
expression 18 decreased by 3 times d can be 30. FINANCIAL LITERACY Jocelyn makes x dollars
represented by the algebraic expression 18 – 3d. per hour working at the grocery store and n dollars
per hour babysitting. Write an expression that
27. k squared minus 11 describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours.
SOLUTION:
The word squared means a number raised to the SOLUTION:
second power. The word minus suggests subtraction. To write an expression for how much Jocelyn made
So, the verbal expression k squared minus 11 can babysitting, multiply the number of hours she babysat,
be represented by the algebraic expression k2 – 11. 25, by her hourly rate, n. This can be represented by
the algebraic expression 25n.
28. 20 divided by t to the fifth power
To write an expression for how much Jocelyn made
SOLUTION:
working at the grocery store, multiply the number of
The words divided by suggest division. So, the hours she worked, 15, by her hourly rate, x. This can
verbal expression 20 divided by t to the fifth power be represented by the algebraic expression 15x.
can be represented by the algebraic expression .
To write an expression for her total earnings, find the
sum of the amount she earned babysitting and the
29. GEOMETRY The volume of a cylinder is π times amount she earned working at the grocery store.
the radius r squared multiplied by the height. Write This can be written as the expression 25n + 15x.
an expression for the volume.
Write a verbal expression for each algebraic
expression.
2
31. 25 + 6x
SOLUTION:
The expression shows the sum of two terms. The
2
term 6x means six times the square of a number.
SOLUTION:
2
So, the algebraic expression 25 + 6x can be
The words times and multiplied by suggest
described by the verbal expression twenty–five plus
multiplication. So, the volume of a cylinder can be
six times a number squared.
2
written as the algebraic expression πr h.
30. FINANCIAL LITERACY Jocelyn makes x dollars 32. 6f2 + 5f
per hour working at the grocery store and n dollars
SOLUTION:
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours The expression shows the sum of two terms. The
and worked at the grocery store for 15 hours. term 6f2 means six times the square of a number.
The term 5f means five times a number. So, the
SOLUTION:
2
algebraic expression 6f + 5f can be described by the
To write an expression for how much Jocelyn made
verbal expression six times a number squared plus
babysitting, multiply the number of hours she babysat,
five times the number.
25, by her hourly rate, n. This can be represented by
the algebraic expression 25n.
33.
To write an expression for how much Jocelyn made
working at the grocery store, multiply the number of
SOLUTION:
hours she worked, 15, by her hourly rate, x. This can
The expression shows the quotient of two terms. The
be represented by the algebraic expression 15x.
5
term 3a means three times a number that has been
To write an expression for her total earnings, find the raised to the fifth power. So, the algebraic expression
sum of the amount she earned babysitting and the can be described by the verbal expression three
1-1 Vamaroiuanbtl essh ea neadr nEexdp wreosrskiionngs a t the grocery store. times a number raised to the fifth power divided
This can be written as the expression 25n + 15x. by two.
Write a verbal expression for each algebraic 34. CCSS SENSE-MAKING A certain smartphone
expression. family plan costs $55 per month plus additional usage
2 costs. If x is the number of cell phone minutes used
31. 25 + 6x
above the plan amount and y is the number of
SOLUTION: megabytes of data used above the plan amount,
The expression shows the sum of two terms. The interpret the following expressions.
2
term 6x means six times the square of a number.
a. 0.25x
2
So, the algebraic expression 25 + 6x can be
described by the verbal expression twenty–five plus
b. 2y
six times a number squared.
c. 0.25x + 2y + 55
2
32. 6f + 5f
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. The a. Since x is the number of cell phone minutes, then
2 0.25x would be the cost of extra minutes at $0.25 per
term 6f means six times the square of a number.
minute.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the b. Since y is the number of megabytes of data used
verbal expression six times a number squared plus above the plan amount, then 2y would be the cost of
five times the number. extra data used at $2 per megabyte.
c. The expression 0.25x + 2y + 55 represents the
33.
extra minute charges and the extra data usage
charge plus the monthly family plan cost of $55. The
SOLUTION:
expression represents the total monthly cost for the
The expression shows the quotient of two terms. The
family.
5
term 3a means three times a number that has been
raised to the fifth power. So, the algebraic expression
can be described by the verbal expression three
35. DREAMS It is believed that about of our dreams
times a number raised to the fifth power divided
by two. involve people that we know.
34. CCSS SENSE-MAKING A certain smartphone
a. Write an expression to describe the number of
family plan costs $55 per month plus additional usage
dreams that feature people you know if you have d
costs. If x is the number of cell phone minutes used
dreams.
above the plan amount and y is the number of
megabytes of data used above the plan amount,
b. Use the expression you wrote to predict the
interpret the following expressions.
number of dreams that include people you know out
of 28 dreams.
a. 0.25x
b. 2y
SOLUTION:
a. To write an expression to describe the number of
c. 0.25x + 2y + 55
dreams that feature people you know if you have d
dreams, multiply by d or .
SOLUTION:
a. Since x is the number of cell phone minutes, then
0.25x would be the cost of extra minutes at $0.25 per b. To predict the number of dreams that include
minute. people you know out of 28 dreams, replace d with 28
in the expression .
eSolutbio.n SsiMnacneu a yl- iPso twheer endubmybCeogr noefr omegabytes of data used Page4
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
c. The expression 0.25x + 2y + 55 represents the
extra minute charges and the extra data usage
charge plus the monthly family plan cost of $55. The
So, you would predict having 21 dreams that include
expression represents the total monthly cost for the
people you know.
family.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
35. DREAMS It is believed that about of our dreams touchdown.
involve people that we know.
a. Write an expression that describes the number of
points scored on touchdowns and points after
a. Write an expression to describe the number of touchdowns by one team in a game.
dreams that feature people you know if you have d
dreams.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of
b. Use the expression you wrote to predict the touchdowns and points after touchdowns by the
number of dreams that include people you know out winning team.
of 28 dreams.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
SOLUTION:
touchdowns were scored during the game by both
a. To write an expression to describe the number of
teams?
dreams that feature people you know if you have d
SOLUTION:
dreams, multiply by d or .
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
b. To predict the number of dreams that include expression 6T + p, describes the number of points
people you know out of 28 dreams, replace d with 28 scored on touchdowns and points after touchdowns
by one team in a game.
in the expression .
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
c. If a team wins a football game 21–7, then 6T + p
So, you would predict having 21 dreams that include = 28 represents the possible number of touchdowns
people you know. and points after touchdowns that were scored during
the game by both teams.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a Let T = 4 and p = 4.
touchdown.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
So, it is possible that 4 touchdowns and 4 points after
b. If a team wins a football game 27–0, write an
touchdowns were scored during the game by both
equation to represent the possible number of
teams.
touchdowns and points after touchdowns by the
winning team.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
c. If a team wins a football game 21–7, how many with like bases.
possible number of touchdowns and points after
touchdowns were scored during the game by both
a. TABULAR Copy and complete the table.
teams?
SOLUTION:
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
b. ALGEBRAIC Write an equation for the pattern
by one team in a game.
in the table.
b. If a team scores 27 points in a game, then 6T + p
c. VERBAL Make a conjecture about the exponent
= 27 represents the possible number of touchdowns
of the product of two powers with like bases.
and points after touchdowns by the winning team.
SOLUTION:
c. If a team wins a football game 21–7, then 6T + p a.
= 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
the game by both teams.
Let T = 4 and p = 4.
b. The exponent of the product is the sum of the
exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
c. The exponent of the product of two powers is the
So, it is possible that 4 touchdowns and 4 points after sum of the exponents of the powers with the same
touchdowns were scored during the game by both bases.
teams.
38. REASONING Explain the differences between an
37. MULTIPLE REPRESENTATIONS In this algebraic expression and a verbal expression.
problem, you will explore the multiplication of powers
SOLUTION:
with like bases.
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
a. TABULAR Copy and complete the table.
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
b. ALGEBRAIC Write an equation for the pattern
in feet. Then use the variable to write an algebraic
in the table.
expression to represent one of your daily activities.
Describe in words what your expression represents,
c. VERBAL Make a conjecture about the exponent and explain your reasoning.
of the product of two powers with like bases.
SOLUTION:
SOLUTION:
Sample answer: x is the number of minutes it takes to
a. walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing
b. The exponent of the product is the sum of the an algebraic expression for the verbal expression
exponents of the factors. So, the algebraic equation three times the sum of n squared and 3. Is either
2 x (2 + x) of them correct? Explain your reasoning.
10 × 10 = 10 represents the pattern.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
bases.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a SOLUTION:
verbal expression. The expression 2x + 3 is the Consuelo is correct. The verbal expression says that
algebraic expression that represents the verbal the sum of n squared and 3 is multiplied by 3. So,
expression “three more than a double a number”.
parentheses are necessary. James left out the
2
39. OPEN ENDED Define a variable to represent a parentheses around n + 3.
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic 41. CHALLENGE For the cube, x represents a positive
expression to represent one of your daily activities. whole number. Find the value of x such that the
Describe in words what your expression represents, volume of the cube and 6 times the area of one of its
and explain your reasoning. faces have the same value.
SOLUTION:
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk. SOLUTION:
The volume of a cube can be found by multiplying
40. CCSS CRITIQUE Consuelo and James are writing the length times the width times the height. Because
an algebraic expression for the verbal expression the sides of a cube all have the same length, V = x •
three times the sum of n squared and 3. Is either 3
x • x, or x . The area of one of the faces of the cube
of them correct? Explain your reasoning.
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
x x3 = 6x2 Yes/No
4 No
SOLUTION:
Consuelo is correct. The verbal expression says that
6 Yes
the sum of n squared and 3 is multiplied by 3. So,
parentheses are necessary. James left out the
2
parentheses around n + 3.
41. CHALLENGE For the cube, x represents a positive
So, the sides must have a length of 6 for the volume
whole number. Find the value of x such that the
of the cube and 6 times the area of one of its faces
volume of the cube and 6 times the area of one of its
faces have the same value. to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
SOLUTION:
Sample answer: An algebraic expression is a math
The volume of a cube can be found by multiplying
phrase that contains one or more numbers or
the length times the width times the height. Because
variables. To write an algebraic expression from real
the sides of a cube all have the same length, V = x •
world situation, first assign variables. Then determine
3
x • x, or x . The area of one of the faces of the cube the arithmetic operations done on the variables.
can be found by multiplying the length times the Finally, put the terms in order.
2
width. So, A = x • x, or x .
43. Which expression best represents the volume of the
cube?
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
x x3 = 6x2 Yes/No
4 No
A the product of three and five
6 Yes
B three to the fifth power
C three squared
D three cubed
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces SOLUTION:
to have the same value. The volume of a cube can be found by multiplying
the length times the width times the height. Because
42. WRITING IN MATH Describe how to write an the sides of a cube all have the same length, V = x •
algebraic expression from a real–world situation. 3
x • x, or x . Because the length of each side is 3
Include a definition of algebraic expression in your
units, the expression three cubed best represents the
own words.
volume of the cube.
SOLUTION:
So, Choice D is the correct answer.
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or
44. Which expression best represents the perimeter of
variables. To write an algebraic expression from real
the rectangle?
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
Finally, put the terms in order.
43. Which expression best represents the volume of the
cube?
F 2lw
G l + w
H 2l + 2w
J 4(l + w)
A the product of three and five
SOLUTION:
To find the perimeter of a rectangle, find the sum of
B three to the fifth power
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
C three squared rectangle.
D three cubed Choice H is the correct answer.
SOLUTION:
45. SHORT RESPONSE The yards of fabric needed
The volume of a cube can be found by multiplying
to make curtains is 3 times the length of a window in
the length times the width times the height. Because
inches, divided by 36. Write an expression that
the sides of a cube all have the same length, V = x •
represents the yards of fabric needed in terms of the
3
x • x, or x . Because the length of each side is 3 length of the window l.
units, the expression three cubed best represents the
volume of the cube. SOLUTION:
The word times suggests multiplication and the words
So, Choice D is the correct answer.
divided by suggest division. So, represents the
44. Which expression best represents the perimeter of
yards of fabric needed in terms of the length of the
the rectangle? window l.
46. GEOMETRY Find the area of the rectangle.
F 2lw
G l + w
A 14 square meters
H 2l + 2w
B 16 square meters
J 4(l + w)
C 50 square meters
SOLUTION:
To find the perimeter of a rectangle, find the sum of
D 60 square meters
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the SOLUTION:
rectangle.
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
So, the area of the rectangle is 16 square meters.
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
Choice B is the correct answer.
represents the yards of fabric needed in terms of the
length of the window l.
47. AMUSEMENT PARKS A roller coaster
SOLUTION: enthusiast club took a poll to see what each
The word times suggests multiplication and the words member’s favorite ride was.. Make a bar graph of
the results.
divided by suggest division. So, represents the
yards of fabric needed in terms of the length of the Our Favorite Rides
window l. Number
Ride
of Votes
46. GEOMETRY Find the area of the rectangle. Big Plunge 5
Twisting
22
Time
The Shiner 16
Raging Bull 9
The
25
Bat
A 14 square meters
Teaser 6
The
B 16 square meters 12
Adventure
C 50 square meters
SOLUTION:
D 60 square meters Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted
SOLUTION: for each rollercoaster. The horizontal scale identifies
the roller coaster chosen.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
48. SPORTS The results for an annual 5K race are
Our Favorite Rides shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
Number
Ride the box-and-whisker plot tells about the times for the
of Votes
race.
Big Plunge 5
Twisting
22
Time
The Shiner 16
Raging Bull 9
The
25
Bat
Teaser 6
The
12
Adventure
SOLUTION:
Order the data from least to greatest. The times in
SOLUTION: order from least to greatest are 14:48, 14:58, 15:06,
Draw a bar to represent each roller coaster. The 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
vertical scale is the number of members who voted 20:47, 20:49, 21:35.
for each rollercoaster. The horizontal scale identifies The times are given in minutes and seconds. Rewrite
the roller coaster chosen. the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898
15:06 906
15:48 948
15:54 954
16:10 970
48. SPORTS The results for an annual 5K race are 16:30 990
shown below. Make a box-and-whisker plot for the 19:27 1167
data. Write a sentence describing what the length of 19:58 1198
the box-and-whisker plot tells about the times for the 20:21 1221
race. 20:39 1239
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
Q = 1239
SOLUTION: 3
Order the data from least to greatest. The times in
There are no outliers.
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Find the mean, median, and mode for each set
of data.
Time in Time in
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
Min:Sec Seconds
14:48 888 SOLUTION:
14:58 898
15:06 906
15:48 948
15:54 954
16:10 970
So, the mean is 5.6.
16:30 990
19:27 1167
Order the data from least to greatest.
19:58 1198 {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
20:21 1221
Because there is an even number of data, the median
20:39 1239 is the mean of 6 and 7.
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
So, the median is 6.5.
Q1 = 948
Q = 1078.5 The number 7 appears most often, so the mode is 7.
2
Q = 1239
3
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
There are no outliers. SOLUTION:
So, the mean is 0.4.
Find the mean, median, and mode for each set Order the data from least to greatest.
of data. {–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION: Because there is an even number of data, the median
is the mean of 0 and 0.
So, the mean is 5.6.
So, the median is 0.
Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
The numbers 0 and –1 both occur most often, so the
Because there is an even number of data, the median
is the mean of 6 and 7. modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
So, the median is 6.5.
The number 7 appears most often, so the mode is 7. So, the mean is 15.25.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0} Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
SOLUTION:
Because there is an even number of data, the median
is the mean of 15 and 16.
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
So, the median is 15.5.
Because there is an even number of data, the median
is the mean of 0 and 0. The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
The area of Lisa’s trampoline is 72 square feet.
SOLUTION:
Find each product or quotient.
53.
SOLUTION:
So, the mean is 15.25.
Multiply the numerators and denominators.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
54.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
55.
SOLUTION:
SOLUTION:
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
SOLUTION: Evaluate each expression.
Multiply the numerators and denominators.
56.
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
54.
SOLUTION:
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
55.
SOLUTION:
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
Evaluate each expression. 59. 10.34 + 14.27
56. SOLUTION:
10.34 + 14.27 = 24.61
SOLUTION: 60.
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
61. 37.02 – 15.86
58.
SOLUTION:
37.02 – 15.86 = 21.16
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
Write a verbal expression for each algebraic
4. the sum of a number and 14
expression.
1. 2m SOLUTION:
SOLUTION: Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
Because the 2 and the m are written next to each
number and 14 can be represented by the algebraic
other, they are being multiplied. So, the verbal
expression n + 14.
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
5. 6 less a number t
2. SOLUTION:
The word less suggests subtraction. So, the verbal
SOLUTION: expression 6 less a number t can be represented by
The expression shows the product of the factors the algebraic expression 6 – t.
4 4
and r . The factor r represents a number raised 6. 7 more than 11 times a number
to the fourth power. So, the verbal expression two SOLUTION:
thirds times r raised to the fourth power can be Let n represent a number. The words more than
used to describe the algebraic expression . suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than
3. a2 – 18b 11 times a number can be represented by the
algebraic expression 11n + 7.
SOLUTION:
7. 1 minus the quotient of r and 7
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b SOLUTION:
represents 18 times b. So, the verbal expression a The word minus suggests subtraction and the word
squared minus 18 times b can be used to describe quotient suggests division. So, the verbal expression
2 1 minus the quotient of r and 7 can be represented
the algebraic expression a – 18b.
by the algebraic expression .
Write an algebraic expression for each verbal
expression. 8. two fifths of a number j squared
4. the sum of a number and 14
SOLUTION:
SOLUTION: The words two–fifths of a number suggest
Let n represent a number. The word sum suggests multiplication. The squared means to raise to the
addition. So, the verbal expression the sum of a second power. So, the verbal expression two–fifths
number and 14 can be represented by the algebraic of a number j squared can be represented by the
expression n + 14. algebraic expression .
5. 6 less a number t 9. n cubed increased by 5
SOLUTION: SOLUTION:
The word less suggests subtraction. So, the verbal The word cubed means to raise to the third power.
expression 6 less a number t can be represented by The words increased by suggest addition. So, the
the algebraic expression 6 – t. verbal expression n cubed increased by 5 can be
3
represented by the algebraic expression n + 5.
6. 7 more than 11 times a number
SOLUTION: 10. GROCERIES Mr. Bailey purchased some
groceries that cost d dollars. He paid with a $50 bill.
Let n represent a number. The words more than
Write an expression for the amount of change he will
suggest addition and the word times suggests
receive.
multiplication. So, the verbal expression 7 more than
11 times a number can be represented by the SOLUTION:
algebraic expression 11n + 7.
To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
7. 1 minus the quotient of r and 7
$50. So, Mr. Bailey will receive 50 – d in change.
SOLUTION:
Write a verbal expression for each algebraic
The word minus suggests subtraction and the word
expression.
quotient suggests division. So, the verbal expression
11. 4q
1 minus the quotient of r and 7 can be represented
by the algebraic expression . SOLUTION:
Because 4 and q are written next to each other, they
8. two fifths of a number j squared are being multiplied. So, the verbal expression four
times a number q can be used to describe the
SOLUTION:
algebraic expression 4q.
The words two–fifths of a number suggest
multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths 12.
of a number j squared can be represented by the
algebraic expression . SOLUTION:
Because and y are written next to each other,
9. n cubed increased by 5
they are being multiplied. So, the verbal expression
SOLUTION:
one eighth of a number y can be used to describe
The word cubed means to raise to the third power.
the algebraic expression .
The words increased by suggest addition. So, the
verbal expression n cubed increased by 5 can be
3 13. 15 + r
represented by the algebraic expression n + 5.
SOLUTION:
10. GROCERIES Mr. Bailey purchased some
The expression shows the sum of two terms. So, the
groceries that cost d dollars. He paid with a $50 bill.
verbal expression 15 plus r can be used to describe
Write an expression for the amount of change he will
the algebraic expression 15 + r.
receive.
SOLUTION: 14. w – 24
To find the amount of change Mr. Bailey will
SOLUTION:
receive, subtract the cost of the groceries, d, from
The expression shows the difference of two terms.
$50. So, Mr. Bailey will receive 50 – d in change.
So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24.
Write a verbal expression for each algebraic
expression.
2
11. 4q 15. 3x
SOLUTION: SOLUTION:
Because 4 and q are written next to each other, they The expression shows the product of the factors 3
2 2
are being multiplied. So, the verbal expression four and x . The factor x represents a number raised to
times a number q can be used to describe the the second power. So, the verbal expression 3 times
algebraic expression 4q. x squared can be used to describe the algebraic
2
expression 3x .
12.
16.
SOLUTION:
SOLUTION:
Because and y are written next to each other,
The expression shows the quotient of two terms. The
they are being multiplied. So, the verbal expression 4
term r represents a number raised to the fourth
one eighth of a number y can be used to describe
power. So, the verbal expression r to the fourth
the algebraic expression .
power divided by 9 can be used to describe the
algebraic expression .
13. 15 + r
SOLUTION:
17. 2a + 6
The expression shows the sum of two terms. So, the
verbal expression 15 plus r can be used to describe SOLUTION:
the algebraic expression 15 + r. The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
14. w – 24 verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
SOLUTION:
2a + 6.
The expression shows the difference of two terms.
So, the verbal expression w minus 24 can be used to 18. r4 ∙ t3
describe the algebraic expression w – 24.
SOLUTION:
2
15. 3x The expression shows the product of two factors.
4
The factor r represents a number raised to the
SOLUTION:
3
The expression shows the product of the factors 3 fourth power. The factor t represents a number
2 2 raised to the third power. So, the verbal expression
and x . The factor x represents a number raised to
the product of a number r raised to the fourth
the second power. So, the verbal expression 3 times
power and a number t cubed can be used to
x squared can be used to describe the algebraic
2 describe the algebraic expression r4 ∙ t3.
expression 3x .
Write an algebraic expression for each verbal
16. expression.
19. x more than 7
SOLUTION:
SOLUTION:
The expression shows the quotient of two terms. The
The words more than suggest addition. So, the
4
term r represents a number raised to the fourth verbal expression x more than 7 can be represented
power. So, the verbal expression r to the fourth by the algebraic expression 7 + x.
power divided by 9 can be used to describe the
20. a number less 35
algebraic expression .
SOLUTION:
17. 2a + 6 Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less
SOLUTION:
35 can be represented by the algebraic expression n
The expression shows the sum of two terms. The
– 35.
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times 21. 5 times a number
a can be used to describe the algebraic expression
2a + 6. SOLUTION:
Let n represent a number. The word times suggests
18. r4 ∙ t3 multiplication. So, the verbal expression 5 times a
number can be represented by the algebraic
SOLUTION:
expression 5n.
The expression shows the product of two factors.
4 22. one third of a number
The factor r represents a number raised to the
3
fourth power. The factor t represents a number SOLUTION:
raised to the third power. So, the verbal expression Let n represent a number. The words one third of a
the product of a number r raised to the fourth number suggest multiplication. So, the verbal
power and a number t cubed can be used to expression one third of a number can be
describe the algebraic expression r4 ∙ t3.
represented by the algebraic expression .
Write an algebraic expression for each verbal
expression.
23. f divided by 10
19. x more than 7
SOLUTION:
SOLUTION:
The words divided by suggest division. So, the
The words more than suggest addition. So, the verbal expression f divided by 10 can be
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x. represented by the algebraic expression .
20. a number less 35 24. the quotient of 45 and r
SOLUTION: SOLUTION:
Let n represent a number. The word less suggests The word quotient suggests division. So, the verbal
subtraction. So, the verbal expression a number less expression the quotient of 45 and r can be
35 can be represented by the algebraic expression n
represented by the algebraic expression .
– 35.
21. 5 times a number 25. three times a number plus 16
SOLUTION: SOLUTION:
Let n represent a number. The word times suggests Let n represent a number. The word times suggests
multiplication. So, the verbal expression 5 times a multiplication, and the word plus suggests addition.
number can be represented by the algebraic So, the verbal expression three times a number plus
expression 5n. 16 can be represented by the algebraic expression
3n + 16.
22. one third of a number
26. 18 decreased by 3 times d
SOLUTION:
Let n represent a number. The words one third of a SOLUTION:
number suggest multiplication. So, the verbal The word decreased suggests subtraction, and the
expression one third of a number can be word times suggests multiplication. So, the verbal
expression 18 decreased by 3 times d can be
represented by the algebraic expression .
represented by the algebraic expression 18 – 3d.
23. f divided by 10 27. k squared minus 11
SOLUTION:
SOLUTION:
The word squared means a number raised to the
The words divided by suggest division. So, the
second power. The word minus suggests subtraction.
verbal expression f divided by 10 can be
So, the verbal expression k squared minus 11 can
represented by the algebraic expression . 2
be represented by the algebraic expression k – 11.
24. the quotient of 45 and r 28. 20 divided by t to the fifth power
SOLUTION: SOLUTION:
The word quotient suggests division. So, the verbal The words divided by suggest division. So, the
expression the quotient of 45 and r can be verbal expression 20 divided by t to the fifth power
represented by the algebraic expression . can be represented by the algebraic expression .
25. three times a number plus 16 29. GEOMETRY The volume of a cylinder is π times
the radius r squared multiplied by the height. Write
SOLUTION: an expression for the volume.
Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
So, the verbal expression three times a number plus
16 can be represented by the algebraic expression
3n + 16.
26. 18 decreased by 3 times d
SOLUTION:
SOLUTION:
The words times and multiplied by suggest
The word decreased suggests subtraction, and the
multiplication. So, the volume of a cylinder can be
word times suggests multiplication. So, the verbal
2
written as the algebraic expression πr h.
expression 18 decreased by 3 times d can be
represented by the algebraic expression 18 – 3d.
30. FINANCIAL LITERACY Jocelyn makes x dollars
27. k squared minus 11 per hour working at the grocery store and n dollars
per hour babysitting. Write an expression that
SOLUTION: describes her earnings if she babysat for 25 hours
The word squared means a number raised to the and worked at the grocery store for 15 hours.
second power. The word minus suggests subtraction.
SOLUTION:
So, the verbal expression k squared minus 11 can
To write an expression for how much Jocelyn made
2
be represented by the algebraic expression k – 11.
babysitting, multiply the number of hours she babysat,
25, by her hourly rate, n. This can be represented by
28. 20 divided by t to the fifth power
the algebraic expression 25n.
SOLUTION:
The words divided by suggest division. So, the To write an expression for how much Jocelyn made
verbal expression 20 divided by t to the fifth power working at the grocery store, multiply the number of
hours she worked, 15, by her hourly rate, x. This can
can be represented by the algebraic expression . be represented by the algebraic expression 15x.
To write an expression for her total earnings, find the
29. GEOMETRY The volume of a cylinder is π times
sum of the amount she earned babysitting and the
the radius r squared multiplied by the height. Write
amount she earned working at the grocery store.
an expression for the volume.
This can be written as the expression 25n + 15x.
Write a verbal expression for each algebraic
expression.
2
31. 25 + 6x
SOLUTION:
The expression shows the sum of two terms. The
SOLUTION:
2
The words times and multiplied by suggest term 6x means six times the square of a number.
2
multiplication. So, the volume of a cylinder can be So, the algebraic expression 25 + 6x can be
written as the algebraic expression πr2h. described by the verbal expression twenty–five plus
six times a number squared.
30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
2
32. 6f + 5f
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours SOLUTION:
and worked at the grocery store for 15 hours.
The expression shows the sum of two terms. The
2
SOLUTION: term 6f means six times the square of a number.
To write an expression for how much Jocelyn made The term 5f means five times a number. So, the
babysitting, multiply the number of hours she babysat, algebraic expression 6f2 + 5f can be described by the
25, by her hourly rate, n. This can be represented by verbal expression six times a number squared plus
the algebraic expression 25n. five times the number.
To write an expression for how much Jocelyn made
working at the grocery store, multiply the number of 33.
hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x. SOLUTION:
The expression shows the quotient of two terms. The
To write an expression for her total earnings, find the 5
term 3a means three times a number that has been
sum of the amount she earned babysitting and the
raised to the fifth power. So, the algebraic expression
amount she earned working at the grocery store.
can be described by the verbal expression three
This can be written as the expression 25n + 15x.
times a number raised to the fifth power divided
by two.
Write a verbal expression for each algebraic
expression.
34. CCSS SENSE-MAKING A certain smartphone
2
31. 25 + 6x family plan costs $55 per month plus additional usage
costs. If x is the number of cell phone minutes used
SOLUTION:
above the plan amount and y is the number of
The expression shows the sum of two terms. The
megabytes of data used above the plan amount,
2
term 6x means six times the square of a number. interpret the following expressions.
So, the algebraic expression 25 + 6x2 can be
described by the verbal expression twenty–five plus a. 0.25x
six times a number squared.
b. 2y
2
32. 6f + 5f c. 0.25x + 2y + 55
SOLUTION:
The expression shows the sum of two terms. The SOLUTION:
2
term 6f means six times the square of a number. a. Since x is the number of cell phone minutes, then
The term 5f means five times a number. So, the 0.25x would be the cost of extra minutes at $0.25 per
2 minute.
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number. b. Since y is the number of megabytes of data used
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
33.
c. The expression 0.25x + 2y + 55 represents the
SOLUTION: extra minute charges and the extra data usage
The expression shows the quotient of two terms. The charge plus the monthly family plan cost of $55. The
5 expression represents the total monthly cost for the
term 3a means three times a number that has been
family.
raised to the fifth power. So, the algebraic expression
can be described by the verbal expression three
times a number raised to the fifth power divided
by two. 35. DREAMS It is believed that about of our dreams
34. CCSS SENSE-MAKING A certain smartphone involve people that we know.
family plan costs $55 per month plus additional usage
costs. If x is the number of cell phone minutes used a. Write an expression to describe the number of
above the plan amount and y is the number of dreams that feature people you know if you have d
megabytes of data used above the plan amount, dreams.
interpret the following expressions.
b. Use the expression you wrote to predict the
a. 0.25x number of dreams that include people you know out
of 28 dreams.
b. 2y
c. 0.25x + 2y + 55 SOLUTION:
a. To write an expression to describe the number of
dreams that feature people you know if you have d
SOLUTION:
a. Since x is the number of cell phone minutes, then dreams, multiply by d or .
0.25x would be the cost of extra minutes at $0.25 per
minute.
b. To predict the number of dreams that include
b. Since y is the number of megabytes of data used people you know out of 28 dreams, replace d with 28
above the plan amount, then 2y would be the cost of in the expression .
extra data used at $2 per megabyte.
c. The expression 0.25x + 2y + 55 represents the
extra minute charges and the extra data usage
charge plus the monthly family plan cost of $55. The
expression represents the total monthly cost for the
1-1 Vfaamriilayb. l es and Expressions So, you would predict having 21 dreams that include
people you know.
36. SPORTS In football, a touchdown is awarded 6
35. DREAMS It is believed that about of our dreams
points and the team can then try for a point after a
involve people that we know. touchdown.
a. Write an expression to describe the number of a. Write an expression that describes the number of
dreams that feature people you know if you have d points scored on touchdowns and points after
dreams. touchdowns by one team in a game.
b. Use the expression you wrote to predict the b. If a team wins a football game 27–0, write an
number of dreams that include people you know out equation to represent the possible number of
of 28 dreams. touchdowns and points after touchdowns by the
winning team.
SOLUTION: c. If a team wins a football game 21–7, how many
a. To write an expression to describe the number of possible number of touchdowns and points after
dreams that feature people you know if you have d touchdowns were scored during the game by both
teams?
dreams, multiply by d or .
SOLUTION:
a. Let T be the number of touchdowns and p be the
b. To predict the number of dreams that include
number of points scored after touchdowns. So, the
people you know out of 28 dreams, replace d with 28 expression 6T + p, describes the number of points
in the expression . scored on touchdowns and points after touchdowns
by one team in a game.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
So, you would predict having 21 dreams that include
c. If a team wins a football game 21–7, then 6T + p
people you know.
= 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
36. SPORTS In football, a touchdown is awarded 6
the game by both teams.
points and the team can then try for a point after a
touchdown.
Let T = 4 and p = 4.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of So, it is possible that 4 touchdowns and 4 points after
touchdowns and points after touchdowns by the touchdowns were scored during the game by both
winning team. teams.
c. If a team wins a football game 21–7, how many 37. MULTIPLE REPRESENTATIONS In this
possible number of touchdowns and points after problem, you will explore the multiplication of powers
touchdowns were scored during the game by both with like bases.
teams?
a. TABULAR Copy and complete the table.
SOLUTION:
a. Let T be the number of touchdowns and p be the
eSolutniounmsMbeanr uoafl -pPooiwnetsre sdcboyreCdog anfetreor touchdowns. So, the Page5
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
b. ALGEBRAIC Write an equation for the pattern
b. If a team scores 27 points in a game, then 6T + p in the table.
= 27 represents the possible number of touchdowns
and points after touchdowns by the winning team. c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
c. If a team wins a football game 21–7, then 6T + p
SOLUTION:
= 28 represents the possible number of touchdowns
a.
and points after touchdowns that were scored during
the game by both teams.
Let T = 4 and p = 4.
b. The exponent of the product is the sum of the
exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
So, it is possible that 4 touchdowns and 4 points after
c. The exponent of the product of two powers is the
touchdowns were scored during the game by both
sum of the exponents of the powers with the same
teams.
bases.
37. MULTIPLE REPRESENTATIONS In this
38. REASONING Explain the differences between an
problem, you will explore the multiplication of powers
algebraic expression and a verbal expression.
with like bases.
SOLUTION:
a. TABULAR Copy and complete the table. Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
b. ALGEBRAIC Write an equation for the pattern 39. OPEN ENDED Define a variable to represent a
in the table. real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
c. VERBAL Make a conjecture about the exponent
Describe in words what your expression represents,
of the product of two powers with like bases.
and explain your reasoning.
SOLUTION:
SOLUTION:
a.
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
b. The exponent of the product is the sum of the
40. CCSS CRITIQUE Consuelo and James are writing
exponents of the factors. So, the algebraic equation
an algebraic expression for the verbal expression
2 x (2 + x)
10 × 10 = 10 represents the pattern. three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
bases.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
SOLUTION:
algebraic expression that represents the verbal
expression “three more than a double a number”. Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So,
39. OPEN ENDED Define a variable to represent a parentheses are necessary. James left out the
real-life quantity, such as time in minutes or distance 2
parentheses around n + 3.
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
41. CHALLENGE For the cube, x represents a positive
Describe in words what your expression represents,
whole number. Find the value of x such that the
and explain your reasoning.
volume of the cube and 6 times the area of one of its
faces have the same value.
SOLUTION:
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
SOLUTION:
40. CCSS CRITIQUE Consuelo and James are writing
The volume of a cube can be found by multiplying
an algebraic expression for the verbal expression
the length times the width times the height. Because
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning. the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
x x3 = 6x2 Yes/No
4 No
SOLUTION:
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So,
parentheses are necessary. James left out the 6 Yes
2
parentheses around n + 3.
41. CHALLENGE For the cube, x represents a positive
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
So, the sides must have a length of 6 for the volume
faces have the same value.
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
The volume of a cube can be found by multiplying SOLUTION:
the length times the width times the height. Because Sample answer: An algebraic expression is a math
the sides of a cube all have the same length, V = x • phrase that contains one or more numbers or
3 variables. To write an algebraic expression from real
x • x, or x . The area of one of the faces of the cube
world situation, first assign variables. Then determine
can be found by multiplying the length times the
the arithmetic operations done on the variables.
2
width. So, A = x • x, or x . Finally, put the terms in order.
To find the value of x such that the volume of the 43. Which expression best represents the volume of the
cube and 6 times the area of one of its faces have cube?
the same value, find a value for x such that x3 = 6x2.
x x3 = 6x2 Yes/No
4 No
6 Yes A the product of three and five
B three to the fifth power
C three squared
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces D three cubed
to have the same value.
SOLUTION:
42. WRITING IN MATH Describe how to write an The volume of a cube can be found by multiplying
algebraic expression from a real–world situation. the length times the width times the height. Because
Include a definition of algebraic expression in your the sides of a cube all have the same length, V = x •
own words. x • x, or x3. Because the length of each side is 3
units, the expression three cubed best represents the
SOLUTION:
volume of the cube.
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or So, Choice D is the correct answer.
variables. To write an algebraic expression from real
world situation, first assign variables. Then determine 44. Which expression best represents the perimeter of
the arithmetic operations done on the variables. the rectangle?
Finally, put the terms in order.
43. Which expression best represents the volume of the
cube?
F 2lw
G l + w
H 2l + 2w
A the product of three and five
J 4(l + w)
B three to the fifth power SOLUTION:
To find the perimeter of a rectangle, find the sum of
C three squared twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
rectangle.
D three cubed
SOLUTION: Choice H is the correct answer.
The volume of a cube can be found by multiplying
the length times the width times the height. Because 45. SHORT RESPONSE The yards of fabric needed
the sides of a cube all have the same length, V = x • to make curtains is 3 times the length of a window in
3 inches, divided by 36. Write an expression that
x • x, or x . Because the length of each side is 3
represents the yards of fabric needed in terms of the
units, the expression three cubed best represents the
length of the window l.
volume of the cube.
SOLUTION:
So, Choice D is the correct answer.
The word times suggests multiplication and the words
44. Which expression best represents the perimeter of
divided by suggest division. So, represents the
the rectangle?
yards of fabric needed in terms of the length of the
window l.
46. GEOMETRY Find the area of the rectangle.
F 2lw
G l + w
H 2l + 2w
A 14 square meters
J 4(l + w)
B 16 square meters
SOLUTION:
To find the perimeter of a rectangle, find the sum of C 50 square meters
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the D 60 square meters
rectangle.
SOLUTION:
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that So, the area of the rectangle is 16 square meters.
represents the yards of fabric needed in terms of the
length of the window l. Choice B is the correct answer.
SOLUTION:
47. AMUSEMENT PARKS A roller coaster
The word times suggests multiplication and the words
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
divided by suggest division. So, represents the
the results.
yards of fabric needed in terms of the length of the
window l.
Our Favorite Rides
Number
46. GEOMETRY Find the area of the rectangle. Ride
of Votes
Big Plunge 5
Twisting
22
Time
The Shiner 16
Raging Bull 9
A 14 square meters The
25
Bat
B 16 square meters Teaser 6
The
12
C 50 square meters Adventure
D 60 square meters SOLUTION:
Draw a bar to represent each roller coaster. The
SOLUTION:
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
the roller coaster chosen.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
Our Favorite Rides
48. SPORTS The results for an annual 5K race are
Number
Ride shown below. Make a box-and-whisker plot for the
of Votes
data. Write a sentence describing what the length of
Big Plunge 5 the box-and-whisker plot tells about the times for the
Twisting race.
22
Time
The Shiner 16
Raging Bull 9
The
25
Bat
Teaser 6
The
12
Adventure
SOLUTION:
SOLUTION:
Order the data from least to greatest. The times in
Draw a bar to represent each roller coaster. The
order from least to greatest are 14:48, 14:58, 15:06,
vertical scale is the number of members who voted
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
for each rollercoaster. The horizontal scale identifies
20:47, 20:49, 21:35.
the roller coaster chosen.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898
15:06 906
15:48 948
15:54 954
48. SPORTS The results for an annual 5K race are
16:10 970
shown below. Make a box-and-whisker plot for the
16:30 990
data. Write a sentence describing what the length of
19:27 1167
the box-and-whisker plot tells about the times for the
race. 19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
SOLUTION: 2
Order the data from least to greatest. The times in Q = 1239
3
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, There are no outliers.
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Time in Time in Find the mean, median, and mode for each set
Min:Sec Seconds of data.
14:48 888 49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
14:58 898
SOLUTION:
15:06 906
15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167 So, the mean is 5.6.
19:58 1198
20:21 1221 Order the data from least to greatest.
20:39 1239 {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median
20:47 1247
is the mean of 6 and 7.
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1 So, the median is 6.5.
Q = 1078.5
2
Q = 1239 The number 7 appears most often, so the mode is 7.
3
There are no outliers. 50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
SOLUTION:
So, the mean is 0.4.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8} Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
SOLUTION:
Because there is an even number of data, the median
is the mean of 0 and 0.
So, the mean is 5.6.
Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}. So, the median is 0.
Because there is an even number of data, the median
is the mean of 6 and 7.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
So, the median is 6.5.
The number 7 appears most often, so the mode is 7.
So, the mean is 15.25.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
Order the data from least to greatest.
SOLUTION:
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Because there is an even number of data, the median So, the median is 15.5.
is the mean of 0 and 0.
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
So, the median is 0.
SOLUTION:
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
So, the mean is 15.25.
SOLUTION:
Order the data from least to greatest. Multiply the numerators and denominators.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
54.
So, the median is 15.5.
SOLUTION:
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
55.
SOLUTION:
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
SOLUTION:
Multiply the numerators and denominators.
Evaluate each expression.
56.
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
54.
SOLUTION:
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
55.
58.
SOLUTION:
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
Evaluate each expression.
56. 59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
60.
with denominators of 45 then add the numerators.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
61. 37.02 – 15.86
SOLUTION:
SOLUTION:
37.02 – 15.86 = 21.16
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
Write a verbal expression for each algebraic
The word less suggests subtraction. So, the verbal
expression.
expression 6 less a number t can be represented by
1. 2m
the algebraic expression 6 – t.
SOLUTION:
6. 7 more than 11 times a number
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal SOLUTION:
expression the product of 2 and m can be used to
Let n represent a number. The words more than
describe the algebraic expression 2m.
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than
2. 11 times a number can be represented by the
algebraic expression 11n + 7.
SOLUTION:
7. 1 minus the quotient of r and 7
The expression shows the product of the factors
SOLUTION:
4 4
and r . The factor r represents a number raised
The word minus suggests subtraction and the word
to the fourth power. So, the verbal expression two quotient suggests division. So, the verbal expression
thirds times r raised to the fourth power can be 1 minus the quotient of r and 7 can be represented
by the algebraic expression .
used to describe the algebraic expression .
8. two fifths of a number j squared
3. a2 – 18b
SOLUTION:
SOLUTION:
The words two–fifths of a number suggest
The expression shows the difference of two terms. multiplication. The squared means to raise to the
The term a2 represents a squared. The term 18b second power. So, the verbal expression two–fifths
represents 18 times b. So, the verbal expression a of a number j squared can be represented by the
squared minus 18 times b can be used to describe algebraic expression .
2
the algebraic expression a – 18b.
9. n cubed increased by 5
Write an algebraic expression for each verbal
expression. SOLUTION:
4. the sum of a number and 14 The word cubed means to raise to the third power.
The words increased by suggest addition. So, the
SOLUTION:
verbal expression n cubed increased by 5 can be
Let n represent a number. The word sum suggests 3
represented by the algebraic expression n + 5.
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
10. GROCERIES Mr. Bailey purchased some
expression n + 14.
groceries that cost d dollars. He paid with a $50 bill.
Write an expression for the amount of change he will
5. 6 less a number t receive.
SOLUTION: SOLUTION:
The word less suggests subtraction. So, the verbal To find the amount of change Mr. Bailey will
expression 6 less a number t can be represented by receive, subtract the cost of the groceries, d, from
the algebraic expression 6 – t. $50. So, Mr. Bailey will receive 50 – d in change.
6. 7 more than 11 times a number Write a verbal expression for each algebraic
expression.
SOLUTION:
11. 4q
Let n represent a number. The words more than
SOLUTION:
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than Because 4 and q are written next to each other, they
11 times a number can be represented by the are being multiplied. So, the verbal expression four
algebraic expression 11n + 7. times a number q can be used to describe the
algebraic expression 4q.
7. 1 minus the quotient of r and 7
SOLUTION: 12.
The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression SOLUTION:
1 minus the quotient of r and 7 can be represented
by the algebraic expression . Because and y are written next to each other,
they are being multiplied. So, the verbal expression
8. two fifths of a number j squared one eighth of a number y can be used to describe
the algebraic expression .
SOLUTION:
The words two–fifths of a number suggest
multiplication. The squared means to raise to the 13. 15 + r
second power. So, the verbal expression two–fifths
SOLUTION:
of a number j squared can be represented by the
The expression shows the sum of two terms. So, the
algebraic expression .
verbal expression 15 plus r can be used to describe
the algebraic expression 15 + r.
9. n cubed increased by 5
14. w – 24
SOLUTION:
The word cubed means to raise to the third power. SOLUTION:
The words increased by suggest addition. So, the
The expression shows the difference of two terms.
verbal expression n cubed increased by 5 can be
So, the verbal expression w minus 24 can be used to
represented by the algebraic expression n3 + 5. describe the algebraic expression w – 24.
10. GROCERIES Mr. Bailey purchased some 2
15. 3x
groceries that cost d dollars. He paid with a $50 bill.
Write an expression for the amount of change he will SOLUTION:
receive. The expression shows the product of the factors 3
2 2
and x . The factor x represents a number raised to
SOLUTION:
the second power. So, the verbal expression 3 times
To find the amount of change Mr. Bailey will
x squared can be used to describe the algebraic
receive, subtract the cost of the groceries, d, from
2
$50. So, Mr. Bailey will receive 50 – d in change. expression 3x .
Write a verbal expression for each algebraic
16.
expression.
11. 4q
SOLUTION:
SOLUTION:
The expression shows the quotient of two terms. The
Because 4 and q are written next to each other, they
4
term r represents a number raised to the fourth
are being multiplied. So, the verbal expression four
power. So, the verbal expression r to the fourth
times a number q can be used to describe the
power divided by 9 can be used to describe the
algebraic expression 4q.
algebraic expression .
12.
17. 2a + 6
SOLUTION:
SOLUTION:
Because and y are written next to each other, The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
they are being multiplied. So, the verbal expression
verbal expression 6 more than the product 2 times
one eighth of a number y can be used to describe
a can be used to describe the algebraic expression
the algebraic expression . 2a + 6.
13. 15 + r 18. r4 ∙ t3
SOLUTION: SOLUTION:
The expression shows the sum of two terms. So, the The expression shows the product of two factors.
verbal expression 15 plus r can be used to describe The factor r4 represents a number raised to the
the algebraic expression 15 + r. 3
fourth power. The factor t represents a number
raised to the third power. So, the verbal expression
14. w – 24
the product of a number r raised to the fourth
SOLUTION: power and a number t cubed can be used to
The expression shows the difference of two terms. describe the algebraic expression r4 ∙ t3.
So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24. Write an algebraic expression for each verbal
expression.
15. 3x2 19. x more than 7
SOLUTION: SOLUTION:
The expression shows the product of the factors 3 The words more than suggest addition. So, the
2 2 verbal expression x more than 7 can be represented
and x . The factor x represents a number raised to
by the algebraic expression 7 + x.
the second power. So, the verbal expression 3 times
x squared can be used to describe the algebraic
2 20. a number less 35
expression 3x .
SOLUTION:
16. Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less
35 can be represented by the algebraic expression n
SOLUTION:
– 35.
The expression shows the quotient of two terms. The
4
term r represents a number raised to the fourth 21. 5 times a number
power. So, the verbal expression r to the fourth
SOLUTION:
power divided by 9 can be used to describe the
Let n represent a number. The word times suggests
algebraic expression . multiplication. So, the verbal expression 5 times a
number can be represented by the algebraic
17. 2a + 6 expression 5n.
SOLUTION: 22. one third of a number
The expression shows the sum of two terms. The
SOLUTION:
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times Let n represent a number. The words one third of a
a can be used to describe the algebraic expression number suggest multiplication. So, the verbal
2a + 6. expression one third of a number can be
represented by the algebraic expression .
18. r4 ∙ t3
SOLUTION: 23. f divided by 10
The expression shows the product of two factors.
SOLUTION:
4
The factor r represents a number raised to the
The words divided by suggest division. So, the
3
fourth power. The factor t represents a number verbal expression f divided by 10 can be
raised to the third power. So, the verbal expression
the product of a number r raised to the fourth represented by the algebraic expression .
power and a number t cubed can be used to
describe the algebraic expression r4 ∙ t3. 24. the quotient of 45 and r
Write an algebraic expression for each verbal SOLUTION:
expression. The word quotient suggests division. So, the verbal
19. x more than 7 expression the quotient of 45 and r can be
SOLUTION: represented by the algebraic expression .
The words more than suggest addition. So, the
verbal expression x more than 7 can be represented 25. three times a number plus 16
by the algebraic expression 7 + x.
SOLUTION:
Let n represent a number. The word times suggests
20. a number less 35
multiplication, and the word plus suggests addition.
SOLUTION: So, the verbal expression three times a number plus
Let n represent a number. The word less suggests 16 can be represented by the algebraic expression
subtraction. So, the verbal expression a number less 3n + 16.
35 can be represented by the algebraic expression n
26. 18 decreased by 3 times d
– 35.
SOLUTION:
21. 5 times a number
The word decreased suggests subtraction, and the
SOLUTION: word times suggests multiplication. So, the verbal
expression 18 decreased by 3 times d can be
Let n represent a number. The word times suggests
represented by the algebraic expression 18 – 3d.
multiplication. So, the verbal expression 5 times a
number can be represented by the algebraic
27. k squared minus 11
expression 5n.
SOLUTION:
22. one third of a number
The word squared means a number raised to the
SOLUTION: second power. The word minus suggests subtraction.
So, the verbal expression k squared minus 11 can
Let n represent a number. The words one third of a
2
number suggest multiplication. So, the verbal be represented by the algebraic expression k – 11.
expression one third of a number can be
28. 20 divided by t to the fifth power
represented by the algebraic expression .
SOLUTION:
The words divided by suggest division. So, the
23. f divided by 10
verbal expression 20 divided by t to the fifth power
SOLUTION:
can be represented by the algebraic expression .
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
29. GEOMETRY The volume of a cylinder is π times
represented by the algebraic expression . the radius r squared multiplied by the height. Write
an expression for the volume.
24. the quotient of 45 and r
SOLUTION:
The word quotient suggests division. So, the verbal
expression the quotient of 45 and r can be
represented by the algebraic expression .
SOLUTION:
25. three times a number plus 16 The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be
SOLUTION: 2
written as the algebraic expression πr h.
Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition. 30. FINANCIAL LITERACY Jocelyn makes x dollars
So, the verbal expression three times a number plus per hour working at the grocery store and n dollars
16 can be represented by the algebraic expression per hour babysitting. Write an expression that
3n + 16. describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours.
26. 18 decreased by 3 times d
SOLUTION:
SOLUTION:
To write an expression for how much Jocelyn made
The word decreased suggests subtraction, and the
babysitting, multiply the number of hours she babysat,
word times suggests multiplication. So, the verbal
25, by her hourly rate, n. This can be represented by
expression 18 decreased by 3 times d can be
the algebraic expression 25n.
represented by the algebraic expression 18 – 3d.
To write an expression for how much Jocelyn made
27. k squared minus 11
working at the grocery store, multiply the number of
SOLUTION: hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x.
The word squared means a number raised to the
second power. The word minus suggests subtraction.
To write an expression for her total earnings, find the
So, the verbal expression k squared minus 11 can
sum of the amount she earned babysitting and the
2
be represented by the algebraic expression k – 11.
amount she earned working at the grocery store.
This can be written as the expression 25n + 15x.
28. 20 divided by t to the fifth power
Write a verbal expression for each algebraic
SOLUTION:
expression.
The words divided by suggest division. So, the
2
verbal expression 20 divided by t to the fifth power 31. 25 + 6x
can be represented by the algebraic expression . SOLUTION:
The expression shows the sum of two terms. The
2
29. GEOMETRY The volume of a cylinder is π times term 6x means six times the square of a number.
2
the radius r squared multiplied by the height. Write So, the algebraic expression 25 + 6x can be
an expression for the volume. described by the verbal expression twenty–five plus
six times a number squared.
2
32. 6f + 5f
SOLUTION:
The expression shows the sum of two terms. The
2
SOLUTION: term 6f means six times the square of a number.
The words times and multiplied by suggest The term 5f means five times a number. So, the
multiplication. So, the volume of a cylinder can be algebraic expression 6f2 + 5f can be described by the
written as the algebraic expression πr2h. verbal expression six times a number squared plus
five times the number.
30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
33.
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours. SOLUTION:
The expression shows the quotient of two terms. The
SOLUTION:
5
term 3a means three times a number that has been
To write an expression for how much Jocelyn made
raised to the fifth power. So, the algebraic expression
babysitting, multiply the number of hours she babysat,
can be described by the verbal expression three
25, by her hourly rate, n. This can be represented by
times a number raised to the fifth power divided
the algebraic expression 25n.
by two.
To write an expression for how much Jocelyn made
34. CCSS SENSE-MAKING A certain smartphone
working at the grocery store, multiply the number of
family plan costs $55 per month plus additional usage
hours she worked, 15, by her hourly rate, x. This can
costs. If x is the number of cell phone minutes used
be represented by the algebraic expression 15x.
above the plan amount and y is the number of
megabytes of data used above the plan amount,
To write an expression for her total earnings, find the
interpret the following expressions.
sum of the amount she earned babysitting and the
amount she earned working at the grocery store.
a. 0.25x
This can be written as the expression 25n + 15x.
b. 2y
Write a verbal expression for each algebraic
expression.
c. 0.25x + 2y + 55
2
31. 25 + 6x
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. The
a. Since x is the number of cell phone minutes, then
2
term 6x means six times the square of a number. 0.25x would be the cost of extra minutes at $0.25 per
So, the algebraic expression 25 + 6x2 can be minute.
described by the verbal expression twenty–five plus
six times a number squared. b. Since y is the number of megabytes of data used
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
2
32. 6f + 5f
c. The expression 0.25x + 2y + 55 represents the
SOLUTION:
extra minute charges and the extra data usage
The expression shows the sum of two terms. The
charge plus the monthly family plan cost of $55. The
2
term 6f means six times the square of a number. expression represents the total monthly cost for the
The term 5f means five times a number. So, the family.
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number. 35. DREAMS It is believed that about of our dreams
involve people that we know.
33.
a. Write an expression to describe the number of
SOLUTION:
dreams that feature people you know if you have d
The expression shows the quotient of two terms. The dreams.
term 3a5 means three times a number that has been
raised to the fifth power. So, the algebraic expression b. Use the expression you wrote to predict the
can be described by the verbal expression three number of dreams that include people you know out
times a number raised to the fifth power divided of 28 dreams.
by two.
34. CCSS SENSE-MAKING A certain smartphone SOLUTION:
family plan costs $55 per month plus additional usage a. To write an expression to describe the number of
costs. If x is the number of cell phone minutes used dreams that feature people you know if you have d
above the plan amount and y is the number of
megabytes of data used above the plan amount, dreams, multiply by d or .
interpret the following expressions.
b. To predict the number of dreams that include
a. 0.25x
people you know out of 28 dreams, replace d with 28
b. 2y in the expression .
c. 0.25x + 2y + 55
SOLUTION:
a. Since x is the number of cell phone minutes, then
0.25x would be the cost of extra minutes at $0.25 per So, you would predict having 21 dreams that include
minute. people you know.
b. Since y is the number of megabytes of data used 36. SPORTS In football, a touchdown is awarded 6
above the plan amount, then 2y would be the cost of points and the team can then try for a point after a
extra data used at $2 per megabyte. touchdown.
c. The expression 0.25x + 2y + 55 represents the a. Write an expression that describes the number of
extra minute charges and the extra data usage points scored on touchdowns and points after
charge plus the monthly family plan cost of $55. The touchdowns by one team in a game.
expression represents the total monthly cost for the
family. b. If a team wins a football game 27–0, write an
equation to represent the possible number of
touchdowns and points after touchdowns by the
35. DREAMS It is believed that about of our dreams winning team.
involve people that we know.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
a. Write an expression to describe the number of touchdowns were scored during the game by both
dreams that feature people you know if you have d teams?
dreams.
SOLUTION:
b. Use the expression you wrote to predict the a. Let T be the number of touchdowns and p be the
number of dreams that include people you know out number of points scored after touchdowns. So, the
of 28 dreams. expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
SOLUTION:
a. To write an expression to describe the number of b. If a team scores 27 points in a game, then 6T + p
dreams that feature people you know if you have d = 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
dreams, multiply by d or .
c. If a team wins a football game 21–7, then 6T + p
b. To predict the number of dreams that include = 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
people you know out of 28 dreams, replace d with 28
the game by both teams.
in the expression .
Let T = 4 and p = 4.
So, you would predict having 21 dreams that include
people you know.
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
36. SPORTS In football, a touchdown is awarded 6
teams.
points and the team can then try for a point after a
touchdown.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
a. Write an expression that describes the number of with like bases.
points scored on touchdowns and points after
touchdowns by one team in a game.
a. TABULAR Copy and complete the table.
b. If a team wins a football game 27–0, write an
equation to represent the possible number of
touchdowns and points after touchdowns by the
winning team.
c. If a team wins a football game 21–7, how many b. ALGEBRAIC Write an equation for the pattern
possible number of touchdowns and points after in the table.
touchdowns were scored during the game by both
teams? c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a. Let T be the number of touchdowns and p be the SOLUTION:
number of points scored after touchdowns. So, the a.
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns b. The exponent of the product is the sum of the
and points after touchdowns by the winning team. exponents of the factors. So, the algebraic equation
102 × 10x = 10(2 + x) represents the pattern.
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns c. The exponent of the product of two powers is the
and points after touchdowns that were scored during
sum of the exponents of the powers with the same
the game by both teams. bases.
Let T = 4 and p = 4. 38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
So, it is possible that 4 touchdowns and 4 points after verbal expression. The expression 2x + 3 is the
1-1 Vtoaurcihadbolewsn asn wde Erex spcroeressdi odnursi ng the game by both algebraic expression that represents the verbal
teams. expression “three more than a double a number”.
37. MULTIPLE REPRESENTATIONS In this 39. OPEN ENDED Define a variable to represent a
problem, you will explore the multiplication of powers real-life quantity, such as time in minutes or distance
with like bases. in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
a. TABULAR Copy and complete the table. Describe in words what your expression represents,
and explain your reasoning.
SOLUTION:
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
b. ALGEBRAIC Write an equation for the pattern
and I take my dog on a 15 minute walk.
in the table.
40. CCSS CRITIQUE Consuelo and James are writing
c. VERBAL Make a conjecture about the exponent
an algebraic expression for the verbal expression
of the product of two powers with like bases.
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
SOLUTION:
a.
b. The exponent of the product is the sum of the
exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
c. The exponent of the product of two powers is the
SOLUTION:
sum of the exponents of the powers with the same
Consuelo is correct. The verbal expression says that
bases.
the sum of n squared and 3 is multiplied by 3. So,
38. REASONING Explain the differences between an parentheses are necessary. James left out the
algebraic expression and a verbal expression. parentheses around n2 + 3.
SOLUTION:
41. CHALLENGE For the cube, x represents a positive
Algebraic expressions include variables, numbers, whole number. Find the value of x such that the
and symbols. Verbal expressions contain words. For volume of the cube and 6 times the area of one of its
example, “three more than a double a number” is a faces have the same value.
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic SOLUTION:
expression to represent one of your daily activities.
The volume of a cube can be found by multiplying
Describe in words what your expression represents,
the length times the width times the height. Because
and explain your reasoning.
the sides of a cube all have the same length, V = x •
3
SOLUTION: x • x, or x . The area of one of the faces of the cube
Sample answer: x is the number of minutes it takes to can be found by multiplying the length times the
walk between my house and school. 2x + 15 width. So, A = x • x, or x2.
eSolutionsManual-PoweredbyCognero Page6
represents the amount of time in minutes I spend
walking each day since I walk to and from school To find the value of x such that the volume of the
and I take my dog on a 15 minute walk. cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either x x3 = 6x2 Yes/No
of them correct? Explain your reasoning. 4 No
6 Yes
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
SOLUTION: to have the same value.
Consuelo is correct. The verbal expression says that
42. WRITING IN MATH Describe how to write an
the sum of n squared and 3 is multiplied by 3. So,
algebraic expression from a real–world situation.
parentheses are necessary. James left out the
Include a definition of algebraic expression in your
2
parentheses around n + 3. own words.
41. CHALLENGE For the cube, x represents a positive SOLUTION:
whole number. Find the value of x such that the Sample answer: An algebraic expression is a math
volume of the cube and 6 times the area of one of its phrase that contains one or more numbers or
faces have the same value. variables. To write an algebraic expression from real
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
Finally, put the terms in order.
43. Which expression best represents the volume of the
cube?
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
A the product of three and five
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
B three to the fifth power
the same value, find a value for x such that x3 = 6x2.
C three squared
x x3 = 6x2 Yes/No
4 No D three cubed
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
6 Yes
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
So, the sides must have a length of 6 for the volume So, Choice D is the correct answer.
of the cube and 6 times the area of one of its faces
to have the same value. 44. Which expression best represents the perimeter of
the rectangle?
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
F 2lw
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or
variables. To write an algebraic expression from real G l + w
world situation, first assign variables. Then determine
the arithmetic operations done on the variables. H 2l + 2w
Finally, put the terms in order.
J 4(l + w)
43. Which expression best represents the volume of the
cube? SOLUTION:
To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
rectangle.
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
A the product of three and five
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
B three to the fifth power represents the yards of fabric needed in terms of the
length of the window l.
C three squared
SOLUTION:
The word times suggests multiplication and the words
D three cubed
divided by suggest division. So, represents the
SOLUTION:
The volume of a cube can be found by multiplying yards of fabric needed in terms of the length of the
the length times the width times the height. Because window l.
the sides of a cube all have the same length, V = x •
3 46. GEOMETRY Find the area of the rectangle.
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
A 14 square meters
B 16 square meters
C 50 square meters
F 2lw
D 60 square meters
G l + w SOLUTION:
H 2l + 2w
J 4(l + w)
So, the area of the rectangle is 16 square meters.
SOLUTION:
To find the perimeter of a rectangle, find the sum of Choice B is the correct answer.
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the 47. AMUSEMENT PARKS A roller coaster
rectangle. enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
Choice H is the correct answer. the results.
45. SHORT RESPONSE The yards of fabric needed
Our Favorite Rides
to make curtains is 3 times the length of a window in
Number
inches, divided by 36. Write an expression that Ride
of Votes
represents the yards of fabric needed in terms of the
length of the window l. Big Plunge 5
Twisting
SOLUTION: Time 22
The word times suggests multiplication and the words The Shiner 16
divided by suggest division. So, represents the Raging Bull 9
The
yards of fabric needed in terms of the length of the 25
Bat
window l.
Teaser 6
46. GEOMETRY Find the area of the rectangle. The
12
Adventure
SOLUTION:
Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
A 14 square meters the roller coaster chosen.
B 16 square meters
C 50 square meters
D 60 square meters
SOLUTION:
48. SPORTS The results for an annual 5K race are
shown below. Make a box-and-whisker plot for the
So, the area of the rectangle is 16 square meters.
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
Choice B is the correct answer.
race.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
Our Favorite Rides
Number
Ride
of Votes
Big Plunge 5
Twisting SOLUTION:
22
Time
Order the data from least to greatest. The times in
The Shiner 16 order from least to greatest are 14:48, 14:58, 15:06,
Raging Bull 9 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35.
The
25 The times are given in minutes and seconds. Rewrite
Bat
the times so that they are in seconds by multiplying
Teaser 6
the number of minutes by 60 and then adding the
The
12 seconds.
Adventure
Time in Time in
SOLUTION: Min:Sec Seconds
Draw a bar to represent each roller coaster. The 14:48 888
vertical scale is the number of members who voted 14:58 898
for each rollercoaster. The horizontal scale identifies 15:06 906
the roller coaster chosen. 15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167
19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
48. SPORTS The results for an annual 5K race are Then determine the quartiles.
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of Q = 948
1
the box-and-whisker plot tells about the times for the
race. Q2 = 1078.5
Q = 1239
3
There are no outliers.
Find the mean, median, and mode for each set
SOLUTION:
of data.
Order the data from least to greatest. The times in 49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, SOLUTION:
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
So, the mean is 5.6.
Time in Time in
Order the data from least to greatest.
Min:Sec Seconds
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
14:48 888
Because there is an even number of data, the median
14:58 898 is the mean of 6 and 7.
15:06 906
15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167
So, the median is 6.5.
19:58 1198
20:21 1221
The number 7 appears most often, so the mode is 7.
20:39 1239
20:47 1247
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
20:49 1249
21:35 1295 SOLUTION:
Then determine the quartiles.
Q = 948
1
So, the mean is 0.4.
Q = 1078.5
2
Q = 1239 Order the data from least to greatest.
3
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
There are no outliers.
Because there is an even number of data, the median
is the mean of 0 and 0.
Find the mean, median, and mode for each set
of data.
So, the median is 0.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION: The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
So, the mean is 5.6.
Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median So, the mean is 15.25.
is the mean of 6 and 7.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
So, the median is 6.5.
The number 7 appears most often, so the mode is 7.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
So, the median is 15.5.
SOLUTION:
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
So, the mean is 0.4.
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5} SOLUTION:
Because there is an even number of data, the median
is the mean of 0 and 0.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the SOLUTION:
modes are 0 and –1. Multiply the numerators and denominators.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
54.
So, the mean is 15.25.
Order the data from least to greatest. SOLUTION:
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16.
55.
So, the median is 15.5.
The number 24 appears most often, so the mode is SOLUTION:
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
Evaluate each expression.
56.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient. SOLUTION:
53. The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
SOLUTION:
Multiply the numerators and denominators.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
54.
58.
SOLUTION:
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
55.
59. 10.34 + 14.27
SOLUTION:
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
Evaluate each expression. The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
56.
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
The word less suggests subtraction. So, the verbal
expression 6 less a number t can be represented by
the algebraic expression 6 – t.
6. 7 more than 11 times a number
SOLUTION:
Let n represent a number. The words more than
Write a verbal expression for each algebraic
suggest addition and the word times suggests
expression.
multiplication. So, the verbal expression 7 more than
1. 2m
11 times a number can be represented by the
algebraic expression 11n + 7.
SOLUTION:
Because the 2 and the m are written next to each
7. 1 minus the quotient of r and 7
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to SOLUTION:
describe the algebraic expression 2m. The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression
1 minus the quotient of r and 7 can be represented
2.
by the algebraic expression .
SOLUTION:
8. two fifths of a number j squared
The expression shows the product of the factors
SOLUTION:
4 4
and r . The factor r represents a number raised
The words two–fifths of a number suggest
to the fourth power. So, the verbal expression two multiplication. The squared means to raise to the
thirds times r raised to the fourth power can be second power. So, the verbal expression two–fifths
of a number j squared can be represented by the
used to describe the algebraic expression .
algebraic expression .
3. a2 – 18b
9. n cubed increased by 5
SOLUTION:
SOLUTION:
The expression shows the difference of two terms.
The word cubed means to raise to the third power.
The term a2 represents a squared. The term 18b
The words increased by suggest addition. So, the
represents 18 times b. So, the verbal expression a
verbal expression n cubed increased by 5 can be
squared minus 18 times b can be used to describe
3
represented by the algebraic expression n + 5.
2
the algebraic expression a – 18b.
10. GROCERIES Mr. Bailey purchased some
Write an algebraic expression for each verbal
groceries that cost d dollars. He paid with a $50 bill.
expression.
Write an expression for the amount of change he will
4. the sum of a number and 14 receive.
SOLUTION:
SOLUTION:
Let n represent a number. The word sum suggests To find the amount of change Mr. Bailey will
addition. So, the verbal expression the sum of a receive, subtract the cost of the groceries, d, from
number and 14 can be represented by the algebraic $50. So, Mr. Bailey will receive 50 – d in change.
expression n + 14.
Write a verbal expression for each algebraic
expression.
5. 6 less a number t
11. 4q
SOLUTION:
SOLUTION:
The word less suggests subtraction. So, the verbal
Because 4 and q are written next to each other, they
expression 6 less a number t can be represented by
are being multiplied. So, the verbal expression four
the algebraic expression 6 – t.
times a number q can be used to describe the
6. 7 more than 11 times a number algebraic expression 4q.
SOLUTION:
12.
Let n represent a number. The words more than
suggest addition and the word times suggests
SOLUTION:
multiplication. So, the verbal expression 7 more than
11 times a number can be represented by the
Because and y are written next to each other,
algebraic expression 11n + 7.
they are being multiplied. So, the verbal expression
7. 1 minus the quotient of r and 7
one eighth of a number y can be used to describe
SOLUTION: the algebraic expression .
The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression 13. 15 + r
1 minus the quotient of r and 7 can be represented
SOLUTION:
by the algebraic expression .
The expression shows the sum of two terms. So, the
8. two fifths of a number j squared verbal expression 15 plus r can be used to describe
the algebraic expression 15 + r.
SOLUTION:
The words two–fifths of a number suggest 14. w – 24
multiplication. The squared means to raise to the
SOLUTION:
second power. So, the verbal expression two–fifths
The expression shows the difference of two terms.
of a number j squared can be represented by the
So, the verbal expression w minus 24 can be used to
algebraic expression .
describe the algebraic expression w – 24.
9. n cubed increased by 5 2
15. 3x
SOLUTION:
SOLUTION:
The word cubed means to raise to the third power.
The expression shows the product of the factors 3
The words increased by suggest addition. So, the
2 2
and x . The factor x represents a number raised to
verbal expression n cubed increased by 5 can be
the second power. So, the verbal expression 3 times
3
represented by the algebraic expression n + 5.
x squared can be used to describe the algebraic
2
10. GROCERIES Mr. Bailey purchased some expression 3x .
groceries that cost d dollars. He paid with a $50 bill.
Write an expression for the amount of change he will
16.
receive.
SOLUTION: SOLUTION:
To find the amount of change Mr. Bailey will The expression shows the quotient of two terms. The
receive, subtract the cost of the groceries, d, from 4
term r represents a number raised to the fourth
$50. So, Mr. Bailey will receive 50 – d in change.
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the
Write a verbal expression for each algebraic
expression. algebraic expression .
11. 4q
SOLUTION: 17. 2a + 6
Because 4 and q are written next to each other, they
SOLUTION:
are being multiplied. So, the verbal expression four
The expression shows the sum of two terms. The
times a number q can be used to describe the
term 2a represents the product of 2 and a. So, the
algebraic expression 4q.
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
12. 2a + 6.
SOLUTION: 18. r4 ∙ t3
Because and y are written next to each other, SOLUTION:
The expression shows the product of two factors.
they are being multiplied. So, the verbal expression
4
The factor r represents a number raised to the
one eighth of a number y can be used to describe
3
the algebraic expression . fourth power. The factor t represents a number
raised to the third power. So, the verbal expression
the product of a number r raised to the fourth
13. 15 + r
power and a number t cubed can be used to
SOLUTION: describe the algebraic expression r4 ∙ t3.
The expression shows the sum of two terms. So, the
verbal expression 15 plus r can be used to describe Write an algebraic expression for each verbal
the algebraic expression 15 + r. expression.
19. x more than 7
14. w – 24
SOLUTION:
SOLUTION: The words more than suggest addition. So, the
The expression shows the difference of two terms. verbal expression x more than 7 can be represented
So, the verbal expression w minus 24 can be used to by the algebraic expression 7 + x.
describe the algebraic expression w – 24.
20. a number less 35
2
15. 3x
SOLUTION:
SOLUTION:
Let n represent a number. The word less suggests
The expression shows the product of the factors 3
subtraction. So, the verbal expression a number less
2 2
and x . The factor x represents a number raised to 35 can be represented by the algebraic expression n
the second power. So, the verbal expression 3 times – 35.
x squared can be used to describe the algebraic
2 21. 5 times a number
expression 3x .
SOLUTION:
16. Let n represent a number. The word times suggests
multiplication. So, the verbal expression 5 times a
number can be represented by the algebraic
SOLUTION:
expression 5n.
The expression shows the quotient of two terms. The
4
term r represents a number raised to the fourth 22. one third of a number
power. So, the verbal expression r to the fourth
SOLUTION:
power divided by 9 can be used to describe the
Let n represent a number. The words one third of a
algebraic expression . number suggest multiplication. So, the verbal
expression one third of a number can be
17. 2a + 6
represented by the algebraic expression .
SOLUTION:
The expression shows the sum of two terms. The 23. f divided by 10
term 2a represents the product of 2 and a. So, the
SOLUTION:
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression The words divided by suggest division. So, the
2a + 6. verbal expression f divided by 10 can be
18. r4 ∙ t3 represented by the algebraic expression .
SOLUTION: 24. the quotient of 45 and r
The expression shows the product of two factors.
SOLUTION:
4
The factor r represents a number raised to the
The word quotient suggests division. So, the verbal
3
fourth power. The factor t represents a number expression the quotient of 45 and r can be
raised to the third power. So, the verbal expression
the product of a number r raised to the fourth represented by the algebraic expression .
power and a number t cubed can be used to
describe the algebraic expression r4 ∙ t3. 25. three times a number plus 16
Write an algebraic expression for each verbal SOLUTION:
expression. Let n represent a number. The word times suggests
19. x more than 7 multiplication, and the word plus suggests addition.
So, the verbal expression three times a number plus
SOLUTION:
16 can be represented by the algebraic expression
The words more than suggest addition. So, the 3n + 16.
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x. 26. 18 decreased by 3 times d
SOLUTION:
20. a number less 35
The word decreased suggests subtraction, and the
word times suggests multiplication. So, the verbal
SOLUTION:
expression 18 decreased by 3 times d can be
Let n represent a number. The word less suggests
represented by the algebraic expression 18 – 3d.
subtraction. So, the verbal expression a number less
35 can be represented by the algebraic expression n
27. k squared minus 11
– 35.
SOLUTION:
21. 5 times a number
The word squared means a number raised to the
second power. The word minus suggests subtraction.
SOLUTION:
So, the verbal expression k squared minus 11 can
Let n represent a number. The word times suggests
2
multiplication. So, the verbal expression 5 times a be represented by the algebraic expression k – 11.
number can be represented by the algebraic
expression 5n. 28. 20 divided by t to the fifth power
SOLUTION:
22. one third of a number
The words divided by suggest division. So, the
SOLUTION: verbal expression 20 divided by t to the fifth power
Let n represent a number. The words one third of a
can be represented by the algebraic expression .
number suggest multiplication. So, the verbal
expression one third of a number can be
29. GEOMETRY The volume of a cylinder is π times
represented by the algebraic expression . the radius r squared multiplied by the height. Write
an expression for the volume.
23. f divided by 10
SOLUTION:
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
represented by the algebraic expression .
SOLUTION:
24. the quotient of 45 and r The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be
SOLUTION: 2
written as the algebraic expression πr h.
The word quotient suggests division. So, the verbal
expression the quotient of 45 and r can be 30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
represented by the algebraic expression .
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours
25. three times a number plus 16 and worked at the grocery store for 15 hours.
SOLUTION: SOLUTION:
Let n represent a number. The word times suggests To write an expression for how much Jocelyn made
multiplication, and the word plus suggests addition. babysitting, multiply the number of hours she babysat,
So, the verbal expression three times a number plus 25, by her hourly rate, n. This can be represented by
16 can be represented by the algebraic expression the algebraic expression 25n.
3n + 16.
To write an expression for how much Jocelyn made
26. 18 decreased by 3 times d working at the grocery store, multiply the number of
hours she worked, 15, by her hourly rate, x. This can
SOLUTION:
be represented by the algebraic expression 15x.
The word decreased suggests subtraction, and the
word times suggests multiplication. So, the verbal
To write an expression for her total earnings, find the
expression 18 decreased by 3 times d can be
sum of the amount she earned babysitting and the
represented by the algebraic expression 18 – 3d.
amount she earned working at the grocery store.
This can be written as the expression 25n + 15x.
27. k squared minus 11
Write a verbal expression for each algebraic
SOLUTION:
expression.
The word squared means a number raised to the
2
second power. The word minus suggests subtraction. 31. 25 + 6x
So, the verbal expression k squared minus 11 can
SOLUTION:
2
be represented by the algebraic expression k – 11.
The expression shows the sum of two terms. The
2
28. 20 divided by t to the fifth power term 6x means six times the square of a number.
2
So, the algebraic expression 25 + 6x can be
SOLUTION:
described by the verbal expression twenty–five plus
The words divided by suggest division. So, the
six times a number squared.
verbal expression 20 divided by t to the fifth power
can be represented by the algebraic expression . 2
32. 6f + 5f
SOLUTION:
29. GEOMETRY The volume of a cylinder is π times
The expression shows the sum of two terms. The
the radius r squared multiplied by the height. Write
2
an expression for the volume. term 6f means six times the square of a number.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number.
33.
SOLUTION:
The words times and multiplied by suggest SOLUTION:
multiplication. So, the volume of a cylinder can be The expression shows the quotient of two terms. The
written as the algebraic expression πr2h. 5
term 3a means three times a number that has been
raised to the fifth power. So, the algebraic expression
30. FINANCIAL LITERACY Jocelyn makes x dollars
can be described by the verbal expression three
per hour working at the grocery store and n dollars
times a number raised to the fifth power divided
per hour babysitting. Write an expression that
by two.
describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours. 34. CCSS SENSE-MAKING A certain smartphone
family plan costs $55 per month plus additional usage
SOLUTION:
costs. If x is the number of cell phone minutes used
To write an expression for how much Jocelyn made
above the plan amount and y is the number of
babysitting, multiply the number of hours she babysat,
megabytes of data used above the plan amount,
25, by her hourly rate, n. This can be represented by
interpret the following expressions.
the algebraic expression 25n.
a. 0.25x
To write an expression for how much Jocelyn made
working at the grocery store, multiply the number of
b. 2y
hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x.
c. 0.25x + 2y + 55
To write an expression for her total earnings, find the
sum of the amount she earned babysitting and the
SOLUTION:
amount she earned working at the grocery store.
a. Since x is the number of cell phone minutes, then
This can be written as the expression 25n + 15x.
0.25x would be the cost of extra minutes at $0.25 per
minute.
Write a verbal expression for each algebraic
expression.
b. Since y is the number of megabytes of data used
2
31. 25 + 6x
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
SOLUTION:
The expression shows the sum of two terms. The
c. The expression 0.25x + 2y + 55 represents the
2
term 6x means six times the square of a number.
extra minute charges and the extra data usage
2
So, the algebraic expression 25 + 6x can be charge plus the monthly family plan cost of $55. The
described by the verbal expression twenty–five plus expression represents the total monthly cost for the
six times a number squared. family.
2
32. 6f + 5f
35. DREAMS It is believed that about of our dreams
SOLUTION:
involve people that we know.
The expression shows the sum of two terms. The
2
term 6f means six times the square of a number.
a. Write an expression to describe the number of
The term 5f means five times a number. So, the
dreams that feature people you know if you have d
2
algebraic expression 6f + 5f can be described by the dreams.
verbal expression six times a number squared plus
five times the number.
b. Use the expression you wrote to predict the
number of dreams that include people you know out
33. of 28 dreams.
SOLUTION:
SOLUTION:
The expression shows the quotient of two terms. The
a. To write an expression to describe the number of
5
term 3a means three times a number that has been dreams that feature people you know if you have d
raised to the fifth power. So, the algebraic expression
can be described by the verbal expression three dreams, multiply by d or .
times a number raised to the fifth power divided
by two.
b. To predict the number of dreams that include
34. CCSS SENSE-MAKING A certain smartphone people you know out of 28 dreams, replace d with 28
family plan costs $55 per month plus additional usage in the expression .
costs. If x is the number of cell phone minutes used
above the plan amount and y is the number of
megabytes of data used above the plan amount,
interpret the following expressions.
a. 0.25x
So, you would predict having 21 dreams that include
b. 2y people you know.
c. 0.25x + 2y + 55 36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
touchdown.
SOLUTION:
a. Since x is the number of cell phone minutes, then a. Write an expression that describes the number of
0.25x would be the cost of extra minutes at $0.25 per
points scored on touchdowns and points after
minute. touchdowns by one team in a game.
b. Since y is the number of megabytes of data used
b. If a team wins a football game 27–0, write an
above the plan amount, then 2y would be the cost of
equation to represent the possible number of
extra data used at $2 per megabyte.
touchdowns and points after touchdowns by the
winning team.
c. The expression 0.25x + 2y + 55 represents the
extra minute charges and the extra data usage
c. If a team wins a football game 21–7, how many
charge plus the monthly family plan cost of $55. The
possible number of touchdowns and points after
expression represents the total monthly cost for the
touchdowns were scored during the game by both
family.
teams?
SOLUTION:
35. DREAMS It is believed that about of our dreams a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
involve people that we know.
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
a. Write an expression to describe the number of by one team in a game.
dreams that feature people you know if you have d
dreams.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
b. Use the expression you wrote to predict the and points after touchdowns by the winning team.
number of dreams that include people you know out
of 28 dreams.
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
SOLUTION:
the game by both teams.
a. To write an expression to describe the number of
dreams that feature people you know if you have d
Let T = 4 and p = 4.
dreams, multiply by d or .
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
in the expression .
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
teams.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
with like bases.
So, you would predict having 21 dreams that include
people you know.
a. TABULAR Copy and complete the table.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
touchdown.
a. Write an expression that describes the number of
points scored on touchdowns and points after
b. ALGEBRAIC Write an equation for the pattern
touchdowns by one team in a game.
in the table.
b. If a team wins a football game 27–0, write an
c. VERBAL Make a conjecture about the exponent
equation to represent the possible number of
of the product of two powers with like bases.
touchdowns and points after touchdowns by the
winning team. SOLUTION:
a.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
teams?
SOLUTION:
b. The exponent of the product is the sum of the
a. Let T be the number of touchdowns and p be the
exponents of the factors. So, the algebraic equation
number of points scored after touchdowns. So, the
2 x (2 + x)
10 × 10 = 10 represents the pattern.
expression 6T + p, describes the number of points
scored on touchdowns and points after touchdowns
by one team in a game. c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
bases.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
38. REASONING Explain the differences between an
and points after touchdowns by the winning team.
algebraic expression and a verbal expression.
c. If a team wins a football game 21–7, then 6T + p SOLUTION:
= 28 represents the possible number of touchdowns Algebraic expressions include variables, numbers,
and points after touchdowns that were scored during and symbols. Verbal expressions contain words. For
the game by both teams. example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
Let T = 4 and p = 4. algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
So, it is possible that 4 touchdowns and 4 points after Describe in words what your expression represents,
touchdowns were scored during the game by both and explain your reasoning.
teams.
SOLUTION:
37. MULTIPLE REPRESENTATIONS In this Sample answer: x is the number of minutes it takes to
problem, you will explore the multiplication of powers walk between my house and school. 2x + 15
with like bases. represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
a. TABULAR Copy and complete the table.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
b. ALGEBRAIC Write an equation for the pattern
in the table.
c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a.
SOLUTION:
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So,
b. The exponent of the product is the sum of the parentheses are necessary. James left out the
exponents of the factors. So, the algebraic equation
2
parentheses around n + 3.
2 x (2 + x)
10 × 10 = 10 represents the pattern.
41. CHALLENGE For the cube, x represents a positive
c. The exponent of the product of two powers is the whole number. Find the value of x such that the
sum of the exponents of the powers with the same volume of the cube and 6 times the area of one of its
bases. faces have the same value.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
SOLUTION:
example, “three more than a double a number” is a
The volume of a cube can be found by multiplying
verbal expression. The expression 2x + 3 is the
the length times the width times the height. Because
algebraic expression that represents the verbal
the sides of a cube all have the same length, V = x •
expression “three more than a double a number”.
3
x • x, or x . The area of one of the faces of the cube
39. OPEN ENDED Define a variable to represent a can be found by multiplying the length times the
real-life quantity, such as time in minutes or distance width. So, A = x • x, or x2.
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities. To find the value of x such that the volume of the
Describe in words what your expression represents, cube and 6 times the area of one of its faces have
and explain your reasoning. the same value, find a value for x such that x3 = 6x2.
SOLUTION:
Sample answer: x is the number of minutes it takes to x x3 = 6x2 Yes/No
walk between my house and school. 2x + 15 4 No
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing 6 Yes
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
Sample answer: An algebraic expression is a math
SOLUTION:
phrase that contains one or more numbers or
Consuelo is correct. The verbal expression says that
variables. To write an algebraic expression from real
the sum of n squared and 3 is multiplied by 3. So,
world situation, first assign variables. Then determine
parentheses are necessary. James left out the
1-1 Variables and Expressions the arithmetic operations done on the variables.
parentheses around n2 + 3. Finally, put the terms in order.
41. CHALLENGE For the cube, x represents a positive 43. Which expression best represents the volume of the
whole number. Find the value of x such that the cube?
volume of the cube and 6 times the area of one of its
faces have the same value.
SOLUTION: A the product of three and five
The volume of a cube can be found by multiplying
the length times the width times the height. Because B three to the fifth power
the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube C three squared
can be found by multiplying the length times the
2
width. So, A = x • x, or x . D three cubed
SOLUTION:
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have The volume of a cube can be found by multiplying
the same value, find a value for x such that x3 = 6x2. the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
x x3 = 6x2 Yes/No
units, the expression three cubed best represents the
4 No volume of the cube.
So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
6 Yes
the rectangle?
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
F 2lw
to have the same value.
42. WRITING IN MATH Describe how to write an G l + w
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your H 2l + 2w
own words.
J 4(l + w)
SOLUTION:
Sample answer: An algebraic expression is a math SOLUTION:
phrase that contains one or more numbers or
To find the perimeter of a rectangle, find the sum of
variables. To write an algebraic expression from real
twice the length and twice the width. The expression
world situation, first assign variables. Then determine
2l + 2w best represents the perimeter of the
the arithmetic operations done on the variables.
rectangle.
Finally, put the terms in order.
Choice H is the correct answer.
43. Which expression best represents the volume of the
cube?
eSolutionsManual-PoweredbyCognero 45. SHORT RESPONSE The yards of fabric needPeadg e7
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the
length of the window l.
SOLUTION:
The word times suggests multiplication and the words
A the product of three and five divided by suggest division. So, represents the
yards of fabric needed in terms of the length of the
B three to the fifth power window l.
C three squared 46. GEOMETRY Find the area of the rectangle.
D three cubed
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
A 14 square meters
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
B 16 square meters
volume of the cube.
So, Choice D is the correct answer. C 50 square meters
44. Which expression best represents the perimeter of D 60 square meters
the rectangle?
SOLUTION:
F 2lw So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
G l + w
47. AMUSEMENT PARKS A roller coaster
H 2l + 2w
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
J 4(l + w) the results.
SOLUTION:
Our Favorite Rides
To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression Number
Ride
2l + 2w best represents the perimeter of the of Votes
rectangle. Big Plunge 5
Twisting
22
Choice H is the correct answer. Time
The Shiner 16
45. SHORT RESPONSE The yards of fabric needed
Raging Bull 9
to make curtains is 3 times the length of a window in
The
inches, divided by 36. Write an expression that 25
Bat
represents the yards of fabric needed in terms of the
length of the window l. Teaser 6
The
SOLUTION: 12
Adventure
The word times suggests multiplication and the words
divided by suggest division. So, represents the SOLUTION:
Draw a bar to represent each roller coaster. The
yards of fabric needed in terms of the length of the
window l. vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
46. GEOMETRY Find the area of the rectangle. the roller coaster chosen.
A 14 square meters
B 16 square meters
C 50 square meters 48. SPORTS The results for an annual 5K race are
shown below. Make a box-and-whisker plot for the
D 60 square meters data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
SOLUTION:
race.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of SOLUTION:
the results. Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
Our Favorite Rides 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35.
Number
Ride of Votes The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
Big Plunge 5
the number of minutes by 60 and then adding the
Twisting
22 seconds.
Time
The Shiner 16
Time in Time in
Raging Bull 9 Min:Sec Seconds
The 14:48 888
25
Bat 14:58 898
Teaser 6 15:06 906
The 15:48 948
12
Adventure 15:54 954
16:10 970
16:30 990
SOLUTION:
19:27 1167
Draw a bar to represent each roller coaster. The
19:58 1198
vertical scale is the number of members who voted
20:21 1221
for each rollercoaster. The horizontal scale identifies
20:39 1239
the roller coaster chosen.
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
Q = 1239
3
48. SPORTS The results for an annual 5K race are There are no outliers.
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
race.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
So, the mean is 5.6.
Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
Order the data from least to greatest.
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
20:47, 20:49, 21:35.
Because there is an even number of data, the median
The times are given in minutes and seconds. Rewrite
is the mean of 6 and 7.
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Time in Time in
Min:Sec Seconds
14:48 888 So, the median is 6.5.
14:58 898
15:06 906 The number 7 appears most often, so the mode is 7.
15:48 948
15:54 954
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
16:10 970
16:30 990 SOLUTION:
19:27 1167
19:58 1198
20:21 1221
20:39 1239
So, the mean is 0.4.
20:47 1247
20:49 1249
Order the data from least to greatest.
21:35 1295
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Then determine the quartiles.
Because there is an even number of data, the median
is the mean of 0 and 0.
Q = 948
1
Q = 1078.5
2
Q = 1239
3
There are no outliers.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
Find the mean, median, and mode for each set SOLUTION:
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
So, the mean is 5.6.
Because there is an even number of data, the median
Order the data from least to greatest.
is the mean of 15 and 16.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median
is the mean of 6 and 7.
So, the median is 15.5.
So, the median is 6.5. The number 24 appears most often, so the mode is
24.
The number 7 appears most often, so the mode is 7.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0} trampoline in square feet?
SOLUTION: SOLUTION:
So, the mean is 0.4.
The area of Lisa’s trampoline is 72 square feet.
Order the data from least to greatest.
Find each product or quotient.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
53.
Because there is an even number of data, the median
is the mean of 0 and 0.
SOLUTION:
Multiply the numerators and denominators.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1. 54.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION: SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
55.
Because there is an even number of data, the median
is the mean of 15 and 16.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24. Evaluate each expression.
52. SPORTS Lisa has a rectangular trampoline that is 6 56.
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
57. 5.67 – 4.21
SOLUTION:
SOLUTION:
5.67 – 4.21 = 1.46
Multiply the numerators and denominators.
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
54. with a common denominator of 6.
SOLUTION:
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
55. 60.
SOLUTION: SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
Evaluate each expression.
56.
61. 37.02 – 15.86
SOLUTION:
SOLUTION:
37.02 – 15.86 = 21.16
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
Write a verbal expression for each algebraic expression n + 14.
expression.
1. 2m
5. 6 less a number t
SOLUTION:
SOLUTION:
Because the 2 and the m are written next to each
The word less suggests subtraction. So, the verbal
other, they are being multiplied. So, the verbal
expression 6 less a number t can be represented by
expression the product of 2 and m can be used to
the algebraic expression 6 – t.
describe the algebraic expression 2m.
6. 7 more than 11 times a number
2.
SOLUTION:
Let n represent a number. The words more than
SOLUTION:
suggest addition and the word times suggests
The expression shows the product of the factors
multiplication. So, the verbal expression 7 more than
4 4 11 times a number can be represented by the
and r . The factor r represents a number raised
algebraic expression 11n + 7.
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be 7. 1 minus the quotient of r and 7
used to describe the algebraic expression . SOLUTION:
The word minus suggests subtraction and the word
3. a2 – 18b quotient suggests division. So, the verbal expression
1 minus the quotient of r and 7 can be represented
SOLUTION:
by the algebraic expression .
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b 8. two fifths of a number j squared
represents 18 times b. So, the verbal expression a
SOLUTION:
squared minus 18 times b can be used to describe
2 The words two–fifths of a number suggest
the algebraic expression a – 18b.
multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
Write an algebraic expression for each verbal
of a number j squared can be represented by the
expression.
4. the sum of a number and 14 algebraic expression .
SOLUTION:
9. n cubed increased by 5
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a SOLUTION:
number and 14 can be represented by the algebraic The word cubed means to raise to the third power.
expression n + 14. The words increased by suggest addition. So, the
verbal expression n cubed increased by 5 can be
3
5. 6 less a number t represented by the algebraic expression n + 5.
SOLUTION: 10. GROCERIES Mr. Bailey purchased some
The word less suggests subtraction. So, the verbal groceries that cost d dollars. He paid with a $50 bill.
expression 6 less a number t can be represented by Write an expression for the amount of change he will
the algebraic expression 6 – t. receive.
6. 7 more than 11 times a number SOLUTION:
To find the amount of change Mr. Bailey will
SOLUTION:
receive, subtract the cost of the groceries, d, from
Let n represent a number. The words more than $50. So, Mr. Bailey will receive 50 – d in change.
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than Write a verbal expression for each algebraic
11 times a number can be represented by the expression.
algebraic expression 11n + 7. 11. 4q
SOLUTION:
7. 1 minus the quotient of r and 7
Because 4 and q are written next to each other, they
SOLUTION:
are being multiplied. So, the verbal expression four
The word minus suggests subtraction and the word times a number q can be used to describe the
quotient suggests division. So, the verbal expression algebraic expression 4q.
1 minus the quotient of r and 7 can be represented
by the algebraic expression .
12.
8. two fifths of a number j squared
SOLUTION:
SOLUTION:
Because and y are written next to each other,
The words two–fifths of a number suggest
multiplication. The squared means to raise to the
they are being multiplied. So, the verbal expression
second power. So, the verbal expression two–fifths
one eighth of a number y can be used to describe
of a number j squared can be represented by the
the algebraic expression .
algebraic expression .
13. 15 + r
9. n cubed increased by 5
SOLUTION:
SOLUTION:
The expression shows the sum of two terms. So, the
The word cubed means to raise to the third power.
verbal expression 15 plus r can be used to describe
The words increased by suggest addition. So, the
the algebraic expression 15 + r.
verbal expression n cubed increased by 5 can be
3
represented by the algebraic expression n + 5. 14. w – 24
10. GROCERIES Mr. Bailey purchased some SOLUTION:
groceries that cost d dollars. He paid with a $50 bill. The expression shows the difference of two terms.
Write an expression for the amount of change he will So, the verbal expression w minus 24 can be used to
receive. describe the algebraic expression w – 24.
SOLUTION: 2
15. 3x
To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from SOLUTION:
$50. So, Mr. Bailey will receive 50 – d in change. The expression shows the product of the factors 3
2 2
and x . The factor x represents a number raised to
Write a verbal expression for each algebraic
the second power. So, the verbal expression 3 times
expression.
x squared can be used to describe the algebraic
11. 4q
2
expression 3x .
SOLUTION:
Because 4 and q are written next to each other, they
are being multiplied. So, the verbal expression four 16.
times a number q can be used to describe the
algebraic expression 4q. SOLUTION:
The expression shows the quotient of two terms. The
4
12. term r represents a number raised to the fourth
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the
SOLUTION:
algebraic expression .
Because and y are written next to each other,
they are being multiplied. So, the verbal expression
17. 2a + 6
one eighth of a number y can be used to describe
the algebraic expression . SOLUTION:
The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
13. 15 + r
verbal expression 6 more than the product 2 times
SOLUTION: a can be used to describe the algebraic expression
The expression shows the sum of two terms. So, the 2a + 6.
verbal expression 15 plus r can be used to describe
the algebraic expression 15 + r. 18. r4 ∙ t3
SOLUTION:
14. w – 24
The expression shows the product of two factors.
SOLUTION: 4
The factor r represents a number raised to the
The expression shows the difference of two terms.
3
fourth power. The factor t represents a number
So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24. raised to the third power. So, the verbal expression
the product of a number r raised to the fourth
2 power and a number t cubed can be used to
15. 3x
describe the algebraic expression r4 ∙ t3.
SOLUTION:
The expression shows the product of the factors 3 Write an algebraic expression for each verbal
2 2 expression.
and x . The factor x represents a number raised to
19. x more than 7
the second power. So, the verbal expression 3 times
x squared can be used to describe the algebraic
SOLUTION:
2
expression 3x . The words more than suggest addition. So, the
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x.
16.
20. a number less 35
SOLUTION:
The expression shows the quotient of two terms. The SOLUTION:
4
term r represents a number raised to the fourth Let n represent a number. The word less suggests
power. So, the verbal expression r to the fourth subtraction. So, the verbal expression a number less
power divided by 9 can be used to describe the 35 can be represented by the algebraic expression n
– 35.
algebraic expression .
21. 5 times a number
17. 2a + 6
SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
The expression shows the sum of two terms. The multiplication. So, the verbal expression 5 times a
term 2a represents the product of 2 and a. So, the number can be represented by the algebraic
verbal expression 6 more than the product 2 times expression 5n.
a can be used to describe the algebraic expression
2a + 6. 22. one third of a number
SOLUTION:
18. r4 ∙ t3
Let n represent a number. The words one third of a
SOLUTION: number suggest multiplication. So, the verbal
expression one third of a number can be
The expression shows the product of two factors.
4
The factor r represents a number raised to the represented by the algebraic expression .
3
fourth power. The factor t represents a number
raised to the third power. So, the verbal expression 23. f divided by 10
the product of a number r raised to the fourth
power and a number t cubed can be used to SOLUTION:
describe the algebraic expression r4 ∙ t3. The words divided by suggest division. So, the
verbal expression f divided by 10 can be
Write an algebraic expression for each verbal
represented by the algebraic expression .
expression.
19. x more than 7
24. the quotient of 45 and r
SOLUTION:
SOLUTION:
The words more than suggest addition. So, the
verbal expression x more than 7 can be represented The word quotient suggests division. So, the verbal
by the algebraic expression 7 + x. expression the quotient of 45 and r can be
represented by the algebraic expression .
20. a number less 35
SOLUTION: 25. three times a number plus 16
Let n represent a number. The word less suggests
SOLUTION:
subtraction. So, the verbal expression a number less
Let n represent a number. The word times suggests
35 can be represented by the algebraic expression n
multiplication, and the word plus suggests addition.
– 35.
So, the verbal expression three times a number plus
21. 5 times a number 16 can be represented by the algebraic expression
3n + 16.
SOLUTION:
Let n represent a number. The word times suggests 26. 18 decreased by 3 times d
multiplication. So, the verbal expression 5 times a
SOLUTION:
number can be represented by the algebraic
The word decreased suggests subtraction, and the
expression 5n.
word times suggests multiplication. So, the verbal
expression 18 decreased by 3 times d can be
22. one third of a number
represented by the algebraic expression 18 – 3d.
SOLUTION:
Let n represent a number. The words one third of a 27. k squared minus 11
number suggest multiplication. So, the verbal
SOLUTION:
expression one third of a number can be
The word squared means a number raised to the
represented by the algebraic expression . second power. The word minus suggests subtraction.
So, the verbal expression k squared minus 11 can
2
be represented by the algebraic expression k – 11.
23. f divided by 10
SOLUTION: 28. 20 divided by t to the fifth power
The words divided by suggest division. So, the
SOLUTION:
verbal expression f divided by 10 can be
The words divided by suggest division. So, the
verbal expression 20 divided by t to the fifth power
represented by the algebraic expression .
can be represented by the algebraic expression .
24. the quotient of 45 and r
SOLUTION: 29. GEOMETRY The volume of a cylinder is π times
The word quotient suggests division. So, the verbal the radius r squared multiplied by the height. Write
expression the quotient of 45 and r can be an expression for the volume.
represented by the algebraic expression .
25. three times a number plus 16
SOLUTION:
Let n represent a number. The word times suggests
SOLUTION:
multiplication, and the word plus suggests addition.
So, the verbal expression three times a number plus The words times and multiplied by suggest
16 can be represented by the algebraic expression multiplication. So, the volume of a cylinder can be
3n + 16. written as the algebraic expression πr2h.
26. 18 decreased by 3 times d 30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
SOLUTION:
per hour babysitting. Write an expression that
The word decreased suggests subtraction, and the
describes her earnings if she babysat for 25 hours
word times suggests multiplication. So, the verbal
and worked at the grocery store for 15 hours.
expression 18 decreased by 3 times d can be
represented by the algebraic expression 18 – 3d. SOLUTION:
To write an expression for how much Jocelyn made
27. k squared minus 11
babysitting, multiply the number of hours she babysat,
25, by her hourly rate, n. This can be represented by
SOLUTION:
the algebraic expression 25n.
The word squared means a number raised to the
second power. The word minus suggests subtraction.
To write an expression for how much Jocelyn made
So, the verbal expression k squared minus 11 can
working at the grocery store, multiply the number of
2
be represented by the algebraic expression k – 11. hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x.
28. 20 divided by t to the fifth power
To write an expression for her total earnings, find the
SOLUTION:
sum of the amount she earned babysitting and the
The words divided by suggest division. So, the
amount she earned working at the grocery store.
verbal expression 20 divided by t to the fifth power
This can be written as the expression 25n + 15x.
can be represented by the algebraic expression .
Write a verbal expression for each algebraic
expression.
29. GEOMETRY The volume of a cylinder is π times 2
31. 25 + 6x
the radius r squared multiplied by the height. Write
an expression for the volume. SOLUTION:
The expression shows the sum of two terms. The
2
term 6x means six times the square of a number.
2
So, the algebraic expression 25 + 6x can be
described by the verbal expression twenty–five plus
six times a number squared.
SOLUTION:
2
32. 6f + 5f
The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be SOLUTION:
2
written as the algebraic expression πr h. The expression shows the sum of two terms. The
2
term 6f means six times the square of a number.
30. FINANCIAL LITERACY Jocelyn makes x dollars
The term 5f means five times a number. So, the
per hour working at the grocery store and n dollars
2
per hour babysitting. Write an expression that algebraic expression 6f + 5f can be described by the
describes her earnings if she babysat for 25 hours verbal expression six times a number squared plus
and worked at the grocery store for 15 hours. five times the number.
SOLUTION:
33.
To write an expression for how much Jocelyn made
babysitting, multiply the number of hours she babysat,
25, by her hourly rate, n. This can be represented by SOLUTION:
the algebraic expression 25n. The expression shows the quotient of two terms. The
5
term 3a means three times a number that has been
To write an expression for how much Jocelyn made
raised to the fifth power. So, the algebraic expression
working at the grocery store, multiply the number of
can be described by the verbal expression three
hours she worked, 15, by her hourly rate, x. This can
times a number raised to the fifth power divided
be represented by the algebraic expression 15x. by two.
To write an expression for her total earnings, find the 34. CCSS SENSE-MAKING A certain smartphone
sum of the amount she earned babysitting and the family plan costs $55 per month plus additional usage
amount she earned working at the grocery store. costs. If x is the number of cell phone minutes used
This can be written as the expression 25n + 15x. above the plan amount and y is the number of
megabytes of data used above the plan amount,
Write a verbal expression for each algebraic
interpret the following expressions.
expression.
2
31. 25 + 6x a. 0.25x
SOLUTION:
b. 2y
The expression shows the sum of two terms. The
2
term 6x means six times the square of a number. c. 0.25x + 2y + 55
So, the algebraic expression 25 + 6x2 can be
described by the verbal expression twenty–five plus
SOLUTION:
six times a number squared.
a. Since x is the number of cell phone minutes, then
0.25x would be the cost of extra minutes at $0.25 per
32. 6f2 + 5f minute.
SOLUTION:
b. Since y is the number of megabytes of data used
The expression shows the sum of two terms. The above the plan amount, then 2y would be the cost of
term 6f2 means six times the square of a number. extra data used at $2 per megabyte.
The term 5f means five times a number. So, the
2 c. The expression 0.25x + 2y + 55 represents the
algebraic expression 6f + 5f can be described by the
extra minute charges and the extra data usage
verbal expression six times a number squared plus
charge plus the monthly family plan cost of $55. The
five times the number.
expression represents the total monthly cost for the
family.
33.
SOLUTION:
35. DREAMS It is believed that about of our dreams
The expression shows the quotient of two terms. The
5 involve people that we know.
term 3a means three times a number that has been
raised to the fifth power. So, the algebraic expression
a. Write an expression to describe the number of
can be described by the verbal expression three
dreams that feature people you know if you have d
times a number raised to the fifth power divided
by two. dreams.
34. CCSS SENSE-MAKING A certain smartphone b. Use the expression you wrote to predict the
family plan costs $55 per month plus additional usage number of dreams that include people you know out
costs. If x is the number of cell phone minutes used of 28 dreams.
above the plan amount and y is the number of
megabytes of data used above the plan amount,
SOLUTION:
interpret the following expressions.
a. To write an expression to describe the number of
a. 0.25x dreams that feature people you know if you have d
dreams, multiply by d or .
b. 2y
c. 0.25x + 2y + 55
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
SOLUTION: in the expression .
a. Since x is the number of cell phone minutes, then
0.25x would be the cost of extra minutes at $0.25 per
minute.
b. Since y is the number of megabytes of data used
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte. So, you would predict having 21 dreams that include
people you know.
c. The expression 0.25x + 2y + 55 represents the
36. SPORTS In football, a touchdown is awarded 6
extra minute charges and the extra data usage
points and the team can then try for a point after a
charge plus the monthly family plan cost of $55. The
touchdown.
expression represents the total monthly cost for the
family.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
35. DREAMS It is believed that about of our dreams
involve people that we know. b. If a team wins a football game 27–0, write an
equation to represent the possible number of
a. Write an expression to describe the number of touchdowns and points after touchdowns by the
dreams that feature people you know if you have d winning team.
dreams.
c. If a team wins a football game 21–7, how many
b. Use the expression you wrote to predict the possible number of touchdowns and points after
number of dreams that include people you know out touchdowns were scored during the game by both
of 28 dreams. teams?
SOLUTION:
SOLUTION: a. Let T be the number of touchdowns and p be the
a. To write an expression to describe the number of number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points
dreams that feature people you know if you have d
scored on touchdowns and points after touchdowns
dreams, multiply by d or . by one team in a game.
b. If a team scores 27 points in a game, then 6T + p
b. To predict the number of dreams that include
= 27 represents the possible number of touchdowns
people you know out of 28 dreams, replace d with 28 and points after touchdowns by the winning team.
in the expression .
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
the game by both teams.
Let T = 4 and p = 4.
So, you would predict having 21 dreams that include
people you know.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
touchdown.
So, it is possible that 4 touchdowns and 4 points after
a. Write an expression that describes the number of touchdowns were scored during the game by both
points scored on touchdowns and points after teams.
touchdowns by one team in a game.
37. MULTIPLE REPRESENTATIONS In this
b. If a team wins a football game 27–0, write an problem, you will explore the multiplication of powers
equation to represent the possible number of with like bases.
touchdowns and points after touchdowns by the
winning team. a. TABULAR Copy and complete the table.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
teams?
SOLUTION: b. ALGEBRAIC Write an equation for the pattern
a. Let T be the number of touchdowns and p be the in the table.
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points c. VERBAL Make a conjecture about the exponent
scored on touchdowns and points after touchdowns of the product of two powers with like bases.
by one team in a game.
SOLUTION:
a.
b. If a team scores 27 points in a game, then 6T + p
= 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
c. If a team wins a football game 21–7, then 6T + p
= 28 represents the possible number of touchdowns
b. The exponent of the product is the sum of the
and points after touchdowns that were scored during
exponents of the factors. So, the algebraic equation
the game by both teams.
2 x (2 + x)
10 × 10 = 10 represents the pattern.
Let T = 4 and p = 4.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
bases.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
So, it is possible that 4 touchdowns and 4 points after
SOLUTION:
touchdowns were scored during the game by both
teams. Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
37. MULTIPLE REPRESENTATIONS In this example, “three more than a double a number” is a
problem, you will explore the multiplication of powers verbal expression. The expression 2x + 3 is the
with like bases. algebraic expression that represents the verbal
expression “three more than a double a number”.
a. TABULAR Copy and complete the table.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
Describe in words what your expression represents,
and explain your reasoning.
b. ALGEBRAIC Write an equation for the pattern
SOLUTION:
in the table.
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
c. VERBAL Make a conjecture about the exponent represents the amount of time in minutes I spend
of the product of two powers with like bases. walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
SOLUTION:
a.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
b. The exponent of the product is the sum of the
exponents of the factors. So, the algebraic equation
2 x (2 + x)
10 × 10 = 10 represents the pattern.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
bases.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression. SOLUTION:
Consuelo is correct. The verbal expression says that
SOLUTION:
the sum of n squared and 3 is multiplied by 3. So,
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For parentheses are necessary. James left out the
example, “three more than a double a number” is a parentheses around n2 + 3.
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal 41. CHALLENGE For the cube, x represents a positive
expression “three more than a double a number”. whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
39. OPEN ENDED Define a variable to represent a faces have the same value.
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
Describe in words what your expression represents,
and explain your reasoning.
SOLUTION:
SOLUTION:
Sample answer: x is the number of minutes it takes to
The volume of a cube can be found by multiplying
walk between my house and school. 2x + 15
the length times the width times the height. Because
represents the amount of time in minutes I spend
the sides of a cube all have the same length, V = x •
walking each day since I walk to and from school
3
and I take my dog on a 15 minute walk. x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
40. CCSS CRITIQUE Consuelo and James are writing width. So, A = x • x, or x2.
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either To find the value of x such that the volume of the
of them correct? Explain your reasoning. cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
x x3 = 6x2 Yes/No
4 No
6 Yes
SOLUTION:
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So,
So, the sides must have a length of 6 for the volume
parentheses are necessary. James left out the
of the cube and 6 times the area of one of its faces
2
parentheses around n + 3. to have the same value.
41. CHALLENGE For the cube, x represents a positive 42. WRITING IN MATH Describe how to write an
whole number. Find the value of x such that the algebraic expression from a real–world situation.
volume of the cube and 6 times the area of one of its Include a definition of algebraic expression in your
faces have the same value. own words.
SOLUTION:
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or
variables. To write an algebraic expression from real
world situation, first assign variables. Then determine
SOLUTION: the arithmetic operations done on the variables.
Finally, put the terms in order.
The volume of a cube can be found by multiplying
the length times the width times the height. Because
43. Which expression best represents the volume of the
the sides of a cube all have the same length, V = x •
cube?
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the
2
width. So, A = x • x, or x .
To find the value of x such that the volume of the
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2.
A the product of three and five
x x3 = 6x2 Yes/No
4 No
B three to the fifth power
C three squared
6 Yes
D three cubed
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
So, the sides must have a length of 6 for the volume
3
of the cube and 6 times the area of one of its faces x • x, or x . Because the length of each side is 3
to have the same value. units, the expression three cubed best represents the
volume of the cube.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation. So, Choice D is the correct answer.
Include a definition of algebraic expression in your
own words. 44. Which expression best represents the perimeter of
the rectangle?
SOLUTION:
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or
variables. To write an algebraic expression from real
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
F 2lw
Finally, put the terms in order.
43. Which expression best represents the volume of the G l + w
cube?
H 2l + 2w
J 4(l + w)
SOLUTION:
To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
A the product of three and five rectangle.
B three to the fifth power Choice H is the correct answer.
C three squared 45. SHORT RESPONSE The yards of fabric needed
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
D three cubed
represents the yards of fabric needed in terms of the
SOLUTION: length of the window l.
The volume of a cube can be found by multiplying
SOLUTION:
the length times the width times the height. Because
The word times suggests multiplication and the words
the sides of a cube all have the same length, V = x •
x • x, or x3. Because the length of each side is 3 divided by suggest division. So, represents the
units, the expression three cubed best represents the
yards of fabric needed in terms of the length of the
volume of the cube.
window l.
So, Choice D is the correct answer.
46. GEOMETRY Find the area of the rectangle.
44. Which expression best represents the perimeter of
the rectangle?
A 14 square meters
F 2lw
B 16 square meters
G l + w
C 50 square meters
H 2l + 2w
D 60 square meters
J 4(l + w) SOLUTION:
SOLUTION:
To find the perimeter of a rectangle, find the sum of
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
rectangle. So, the area of the rectangle is 16 square meters.
1-1 V ariables and Expressions
Choice H is the correct answer. Choice B is the correct answer.
45. SHORT RESPONSE The yards of fabric needed 47. AMUSEMENT PARKS A roller coaster
to make curtains is 3 times the length of a window in enthusiast club took a poll to see what each
inches, divided by 36. Write an expression that member’s favorite ride was.. Make a bar graph of
represents the yards of fabric needed in terms of the the results.
length of the window l.
Our Favorite Rides
SOLUTION:
Number
The word times suggests multiplication and the words Ride
of Votes
divided by suggest division. So, represents the Big Plunge 5
Twisting
yards of fabric needed in terms of the length of the 22
Time
window l.
The Shiner 16
46. GEOMETRY Find the area of the rectangle. Raging Bull 9
The
25
Bat
Teaser 6
The
12
Adventure
A 14 square meters
SOLUTION:
Draw a bar to represent each roller coaster. The
B 16 square meters
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
C 50 square meters
the roller coaster chosen.
D 60 square meters
SOLUTION:
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
48. SPORTS The results for an annual 5K race are
47. AMUSEMENT PARKS A roller coaster
shown below. Make a box-and-whisker plot for the
enthusiast club took a poll to see what each
data. Write a sentence describing what the length of
member’s favorite ride was.. Make a bar graph of
the box-and-whisker plot tells about the times for the
the results.
race.
Our Favorite Rides
Number
Ride
of Votes
Big Plunge 5
Twisting
22
Time
The Shiner 16
Raging Bull 9
eSolutionsManual-PoweredbyCognero Page8
The
25 SOLUTION:
Bat
Order the data from least to greatest. The times in
Teaser 6
order from least to greatest are 14:48, 14:58, 15:06,
The 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
12
Adventure 20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
SOLUTION:
the number of minutes by 60 and then adding the
Draw a bar to represent each roller coaster. The
seconds.
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
Time in Time in
the roller coaster chosen.
Min:Sec Seconds
14:48 888
14:58 898
15:06 906
15:48 948
15:54 954
16:10 970
16:30 990
19:27 1167
19:58 1198
20:21 1221
20:39 1239
48. SPORTS The results for an annual 5K race are
20:47 1247
shown below. Make a box-and-whisker plot for the
20:49 1249
data. Write a sentence describing what the length of
21:35 1295
the box-and-whisker plot tells about the times for the
race.
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
Q = 1239
3
There are no outliers.
SOLUTION:
Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35. Find the mean, median, and mode for each set
The times are given in minutes and seconds. Rewrite of data.
the times so that they are in seconds by multiplying 49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
the number of minutes by 60 and then adding the
SOLUTION:
seconds.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898 So, the mean is 5.6.
15:06 906
15:48 948 Order the data from least to greatest.
15:54 954 {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
16:10 970 Because there is an even number of data, the median
16:30 990 is the mean of 6 and 7.
19:27 1167
19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249 So, the median is 6.5.
21:35 1295
The number 7 appears most often, so the mode is 7.
Then determine the quartiles.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
Q = 948
1
SOLUTION:
Q = 1078.5
2
Q = 1239
3
There are no outliers.
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Because there is an even number of data, the median
is the mean of 0 and 0.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
So, the mean is 5.6. modes are 0 and –1.
Order the data from least to greatest. 51. {17, 24, 16, 3, 12, 11, 24, 15}
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
SOLUTION:
Because there is an even number of data, the median
is the mean of 6 and 7.
So, the mean is 15.25.
So, the median is 6.5. Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
The number 7 appears most often, so the mode is 7.
Because there is an even number of data, the median
is the mean of 15 and 16.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
SOLUTION:
So, the median is 15.5.
So, the mean is 0.4.
The number 24 appears most often, so the mode is
Order the data from least to greatest. 24.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
52. SPORTS Lisa has a rectangular trampoline that is 6
Because there is an even number of data, the median feet long and 12 feet wide. What is the area of her
is the mean of 0 and 0. trampoline in square feet?
SOLUTION:
So, the median is 0.
The area of Lisa’s trampoline is 72 square feet.
The numbers 0 and –1 both occur most often, so the
Find each product or quotient.
modes are 0 and –1.
53.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
SOLUTION:
Multiply the numerators and denominators.
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
54.
Because there is an even number of data, the median
is the mean of 15 and 16.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
55.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet? SOLUTION:
SOLUTION:
The area of Lisa’s trampoline is 72 square feet.
Evaluate each expression.
Find each product or quotient.
53.
56.
SOLUTION:
SOLUTION:
Multiply the numerators and denominators.
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
54.
57. 5.67 – 4.21
SOLUTION: SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
55. with a common denominator of 6.
SOLUTION:
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
Evaluate each expression.
60.
56.
SOLUTION:
SOLUTION: The LCD for 12 and 36 is 36. Rewrite the fractions
The LCD for 5 and 9 is 45. Rewrite each fraction with a common denominator of 36.
with denominators of 45 then add the numerators.
57. 5.67 – 4.21
61. 37.02 – 15.86
SOLUTION:
SOLUTION:
5.67 – 4.21 = 1.46
37.02 – 15.86 = 21.16
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
The word less suggests subtraction. So, the verbal
expression 6 less a number t can be represented by
the algebraic expression 6 – t.
6. 7 more than 11 times a number
SOLUTION:
Let n represent a number. The words more than
suggest addition and the word times suggests
Write a verbal expression for each algebraic multiplication. So, the verbal expression 7 more than
expression. 11 times a number can be represented by the
1. 2m algebraic expression 11n + 7.
SOLUTION: 7. 1 minus the quotient of r and 7
Because the 2 and the m are written next to each
SOLUTION:
other, they are being multiplied. So, the verbal
The word minus suggests subtraction and the word
expression the product of 2 and m can be used to
quotient suggests division. So, the verbal expression
describe the algebraic expression 2m.
1 minus the quotient of r and 7 can be represented
by the algebraic expression .
2.
8. two fifths of a number j squared
SOLUTION:
SOLUTION:
The expression shows the product of the factors
The words two–fifths of a number suggest
4 4
and r . The factor r represents a number raised multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
to the fourth power. So, the verbal expression two
of a number j squared can be represented by the
thirds times r raised to the fourth power can be
algebraic expression .
used to describe the algebraic expression .
9. n cubed increased by 5
3. a2 – 18b
SOLUTION:
SOLUTION:
The word cubed means to raise to the third power.
The expression shows the difference of two terms. The words increased by suggest addition. So, the
The term a2 represents a squared. The term 18b verbal expression n cubed increased by 5 can be
represents 18 times b. So, the verbal expression a represented by the algebraic expression n3 + 5.
squared minus 18 times b can be used to describe
2 10. GROCERIES Mr. Bailey purchased some
the algebraic expression a – 18b.
groceries that cost d dollars. He paid with a $50 bill.
Write an algebraic expression for each verbal Write an expression for the amount of change he will
expression. receive.
4. the sum of a number and 14
SOLUTION:
SOLUTION: To find the amount of change Mr. Bailey will
Let n represent a number. The word sum suggests receive, subtract the cost of the groceries, d, from
addition. So, the verbal expression the sum of a $50. So, Mr. Bailey will receive 50 – d in change.
number and 14 can be represented by the algebraic
Write a verbal expression for each algebraic
expression n + 14.
expression.
11. 4q
5. 6 less a number t
SOLUTION:
SOLUTION:
Because 4 and q are written next to each other, they
The word less suggests subtraction. So, the verbal are being multiplied. So, the verbal expression four
expression 6 less a number t can be represented by times a number q can be used to describe the
the algebraic expression 6 – t. algebraic expression 4q.
6. 7 more than 11 times a number
12.
SOLUTION:
Let n represent a number. The words more than SOLUTION:
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than Because and y are written next to each other,
11 times a number can be represented by the
algebraic expression 11n + 7. they are being multiplied. So, the verbal expression
one eighth of a number y can be used to describe
7. 1 minus the quotient of r and 7 the algebraic expression .
SOLUTION:
13. 15 + r
The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression SOLUTION:
1 minus the quotient of r and 7 can be represented
The expression shows the sum of two terms. So, the
by the algebraic expression .
verbal expression 15 plus r can be used to describe
the algebraic expression 15 + r.
8. two fifths of a number j squared
14. w – 24
SOLUTION:
The words two–fifths of a number suggest SOLUTION:
multiplication. The squared means to raise to the
The expression shows the difference of two terms.
second power. So, the verbal expression two–fifths
So, the verbal expression w minus 24 can be used to
of a number j squared can be represented by the describe the algebraic expression w – 24.
algebraic expression .
2
15. 3x
9. n cubed increased by 5
SOLUTION:
SOLUTION: The expression shows the product of the factors 3
The word cubed means to raise to the third power. 2 2
and x . The factor x represents a number raised to
The words increased by suggest addition. So, the
the second power. So, the verbal expression 3 times
verbal expression n cubed increased by 5 can be
x squared can be used to describe the algebraic
represented by the algebraic expression n3 + 5. 2
expression 3x .
10. GROCERIES Mr. Bailey purchased some
groceries that cost d dollars. He paid with a $50 bill. 16.
Write an expression for the amount of change he will
receive. SOLUTION:
The expression shows the quotient of two terms. The
SOLUTION:
4
To find the amount of change Mr. Bailey will term r represents a number raised to the fourth
receive, subtract the cost of the groceries, d, from power. So, the verbal expression r to the fourth
$50. So, Mr. Bailey will receive 50 – d in change. power divided by 9 can be used to describe the
Write a verbal expression for each algebraic algebraic expression .
expression.
11. 4q 17. 2a + 6
SOLUTION: SOLUTION:
Because 4 and q are written next to each other, they The expression shows the sum of two terms. The
are being multiplied. So, the verbal expression four term 2a represents the product of 2 and a. So, the
times a number q can be used to describe the verbal expression 6 more than the product 2 times
algebraic expression 4q. a can be used to describe the algebraic expression
2a + 6.
12.
18. r4 ∙ t3
SOLUTION:
SOLUTION:
The expression shows the product of two factors.
Because and y are written next to each other,
4
The factor r represents a number raised to the
they are being multiplied. So, the verbal expression
3
fourth power. The factor t represents a number
one eighth of a number y can be used to describe
raised to the third power. So, the verbal expression
the algebraic expression .
the product of a number r raised to the fourth
power and a number t cubed can be used to
13. 15 + r describe the algebraic expression r4 ∙ t3.
SOLUTION:
Write an algebraic expression for each verbal
The expression shows the sum of two terms. So, the
expression.
verbal expression 15 plus r can be used to describe
19. x more than 7
the algebraic expression 15 + r.
SOLUTION:
14. w – 24
The words more than suggest addition. So, the
verbal expression x more than 7 can be represented
SOLUTION:
by the algebraic expression 7 + x.
The expression shows the difference of two terms.
So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24. 20. a number less 35
SOLUTION:
2
15. 3x
Let n represent a number. The word less suggests
SOLUTION: subtraction. So, the verbal expression a number less
The expression shows the product of the factors 3 35 can be represented by the algebraic expression n
2 2 – 35.
and x . The factor x represents a number raised to
the second power. So, the verbal expression 3 times
21. 5 times a number
x squared can be used to describe the algebraic
2 SOLUTION:
expression 3x .
Let n represent a number. The word times suggests
multiplication. So, the verbal expression 5 times a
16.
number can be represented by the algebraic
expression 5n.
SOLUTION:
The expression shows the quotient of two terms. The 22. one third of a number
4
term r represents a number raised to the fourth SOLUTION:
power. So, the verbal expression r to the fourth
Let n represent a number. The words one third of a
power divided by 9 can be used to describe the
number suggest multiplication. So, the verbal
expression one third of a number can be
algebraic expression .
represented by the algebraic expression .
17. 2a + 6
SOLUTION: 23. f divided by 10
The expression shows the sum of two terms. The
SOLUTION:
term 2a represents the product of 2 and a. So, the
The words divided by suggest division. So, the
verbal expression 6 more than the product 2 times
verbal expression f divided by 10 can be
a can be used to describe the algebraic expression
2a + 6. represented by the algebraic expression .
18. r4 ∙ t3
24. the quotient of 45 and r
SOLUTION:
SOLUTION:
The expression shows the product of two factors.
The word quotient suggests division. So, the verbal
4
The factor r represents a number raised to the expression the quotient of 45 and r can be
3
fourth power. The factor t represents a number
represented by the algebraic expression .
raised to the third power. So, the verbal expression
the product of a number r raised to the fourth
power and a number t cubed can be used to 25. three times a number plus 16
describe the algebraic expression r4 ∙ t3. SOLUTION:
Let n represent a number. The word times suggests
Write an algebraic expression for each verbal
multiplication, and the word plus suggests addition.
expression.
So, the verbal expression three times a number plus
19. x more than 7
16 can be represented by the algebraic expression
SOLUTION: 3n + 16.
The words more than suggest addition. So, the
26. 18 decreased by 3 times d
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x. SOLUTION:
The word decreased suggests subtraction, and the
20. a number less 35 word times suggests multiplication. So, the verbal
expression 18 decreased by 3 times d can be
SOLUTION:
represented by the algebraic expression 18 – 3d.
Let n represent a number. The word less suggests
subtraction. So, the verbal expression a number less 27. k squared minus 11
35 can be represented by the algebraic expression n
SOLUTION:
– 35.
The word squared means a number raised to the
21. 5 times a number second power. The word minus suggests subtraction.
So, the verbal expression k squared minus 11 can
SOLUTION:
2
be represented by the algebraic expression k – 11.
Let n represent a number. The word times suggests
multiplication. So, the verbal expression 5 times a 28. 20 divided by t to the fifth power
number can be represented by the algebraic
expression 5n. SOLUTION:
The words divided by suggest division. So, the
22. one third of a number verbal expression 20 divided by t to the fifth power
SOLUTION:
can be represented by the algebraic expression .
Let n represent a number. The words one third of a
number suggest multiplication. So, the verbal
29. GEOMETRY The volume of a cylinder is π times
expression one third of a number can be
the radius r squared multiplied by the height. Write
represented by the algebraic expression . an expression for the volume.
23. f divided by 10
SOLUTION:
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
SOLUTION:
represented by the algebraic expression .
The words times and multiplied by suggest
multiplication. So, the volume of a cylinder can be
24. the quotient of 45 and r
2
written as the algebraic expression πr h.
SOLUTION:
The word quotient suggests division. So, the verbal 30. FINANCIAL LITERACY Jocelyn makes x dollars
expression the quotient of 45 and r can be per hour working at the grocery store and n dollars
per hour babysitting. Write an expression that
represented by the algebraic expression . describes her earnings if she babysat for 25 hours
and worked at the grocery store for 15 hours.
25. three times a number plus 16
SOLUTION:
SOLUTION: To write an expression for how much Jocelyn made
babysitting, multiply the number of hours she babysat,
Let n represent a number. The word times suggests
25, by her hourly rate, n. This can be represented by
multiplication, and the word plus suggests addition.
the algebraic expression 25n.
So, the verbal expression three times a number plus
16 can be represented by the algebraic expression
To write an expression for how much Jocelyn made
3n + 16.
working at the grocery store, multiply the number of
26. 18 decreased by 3 times d hours she worked, 15, by her hourly rate, x. This can
be represented by the algebraic expression 15x.
SOLUTION:
The word decreased suggests subtraction, and the To write an expression for her total earnings, find the
word times suggests multiplication. So, the verbal sum of the amount she earned babysitting and the
expression 18 decreased by 3 times d can be amount she earned working at the grocery store.
represented by the algebraic expression 18 – 3d. This can be written as the expression 25n + 15x.
27. k squared minus 11 Write a verbal expression for each algebraic
expression.
SOLUTION:
2
31. 25 + 6x
The word squared means a number raised to the
second power. The word minus suggests subtraction. SOLUTION:
So, the verbal expression k squared minus 11 can
The expression shows the sum of two terms. The
be represented by the algebraic expression k2 – 11. term 6x2 means six times the square of a number.
2
So, the algebraic expression 25 + 6x can be
28. 20 divided by t to the fifth power
described by the verbal expression twenty–five plus
SOLUTION: six times a number squared.
The words divided by suggest division. So, the
verbal expression 20 divided by t to the fifth power
2
32. 6f + 5f
can be represented by the algebraic expression .
SOLUTION:
The expression shows the sum of two terms. The
29. GEOMETRY The volume of a cylinder is π times 2
term 6f means six times the square of a number.
the radius r squared multiplied by the height. Write
The term 5f means five times a number. So, the
an expression for the volume.
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number.
33.
SOLUTION: SOLUTION:
The words times and multiplied by suggest The expression shows the quotient of two terms. The
multiplication. So, the volume of a cylinder can be term 3a5 means three times a number that has been
written as the algebraic expression πr2h. raised to the fifth power. So, the algebraic expression
can be described by the verbal expression three
30. FINANCIAL LITERACY Jocelyn makes x dollars times a number raised to the fifth power divided
per hour working at the grocery store and n dollars by two.
per hour babysitting. Write an expression that
describes her earnings if she babysat for 25 hours 34. CCSS SENSE-MAKING A certain smartphone
and worked at the grocery store for 15 hours. family plan costs $55 per month plus additional usage
costs. If x is the number of cell phone minutes used
SOLUTION: above the plan amount and y is the number of
To write an expression for how much Jocelyn made megabytes of data used above the plan amount,
babysitting, multiply the number of hours she babysat, interpret the following expressions.
25, by her hourly rate, n. This can be represented by
the algebraic expression 25n. a. 0.25x
To write an expression for how much Jocelyn made b. 2y
working at the grocery store, multiply the number of
hours she worked, 15, by her hourly rate, x. This can c. 0.25x + 2y + 55
be represented by the algebraic expression 15x.
To write an expression for her total earnings, find the SOLUTION:
sum of the amount she earned babysitting and the a. Since x is the number of cell phone minutes, then
amount she earned working at the grocery store. 0.25x would be the cost of extra minutes at $0.25 per
This can be written as the expression 25n + 15x. minute.
Write a verbal expression for each algebraic b. Since y is the number of megabytes of data used
expression.
above the plan amount, then 2y would be the cost of
2
31. 25 + 6x extra data used at $2 per megabyte.
SOLUTION:
c. The expression 0.25x + 2y + 55 represents the
The expression shows the sum of two terms. The extra minute charges and the extra data usage
term 6x2 means six times the square of a number. charge plus the monthly family plan cost of $55. The
2 expression represents the total monthly cost for the
So, the algebraic expression 25 + 6x can be
family.
described by the verbal expression twenty–five plus
six times a number squared.
35. DREAMS It is believed that about of our dreams
2
32. 6f + 5f
involve people that we know.
SOLUTION:
The expression shows the sum of two terms. The
a. Write an expression to describe the number of
2
term 6f means six times the square of a number. dreams that feature people you know if you have d
The term 5f means five times a number. So, the dreams.
algebraic expression 6f2 + 5f can be described by the
verbal expression six times a number squared plus b. Use the expression you wrote to predict the
five times the number. number of dreams that include people you know out
of 28 dreams.
33.
SOLUTION:
SOLUTION: a. To write an expression to describe the number of
The expression shows the quotient of two terms. The dreams that feature people you know if you have d
5
term 3a means three times a number that has been
dreams, multiply by d or .
raised to the fifth power. So, the algebraic expression
can be described by the verbal expression three
times a number raised to the fifth power divided b. To predict the number of dreams that include
by two.
people you know out of 28 dreams, replace d with 28
34. CCSS SENSE-MAKING A certain smartphone in the expression .
family plan costs $55 per month plus additional usage
costs. If x is the number of cell phone minutes used
above the plan amount and y is the number of
megabytes of data used above the plan amount,
interpret the following expressions.
So, you would predict having 21 dreams that include
a. 0.25x
people you know.
b. 2y
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
c. 0.25x + 2y + 55
touchdown.
SOLUTION: a. Write an expression that describes the number of
a. Since x is the number of cell phone minutes, then points scored on touchdowns and points after
0.25x would be the cost of extra minutes at $0.25 per touchdowns by one team in a game.
minute.
b. If a team wins a football game 27–0, write an
b. Since y is the number of megabytes of data used equation to represent the possible number of
above the plan amount, then 2y would be the cost of touchdowns and points after touchdowns by the
extra data used at $2 per megabyte. winning team.
c. The expression 0.25x + 2y + 55 represents the c. If a team wins a football game 21–7, how many
extra minute charges and the extra data usage possible number of touchdowns and points after
charge plus the monthly family plan cost of $55. The touchdowns were scored during the game by both
expression represents the total monthly cost for the teams?
family.
SOLUTION:
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
35. DREAMS It is believed that about of our dreams
expression 6T + p, describes the number of points
involve people that we know. scored on touchdowns and points after touchdowns
by one team in a game.
a. Write an expression to describe the number of
dreams that feature people you know if you have d b. If a team scores 27 points in a game, then 6T + p
dreams. = 27 represents the possible number of touchdowns
and points after touchdowns by the winning team.
b. Use the expression you wrote to predict the
number of dreams that include people you know out c. If a team wins a football game 21–7, then 6T + p
of 28 dreams. = 28 represents the possible number of touchdowns
and points after touchdowns that were scored during
the game by both teams.
SOLUTION:
a. To write an expression to describe the number of Let T = 4 and p = 4.
dreams that feature people you know if you have d
dreams, multiply by d or .
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
So, it is possible that 4 touchdowns and 4 points after
in the expression . touchdowns were scored during the game by both
teams.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers
with like bases.
So, you would predict having 21 dreams that include a. TABULAR Copy and complete the table.
people you know.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a
touchdown.
a. Write an expression that describes the number of b. ALGEBRAIC Write an equation for the pattern
points scored on touchdowns and points after in the table.
touchdowns by one team in a game.
c. VERBAL Make a conjecture about the exponent
b. If a team wins a football game 27–0, write an of the product of two powers with like bases.
equation to represent the possible number of
SOLUTION:
touchdowns and points after touchdowns by the
winning team. a.
c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
teams?
b. The exponent of the product is the sum of the
SOLUTION: exponents of the factors. So, the algebraic equation
a. Let T be the number of touchdowns and p be the 102 × 10x = 10(2 + x) represents the pattern.
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points
c. The exponent of the product of two powers is the
scored on touchdowns and points after touchdowns
sum of the exponents of the powers with the same
by one team in a game.
bases.
b. If a team scores 27 points in a game, then 6T + p 38. REASONING Explain the differences between an
= 27 represents the possible number of touchdowns algebraic expression and a verbal expression.
and points after touchdowns by the winning team.
SOLUTION:
Algebraic expressions include variables, numbers,
c. If a team wins a football game 21–7, then 6T + p
and symbols. Verbal expressions contain words. For
= 28 represents the possible number of touchdowns
example, “three more than a double a number” is a
and points after touchdowns that were scored during
verbal expression. The expression 2x + 3 is the
the game by both teams.
algebraic expression that represents the verbal
expression “three more than a double a number”.
Let T = 4 and p = 4.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
in feet. Then use the variable to write an algebraic
expression to represent one of your daily activities.
Describe in words what your expression represents,
and explain your reasoning.
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both SOLUTION:
teams.
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
37. MULTIPLE REPRESENTATIONS In this
represents the amount of time in minutes I spend
problem, you will explore the multiplication of powers
walking each day since I walk to and from school
with like bases.
and I take my dog on a 15 minute walk.
a. TABULAR Copy and complete the table. 40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
b. ALGEBRAIC Write an equation for the pattern
in the table.
c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a.
SOLUTION:
Consuelo is correct. The verbal expression says that
the sum of n squared and 3 is multiplied by 3. So,
parentheses are necessary. James left out the
b. The exponent of the product is the sum of the parentheses around n2 + 3.
exponents of the factors. So, the algebraic equation
102 × 10x = 10(2 + x) represents the pattern. 41. CHALLENGE For the cube, x represents a positive
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
c. The exponent of the product of two powers is the
faces have the same value.
sum of the exponents of the powers with the same
bases.
38. REASONING Explain the differences between an
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers, SOLUTION:
and symbols. Verbal expressions contain words. For The volume of a cube can be found by multiplying
example, “three more than a double a number” is a the length times the width times the height. Because
verbal expression. The expression 2x + 3 is the the sides of a cube all have the same length, V = x •
algebraic expression that represents the verbal 3
x • x, or x . The area of one of the faces of the cube
expression “three more than a double a number”.
can be found by multiplying the length times the
2
39. OPEN ENDED Define a variable to represent a width. So, A = x • x, or x .
real-life quantity, such as time in minutes or distance
To find the value of x such that the volume of the
in feet. Then use the variable to write an algebraic
cube and 6 times the area of one of its faces have
expression to represent one of your daily activities.
Describe in words what your expression represents, the same value, find a value for x such that x3 = 6x2.
and explain your reasoning.
SOLUTION: x x3 = 6x2 Yes/No
Sample answer: x is the number of minutes it takes to 4 No
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
6 Yes
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
of them correct? Explain your reasoning.
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or
SOLUTION:
variables. To write an algebraic expression from real
Consuelo is correct. The verbal expression says that world situation, first assign variables. Then determine
the sum of n squared and 3 is multiplied by 3. So, the arithmetic operations done on the variables.
parentheses are necessary. James left out the Finally, put the terms in order.
2
parentheses around n + 3.
43. Which expression best represents the volume of the
cube?
41. CHALLENGE For the cube, x represents a positive
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
faces have the same value.
A the product of three and five
SOLUTION:
The volume of a cube can be found by multiplying B three to the fifth power
the length times the width times the height. Because
the sides of a cube all have the same length, V = x • C three squared
3
x • x, or x . The area of one of the faces of the cube
can be found by multiplying the length times the D three cubed
2
width. So, A = x • x, or x . SOLUTION:
The volume of a cube can be found by multiplying
To find the value of x such that the volume of the
the length times the width times the height. Because
cube and 6 times the area of one of its faces have
the sides of a cube all have the same length, V = x •
the same value, find a value for x such that x3 = 6x2. 3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
x x3 = 6x2 Yes/No volume of the cube.
4 No
So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
6 Yes
So, the sides must have a length of 6 for the volume F 2lw
of the cube and 6 times the area of one of its faces
to have the same value. G l + w
42. WRITING IN MATH Describe how to write an
H 2l + 2w
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words. J 4(l + w)
SOLUTION: SOLUTION:
Sample answer: An algebraic expression is a math To find the perimeter of a rectangle, find the sum of
phrase that contains one or more numbers or twice the length and twice the width. The expression
variables. To write an algebraic expression from real 2l + 2w best represents the perimeter of the
world situation, first assign variables. Then determine rectangle.
the arithmetic operations done on the variables.
Finally, put the terms in order. Choice H is the correct answer.
43. Which expression best represents the volume of the 45. SHORT RESPONSE The yards of fabric needed
cube? to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the
length of the window l.
SOLUTION:
The word times suggests multiplication and the words
divided by suggest division. So, represents the
A the product of three and five yards of fabric needed in terms of the length of the
window l.
B three to the fifth power
46. GEOMETRY Find the area of the rectangle.
C three squared
D three cubed
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because A 14 square meters
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3 B 16 square meters
units, the expression three cubed best represents the
volume of the cube.
C 50 square meters
So, Choice D is the correct answer.
D 60 square meters
44. Which expression best represents the perimeter of
SOLUTION:
the rectangle?
So, the area of the rectangle is 16 square meters.
F 2lw
Choice B is the correct answer.
G l + w 47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
H 2l + 2w member’s favorite ride was.. Make a bar graph of
the results.
J 4(l + w)
Our Favorite Rides
SOLUTION:
Number
To find the perimeter of a rectangle, find the sum of Ride
of Votes
twice the length and twice the width. The expression
Big Plunge 5
2l + 2w best represents the perimeter of the
Twisting
rectangle. 22
Time
Choice H is the correct answer. The Shiner 16
Raging Bull 9
45. SHORT RESPONSE The yards of fabric needed The
25
to make curtains is 3 times the length of a window in Bat
inches, divided by 36. Write an expression that
Teaser 6
represents the yards of fabric needed in terms of the
The
length of the window l. 12
Adventure
SOLUTION:
The word times suggests multiplication and the words
SOLUTION:
divided by suggest division. So, represents the Draw a bar to represent each roller coaster. The
vertical scale is the number of members who voted
yards of fabric needed in terms of the length of the
for each rollercoaster. The horizontal scale identifies
window l. the roller coaster chosen.
46. GEOMETRY Find the area of the rectangle.
A 14 square meters
B 16 square meters
48. SPORTS The results for an annual 5K race are
C 50 square meters shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
D 60 square meters
race.
SOLUTION:
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
SOLUTION:
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of Order the data from least to greatest. The times in
the results. order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
20:47, 20:49, 21:35.
Our Favorite Rides
The times are given in minutes and seconds. Rewrite
Number
Ride the times so that they are in seconds by multiplying
of Votes
the number of minutes by 60 and then adding the
Big Plunge 5 seconds.
Twisting
22
Time Time in Time in
The Shiner 16 Min:Sec Seconds
14:48 888
Raging Bull 9
14:58 898
The
25 15:06 906
Bat
15:48 948
Teaser 6
15:54 954
The
12 16:10 970
Adventure
16:30 990
19:27 1167
SOLUTION: 19:58 1198
Draw a bar to represent each roller coaster. The 20:21 1221
vertical scale is the number of members who voted 20:39 1239
for each rollercoaster. The horizontal scale identifies 20:47 1247
the roller coaster chosen. 20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
Q = 1239
3
There are no outliers.
48. SPORTS The results for an annual 5K race are
shown below. Make a box-and-whisker plot for the
1-1 Vdaatrai.a Wblreiste a an sde nEtxepncree sdseisocnrsib ing what the length of
the box-and-whisker plot tells about the times for the
race.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
So, the mean is 5.6.
SOLUTION:
Order the data from least to greatest. The times in Order the data from least to greatest.
order from least to greatest are 14:48, 14:58, 15:06, {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, Because there is an even number of data, the median
20:47, 20:49, 21:35. is the mean of 6 and 7.
The times are given in minutes and seconds. Rewrite
the times so that they are in seconds by multiplying
the number of minutes by 60 and then adding the
seconds.
Time in Time in
So, the median is 6.5.
Min:Sec Seconds
14:48 888
The number 7 appears most often, so the mode is 7.
14:58 898
15:06 906
15:48 948 50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
15:54 954
SOLUTION:
16:10 970
16:30 990
19:27 1167
19:58 1198
20:21 1221 So, the mean is 0.4.
20:39 1239
20:47 1247 Order the data from least to greatest.
20:49 1249 {–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
21:35 1295
Because there is an even number of data, the median
Then determine the quartiles. is the mean of 0 and 0.
Q = 948
1
Q = 1078.5
2
Q = 1239
3
So, the median is 0.
There are no outliers.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
SOLUTION:
eSolutFioinnsdM athnuea lm-Peoawne,r emdebydCiaongn, earond mode for each set Page9
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
So, the mean is 5.6.
Because there is an even number of data, the median
is the mean of 15 and 16.
Order the data from least to greatest.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median
is the mean of 6 and 7.
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
So, the median is 6.5.
52. SPORTS Lisa has a rectangular trampoline that is 6
The number 7 appears most often, so the mode is 7. feet long and 12 feet wide. What is the area of her
trampoline in square feet?
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
SOLUTION:
SOLUTION:
The area of Lisa’s trampoline is 72 square feet.
So, the mean is 0.4.
Find each product or quotient.
Order the data from least to greatest. 53.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Because there is an even number of data, the median
SOLUTION:
is the mean of 0 and 0.
Multiply the numerators and denominators.
So, the median is 0.
54.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15} SOLUTION:
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest. 55.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
SOLUTION:
is the mean of 15 and 16.
So, the median is 15.5.
Evaluate each expression.
The number 24 appears most often, so the mode is
24.
56.
52. SPORTS Lisa has a rectangular trampoline that is 6
feet long and 12 feet wide. What is the area of her
trampoline in square feet? SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
SOLUTION: with denominators of 45 then add the numerators.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53. 57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
SOLUTION:
Multiply the numerators and denominators.
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
54.
SOLUTION:
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
55.
SOLUTION:
SOLUTION: The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
Evaluate each expression.
56. 61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Write a verbal expression for each algebraic
expression.
1. 2m
SOLUTION:
Because the 2 and the m are written next to each
other, they are being multiplied. So, the verbal
expression the product of 2 and m can be used to
describe the algebraic expression 2m.
2.
SOLUTION:
The expression shows the product of the factors
4 4
and r . The factor r represents a number raised
to the fourth power. So, the verbal expression two
thirds times r raised to the fourth power can be
used to describe the algebraic expression .
3. a2 – 18b
SOLUTION:
The expression shows the difference of two terms.
The term a2 represents a squared. The term 18b
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe
2
the algebraic expression a – 18b.
Write an algebraic expression for each verbal
expression.
4. the sum of a number and 14
SOLUTION:
Let n represent a number. The word sum suggests
addition. So, the verbal expression the sum of a
number and 14 can be represented by the algebraic
expression n + 14.
5. 6 less a number t
SOLUTION:
The word less suggests subtraction. So, the verbal
expression 6 less a number t can be represented by
the algebraic expression 6 – t.
6. 7 more than 11 times a number
Write a verbal expression for each algebraic
SOLUTION:
expression.
1. 2m Let n represent a number. The words more than
suggest addition and the word times suggests
SOLUTION: multiplication. So, the verbal expression 7 more than
Because the 2 and the m are written next to each 11 times a number can be represented by the
other, they are being multiplied. So, the verbal algebraic expression 11n + 7.
expression the product of 2 and m can be used to
describe the algebraic expression 2m. 7. 1 minus the quotient of r and 7
SOLUTION:
2. The word minus suggests subtraction and the word
quotient suggests division. So, the verbal expression
SOLUTION: 1 minus the quotient of r and 7 can be represented
The expression shows the product of the factors by the algebraic expression .
4 4
and r . The factor r represents a number raised 8. two fifths of a number j squared
to the fourth power. So, the verbal expression two SOLUTION:
thirds times r raised to the fourth power can be
The words two–fifths of a number suggest
used to describe the algebraic expression . multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
3. a2 – 18b of a number j squared can be represented by the
algebraic expression .
SOLUTION:
The expression shows the difference of two terms. 9. n cubed increased by 5
The term a2 represents a squared. The term 18b
SOLUTION:
represents 18 times b. So, the verbal expression a
squared minus 18 times b can be used to describe The word cubed means to raise to the third power.
2 The words increased by suggest addition. So, the
the algebraic expression a – 18b.
verbal expression n cubed increased by 5 can be
3
Write an algebraic expression for each verbal represented by the algebraic expression n + 5.
expression.
4. the sum of a number and 14 10. GROCERIES Mr. Bailey purchased some
groceries that cost d dollars. He paid with a $50 bill.
SOLUTION: Write an expression for the amount of change he will
Let n represent a number. The word sum suggests receive.
addition. So, the verbal expression the sum of a
SOLUTION:
number and 14 can be represented by the algebraic
expression n + 14. To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
$50. So, Mr. Bailey will receive 50 – d in change.
5. 6 less a number t
Write a verbal expression for each algebraic
SOLUTION:
expression.
The word less suggests subtraction. So, the verbal
11. 4q
expression 6 less a number t can be represented by
the algebraic expression 6 – t. SOLUTION:
Because 4 and q are written next to each other, they
6. 7 more than 11 times a number
are being multiplied. So, the verbal expression four
times a number q can be used to describe the
SOLUTION:
algebraic expression 4q.
Let n represent a number. The words more than
suggest addition and the word times suggests
multiplication. So, the verbal expression 7 more than 12.
11 times a number can be represented by the
algebraic expression 11n + 7. SOLUTION:
7. 1 minus the quotient of r and 7 Because and y are written next to each other,
SOLUTION: they are being multiplied. So, the verbal expression
The word minus suggests subtraction and the word one eighth of a number y can be used to describe
quotient suggests division. So, the verbal expression the algebraic expression .
1 minus the quotient of r and 7 can be represented
by the algebraic expression .
13. 15 + r
8. two fifths of a number j squared SOLUTION:
The expression shows the sum of two terms. So, the
SOLUTION:
verbal expression 15 plus r can be used to describe
The words two–fifths of a number suggest
the algebraic expression 15 + r.
multiplication. The squared means to raise to the
second power. So, the verbal expression two–fifths
14. w – 24
of a number j squared can be represented by the
algebraic expression . SOLUTION:
The expression shows the difference of two terms.
9. n cubed increased by 5 So, the verbal expression w minus 24 can be used to
describe the algebraic expression w – 24.
SOLUTION:
The word cubed means to raise to the third power. 15. 3x2
The words increased by suggest addition. So, the
SOLUTION:
verbal expression n cubed increased by 5 can be
3 The expression shows the product of the factors 3
represented by the algebraic expression n + 5.
2 2
and x . The factor x represents a number raised to
10. GROCERIES Mr. Bailey purchased some the second power. So, the verbal expression 3 times
groceries that cost d dollars. He paid with a $50 bill. x squared can be used to describe the algebraic
Write an expression for the amount of change he will 2
expression 3x .
receive.
SOLUTION:
16.
To find the amount of change Mr. Bailey will
receive, subtract the cost of the groceries, d, from
SOLUTION:
$50. So, Mr. Bailey will receive 50 – d in change.
The expression shows the quotient of two terms. The
4
Write a verbal expression for each algebraic term r represents a number raised to the fourth
expression. power. So, the verbal expression r to the fourth
11. 4q power divided by 9 can be used to describe the
SOLUTION: algebraic expression .
Because 4 and q are written next to each other, they
are being multiplied. So, the verbal expression four
17. 2a + 6
times a number q can be used to describe the
algebraic expression 4q. SOLUTION:
The expression shows the sum of two terms. The
term 2a represents the product of 2 and a. So, the
12.
verbal expression 6 more than the product 2 times
a can be used to describe the algebraic expression
SOLUTION:
2a + 6.
Because and y are written next to each other,
18. r4 ∙ t3
they are being multiplied. So, the verbal expression
SOLUTION:
one eighth of a number y can be used to describe
the algebraic expression . The expression shows the product of two factors.
4
The factor r represents a number raised to the
3
13. 15 + r fourth power. The factor t represents a number
raised to the third power. So, the verbal expression
SOLUTION:
the product of a number r raised to the fourth
The expression shows the sum of two terms. So, the power and a number t cubed can be used to
verbal expression 15 plus r can be used to describe describe the algebraic expression r4 ∙ t3.
the algebraic expression 15 + r.
Write an algebraic expression for each verbal
14. w – 24
expression.
SOLUTION: 19. x more than 7
The expression shows the difference of two terms.
SOLUTION:
So, the verbal expression w minus 24 can be used to
The words more than suggest addition. So, the
describe the algebraic expression w – 24.
verbal expression x more than 7 can be represented
by the algebraic expression 7 + x.
2
15. 3x
SOLUTION: 20. a number less 35
The expression shows the product of the factors 3
SOLUTION:
2 2
and x . The factor x represents a number raised to
Let n represent a number. The word less suggests
the second power. So, the verbal expression 3 times
subtraction. So, the verbal expression a number less
x squared can be used to describe the algebraic
35 can be represented by the algebraic expression n
2
expression 3x . – 35.
21. 5 times a number
16.
SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
The expression shows the quotient of two terms. The multiplication. So, the verbal expression 5 times a
4 number can be represented by the algebraic
term r represents a number raised to the fourth
expression 5n.
power. So, the verbal expression r to the fourth
power divided by 9 can be used to describe the
22. one third of a number
algebraic expression .
SOLUTION:
Let n represent a number. The words one third of a
17. 2a + 6 number suggest multiplication. So, the verbal
expression one third of a number can be
SOLUTION:
The expression shows the sum of two terms. The represented by the algebraic expression .
term 2a represents the product of 2 and a. So, the
verbal expression 6 more than the product 2 times
23. f divided by 10
a can be used to describe the algebraic expression
2a + 6. SOLUTION:
The words divided by suggest division. So, the
18. r4 ∙ t3 verbal expression f divided by 10 can be
SOLUTION:
represented by the algebraic expression .
The expression shows the product of two factors.
4
The factor r represents a number raised to the 24. the quotient of 45 and r
3
fourth power. The factor t represents a number
SOLUTION:
raised to the third power. So, the verbal expression
The word quotient suggests division. So, the verbal
the product of a number r raised to the fourth
expression the quotient of 45 and r can be
power and a number t cubed can be used to
describe the algebraic expression r4 ∙ t3. represented by the algebraic expression .
Write an algebraic expression for each verbal
25. three times a number plus 16
expression.
19. x more than 7 SOLUTION:
SOLUTION: Let n represent a number. The word times suggests
multiplication, and the word plus suggests addition.
The words more than suggest addition. So, the
So, the verbal expression three times a number plus
verbal expression x more than 7 can be represented
16 can be represented by the algebraic expression
by the algebraic expression 7 + x.
3n + 16.
20. a number less 35 26. 18 decreased by 3 times d
SOLUTION: SOLUTION:
Let n represent a number. The word less suggests The word decreased suggests subtraction, and the
subtraction. So, the verbal expression a number less word times suggests multiplication. So, the verbal
35 can be represented by the algebraic expression n expression 18 decreased by 3 times d can be
– 35. represented by the algebraic expression 18 – 3d.
21. 5 times a number 27. k squared minus 11
SOLUTION: SOLUTION:
Let n represent a number. The word times suggests The word squared means a number raised to the
multiplication. So, the verbal expression 5 times a second power. The word minus suggests subtraction.
number can be represented by the algebraic So, the verbal expression k squared minus 11 can
expression 5n. 2
be represented by the algebraic expression k – 11.
22. one third of a number
28. 20 divided by t to the fifth power
SOLUTION:
SOLUTION:
Let n represent a number. The words one third of a The words divided by suggest division. So, the
number suggest multiplication. So, the verbal verbal expression 20 divided by t to the fifth power
expression one third of a number can be
can be represented by the algebraic expression .
represented by the algebraic expression .
29. GEOMETRY The volume of a cylinder is π times
23. f divided by 10 the radius r squared multiplied by the height. Write
an expression for the volume.
SOLUTION:
The words divided by suggest division. So, the
verbal expression f divided by 10 can be
represented by the algebraic expression .
24. the quotient of 45 and r
SOLUTION:
SOLUTION:
The words times and multiplied by suggest
The word quotient suggests division. So, the verbal
multiplication. So, the volume of a cylinder can be
expression the quotient of 45 and r can be
2
written as the algebraic expression πr h.
represented by the algebraic expression .
30. FINANCIAL LITERACY Jocelyn makes x dollars
per hour working at the grocery store and n dollars
25. three times a number plus 16
per hour babysitting. Write an expression that
SOLUTION: describes her earnings if she babysat for 25 hours
Let n represent a number. The word times suggests and worked at the grocery store for 15 hours.
multiplication, and the word plus suggests addition.
SOLUTION:
So, the verbal expression three times a number plus
To write an expression for how much Jocelyn made
16 can be represented by the algebraic expression
babysitting, multiply the number of hours she babysat,
3n + 16.
25, by her hourly rate, n. This can be represented by
the algebraic expression 25n.
26. 18 decreased by 3 times d
SOLUTION: To write an expression for how much Jocelyn made
The word decreased suggests subtraction, and the working at the grocery store, multiply the number of
word times suggests multiplication. So, the verbal hours she worked, 15, by her hourly rate, x. This can
expression 18 decreased by 3 times d can be be represented by the algebraic expression 15x.
represented by the algebraic expression 18 – 3d.
To write an expression for her total earnings, find the
27. k squared minus 11 sum of the amount she earned babysitting and the
amount she earned working at the grocery store.
SOLUTION:
This can be written as the expression 25n + 15x.
The word squared means a number raised to the
second power. The word minus suggests subtraction. Write a verbal expression for each algebraic
So, the verbal expression k squared minus 11 can expression.
2 2
be represented by the algebraic expression k – 11. 31. 25 + 6x
28. 20 divided by t to the fifth power SOLUTION:
The expression shows the sum of two terms. The
SOLUTION:
2
term 6x means six times the square of a number.
The words divided by suggest division. So, the
2
verbal expression 20 divided by t to the fifth power So, the algebraic expression 25 + 6x can be
described by the verbal expression twenty–five plus
can be represented by the algebraic expression . six times a number squared.
29. GEOMETRY The volume of a cylinder is π times 2
32. 6f + 5f
the radius r squared multiplied by the height. Write
an expression for the volume. SOLUTION:
The expression shows the sum of two terms. The
2
term 6f means six times the square of a number.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the
verbal expression six times a number squared plus
five times the number.
SOLUTION:
The words times and multiplied by suggest
33.
multiplication. So, the volume of a cylinder can be
2
written as the algebraic expression πr h.
SOLUTION:
The expression shows the quotient of two terms. The
30. FINANCIAL LITERACY Jocelyn makes x dollars
5
per hour working at the grocery store and n dollars term 3a means three times a number that has been
per hour babysitting. Write an expression that raised to the fifth power. So, the algebraic expression
describes her earnings if she babysat for 25 hours can be described by the verbal expression three
and worked at the grocery store for 15 hours. times a number raised to the fifth power divided
by two.
SOLUTION:
To write an expression for how much Jocelyn made 34. CCSS SENSE-MAKING A certain smartphone
babysitting, multiply the number of hours she babysat, family plan costs $55 per month plus additional usage
25, by her hourly rate, n. This can be represented by costs. If x is the number of cell phone minutes used
the algebraic expression 25n. above the plan amount and y is the number of
megabytes of data used above the plan amount,
To write an expression for how much Jocelyn made interpret the following expressions.
working at the grocery store, multiply the number of
hours she worked, 15, by her hourly rate, x. This can a. 0.25x
be represented by the algebraic expression 15x.
b. 2y
To write an expression for her total earnings, find the
sum of the amount she earned babysitting and the c. 0.25x + 2y + 55
amount she earned working at the grocery store.
This can be written as the expression 25n + 15x.
SOLUTION:
Write a verbal expression for each algebraic a. Since x is the number of cell phone minutes, then
expression. 0.25x would be the cost of extra minutes at $0.25 per
31. 25 + 6x2 minute.
SOLUTION: b. Since y is the number of megabytes of data used
The expression shows the sum of two terms. The above the plan amount, then 2y would be the cost of
term 6x2 means six times the square of a number. extra data used at $2 per megabyte.
2
So, the algebraic expression 25 + 6x can be
c. The expression 0.25x + 2y + 55 represents the
described by the verbal expression twenty–five plus
extra minute charges and the extra data usage
six times a number squared.
charge plus the monthly family plan cost of $55. The
expression represents the total monthly cost for the
32. 6f2 + 5f family.
SOLUTION:
The expression shows the sum of two terms. The
35. DREAMS It is believed that about of our dreams
2
term 6f means six times the square of a number.
involve people that we know.
The term 5f means five times a number. So, the
2
algebraic expression 6f + 5f can be described by the
a. Write an expression to describe the number of
verbal expression six times a number squared plus
five times the number. dreams that feature people you know if you have d
dreams.
33.
b. Use the expression you wrote to predict the
number of dreams that include people you know out
SOLUTION: of 28 dreams.
The expression shows the quotient of two terms. The
5
term 3a means three times a number that has been
SOLUTION:
raised to the fifth power. So, the algebraic expression
a. To write an expression to describe the number of
can be described by the verbal expression three
dreams that feature people you know if you have d
times a number raised to the fifth power divided
by two.
dreams, multiply by d or .
34. CCSS SENSE-MAKING A certain smartphone
family plan costs $55 per month plus additional usage b. To predict the number of dreams that include
costs. If x is the number of cell phone minutes used
people you know out of 28 dreams, replace d with 28
above the plan amount and y is the number of
in the expression .
megabytes of data used above the plan amount,
interpret the following expressions.
a. 0.25x
b. 2y
So, you would predict having 21 dreams that include
c. 0.25x + 2y + 55
people you know.
36. SPORTS In football, a touchdown is awarded 6
SOLUTION:
points and the team can then try for a point after a
a. Since x is the number of cell phone minutes, then
touchdown.
0.25x would be the cost of extra minutes at $0.25 per
minute.
a. Write an expression that describes the number of
points scored on touchdowns and points after
b. Since y is the number of megabytes of data used
touchdowns by one team in a game.
above the plan amount, then 2y would be the cost of
extra data used at $2 per megabyte.
b. If a team wins a football game 27–0, write an
c. The expression 0.25x + 2y + 55 represents the equation to represent the possible number of
extra minute charges and the extra data usage touchdowns and points after touchdowns by the
charge plus the monthly family plan cost of $55. The winning team.
expression represents the total monthly cost for the
family. c. If a team wins a football game 21–7, how many
possible number of touchdowns and points after
touchdowns were scored during the game by both
teams?
35. DREAMS It is believed that about of our dreams
SOLUTION:
involve people that we know.
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
a. Write an expression to describe the number of
expression 6T + p, describes the number of points
dreams that feature people you know if you have d
scored on touchdowns and points after touchdowns
dreams.
by one team in a game.
b. Use the expression you wrote to predict the
b. If a team scores 27 points in a game, then 6T + p
number of dreams that include people you know out
= 27 represents the possible number of touchdowns
of 28 dreams.
and points after touchdowns by the winning team.
SOLUTION: c. If a team wins a football game 21–7, then 6T + p
a. To write an expression to describe the number of = 28 represents the possible number of touchdowns
dreams that feature people you know if you have d and points after touchdowns that were scored during
the game by both teams.
dreams, multiply by d or .
Let T = 4 and p = 4.
b. To predict the number of dreams that include
people you know out of 28 dreams, replace d with 28
in the expression .
So, it is possible that 4 touchdowns and 4 points after
touchdowns were scored during the game by both
teams.
So, you would predict having 21 dreams that include 37. MULTIPLE REPRESENTATIONS In this
people you know. problem, you will explore the multiplication of powers
with like bases.
36. SPORTS In football, a touchdown is awarded 6
points and the team can then try for a point after a a. TABULAR Copy and complete the table.
touchdown.
a. Write an expression that describes the number of
points scored on touchdowns and points after
touchdowns by one team in a game.
b. If a team wins a football game 27–0, write an b. ALGEBRAIC Write an equation for the pattern
in the table.
equation to represent the possible number of
touchdowns and points after touchdowns by the
winning team. c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
c. If a team wins a football game 21–7, how many
SOLUTION:
possible number of touchdowns and points after
a.
touchdowns were scored during the game by both
teams?
SOLUTION:
a. Let T be the number of touchdowns and p be the
number of points scored after touchdowns. So, the
expression 6T + p, describes the number of points b. The exponent of the product is the sum of the
scored on touchdowns and points after touchdowns exponents of the factors. So, the algebraic equation
by one team in a game. 102 × 10x = 10(2 + x) represents the pattern.
b. If a team scores 27 points in a game, then 6T + p c. The exponent of the product of two powers is the
= 27 represents the possible number of touchdowns sum of the exponents of the powers with the same
and points after touchdowns by the winning team. bases.
c. If a team wins a football game 21–7, then 6T + p 38. REASONING Explain the differences between an
= 28 represents the possible number of touchdowns algebraic expression and a verbal expression.
and points after touchdowns that were scored during
SOLUTION:
the game by both teams.
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
Let T = 4 and p = 4.
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
algebraic expression that represents the verbal
expression “three more than a double a number”.
39. OPEN ENDED Define a variable to represent a
real-life quantity, such as time in minutes or distance
So, it is possible that 4 touchdowns and 4 points after
in feet. Then use the variable to write an algebraic
touchdowns were scored during the game by both
expression to represent one of your daily activities.
teams.
Describe in words what your expression represents,
and explain your reasoning.
37. MULTIPLE REPRESENTATIONS In this
problem, you will explore the multiplication of powers SOLUTION:
with like bases.
Sample answer: x is the number of minutes it takes to
walk between my house and school. 2x + 15
a. TABULAR Copy and complete the table. represents the amount of time in minutes I spend
walking each day since I walk to and from school
and I take my dog on a 15 minute walk.
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
b. ALGEBRAIC Write an equation for the pattern of them correct? Explain your reasoning.
in the table.
c. VERBAL Make a conjecture about the exponent
of the product of two powers with like bases.
SOLUTION:
a.
SOLUTION:
b. The exponent of the product is the sum of the
Consuelo is correct. The verbal expression says that
exponents of the factors. So, the algebraic equation
the sum of n squared and 3 is multiplied by 3. So,
2 x (2 + x)
10 × 10 = 10 represents the pattern.
parentheses are necessary. James left out the
2
parentheses around n + 3.
c. The exponent of the product of two powers is the
sum of the exponents of the powers with the same
41. CHALLENGE For the cube, x represents a positive
bases.
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
38. REASONING Explain the differences between an
faces have the same value.
algebraic expression and a verbal expression.
SOLUTION:
Algebraic expressions include variables, numbers,
and symbols. Verbal expressions contain words. For
example, “three more than a double a number” is a
verbal expression. The expression 2x + 3 is the
SOLUTION:
algebraic expression that represents the verbal
expression “three more than a double a number”. The volume of a cube can be found by multiplying
the length times the width times the height. Because
39. OPEN ENDED Define a variable to represent a the sides of a cube all have the same length, V = x •
real-life quantity, such as time in minutes or distance x • x, or x3. The area of one of the faces of the cube
in feet. Then use the variable to write an algebraic can be found by multiplying the length times the
expression to represent one of your daily activities. 2
width. So, A = x • x, or x .
Describe in words what your expression represents,
and explain your reasoning.
To find the value of x such that the volume of the
SOLUTION: cube and 6 times the area of one of its faces have
Sample answer: x is the number of minutes it takes to the same value, find a value for x such that x3 = 6x2.
walk between my house and school. 2x + 15
represents the amount of time in minutes I spend
x x3 = 6x2 Yes/No
walking each day since I walk to and from school
and I take my dog on a 15 minute walk. 4 No
40. CCSS CRITIQUE Consuelo and James are writing
an algebraic expression for the verbal expression
three times the sum of n squared and 3. Is either
6 Yes
of them correct? Explain your reasoning.
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your
own words.
SOLUTION:
Consuelo is correct. The verbal expression says that SOLUTION:
the sum of n squared and 3 is multiplied by 3. So, Sample answer: An algebraic expression is a math
parentheses are necessary. James left out the phrase that contains one or more numbers or
2 variables. To write an algebraic expression from real
parentheses around n + 3.
world situation, first assign variables. Then determine
the arithmetic operations done on the variables.
41. CHALLENGE For the cube, x represents a positive
Finally, put the terms in order.
whole number. Find the value of x such that the
volume of the cube and 6 times the area of one of its
43. Which expression best represents the volume of the
faces have the same value.
cube?
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
A the product of three and five
the sides of a cube all have the same length, V = x •
3
x • x, or x . The area of one of the faces of the cube
B three to the fifth power
can be found by multiplying the length times the
2
width. So, A = x • x, or x . C three squared
To find the value of x such that the volume of the
D three cubed
cube and 6 times the area of one of its faces have
the same value, find a value for x such that x3 = 6x2. SOLUTION:
The volume of a cube can be found by multiplying
x x3 = 6x2 Yes/No the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
4 No
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube.
6 Yes So, Choice D is the correct answer.
44. Which expression best represents the perimeter of
the rectangle?
So, the sides must have a length of 6 for the volume
of the cube and 6 times the area of one of its faces
to have the same value.
42. WRITING IN MATH Describe how to write an F 2lw
algebraic expression from a real–world situation.
Include a definition of algebraic expression in your G l + w
own words.
SOLUTION: H 2l + 2w
Sample answer: An algebraic expression is a math
phrase that contains one or more numbers or J 4(l + w)
variables. To write an algebraic expression from real
SOLUTION:
world situation, first assign variables. Then determine
the arithmetic operations done on the variables. To find the perimeter of a rectangle, find the sum of
Finally, put the terms in order. twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the
43. Which expression best represents the volume of the rectangle.
cube?
Choice H is the correct answer.
45. SHORT RESPONSE The yards of fabric needed
to make curtains is 3 times the length of a window in
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the
length of the window l.
SOLUTION:
A the product of three and five
The word times suggests multiplication and the words
B three to the fifth power divided by suggest division. So, represents the
yards of fabric needed in terms of the length of the
C three squared
window l.
D three cubed 46. GEOMETRY Find the area of the rectangle.
SOLUTION:
The volume of a cube can be found by multiplying
the length times the width times the height. Because
the sides of a cube all have the same length, V = x •
3
x • x, or x . Because the length of each side is 3
units, the expression three cubed best represents the
volume of the cube. A 14 square meters
So, Choice D is the correct answer. B 16 square meters
44. Which expression best represents the perimeter of
C 50 square meters
the rectangle?
D 60 square meters
SOLUTION:
F 2lw
G l + w So, the area of the rectangle is 16 square meters.
H 2l + 2w Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
J 4(l + w)
enthusiast club took a poll to see what each
SOLUTION: member’s favorite ride was.. Make a bar graph of
To find the perimeter of a rectangle, find the sum of the results.
twice the length and twice the width. The expression
2l + 2w best represents the perimeter of the Our Favorite Rides
rectangle. Number
Ride
of Votes
Choice H is the correct answer. Big Plunge 5
Twisting
45. SHORT RESPONSE The yards of fabric needed 22
Time
to make curtains is 3 times the length of a window in
The Shiner 16
inches, divided by 36. Write an expression that
represents the yards of fabric needed in terms of the Raging Bull 9
length of the window l. The
25
Bat
SOLUTION:
Teaser 6
The word times suggests multiplication and the words
The
12
divided by suggest division. So, represents the Adventure
yards of fabric needed in terms of the length of the
window l. SOLUTION:
Draw a bar to represent each roller coaster. The
46. GEOMETRY Find the area of the rectangle. vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies
the roller coaster chosen.
A 14 square meters
B 16 square meters
C 50 square meters
D 60 square meters
48. SPORTS The results for an annual 5K race are
SOLUTION: shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of
the box-and-whisker plot tells about the times for the
race.
So, the area of the rectangle is 16 square meters.
Choice B is the correct answer.
47. AMUSEMENT PARKS A roller coaster
enthusiast club took a poll to see what each
member’s favorite ride was.. Make a bar graph of
the results.
SOLUTION:
Our Favorite Rides
Order the data from least to greatest. The times in
Number order from least to greatest are 14:48, 14:58, 15:06,
Ride
of Votes 15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39,
Big Plunge 5 20:47, 20:49, 21:35.
Twisting The times are given in minutes and seconds. Rewrite
22
Time the times so that they are in seconds by multiplying
The Shiner 16 the number of minutes by 60 and then adding the
seconds.
Raging Bull 9
The
25 Time in Time in
Bat
Min:Sec Seconds
Teaser 6
14:48 888
The
14:58 898
12
Adventure
15:06 906
15:48 948
SOLUTION: 15:54 954
16:10 970
Draw a bar to represent each roller coaster. The
16:30 990
vertical scale is the number of members who voted
for each rollercoaster. The horizontal scale identifies 19:27 1167
the roller coaster chosen. 19:58 1198
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295
Then determine the quartiles.
Q = 948
1
Q = 1078.5
2
48. SPORTS The results for an annual 5K race are Q3 = 1239
shown below. Make a box-and-whisker plot for the
data. Write a sentence describing what the length of There are no outliers.
the box-and-whisker plot tells about the times for the
race.
Find the mean, median, and mode for each set
of data.
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
Order the data from least to greatest. The times in
order from least to greatest are 14:48, 14:58, 15:06,
15:48, 15:54, 16:10, 16:30, 19:27, 19:58, 20:21, 20:39, So, the mean is 5.6.
20:47, 20:49, 21:35.
The times are given in minutes and seconds. Rewrite Order the data from least to greatest.
the times so that they are in seconds by multiplying {2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
the number of minutes by 60 and then adding the Because there is an even number of data, the median
seconds. is the mean of 6 and 7.
Time in Time in
Min:Sec Seconds
14:48 888
14:58 898
15:06 906 So, the median is 6.5.
15:48 948
15:54 954 The number 7 appears most often, so the mode is 7.
16:10 970
16:30 990
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
19:27 1167
19:58 1198 SOLUTION:
20:21 1221
20:39 1239
20:47 1247
20:49 1249
21:35 1295 So, the mean is 0.4.
Then determine the quartiles. Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5}
Q = 948
1
Because there is an even number of data, the median
Q2 = 1078.5 is the mean of 0 and 0.
Q = 1239
3
There are no outliers.
So, the median is 0.
The numbers 0 and –1 both occur most often, so the
modes are 0 and –1.
Find the mean, median, and mode for each set
of data. 51. {17, 24, 16, 3, 12, 11, 24, 15}
49. {7, 6, 5, 7, 4, 8, 2, 2, 7, 8}
SOLUTION:
SOLUTION:
So, the mean is 15.25.
So, the mean is 5.6.
Order the data from least to greatest.
Order the data from least to greatest. {3, 11, 12, 15, 16, 17, 24, 24}.
{2, 2, 4, 5, 6, 7, 7, 7, 8, 8}.
Because there is an even number of data, the median Because there is an even number of data, the median
is the mean of 6 and 7. is the mean of 15 and 16.
So, the median is 6.5. So, the median is 15.5.
The number 7 appears most often, so the mode is 7. The number 24 appears most often, so the mode is
24.
50. {–1, 0, 5, 2, –2, 0 ,–1, 2, –1, 0}
52. SPORTS Lisa has a rectangular trampoline that is 6
SOLUTION: feet long and 12 feet wide. What is the area of her
trampoline in square feet?
SOLUTION:
So, the mean is 0.4.
Order the data from least to greatest.
{–2, –1, –1, –1, 0, 0, 0, 2, 2, 5} The area of Lisa’s trampoline is 72 square feet.
Because there is an even number of data, the median Find each product or quotient.
is the mean of 0 and 0. 53.
SOLUTION:
Multiply the numerators and denominators.
So, the median is 0.
1-1 VTahrei anbumlebs earns d0 Eanxdp –re1s bsoiothn so ccur most often, so the
modes are 0 and –1.
51. {17, 24, 16, 3, 12, 11, 24, 15}
54.
SOLUTION:
SOLUTION:
So, the mean is 15.25.
Order the data from least to greatest.
{3, 11, 12, 15, 16, 17, 24, 24}.
Because there is an even number of data, the median
is the mean of 15 and 16. 55.
SOLUTION:
So, the median is 15.5.
The number 24 appears most often, so the mode is
24.
52. SPORTS Lisa has a rectangular trampoline that is 6
Evaluate each expression.
feet long and 12 feet wide. What is the area of her
trampoline in square feet?
56.
SOLUTION:
SOLUTION:
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators.
The area of Lisa’s trampoline is 72 square feet.
Find each product or quotient.
53.
SOLUTION:
57. 5.67 – 4.21
Multiply the numerators and denominators.
SOLUTION:
5.67 – 4.21 = 1.46
58.
54. SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
eSolutSioOnsLMUanTuIaOl-NPo: w eredbyCognero Page10
59. 10.34 + 14.27
SOLUTION:
55. 10.34 + 14.27 = 24.61
60.
SOLUTION:
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
Evaluate each expression.
56.
SOLUTION:
61. 37.02 – 15.86
The LCD for 5 and 9 is 45. Rewrite each fraction
with denominators of 45 then add the numerators. SOLUTION:
37.02 – 15.86 = 21.16
57. 5.67 – 4.21
SOLUTION:
5.67 – 4.21 = 1.46
58.
SOLUTION:
The LCD for 3 and 6 is 6. Rewrite each fraction
with a common denominator of 6.
59. 10.34 + 14.27
SOLUTION:
10.34 + 14.27 = 24.61
60.
SOLUTION:
The LCD for 12 and 36 is 36. Rewrite the fractions
with a common denominator of 36.
61. 37.02 – 15.86
SOLUTION:
37.02 – 15.86 = 21.16
Description:Which expression best represents the perimeter of the rectangle? enthusiast club took a poll to see what each member's . BOOKS Akira bought one new book for $20 and three used .. ERROR ANALYSIS Tara and Curtis are.
