Math 6

Colonial School District

Summer Math Packet

Answer Key
Promoting a Culture of Collaboration, Innovation and Inspiration

June 2019

Dear Parents/Guardians,

First, we would like to thank you for all of the additional support you offer at home.
Education is a true partnership between school and family that is essential to a child’s
success.

As this school year comes to a close, we wanted to again encourage you to continue to
reinforce and foster the mathematical skills and practices that have been developed this
year by scheduling time for your child to work through this summer math packet. The
activities were selected by our grade level experts with the key mathematical concepts of
the school year in mind. The ultimate goal is to reinforce and strengthen the skills that
will serve as building blocks for future learning.

Wishing you a relaxing, yet exciting, math-filled summer!

Sincerely,

The Curriculum Department

Serving the students of Conshohocken, Plymouth and Whitemarsh

230 Flourtown Road, Plymouth Meeting, PA 19462 – Phone (610) 834-1670 – Fax (610) 834-7535 –
www.colonialsd.org
 Integers
Adding

RULE EXAMPLES
SAME SIGNS
5 + 8 = 13
1. Add.
–5 + (–8) = –13
2. Sum is positive if both are positive;
negative if both are negative.

DIFFERENT SIGNS
5 + (–8) = –3
1. Subtract the absolute values.
–5 + 8 = 3
2. Answer is sign of the integer
with the greater absolute value.

Find each sum.

1. –4 + (–8) 2. 14 + 16 3. –43 + (–12)

–12 30 –55

4. –16 + 11 5. 28 + ( –42) 6. 75 + (–5)

–5 –14 70

7. –49 + (–32) 8. 23 + (–23) 9. 86 + (–18)

–49 86
–81 –32 0 68 –18
–81 68
 Integers
Multiplying & Dividing

RULE EXAMPLES
1. Multiply or divide.
–5 x ( –8) = 40 16 x ( –3) = –48
2. The answer is positive if
the signs are the same 40  4 = 10 –20  10 = –2
(both positive or both negative);
negative if the signs are different
(one positive and one negative).

Find each product or quotient.

1. –3 x (–8) 2. –5 x (–5) 3. –15 x 3

24 25 –45

4. 0 x (–121) 5. –35  ( –7) 6. –65  5

0 5 –13

7. 240  (–4) 8. 36  12 9. (–49  7) x 8

–7 x 8
–60 3
–56
 Integers
Subtracting

RULE EXAMPLES
1. Change the minus sign to a plus. 5 – 8 –9 – (–12)
2. Find the opposite of the 2nd number. = 5 + –8 = –9 + 12
3. Add; using your rules for adding integers. = –3 = 3

Find each difference.

1. 4 – 7 2. –5 – 3 3. –8 – 2

= 4 + –7 = –5 + –3 = –8 + –2

–3 –8 –10

4. –3 – 24 5. 10 – 17 6. 13 – 9

= –3 + –24 = 10 + –17 = 13 + (–9)

–27 –7 4

7. –41 – 37 8. 62 – (–29) 9. –6 – (–6)

= –41 + ( –37) = 62 + +29 = –6 + +6

–41 62 0
–37 29
–78 91
 Integers
Problem Solving

RULE
4–Step Plan for Problem Solving

1. Explore. You need to read the problem and know what information you
have and need and what is asked.

2. Plan. Develop a plan to solve the problem. Choose a strategy.

Often it is helpful to make an estimate.

3. Solve. Carry out your plan.

4. Examine. Be sure to label your answer correctly. Check your answer

By comparing to your estimate.
If the answer does not make sense, make a new plan and try again.

NOTE:
Remember in most cases there is more than one way to solve the problem!

1. Rita opened a checking account with a balance of $150. She wrote

2 checks: $87 and $68. How much money remained in the account?

150 + (–87) + (–68) 150 63

–87 –68
–$5 is left: the account 63 –5
is overdrawn

2. During a space shuttle launch a maneuver is scheduled to begin at

T minus 85 seconds (i.e. 85 seconds before liftoff). If the maneuver
lasts 2 minutes, at what time will the maneuver be complete?

–85 + 2 (60) 60 sec = 1 min

–85 + 120 120
–85
At T + 35 seconds

3. The water level in a tank decreased 10 centimeters in 5 minutes. If

the tank drains at a steady rate, what is the change in the water
level each minute?

10
5 which is 2 cm per min
 Fractions
Adding and Subtracting

RULE EXAMPLE
5 3
1. Find the lowest common 6 + 8
denominator (LCD).
LCD = 24
2. Write equivalent fractions
using the LCD. 5 20
6 = 24
3. Add or subtract the numerators.
Write the sum or difference 3 9
over the LCD. 8 = 24

4. Reduce if necessary. 29 5
24 = 1 24

Find each sum.

2 3 1 3
1. 7 + 8 2. 6 + 5

16 21 37 5 18 23
56 + 56 = 56 30 + 30 = 30

5 2 3 5
3. 16 – 9 4. 4 – 12

45 32 13 9 5 4 1
144 – 144 = 144 12 – 12 = 12 = 3

6 1 3 2
5. 3 7 + 4 8 6. 4 5 – 2 3

48 7 55 9 10
3 56 + 456 = 7 56 4 15 – 2 15 =

24 10 14
3 15 – 2 15 = 1 15
 Fractions
Multiplying

RULE EXAMPLES
3 2 5 3
1. Write any mixed numbers as 10 x 3 38 x 7
improper fractions.
6 29 3
2. Multiply the numerators.
30 8 x 7
3. Multiply the denominators. 87
1 = 56
= 5
4. Reduce if necessary.
31
= 1 56

Find each product.

1 1 2 3
1. 3 x 3 2. 9 x 8

1 6 1
9 72 = 12

3 2 3
3. 10 x 3 4. 1 4 x 7

6 1 7 7 49 1
30 = 5 4 x 1 = 4 = 12 4

4 3 4 1 5
5. 4 5 x 3 4 6. 5 x 3 x 12

6 3 1 1
24 15 24 15 4 1 5 1
5 x 4 = 5 x 4 5 x 3 x 12 = 9
1 1 1 3
6 3
= 1 x 1 = 18
 Fractions
Dividing

RULE EXAMPLES

1. Write any mixed numbers as 3 2 5 3

improper fractions. 10  3 38  7

2. Change the 2nd fraction to 3 3 29 7

its reciprocal. 10 x 2 8 x 3
(i.e. flip it over)
9 203
= 20 = 24
3. Multiply.

4. Reduce if necessary. 11
= 8 24

Find each quotient.

1 1 5 1
1. 3  6 2. 8  16

2
1 6 6 5 16 5 16
3 x 1 = 3 = 2 8 x 1 = 8 x 1 = 10
1

5 3 1
3. 12  16 4. 2  1 4

4
5 16 5 16 20 2 1 5 1 4 8 3
12 x 3 = 12 x 3 = 9 = 39 3  4 = 3 x 5 = 5 = 15
3

1 5 3 3
5. 1 3  2 6 6. 11 4  5 4

2 1
4 17 4 6 8 47 23 47 4 47 1
3  6 = 3 x 17 = 17 4  4 = 4 x 23 = 23 = 223
1 1
 Fractions
Problem Solving

RULE
4–Step Plan for Problem Solving

1. Explore. You need to read the problem and know what information you
have and need and what is asked.

2. Plan. Develop a plan to solve the problem. Choose a strategy.

Often it is helpful to make an estimate.

3. Solve. Carry out your plan.

4. Examine. Be sure to label your answer correctly. Check your answer

By comparing to your estimate.
If the answer does not make sense, make a new plan and try again.

NOTE:
Remember in most cases there is more than one way to solve the problem!

5 1
1. The total length of the bicycle race track is 8 miles. The first 5 mile is
hilly and the rest is flat. What fraction of the course is flat?

5 1 25 8 17
8 – 5 = 40 – 40 = 40

2. The cooking instructions for a turkey recommend roasting the turkey at a low
3 1
temperature for 4 hours for each pound. How long should you cook a 10 2
pound turkey?

1 3 21 3 63 7
102 x 4 = 2 x 4 = 8 = 78

3. In one year 120 students enrolled at a community college. This was

3
5 of the number of students accepted. How many of those accepted
did not enroll?
40
3 120 5
120  5 = 1 x 3 = 200
1
 Decimals
Adding and Subtracting

RULE EXAMPLE
1. Line up the decimal points.
33.4 – 3.82
2. Add zeros if necessary.
33.40
3. Add or subtract.
– 3.82
NOTE: 29.58
Remember to bring down your
decimal point into your answer!

Find each sum or difference.

1. 3.956 + 2.41 2. 0.0589 + 0.278

1 1 1
3.956 .0589
+ 2.41 + .278
6.366 .3369

3. 117 + 105.02 4. 6.788 – 0.2

1
117. 6.788
+ 105.02 – 0.2
222.02 6.588

5. 3.24 – 0.51 6. 117 – 105.0023

2 1 6 9991
3.24 117.0000
– 0.51 – 105.0023
2.73 11.9977
 Decimals
Multiplying

RULE EXAMPLE
1. Multiply as you would whole numbers
62.8 x 0.093
2. Count the number of digits to the right
of the decimal point in each number. 62.8 1 decimal place
x .93 2 decimal places
3. In you answer, count from the right to
the left that number of place and put 1884
your decimal point. 56520
58.404 3 decimal places
NOTE:
Remember, do NOT line up the decimal points
when setting up your problem!

Find each product.

1. 0.6 x 0.8 2. 0.9 x 0.27

0.6 x 0.8 = .48 = .48 0.9 x 0.27 = .243

(1) (2) (1) (2 3)

3. 18.3 x 0.67 4. 7.2 x 5.4

18.3 (1) 7.2 (1)

x .67 (2 3) x 5.4 (2)
1281 288
10980 3600
12.261 38.88
(1 2 3) (1 2)

5. 8.4 x 0.003 6. 0.04 x 0.3

(1 2) (3)

8.4 (1) = .012

.003 (2 3 4)

. 0252
(1 2 3 4)
 Decimals
Dividing

RULE EXAMPLE
1. Change the divisor to a whole number by
moving the decimal point to the right. 3.9  0.13
2. Move the decimal point in the dividend
the same number of places.
Add zeros if necessary. 0.133.9

3. Divide.
30.
NOTE: 13390
39
Remember to bring your decimal point up
0
into your answer!

Find each quotient.

1. 82  0.4 2. 2.38  3.5

205. .68
.4. 82.0. 4. 820. 3.5. 2.3. 80 35. 23.80
–8 – 210
020 280
– 20 – 280

3. 121.8  1.4 4. 0.0092  8

121.8 10 1218 .0092

1.4 x 10 = 14 = 87 8 = .00115

5. 149.73  0.23 6. 2.004  0.2

2.004 10 20.04
0.2 x 10 = 2 = 10.02
651.
.23.14973 23.14973
– 138
117
– 115 
23
– 23
 Decimals
Problem Solving

RULE
4–Step Plan for Problem Solving

1. Explore. You need to read the problem and know what information you
have and need and what is asked.

2. Plan. Develop a plan to solve the problem. Choose a strategy.

Often it is helpful to make an estimate.

3. Solve. Carry out your plan.

4. Examine. Be sure to label your answer correctly. Check your answer

By comparing to your estimate.
If the answer does not make sense, make a new plan and try again.

NOTE:
Remember in most cases there is more than one way to solve the problem!

1. Megan has $80 to spend on clothes for school. After looking at the
ads, she decides to buy two pairs of jeans for $29.99 each and two
tank tops for $8.18 each. Does she have enough money to buy three
new hair clips that are on sale 3 for $10?

No, because 2 x $29.99 = $59.98

2 x $8.18 = $16.36
$76.34 + $10 = $86.34

2. Paula calls her grandparents long distance in California and talks for
45 minutes. The phone company charges $0.05 per half-minute.
How much does the call cost?
45
x .05
2.25

3. Ms. Francis drove her car 427 miles on 15.8 gallons of gas.
a. To the nearest mile, how many miles per gallon is this?
b. What was the cost of the gasoline she used if the price was
$1.96 per gallon?

a) 27. She got 27 mpg. b) 1.96 The cost of gasoline

15.8. 427.0. 27 was $52.92
316 1372
1110 392
1106 52.92
4
 Percent
Conversions

RULE EXAMPLE
FRACTION TO PERCENT 3
8
1. Change the fraction to a decimal.
(numerator  denominator)
3  8 = 0.375
2. Change to decimal to a percent.
(Multiply by 100) 0.375 x 100 = 37.5%

3. Label with a percent sign.

Express each fraction as a percent.

24 2 40 2
1. 25 2. 5 3. 125 4. 3

24  25 = 0.96 2  5 = 0.4 40  125 = 0.32 2  3 = 0.66

.096 x 100 = 96% 0.4 x 100 = 40% 0.32 x 100 = 32% 0.66 x 100 = 66.6%

RULE EXAMPLE

PERCENT TO FRACTION 15%

1. Write the number over 100. 15
(no % symbol) 100
2. Reduce the fraction. 3
= 20

Express each percent as a fraction.

5. 20% 6. 72% 7. 70% 8. 2%

20 1 72 18 70 7 2 2
100 = 5 100 = 25 100 = 10 100 = 50
 Percent
Percent of a Number

RULE EXAMPLE
What number is 25% of 520?
1. Identify the part, whole,
and /or percent. Percent = 25
Whole = 520
2. Plug in the numbers into the (NOTE: Whole is after “of” in the problem)
proportion and solve. 25
Percent Proportion 520 = 100

Part % 100 x __ = 520 x 25

Whole = 100 100 x __ = 13,000
100 x 130 = 13,000

130 is 25% of 520.

Use a proportion to solve each problem (round your answer to the

nearest tenth if necessary).

1. What number is 60% of 72? 2. Find 92% of 120.

60 92
100 = 72 100 = 120
100 x __ = 60 x 72 100 x __ = 92 x 120
100 x __ = 4,320 100 x __ = 11,040
4,320  100 = 43.2 11,040  100 = 110.4

3. 25 is what % of 40? 4. 55 is what % of 60?

25 55
40 = 100 60 = 100
25 x 100 = 40 x __ 55 x 100 = 60 x __
2,500 = 40 x __ 5,500 = 60 x __
2,500  40 = 62.5% 5,500  60 = 91.6% = 91.7%

5. 64 is 50% of what number? 6. 2 is 40% of what number?

64 50 2 40
= 100 = 100

64 x 100 = 50 x __ 2 x 100 = 40 x __
6,400 = 50 x __ 200 = 40 x __
6,400  50 = 128 200  40 = 5
 Percent
Percent of Change

RULE EXAMPLE
Old: 8
1. Find the amount of increase or New: 15
decrease.
15 – 8 = 7 increase
2. Fill in numbers in the proportion:
7 x
8 = 100
Increase/Decrease Amount x
Original = 100 7 x 100 x = 8 x x
700 = 8x
3. Solve to find the % of change.
700  8
88% increase

Use a proportion to solve each problem (round to the nearest whole

percent if necessary).

1. Old: $4 New: $7 2. Old: 36 New: 18

$7 – $4 = $3 increase 36 – 18 = 18 decrease
3 x 18 x
4 = 100 36 = 100
300  4 = 7.5 = 7.5% 1,800  36 = 50%

3. Old: $6.80 New: $8.20 4. Old: $150 New: $126

$8.20 – $6.80 = $1.40 increase 36 – 18 = 18 decrease

1.4 x 24 x
6.8 = 100 150 = 100
140  6.8 = 20.58 = 21% 2,400  1500 = 16%

5. A book is on sale for $14. The original price of the book was $20.
Find the percent of the discount.
20 – 14 = 6
6 x
20 = 100 6 x 100 = 600 600  20 = 30 = 30%
 Percent
Problem Solving

RULE
4–Step Plan for Problem Solving

1. Explore. You need to read the problem and know what information you
have and need and what is asked.

2. Plan. Develop a plan to solve the problem. Choose a strategy.

Often it is helpful to make an estimate.

3. Solve. Carry out your plan.

4. Examine. Be sure to label your answer correctly. Check your answer

By comparing to your estimate.
If the answer does not make sense, make a new plan and try again.

NOTE:
Remember in most cases there is more than one way to solve the problem!

1. Mr. Treed bought his son a new bicycle that cost $198. The store
required a 15% down payment to hold the bike. How much was the
down payment?
15 x
100 = 198

100 x __ = 15 x 198 = 100 x __ = 2,970

2,970  100 = $29.70

2. Twenty-eight of the 131 students in Ms. Martin’s classes received

A’s on the last test. About what percent of the students earned A’s?
x 28
100 = 131

100 x 28 = __ x 131 = 2,800 = __ x 131

2,800  131 = 21.37 = 21%

3. Mrs. Miller bought a new suit that cost $175. She bought it when it
was on sale for 40% off. What was the original price of the suit?
40 175
100 = x
40 x __ = 175 x 100 = 40 x __ = 17,500
17,500  40 = $437.50
 The Language of Algebra

Write a variable expression to represent the word phrase.

1. Steve had an unknown amount of money in his pocket. He x – 23

then lost $23. What is the expression that shows how much
money he has now?

x
2. Adam found a bag of money that he split with 22 friends. 23
What is the expression that shows the amount of money that
each person has? (Don’t forget to include Adam).

3. Rachel found a box with money in it. What is the expression x

for this money?

4. Steve cashed his paycheck and then found $23. What is the x + 23
expression that shows how much money Steve has now?

5. A dog lost 15 pounds. What is the expression that shows the x – 15

dog’s current weight?

6. Ryan weighs 6 times as much as his dog. What is an expression 6n

for Ryan’s weight if you call his dog’s weight n?

7. What is an expression for the value of an unknown number 0.1d

of dimes?

8. Jamie is 7 years older than Nancy. What is an expression for n+7

Jamie’s age if Nancy’s is called n?

9. Fritz is 6 years older than twice his brother’s age. What is an 2n + 6

expression for Fritz’s age if his brother’s age is called n?

10. What is an expression for the circumference of a circle with a 

diameter of n inches?

11. What is an expression for the value of an unknown number of 0.5n

half–dollars?

12. If there are 4 times as many dimes in a pile of coins as there 4n

are nickels, what is the expression for the number of dimes if
you call the number of nickels n?
 Writing Expressions and Equations

The table shows phrases written as mathematical expressions.

Phrase Expression Phrase Expression

8 more than a number 7 subtracted from a number

the sum of 8 and a number h minus 7
x+8 h–7
x plus 8 7 less than a number
x increased by x a number decreased by 7
Phrase Expression Phrase Expression

3 multiplied by n a number divided by 5 t

3 times a number 3n the quotient of t divided by 5 5
the product of n and 3 divide a number by 5

Write each phrase as an algebraic expression. No = here!

1. 12 more than a number x + 12

x
2. The quotient of a number divided by 9 9

3. 4 times a number 4x

4. 15 less than a number x – 15

5. 1 less than the product of 3 and m 3m – 1

6. The product of 4 times a number minus 8 4x – 8

Write each phrase as an algebraic expression. Use = here!

7. A number minus 6 equals 12 x – 6 = 12

8. A number plus 14 equals 25 x + 14 = 25

9. 3 more than 5 times the number of dogs is 18 dogs 5x + 3 = 18

10. 4 times the number of cows plus 2 times the number of ducks is 20 4c + 2d = 20
12
11. 2 less than the quotient of 12 divided by a number is 2 – 2 = 2
x

12. The product of 5 and y added to 3 is 33 5y + 3 = 33

 P E M or D S or A
Easy Applications

The acronym for this order of operations is PEMDAS.

Parentheses Exponents Multiplication Division Addition Subtraction
left to right left to right
A popular expression for remembering this is Please Excuse My Dear Aunt Sally

Directions: Find the numerical value of the following expressions using the correct order of operations.

1. 9 x 5 – 4 + 3 x 4 = ____53________ 2. 12 + 8 x 6  2 x 8 = ______204_______
45 – 4 + 12 = 12 + 48  2 x 8 =
41 + 12 = 12 + 24 x 8 = 12 + 192 =
3. 3 + 6 x 8 – 5 x 2 = ______41______ 4. 7 + 8  4 + 3 – 2 = ______10_______
3 + 48 – 10 = 7 + 2 +3 –2 =
51 – 10 =
5. 22  11 + 12 – 3 = _____11_______ 6. 9 x 8 – 6 x 3 + 7 = _____61________
2 + 12 – 3 = 72 – 18 + 7 =

7. 13 + 5 x 6  2 + 10 = ____38_____ 8. 35  7 x 8 + 2 – 4 x 2 = ___34________
13 + 30  2 + 10 5 x 8 + 2 –8
13 + 15 + 10 40 + 2 – 8
9. 100  5 x 5 + 4 – 9 = _____95_____ 10. 88  11 + 56  8 + 12 – 5 = ___22_____
20 x 5 + 4 – 9 = 8 + 7 + 12 – 5 =
100 + 4 – 9 =

Remember the following facts:

 The fraction bar (–) means division.
 The raised dot (•) means multiplication.
 Numbers written next to parenthesis or parentheses next to each
other also require multiplication.

Directions: Find the numerical value of these expressions.

30 9
11. 5(8) – 5 + 4 x 3 = ____46_______ 12. (7)(9) + 3 – 20 x 3 = ____6______
A 40 – 6 + 12 = 63 + 3 – 60 =
13. 8(9) + 10 • 5 + 8 • 2 = ___138______ 14. 3 + 8 • 10 – 13 x 3 = ____44_______
72 + 50 + 16 = 3 + 80 – 39 =
12 44
15. 17 + 5 – 6 • 4 + 3 = ____2______ 16. 9 + 4 – 8 x 2 + 20 – 3 = _____21____
17 + 5 – 24 + 4 = 9 + 11 – 16 + 20 – 3 =
 Function Table

Complete the table by filling in the missing number. Then, write the equation.

1. x y 2. x y
1 6 6=1+5 11 2 11 – 9 = 2

2 7 7=2+5 12 3 12 – 9 = 3

3 8 8=3+5 13 4 13 – 9 = 4
4 9 9=4+5 14 5 14 – 9 = 5
5 10 10 = 5 + 5 15 6 15 – 9 = 6
Equation: y = x + 5 Equation: x – 9 = y

3. x y 4. x y
12 2 2 = 12  6 1 8 8x1=8
18 3 3 = 18  6 2 16 8 x 2 = 16
24 4 4 = 24  6 3 24 8 x 3 = 24
30 5 5 = 30  6 4 32 8 x 4 = 32
36 6 6 = 36  6 5 40 8 x 5 = 40
x
Equation: y = 6 Equation: 8x = y

5. x y 6. x y
1 1 3x1–2=1 1 6 5x1+1=6
2 4 3x2–2=4 2 11 5 x 2 + 1 = 11
3 7 3x3–2=7 3 16 5 x 3 + 1 = 16
4 10 3 x 4 – 2 = 10 4 21 5 x 4 + 1 = 21
5 13 3 x 5 – 2 = 13 5 26 5 x 5 + 1 = 26
Equation: 3x – 2 = y Equation: 5x + 1 = y
Properties of Operations
VOCABULARY TERMS

Addend A number that is added in an addition expression.

Associative Property of The grouping of addends does not change the

Addition sum: (a + b) + c = a + (b +c).

Associative Property of The grouping of factors does not change the

Multiplication product: (ab) c = a (bc).

Commutative Property The order of addends does not change the sum:
of Addition a + b = b + a.

Commutative Property The order of factors does not change the product:
of Multiplication ab = ba.

Distributive Property The product of a factor and a sum is equal to the

sum of the products: a(b + c) = ab + ac.

Factor A number that divides into another number with

no remainder. When two or more factors are
multiplied, they form a product. For example:
2 x 5 = 10; 2 and 5 are factors, 10 is the product.

Identity Property The sum of any number and 0 is that number:

Addition a + 0 = a.

Identity Property Any number multiplied by one equals that

Multiplication number: a x 1 = a.

Product The result of multiplication.

Sum The result of addition.

Zero Product The product of any number and zero is

Property zero: a x 0 = 0.
Properties of Operations
INDEPENDENT PRACTICE

Fill in the missing number below and tell which property the problem demonstrates.

1. 51 x ___1__ = 51

Property used: ___Identity of multiplication___________________________

2. 71 + (__90___ + 5) = (71 + 90) + 5

Property used: ____Associative______________________________________

3. 115 x _23___ = 23 x 115

Property used: ____Communtative___________________________________

4. 0 + 78 = __78___

Property used: __Identity of Addition_________________________________

5. 17 x (5 x 12) = (__17___ x 5) x 12

Property used: ______Associative____________________________________

6. 54 + 60 = 60 + __54___

Property used: _____Commutative___________________________________

 Equations
One–step Equations

RULE EXAMPLE
1. Look at what has been done
to the variable. X – 15 = 29
+ 15 +15
2. Undo it using the inverse
operation on both sides of the X = 44
equation.
3. Check your answer by  44 – 15 = 29
replacing the variable with
the solution.

Solve.

1. d + 32 = 70 2. 708 = c + 30
– 32 – 32 – 32 – 32
d = 38 678 = c

 38 + 32 = 70  708 = 678 + 30

3. x – 89 = 176 4. x – 36 = 12
+ 89 + 89 + 36 + 36
x = 265 x = 48

 265 – 89 = 176  48 – 36 = 12

5. 6.
5x 225 12n 96
5 = 5 12 = 12
x = 45 n = 8

5 x 45 = 225  12 x 8 = 96

7. n  72 = 360 8. n  12 = 12
n n
72 x 72 = 360 x 72 12 x 12 = 12 x 12

n = 25,920 n = 144

25,920  72 = 360  144  12 = 12

 Translating Word Problems to Equations

Write an equation for each sentence. Solve. Show your work.

1. A number b plus 5 equals 15.

b + 5 = 15
–5 –5
___________________
b = 10 check: 10 + 5 = 15

2. A number r minus 2 is 8.
r –2 = 8
+2 +2
___________________
r = 10 check: 10 – 2 = 8

3. A number w added to 7 is 32.

w + 7 = 32
–7 –7
________________________
w = 25 check: 25 + 7 = 32

4. If 4 is added to the product of 6 and a number t, the result is 76.

6t + 4 = 76
– 4 –4
_______________________
6t = 72
6t 72
6 = 6 check: (6 x 12) + 4 = 76

t = 12 72 + 4 = 76

5. Rebecca completes four addition problems each minute. How many minutes
will it take her to complete 12 problems?
m = minutes
4m 12
4 = 4

m = 3 check: 4 x 3 = 12 It will take Rebecca 3 minutes

to solve 12 problems
6. Melissa spent three hours each day painting her house. She spent a total of
27 hours painting. How many days did she paint?
d = days
4m 12
4 = 4
m = 3 check: 4 x 3 = 12 It will take Rebecca 3 minutes
to solve 12 problems
 Graphing on the Coordinate Plane

Directions:
 Identify the quadrant or axis where the point is located.
 Graph each ordered pair on the coordinate grid.
 Write the letter next to the point.

1. A (–4, –1) 2. B (4, 1) 3. C (3, 0)

III I x-axis

4. D (0, 4) 5. E (2, 2) 6. F (–2, 5)

y-axis I II

7. G (–2, –5) 8. H (–1, 4)

III III

Quadrant II Quadrant I
y
10

–10
–10 10
Quadrant III Quadrant IV
 Measures of Center

Complete the problems shown.

I scored these points in 8 I earned these amounts: I worked these hours at my job:
basketball games: 20, 20, $2.50, $3.75, $6.20, $3.75, 1 1 1
8, 6 2 , 5, 8, 5 2, 7, 7 2, 8.
16, 21, 15, 20, 14, 10. $8.00, $5.75.
How much greater is the mean Which is greatest: the mean,
range = _11____21 – 10_ than the mode? the mode, or the median?

136 Mode is $3.75 Mode = 8

mean = _17________ 8 7 + 7.5
Mean is $29.95 6 = $4.99 Median = 2 =
16 + 20 1
median = 18__ 4.99 7.25 or 74
2
– 3.75
55.5
mode = __20___________ $1.24 greater Mean = 8 = 6.9375

Five baseball players hit these What 4 numbers have a Is there a mode in this data:
many home runs in a season: range of 4, 3, 4, 5, 6, 7, 8?
36, 25, 45, 23, 8. a median of 22,
What is the median for these a mean of 22, and
data? a mode of 22? No, there is no mode

8, 23, 25, 36, 45 20, 22, 22, 24

Students received these test These numbers were on a I have 5 numbers. The mean
scores: 96%, 88%, 52%, lottery ticket: 18, 33, 42, for these numbers is 12.
75%, 82%, 91%, 75%. 17, 26. What is the sum of the
What is the mean? What is the range? numbers?

96 + 88 + 52 + 75 + 82 42 5 • 12 = x
+ 91 + 75 = 559 – 17
25 60 = x
559  7 = 79.86%
The sum is 60
 Geometry Connection: Perimeter

Remember: perimeter refers to the sum (+) of all of the outside edges of a figure.

Find the perimeter of each figure shown or described below.

1. 12 ft 2.
P = 2(10) + 2 (2)
5 ft 10 m P = 20 + 4
P = 24 m

P = 2(5) + 2(12)
P = 10 + 24 2m
P = 34 ft

3. 8 cm 4. 12 ft

6 cm 10 ft 10 ft
10 cm

10 ft 10 ft
P = 6 + 8 + 10
P = 24 cm P = 12 + 4(10)
P = 12 + 40
P = 52 ft

5. rectangle: 6. rectangle:
1 3
l = 6 yards w = 4 yards l = 72 inches w = 68 inches

Find the perimeter of each figure. Measure to the nearest eighth inch. Your answers may vary
due to differences in printing and scanning this sheet.
1
1 12
7. 8. 9.
3 3
4 4 1 1 7 3 3 7
8 8 8 8

3 1
14 1 12
1 1
= 44 in = 3 in 52 in
1
10. Find the perimeter of a square with side 142 inches.
1
P = 4(142) P = 58 in
1 1
11. Find the perimeter of a triangle with sides 4 inches, 82 inches, and 94 inches.
1 1 3
P =4 + 82 + 94 P = 214
 Geometry Connection: Area

Rectangle Parallelogram

width (w) height(h )

16 cm 12 in.

length (l ) 40 cm base (b ) 30 in.

The area of a rectangle equals the The area of a parallelogram equals the
product of its length and its width. product of its base and its height.

A = lw A = bh
A = lw A = bh
A = 40 • 16 A = 30 • 12
A = 640 cm2 A = 360 in2

Find the area of each figure shown or described below.

1. 2. 3.

6 cm 8 yd 7 mm

14 cm 3 yd 12 mm

A = 6 • 14 A = 84cm2 A = 3 • 8 A = 24yd2 A = 12 • 7 A = 84mm2

4. 5. 6.

1
22 3.5 yd 6 ft
8 in

1 5 40
A = 22 • 8 = 2 • 8 A = 2 = 20in2 5 yd 7.5 ft
A = 5 • 3.5 A = 17.5 yd2 A = 7.5 • 6 A = 45ft2
7. parallelogram: b = 15 ft, h = 21 ft
15 • 21 = 315ft2
8. rectangle: l = 7.5 cm, w = 12 cm
7.5 • 12 = 90cm2
9. parallelogram: b = 4.7 m, h = 2.2 m
4.7 • 2.2 = 10.34m2
1 1
10. rectangle: l = 14 yd, w = 2 yd
1 1 5 1 5
14 • 2 = 4 • 2 = 8 yd2
 Geometry Connection: Area

Find the area of each figure. A = b • h or A = l • w

1. 2.
13 • 5 = 65 in2 5 in 2 ft 2 • 10.5 = 21ft2

10.5 ft
13 in

3. 4.
3 mm 14 • 3 = 42mm2
3•3= 9ft2 3 ft
14 mm
3 ft

5. 6.
3.5 • .75 = 2.625ft2 3.5 ft 17 • 8 = 136in2
17 in

0.75 ft
8 in

7. 8.
1 1 1
32 • 14 32 ft 9.8 yd 9.8 • 9.8 = 96.04yd2

3.5 • 1.25
4.375ft2 or
3 1
48ft2 14 ft 9.8 yd

Area of Triangles
Find the area of each triangle. A = (b • h )  2

1. 2. base: 12 ft
13 cm 12(7)  2
1
2(7 • 8) 8 cm height: 7 ft 42ft2

56  2 = 28cm2 7 cm
1
(17 • 6)  2 3. base: 17 m 4. base: 32 in (3.5 •1.625)  2
5
51m2 height: 6m height: 18 in 2.84in2

(3.9)(7.2) 25. base: 3.9 mm 6. base: 7 km (7)(4.2) 2

14.04mm2
height: 7.2 mm height: 4.2 km 14.7km2
 Geometry Connection: Area of Circles
Find the area of each circle shown or described below. Round answers
to the nearest hundredth. A = r2 use 3.14 for 

1. A = 3.14 • 62 2. A = 3.14 • 11.52

A = 3.14 • 36 A = 3.14 •
132.25

6m A = 113.04m2 A = 415.27ft2
23ft

3. 4.
A = 3.14 • 82 3.5 cm A = 3.14 • 3.52
A = 3.14 • 64 A = 3.14 • 12.25
A = 200.96in2 A = 38.47cm2
8 in

5. 6.
A = 3.14 • 162 15 km A = 3.14 • 7.52
A = 3.14 • 256 A = 3.14 • 56.25
A = 803.84ft2 A = 176.63km2
16 ft

Circles and Circumference

Find the circumference of each circle show or described below
C = 2r or C =  • d use 3.14 for 

1. 2.
1
6.5 m C = 2 • 3.14 • 6.5 C = 2 • 3.14 • 32
1
C = 40.82m 32 in C = 21.98in

3. 4.
1 1
210 cm C = 3.14 • 210 7.5 m C = 3.14 • 7.5
C = 6.59cm C = 23.55m
 3
5. d = 84 in. 6. r = 11 ft
3
C = 3.14 • 84 C = 2 • 3.14 • 11
C = 27.48in C = 69.08ft
Surface Area of Prisms SA = 2(l • w) + 2(w • h) + 2(l • h)
Find the surface area of each rectangular prism. Round decimal answers to the nearest tenth.

Volume of Prisms V=L•W•H

Find the volume of each rectangular prism. Round decimal answers to the nearest tenth.

1. 2.

SA = 2(18•15) + 2(15•32) + 2(18•32) = 2,652cm2 SA= 2(8 • 7) + 2(7 • 7) + 2(8 • 7) = 322in2

V = 18 • 15 • 32 = 8,640 cm3 V = 8 • 7 • 7 = 392 in3
3. 4.

6 in. 2.4 m

1.75 m
2
13 0.5 m
4
5 in
2 4 2 4
SA = 2(6 • 13) + 2(5 • 13) + 2(6 • 5) = 32.3 in2 SA= 2(1.75 • 0.5) + 2(0.5 • 2.4) + 2(1.75 • 2.4) = 12.6 m2
4 2
V = 6 • 5 • 13 = 8 in3 V = 1.75 • 0.5 • 2.4 = 2.1 m3

5. length, 8 mm 6. length, 9 ft
width, 12 mm width, 7 ft
height, 10 mm height, 12.5 ft
SA = 2(8 • 12) + 2(12 • 10) + 2(8 • 10) = 592 mm2 SA= 2(9 • 7) + 2(7 • 12.5) + 2(9 • 12.5) = 526 ft2
V = 6 • 12 • 10 = 960 mm3 V = 9 • 7 • 12.5 = 787.5 ft3

7. length, 7.6 in. 8. length, 18.3 cm

width, 8.4 in. width, 27 cm
height, 15 in. height, 21 cm
SA = 2(7.6 • 8.4) + 2(8.4 • 15) + 2(7.6 • 15) SA= 2(18.3 • 27) + 2(27 • 21) + 2(18.3 • 21) =
= 607.7 in2 = 2,890.8 cm2
V = 6 • 12 • 10 = 960 mm3 V = 9 • 7 • 12.5 = 787.5 ft3

9. A cube has sides that are 9.2 inches long. What is the volume of the cube?

All faces of a cube are congruent 9.2 • 9.2 • 9.2 = 778.7 in3

Glencoe Division Macmillian/McGraw-Hill

Histograms

A histogram is a graph that shows how many items occur between

two numbers.

The Springfield Library has books arranged by grade level.

How many books are there for grades 6-11?

Find the number of books for grades 6-8. 250

Find the number of books for grades 9-11. 300
Add to find the books for grades 6-11. 250 + 300 = 550

There are 550 books for grades 6-11.

Use the histogram above to answer each question.

1. How many books are there for grades 3-5? 150

2. Which grade levels have the greatest number of books? 12 or over

3. Which grade levels have the fewest number of books? 3-5

4. How many books are there for students in grade 6 and above? 250 + 300 + 400 = 950

5. How many books are in the Springfield Library? 300 + 150 + 950 = 1,400

Critical Thinking
What percent of all of the books in the histogram are for grades 9 and above?

700
1400 = 50%
Worksheet – BOX–AND–WHISKER PLOTS

A. Use the Box-and-Whisker Plot to Answer Questions 1-5

Ages of Dogs in the Dog Show (years)

1. What is the age of the oldest dog(s) in the show? 8 yrs old

2. What is the median age of the dogs? 4 yrs old

3. What number is the lower quartile? 3 yrs old

4. What is the inter-quartile range? 5 – 3 = 2 yrs

1
5. About what fraction of the dogs are 5 years old or older? 4

B. Use the Box-and-Whisker Plot to Answer Questions 6-10

Algebra Test Scores

6. What is the median of all scores? 75

7. What number is the lower quartile? 70

8. What percent of the scores are between 70% and 80%? 50%

1
9. What fraction of the scores fall between 80% and 95%? 4

10. What is the range of the test scores? 95 – 65 = 30

 Equivalent Ratios

1. 7 14 21 2. 8 16 24 3. 2 4 6
5 10 15 7 14 21 5 10 15

4. 3 6 9 5. 7 14 21 6. 7 14 21
5 10 15 12 24 36 4 8 12

Determine whether the ratios are equivalent.

6 18 12 24 4 16
7. 7 and 21 __Yes__ 8. 11 and 22 __Yes__ 9. 5 and 20 __Yes__

9 11 7 5 5 15
10. 4 and 5 __No___ 11. 9 and 3 __No___ 12. 7 and 21 __Yes__

Use equivalent ratios to find the unknown value.

25 5 50 10 r 2
13. c and 8 c = _40__ 14. c and 11 c = __55_ 15. 14 and 7 r = __4__

r 5 7 49 11 a
16. 44 and 11 r = _20__ 17. 6 and h h = __42__ 18. 12 and 84 a = __77_

Math-Aids.Com
Why Did Bonzo Hit His Sister During the Game?
 5 lb of potatoes for $2.19  60 oz of honey for $4.89
2.19  5 = $ _0.44__ per lb 4.89  60 $ __0.08__ per oz

 200 ft of foil for $6.24  1 dozen roses for $29.75

6.24  200 $ _0.03__ per ft 29.75  12 $ _2.48__ per rose

 36 oz of peanut butter for $4.39  25 greeting cards for $7.95

4.39  36 $ _0.12__ per oz 7.95  25 $ _0.32__ per card

 18 issues of a magazine for $28.90  147 oz of detergent for $9.27

28.90  18 $ _1.61__ per issue 9.27  147 $ _0.06__ per oz

 1 dozen doughnuts for $4.50  7 tennis lessons for $99

4.50  12 $ _0.38__ per doughnut 99  7 $ _14.14_ per lessons

 22 oz of cereal for $3.67  3.5 lb of cheese for $8.94

3.67  22 $ _0.17__ per oz 8.94  3.5 $ _2.55_ per oz
H E H E A R D I T W A S
1.61 0.03 0.41 14.14 0.32 0.08 2.55 2.48 0.19 0.44 0.06 2.67 0.17 0.12 0.38

 14 oz for $0.99 .99  14 = $0.07 per oz  5 pieces for $4.79 4.79  5 = $0.96 per piece

 64 oz for $3.10 3.10  64 = $0.05 per oz  21 pieces for $18.77 18.7721= $0.89 per piece

 165 oz for $0.50 .50  1.65 = $0.30 per oz 30 pieces for $2.59 2.5930 = $0.09 per piece

 8 oz for $1.95 1.95  8 = $0.24 per oz 165 pieces for $7.28 7.28 165 = $0.04 per piece

 A monthly magazine charges $17.40 for a  A sports store pays $380 for a case of 144
one-year subscription (12 issues). The baseball. The store sells the baseballs for
same magazine sells at the newsstand for $4.75 each. How much less is their cost
$2.00 a copy. How much do you save on than their selling cost for each ball?
each issue by buying a subscription?
380  144 = $2.64 4.75
17.40  12 = $1.45 2.00 – 2.64
– 1.45 The cost is $2.11 less
You save $ 0.55
 A season ticket to the Olde Theatre costs  For film and processing, a 36-exposure roll
$76 and admits you to 6 plays. Single tickets of film costs $19.20. A 24-exposure roll
to each play cost $15. How much do you costs $16.40. How much can you save per
save on each play by buying a season ticket? picture by choosing the better buy?

76  6 = $12.67 19.20  36 = 0.53 16.40  24 = 0.68

15.00 – 12.67 = 2.33 0.68 – 0.53 = 0.15
You save $2.33 per play You save $0.15 per picture
A S O C K H E R G A M E
0.24 0.02 2.11 0.96 0.15 0.30 2.16 0.55 0.07 2.33 0.46 0.09 0.05 0.04 0.89