Summer Review
For Students Entering
Math 6
Dear Student,
Welcome to Grade 6! We are going to be learning many new skills in Math 6. In order to
help you be the best math student possible, you need to complete all the problems in this
packet. The work in this packet has been designed to reinforce the skills that you learned in
5th grade. In order for you to get the most out of this packet, it is highly recommended that
you complete a portion of this packet each week of summer vacation. There are 100
questions in all. You should do about 20 problems each week. In this way, the number of
problems will not overwhelm you. This packet will be collected the first week of school
and will count as your first math grade.
Here are some websites that you can use to help you review and practice:
www.mathforum.com This online community includes teachers, students, researchers,
parents and educators who have an interest in math and math education. The site includes
Ask Dr. Math, Problems of the Week, discussion groups and much more.
 www.AAAmath.com
This site is arranged by topic and features explanations of various
mathematical topics, practice problems and fun, and challenging games.
 www.coolmath.com
This fully interactive site and allows you to sharpen basic math
skills, play games, and explore new math concepts.
www.mathisfun.com
understandable way.
A great website that explains all math steps in an easy and
www.khanacademy.com This site will review specific skills by watching videos and then
gives you the opportunity to practice.
www.ixl.com This is a for fee website that will help you improve your math skills.
Your parent must sign you up for this website.
Math 5 Review for Incoming 6th Graders
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WRITING NUMBERS IN WORDS AND DIGITS
In order to read numbers correctly, we need to know the order of each place value. The order is
the following:
one
millions
hundred
thousands
ten
thousands
one
thousands
hundreds
tens
ones
decimal
point

tenths
hundredths
thousandths
1,000,000 is one million 100,000
is one hundred thousand 10,000
is ten thousand
1,000 is one thousand
100 is one hundred
10 is ten
1 is one
0.1 is one tenth
0.01 is one hundredth
0.001 is one thousandth
The number 354.67 is read as three hundred fifty-four and sixty-seven hundredths.
The number 3,500,607.004 is read as three million, five hundred thousand, six hundred seven and four
thousandths.
Please remember the word "and" indicates the location of the decimal point in math and should not be used
anywhere else. For example, it is inappropriate to read 350 as three hundred and fifty, because "and"
means a decimal point. Also, the term "point" in mathematics is a geometry term and should not be used in
naming numbers. For example, 3.5 is not three "point" five, but it is three and five tenths.
ORDERING DECIMALS
To compare decimals and list them from least to greatest, it is easier to compare decimals that are
the same place value. This means that the numbers should have the same number of decimal places.
To do this, we can add additional zeroes to the end of the number to make all the decimals have the same
place value. In addition, when ordering any set of numbers, you must use the proper sign, either <, the less
than sign, or >, the greater than sign.
For example, to put the following in order from least to greatest:
.3, 1.61, .006, .107 is easier to compare as: 0.300, 1.610, 0.006, 0.107
Now, it is easier to determine the decimal with the smallest value: 0.006, 0.107, 0.300, 1.610.
Then return to the original form: 0.006, 0.107, 0.3, 1.61.
Your final answer would be: 0.006 < 0.107 < 0.3 < 1.61
LONG DIVISION



the number to be divided into is known as the dividend
The number which divides the other number is known as the divisor
the answer is called the quotient
And here we go:
4 ÷ 25 = 0 remainder 4
The first digit of the dividend (4)
is divided by the divisor.
The whole number result is placed at the top. Any remainders are
ignored at this point.
Math 5 Review for Incoming 6th Graders
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25 × 0 = 0
The answer from the first operation is multiplied by the divisor.
The result is placed under the number divided into.
4–0=4
Now we subtract the bottom number from the top number.
Bring down the next digit of the dividend.
42 ÷ 25 = 1 remainder 17
Divide this number by the divisor.
The whole number result is placed at the top. Any remainders are
ignored at this point.
25 × 1 = 25
The answer from the above operation is multiplied by the divisor.
The result is placed under the last number divided into.
42 – 25 = 17
Now we subtract the bottom number from the top number.
Bring down the next digit of the dividend.
175 ÷ 25 = 7 remainder 0
Divide this number by the divisor.
The whole number result is placed at the top. Any remainders are
ignored at this point.
25 × 7 = 175
Math 5 Review for Incoming 6th Graders
The answer from the above operation is multiplied by the divisor.
The result is placed under the number divided into.
4
175 – 175 = 0
Now we subtract the bottom number from the top number.
There are no more digits to bring down.
The answer must be 17.
If you have a remainder, write the
remainder with “R”, then the number.
Math 5 Review for Incoming 6th Graders
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LEAST COMMON MULTIPLE (LCM)
The smallest (non-zero) number that is a multiple of two or more numbers.
Least Common Multiple is made up of the words Least, Common and Multiple:
What is a "Multiple" ?
The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5,
etc). Just like the multiplication table.
Here are some examples:
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ...
The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc...
What is a "Common Multiple" ?
When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of
those numbers.
For example, when you write down the multiples of 4 and 5, the common multiples are those that are found in both lists:
The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...
The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...
Notice that 20 and 40 appear in both lists?
So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)
What is the "Least Common Multiple" ?
It is simply the smallest of the common multiples.
In our previous example, the smallest of the common multiples is 20 ...
... so the Least Common Multiple of 4 and 5 is 20.
Finding the Least Common Multiple
It is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.
Math 5 Review for Incoming 6th Graders
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GREATEST COMMON FACTOR
The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fractions!
Let's start with an Example ...
Greatest Common Factor of 12 and 16
 1. Find all the Factors of each number,
 2. Circle the Common factors,
 3. Choose the Greatest of those
So ... what is a "Factor" ?
Factors are the numbers you multiply together to get another number:
A number can have many factors:
Factors of 12 are 1, 2, 3, 4, 6 and 12 ...
... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.
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What is a "Common Factor" ?
Let us say you have worked out the factors of two
numbers:
Example: Factors of 12 and 30
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Then the common factors are those that are found in
both lists:
 Notice that 1, 2, 3 and 6 appear in both lists?
 So, the common factors of 12 and 30 are: 1, 2,
3 and 6 It is a common factor when it is a factor of two
or more numbers. (It is then "common to" those
numbers.)
Here is another example with three numbers:
Example: The common factors of 15, 30 and 105
Factors of 15 are 1, 3, 5, and 15
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
The factors that are common to all three numbers are 1, 3, 5 and 15
In other words, the common factors of 15, 30 and 105 are 1, 3, 5
and 15 What is the "Greatest Common Factor" ?
It is simply the largest of the common factors.
In our previous example, the largest of the common factors is 15, so the Greatest Common Factor of 15, 30
and 105 is 15
The "Greatest Common Factor" is the largest of the common factors (of two or more numbers)
Why is this Useful?
One of the most useful things is when we want to simplify a
fraction: Example: How could we simplify 12/30 ?
Earlier we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the Greatest
Common Factor is 6. So the largest number we can divide both 12 and 30 evenly by is 6.
The Greatest Common Factor of 12 and 30 is 6.
Math 5 Review for Incoming 6th Graders
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ADDING AND SUBTRACTING DECIMALS
To add decimals, follow these steps:



Write down the numbers, one under the other, with the decimal points lined up
Put in zeros so the numbers have the same length.
Then add using column addition, remembering to put the decimal point in the answer. (The sum or difference has the same
number of decimal places as the numbers.
84.9
+ 0. 463
84.900
+ 0.463
85.363
MULTIPLYING DECIMALS
Just follow these steps:


Multiply normally, ignoring the decimal points.
Then put the decimal point in the answer - it will have as many decimal places as the two original numbers combined.
In other words, just count up how many numbers are after the decimal point in both numbers you are multiplying, then the answer
should have that many numbers after its decimal point.
Example: Multiply 0.25 by 0.2
start with: 0.25 × 0.2
multiply without decimal points: 25 × 2 = 50
0.25 has 2 decimal places,
and 0.2 has 1 decimal place,
so the answer has 3 decimal places:
0.050
ADDING AND SUBTRACTING FRACTIONS
There are 4 simple steps to add fractions:
1.
When adding and subtracting fractions, the denominators must be the same. If
they are not the same, you must find a common denominator.
2.
Keep the denominator the same.
3.
Add or subtract the numerators.
4.
Simplify, if you can.
So,
Math 5 Review for Incoming 6th Graders
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MULTIPLYING FRACTIONS
Multiply the tops, multiply the bottoms.
There are 3 simple steps to multiply fractions
1. Multiply the top numbers (the numerators).
2. Multiply the bottom numbers (the denominators).
3. Simplify the fraction if needed.
Example:
MULTIPLYING MIXED NUMBERS
Here are the simple steps for multiplying mixed numbers:
1. Write all mixed numbers as improper fractions.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify, if possible.
DIVIDING FRACTIONS
Example:
Step 1. Turn the second fraction (the divisor)upside down (it becomes a reciprocal) and change the division to multiplication:
Math 5 Review for Incoming 6th Graders
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Math 5 Review for Incoming 6th Graders
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Student Name:
Directions: All work must be done in pencil. All work must be shown either on this paper
or on a separate sheet of paper with problems numbered. Attach your “thinking” paper to
this review. Do not use a calculator.
Read each question. Find the answer.
1. What is the value of the digit 8 in 0.918?
1.
2. What is the value of the digit 7 in 619.473?
2.
3. Write the set of numbers in order from least to greatest:
5.89, 50.74, 5.676, 50.135
3.
4. Compare using <, >,or =.
9.359 
9.370
4.
5. Estimate the difference by rounding to the nearest whole number:
$125.56 $43.22
5.
6. Estimate the sum:
23 + 49 + 62 + 71
6.
7-10. Find the sum or difference.
7.
8,070
 5,093
8.
3.425
+ 9.696
9.
9,003
 314
10. 5.4 + 3.21
11-12. Estimate the product by rounding:
11. 36 x 78 __________________
12.
61 x 43
13-17. Find the product.
13.
5,000
x
80
Math 5 Review for Incoming 6th Graders
14. 643
x 69
15. 6,000
x 50
16. 971
x 34
17. 0.05
x 0.4
12
Find the quotient.
18. 6,349  7 =
19. 5,692 35 =
20. 239.02 34 =
5,473 6 =
22. 1,065 27 =
23. 127.05
21.
21 =
Answer the following questions.
24. A triangle has three sides of equal length. What
type of triangle must it be?
24
25. What solid figure has triangular faces and
one rectangular base?
25.
26. What solid figure has triangular faces and
one pentagonal base?
26.
27-28. The high temperatures for the last five days were 27.
35oF, 45oF, 38oF, 42oF, and 35oF. What was the
mean high temperature for the five days? What was 28.
the mode of the temperatures?
29-30. Ellen’s scores on her math tests were 80, 92, 85,
90, and 83. What was the mean of his scores?
What was the range of his scores?
Math 5 Review for Incoming 6th Graders
29.
30.
13
31. What is the simplest form of
?
31.
32. What is the simplest form of
?
32.
33. What is the greatest common factor of 15 and 12?
33.
34. What is the greatest common factor of 18 and 54?
34.
35. What is the least common multiple of 6 and 8?
35.
36. What is the least common multiple of 4 and 6?
36.
37. What is the least common denominator of
?
37.
38. What is the least common denominator of
?
39. The numerator of a fraction is about the same as
denominator. The fraction that is closest to this
is:
0
½
1
38.
39.
Find the sum or difference. Write your answer in simplest form.
40.
41.
42.
43.
Summer Review for Incoming 6 th Grade Students
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44.
45.
46.
47.
48.
49.
Find the product. Write your answer in simplest form.
50.
51.
52.
53.
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54.
55.
56.
57. Sam had 18 stamps. He gave of the stamps to his
brother. How man stamps did Sam have left?
58.
Nat had a piece of wood that was
57. __________________
long.
He used of it for a project. How many feet of wood
did he use for the project?
59-60. Daniel has 24 soccer cards. He gave of of them to
Marc. How many cards did he give to Marc? How
many cards does Daniel have left?
58. __________________
59.
60.
61.
62.
Summer Review for Incoming 6 th Grade Students
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63. John painted of a room while his friend Bob
painted of the room. How much of the room
did they paint together?
63. ___________________
64-65. In a Men’s Choir, of the choir sing bass,
of the
choir sings baritone. What part of the choir sings
alto? If there are 6 singers who sing bass, how many
sing alto?
64. ___________________
65. ___________________
Finalists in Gymnastics
School
Number of Students
Clemens
10
Hollywood
8
Webster
6
Nichols
8
66-67. Use the table to answer the
questions:
66.What fraction of the finalists were from
Webster School?
66.
67.What fraction of the finalists were either
Hollywood School or Clemens School?
67.
68-70. Use the schedule to answer
the questions:
68. How long is the Crafts class?
Summer Review for Incoming 6 th Grade Students
Activity
Breakfast
Crafts
Swimming
Lunch
Free Time
Camp Schedule
Begins
7:45 a.m.
8:30 a.m.
10:30 a.m.
12:00 noon
12:45 p.m.
Ends
8:15 a.m.
10:15 a.m.
11:45 a.m.
12:45 p.m.
2:00 p.m.
68.
19
69.How much more time is there for swimming than
for lunch?
70. Which activity takes the most time? How do you
know?
69.
70.
71. Explain how you would find the perimeter of a triangle?
72. What is the perimeter of a square with a side of 9 cm?
73. What is the perimeter of a rectangle with a length of
39 feet and width of 17 feet?
72.
73. ____________________
74. What is area? How do you find the area of a rectangle? How do you write the units of
measure with area?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Summer Review for Incoming 6 th Grade Students
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75. What is the area of a square whose side is 7 meters?
75. ____________________
76. What is the area of this rectangle?
7 meters
8 meters
25 cm
5 cm
77. What is the area of this figure?
12 cm
Summer Review for Incoming 6 th Grade Students
13 cm
21
78. David is building a rectangular garden, using 34 feet of
edging. He wants the area of the garden to be exactly
72 square feet. What should he make the dimensions
of his garden?
78.
79.Mr. Cohen bought 32 feet of fencing to build a pen
for his dogs. He wants the pen to have the greatest
possible area. What should the dimensions of the
pen be to give him the greatest possible area?
79.
Adding, Subtracting, Multiplying, and Dividing with Decimals
80. Write 13.45 in expanded form and word form.
81. 9.6 + 4.8
84.
12.6 – 6.9
87.
Summer Review for Incoming 6 th Grade Students
82.
12.65 + 37.08
83.
18.5 + 23.76
85.
16.34 – 8.26
86.
34.5 – 16.92
88.
89.
22
90.
91.
93.
94.
359.4
3
92.
624.4
5
95. A sweatshirt costs $14.99, and socks cost $1.59 a pair.
many of each did she buy?
95.
Lilly
96. Using the digits 3, 4, 8, and 7, how many four-digit
96.
numb
97. Last month the fourth graders read 126 books, the
171 books. This month the fourth graders read
97.
fifth g
Summer Review for Incoming 6 th Grade Students
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140 books, the fifth graders read 163, and the
sixth graders read 149. Which group read the
most books in the two months?
98. Mrs. Washington wants to plant bushes every
5 feet around the edges of her garden. Her garden
is 15 feet long and 20 feet wide. How many
bushes does she need?
98.
99. Jon, Barb, and Ali recycled 140 cans. Barb
than Jon. How many cans did Ali recycle?
99.
recycl
100. Karen made a border using brown and gold tiles.
tiles, and so on, ending with a brown tile. If she
used 20 gold tiles, how many brown tiles did she
use?
100.
She b
101. Five people entered a tournament. Each person
many games were played?
101.
played
102. Kate had 30 books. Rich has 6 more books than
many books does Mark have?
102.
Kate. M
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103 . On Monday Kathy went shopping and spent $9.
.. On Tuesday she earned $6 babysitting. On
Wednesday her aunt gave her $3. On Thursday
she spent $7 and then had $10 left. How much
money did Kathy have before she went shopping
on Monday?
103.
104. Alan is bagging 30 apples and 45 oranges. He wants
What is the greatest number of apples or oranges he
could put into each bag?
104.
105. Julia is making a punch that calls for the same amounts
of apple juice and cranberry juice. Apple juice comes in
8-ounce bottles while cranberry juice comes in 6-ounce
bottles. If she wants to use all the juice she buys, what is
the least number of ounces of juice of each type of juice
she should buy?
105.
to put
106. There are 9 bowls of snacks arranged in a circle. If Nan continues to take one snack
from each fourth bowl, will she take one snack from each bowl before she returns
to the first bowl? Explain your answer.
Summer Review for Incoming 6 th Grade Students
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107. Karla works 4 afternoons a week after school for $6.
an hour. She works 3 hours each afternoon. She must
pay $0.50 for the bus each way to and from her job. How
much does she have left each week after taking the bus?
107. __________________
108. Hope needs to make 24 posters for her campaign for
Student Council. She plans to make 3 posters the
first day and then to make 2 more posters each day
after than she made the day before. How many days
will it take her to make all 24 posters?
108.
109. Talia has a rectangular tablecloth that is 52 inches
by 70 inches. She wants to sew a border around the
edges of the tablecloth. How much border will she
need?
109.
110. Rachel purchased two packages of meat. One
weighed 3.27 pounds and the other weighed
2.85 pounds. How many pounds of meat did
she purchase?
110.
111. Together, Harry and Jen scored 250 points.
Jen scored 40 more points than Harry. How
many points did Jennie score?
111.
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