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Academir Charter School Middle
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2014 Summer Math Packet
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Academir Charter School Middle is committed to fostering continuous student learning, development and achievement.
This
commitment extends through the summer as to prevent the decay of learning. Research indicates that the summer break may
have detrimental learning effects for many students. On average, all students lose skills, particularly in mathematics. It is crucial
that parents are diligent during the summer to ensure their students' educational growth and success. Given the connection
between the importance of sustaining academic skills, summer learning loss and parental involvement, it is reasonable to assume
that structured academic activities during the summer would help mitigate this loss and may even produce gains. In an effort to
promote and reinforce mathematics learning and retention during the summer months, Academir Charter School Middle is
implementing a Summer Math Packet for all grade levels.
The Summer Math Packet is a series of worksheets designed to review basic math skills and keep students thinking mathematically
over the summer break. The skills addressed in the packet are important to your student moving to the next level of mathematics.
Please have your student follow all directions in the packet and have him/her show all work on an organized and labeled notebook
paper, as needed. The students are to bring the completed packet and notebook paper showcasing their calculations of problems
within the packet with them on the first day of school. Academir teachers will review the packet during the first few days, which
will be followed with a diagnostic assessment to gauge students mathematical knowledge and level.
Basic Operations
There are four basic operations in mathematics:
+ (addition)
the sum of 8 and 2 is 10
as in 8 + 2 = 10
— (subtraction)
the difference of 8 and 2 is 6
as in 8 — 2 = 6
x (multiplication) the product of 8 and 2 is 16
as in 8 x 2 = 16
÷ or F(division) the quotient of 8 and 2 is 4
as in 2)8 or 8 ÷ 2 = 4
the quotient of 2 divided into 8 is 4 as in 2)8 = 4
the quotient of 8 divided by 4 is 2 as in 8 ÷ 4 = 2
There are several other words that are used: total (+), remainder (—), times (x), divide
up (÷). In working these problems, read carefully and rewrite the problems so that you
can do them correctly.
Example: Find the sum of 15 and 27.
15
+ 27
Do the indicated operation. Write the answer on the line.
1. Find the sum of 21 and 35
2. Find the product of 5 and 12
3. Find the difference of 23 and 17
4. Find the sum of 567 and 197
5. Find the quotient of 1,900 and 25
6. Find the product of 31 and 306
7. Find the difference of 321 and 176
8. Find the quotient of 7,004 and 68
9. Find the total of 57, 73, and 81
10. What is 19 times 87?
11. What is the remainder of 694 minus 405?
12. What would you have if you divided up 256 into 8 equal shares?
Exponents
Recall that multiplication is repeated addition. So 6 times 2 means six 2's added
together or 2 + 2 + 2 + 2 + 2 + 2 = 12. An exponent is making repeated multiplication
happen, or three to the fourth power multiplied together would be 3 x 3 x 3 x 3 = 81;
thus, exponents makes a fifth operation. An exponent is a superscript number written
to the upper right of a number. If 6 is the base and the 2 is the exponent, we say "6 to
the second power," which is the area of a square with sides of 6 units. So some people
will say, "six squared." A problem would look like 6 2 or 6^2, which appears on
computers and some calculators. The third power of a number such as four to the third
power would look like this-43. So people will say "4 cubed," since the answer would
be the area of a cube with sides of 4 units.
Example: 82 = 8 x 8 = 64
122 = 12 x 12 = 144
10^2 = 10 x 10 = 100
Do the indicated operation. Write your answer on the line.
1. 22 =
6. 102 =
11. 122 =
2. 32 =
7.82 =
12. 152 =
3. 5 2 =
8. 63 =
13. 105 =
4. 24 =
9. 73 =
14 23 x 32 =
5. 33 =
10. 104 =
15. 103 x 104 =
AWL
Order of Operations
The biggest rule in mathematics is the "Order of Operations." Everyone must have the
same rules so that the answers will match. Everyone uses these rules: businesses,
engineers, and scientists. Computers and calculators use the rules as well.
The rules are four steps:
1. Do the operations with parentheses first.
2. Do exponents.
3. Perform multiplication or division in order from left to right.
4. Finally, do addition and subtraction from left to right.
Examples: 1. 5 + (6 x 3) =
5 + 18 = 23
2. 4 + 5 x 8 =
4 + 40 = 44
3. 9 x 5 +3 =
45 + 3 = 15
Do the indicated operations. Write your answer on the line.
1. 7 x 4 —(4 + 7).
9.(15 — 9)x 92 =
2.9 x 6 + 4 =
10. (37 — 21)x(11 + 4)=
3. 15 + 3 4 =
11.12 — 3 + 4 x 8 + 5 — 9 =
4. 5(7 — 3)=
12.2 x 9 + 15 - 3 — 18 =
5.35 — 3 x 7 =
13.2 + 3 — 4 + 5 — 6 + 7 — 8 + 9 x 2 =
6.5 + 7 — 8 =
7. 7 + (8 x 10) — 15=
8. (87 —
+3 =
14. 9 — 8 + 7 — 6 + 5 — 4 + 3 — 2 x 2 =
Prime Numbers
it
Prime numbers are natural numbers greater than one that have exactly two factors, one
and itself. Composite numbers are numbers that have more than two factors.
The Fundamental Theorem of Arithmetic: Every natural number greater than one is
either a prime or a composite. A composite can be expressed uniquely as a
product of primes.
Prime factorization is needed in many problems. To find the prime factorization of a
number, express the number in terms of prime numbers. A list of the first ten prime
numbers is: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Remember the list never stops; there is
no end, and the number one is not a prime number since it only has one factor, itself.
Example: Find the prime factorization of 12.
12 ÷ 2 = 6
6 4- 2 = 3
3 is prime, so the prime factorization will be 12 = 2 x 2 x 3 or 2 2 x 3
The identity property tells you that the product of any number multiplied by 1 is that number.
Find the prime factorization of the following numbers. Write the answer on the line.
1. 14
S. 36
9. 48
2. 18
6. 31
10. 45
3. 8
7. 125
11. 56
4. 27
8. 96
12. 121
Fractions: Adding and Subtracting
Fractions are found on all standardized tests and in all types of applications. Fractions
are how the world deals with parts of a whole. Fractions may be proper fractions,
improper fractions, or mixed numbers. The power of the number one is the secret of
fractions. The number one may take on many different appearances. The equivalent
fractions of one have the same numerator and denominator. So there is a series of
fractions all equal to one that looks like:
1 = = 3 = = = 8 =...
To add or subtract fractions, you must have common denominators. The numerators
are what you add or subtract. Therefore, the denominators are just labels that must be
the same. After having common denominators, add (or subtract) the numerators. That
answer is this number over the common denominator.
Examples:
To add:
2
4
i
3 X 4 X 12
3
"5
3 ____ ). 1
+qX
12
17
12 =
5
To subtract:
-
_5_
1 12
4
2
6 X
12
4X3
12
7
12
Find the sum or difference and reduce to lowest terms. Write the answer on the line.
1. + L1,4- =
5.
2.3+2 =
6.
i +4 =
3. 2
7. ai — 1 14 =
11. 253 —214=
4.214+2 =
8. 51 — =
12. 36— 2i=
9. 10i
=
g-=
10.
5
—
=
1 5g - 4 =
Fractions: Multiplying and Dividing
One way of thinking about multiplication is that the answer is the area of a rectangle. When
fractions are multiplied, the answer is part of a rectangle with sides of the two factors. (Picture
of When multiplying 2 by 4, the answer is 8 .
a square 1 by 1, with the shading of of
3x3
a
a
Example:
4 X 8 = 4 x 8 — 32
To divide fractions, take the reciprocal of the divisor and multiply. This algorithm will give you
an answer, but what does it mean? There are several ways to look at the answer. A square
of that pizza. 4 = :11 ; there
pizza is cut into 4 rows and 4 columns, so one piece would be of
was only of a pizza and you eat a quarter of it. You had 3 pieces.
4
Example:
3 2 3_ 3
8 ÷ 3 = 8 A 2 16
With mixed numbers, change them to improper numbers and follow the Order of Operations.
Examples:
A
4+2 x
n
41
2 4 3.§
7-- 2 x 3 = 6 =
_5_
= 3 X 7 = 21 = 3 =1 3
Find the product or quotient and reduce to lowest terms. Write the answer on the line.
1. 2 X 4 =
2.
4.
x
5 .3=s=
=
=
10.15=4=
7.12x3=
11.122=13=
8.25 x 1 12- =
12. et ÷ 1 14 =
13. What is the area of a poster that is 1
feet by 22 feet?
9.14x5=
14. What is the area of a desktop that is
22 feet by 5 feet?
Integers
Integers are whole numbers and their opposites. Most of us would say they are
positive and negative numbers. Remember that zero is also in the group of integers.
Zero is not positive or negative; it is between the positive and negative numbers.
Ordering integers from smallest to largest (or least to greatest) will take some thinking.
Placing integers on a number line will help.
A real number is any number that can be shown on a number line. Usually these
numbers would be fractions and decimals. Numbers that cannot be shown as fractions
or decimals are called irrational numbers. An irrational number is a number that
cannot be represented by a ratio of two integers. That is because this type of number
does not terminate. It does not repeat the way rational numbers do. Since these
numbers are hard to represent, some are shown with symbols like it or 4-2-.
Write the numbers listed below in order from least to greatest. If the numbers are already in
order, indicate that by writing correct. Write the answer on the line.
1. 7, -9, 15, 3
4. 38, -49, -23, 34
2. 345, 241, -256, 265
5. 289, -321, 192, 305
3. - 15, -29, 38, 57
6. 21, 28, -15, -18
-
-
-
-
Write the numbers listed below in order from least to greatest. Then put the following
sets of numbers on the number line provided. Write the answer on the line.
■-
121110-9 -8-7-6 -5 4 -3 -2 1 01+2 434+5 467+8+9101112
7. - 11, 5, 12
8. 51, -4,
i
9.64, - 10i, 514, 2
10. - 3i, 53, 41, -12, 11
-
11. -2i, - 2i, 11, 34
12. - 818-, 9i, 3i, 2i, 18
-
-
Adding Integers
When adding two numbers together, a popular model to explain this operation is
movement on a number line. Addition of positive numbers moves to the right, and addition
of negative numbers moves to the left. There are two basic rules for sign operations:
1) When adding two integers with the same sign, add the absolute values, and use
that sign for the answer.
2) When adding two integers with different signs, subtract the two numbers—the
lesser absolute value from the greater absolute value. Use the sign of the
greater absolute value for the answer.
Examples: A. Add -5 + (-7) =
Absolute value of -5 is 5 and of -7 is 7, so
5 + 7 = 12 and same signs (negative) = - 12
B. Add 3 + (-7) =
Absolute value of 3 is 3 and of -7 is 7
different signs, so
7 - 3 = 4 and using the sign of the
greater absolute value = -4
Determine the sum of each of the following. Write the answer on the line.
9. 573 + (-336) =
1. 7 + (-9) =
2.19 + 23 =
10. -96 + 69 =
3. -6 + (-31) =
11. 62 + (-48) =
4. -25 + 42 =
12. -1693 + (-2791) =
5. 108 + (-24) =
13. 871 + (-25) + 531 =
6. - 121 + (-25) =
14. 25 + (-91) + 39 =
7. 150 + (-63) =
15. 17 + (-34) + 51 =
8. 524 + 378 =
16. - 19 + 38 + (-57) =
e
Subtracting Integers
Subtraction is the inverse operation of addition. That means the rules will be similar to
addition. If two numbers are added together and the answer is 0, the numbers are called
additive inverses or opposites of each other. So, 5 is the opposite of -5, because their
sum is 0. Using symbols to show this: x+ (-x) = 0. The rule for subtraction will be to
subtract a number, you add the opposite.
A problem like 7 — 4 = 3 could be changed to 7 + ( -4) = 3 and have the same answer.
To subtract a number, add its opposite. For integers a and b, a — b = a + (-b).
Examples:
B. 6 — 11 = 6 + (-11)= -5
D. -12 — (-4) = -12 + 4 = -8
A.5 —(-8). 5 +8 = 13
C. -7 —(-9)= -7 + 9=2
Determine the sum of each of the following. Write the answer on the line.
1. 7 — (-9) =
9. 59 — 79 =
2. 47 — 32 =
10. 128 — 278 =
3. 8 — 15 =
11. 378 — (- 125) =
4. 13 — 34 =
12. - 52 — 52 =
5. -23 — 7 =
13. -168 — 168 =
6. -47 — 57 =
14. 321 — 752 =
7. 17 — (-39) =
15. 5 — 6 — 7 =
8. 28 — (-13) =
16. -12 — (- 14) — (- 16) =
61
Multiplying and Dividing Integers
In multiplying and dividing integers, there are two rules. First, the product of two
integers that have the same sign is positive. Second, the product of two integers that
have different signs is negative.
Examples:
A. (-6) x (-8) = 48
B. (- 36) ÷ 9 = -4
Determine the solution for each of the following. Write the answer on the line.
1. 7 x (-9) =
9. 30 ÷ 15 =
2. (-5) x 12 =
10. (- 120) ÷ 3 =
3. 6 x 18 =
11. 3 x (-4) + (-2) x (-8) =
4. (-7) x 11 =
12.8 x 3 +(-6)÷2 =
5. 10 x (- 31) =
13.10 x(-5)+ 12 ÷ 4 =
6. 288 ÷ 32 =
14. 4 x (- 11)- 3 x (- 12).
7. 60 ÷ (-30) =
15. (-5) x (-20)
8. 25 ÷ (-5) =
16. 24 - (-3) - 30(-3) =
-
-
36 ÷ 6 =
(
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Variables
A variable may be a letter or symbol. A variable written with a number is an expression.
Any operation may be used with a variable. A coefficient is the number that is written
next to a variable. The coefficient is multiplied by whatever the variable is. In the
expression: 2x + 3y, 2 is the coefficient of x and 3 is the coefficient of y. In an expression,
a constant is a quantity that stays the same, a number that is added to or subtracted from
a variable. Writing algebraic expressions from verbal expressions is a needed skill.
Examples: the sum of a number and 5, would be x + 5
the product of a number and 7, would be 7x
6 less than a number, would be x — 6
a number divided by 2, would be
Using x as the variable, write an algebraic expression for each on the line.
1. the product of a number and 12
2. the quotient of a number and 5
3. the total of a number and 16
4. a number decreased by 15
5. a number to the fourth power
6. 3 less than a number
7. twice a number decreased by 12
8. three-fourths of a number plus 25
Write a verbal expression for each on the line.
9. x + 9
10. x — 12
11. 2x — 5
12.
2
+ 18
13. x2 + 9
14. 5x — 13
Distributive Property
In the multiplication of any two numbers, 6 x 15 for example, there are different ways
to determine the answer. We know that 5 x 9 = 45, but 5 x (3 + 6) = 15 + 30 = 45. This
is an example of the Distributive Property.
The Distributive Property states that for any three numbers a, b, and c, a x (b + c) =
(ab) + (ac) or a x (b — c) = (ab) — (ac).
A term is a number, a variable, or a product or quotient of numbers and variables. In
simplifying an expression, you add or subtract like terms together. In adding or
subtracting, you add or subtract the coefficients together. (You do not add exponents
together!)
Example: x x (3 + 5) = 3x + 5x = 8x; 3x and 5x are the like terms. Since they are like
terms, the coefficients can be added or subtracted.
Remove parentheses and add like terms, if possible. Simplify each expression. Write the
answer on the line.
1.15 x(2 +3)=
9.y x (7 + 13) =
2.11 x(3 + 7)=
10.3 x(x+ 5).
3. 6 x (12 + 9) =
11. 15 x (4+x).
4. 8 x (10 + 12) =
12. 5(x + y) =
5. 8 x (20 — 7) =
13. x(15 + 8) =
6. 9 x (30 — 3) =
14. y(10
7. 12 x (30 — 5) =
15. x(x + 9) =
8. x(4 + 8) =
16. x(2x
—
—
y) =
3) =
-
Solving One-step Equations
If you are asked to solve an equation for a variable, you need to find a value for the
variable that makes the equation true.
One-step equations require you to reverse the operation of the problem to find the
value of the variable.
Examples:
Addition Property of Equality: You can add or subtract the same number from both
sides of an equation and the equation will stay balanced.
A.
x + 3=9
B.
x - 5 = 12
x+3-3=9-3
x - 5 + 5 = 12 + 5
(subtract 3 from both sides)
(add 5 to both sides)
x=6
x = 17
Multiplication Property of Equality: You can multiply or divide both sides of an
equation by the same number and the equation will stay balanced.
C. 5x = 25
D.
4=8
5x _- 25
4x4=8x4
5
5
(divide both sides by 5)
(multiply both sides by 4)
x =5
x = 32
Solve the following problems using the Properties of Equality. Show each step. Write the
answer on the line.
1. x + 9 = 12
9. 6x = 18
2. x + 15 = 28
10. 8x = 56
3. x + 5 = 8
11. 15x = -45
4. x + 25 = - 15
12. 3x = 36
5. x - 15 = 21
13. A. = 8
5
6. x - 23 = 17
14. A. = 8
8
7. x - 5 = 8
15.
8. x - 12 = - 17
16.
-
-
-
'4
( 4)
= 24
= - 10
111.
Solving TWo-step Equations
In two-step problems, you work the equation backward. You have the two Properties of
Equality, one for addition/subtraction and one for multiplication/division. Since this is
backward, the Order of Operations is reversed.
Examples:
4x + 6 = 18
A.
x
B.
a
- 7 = -5
x
a - 7 + 7 = -5 + 7
4x + 6 — 6 = 18 — 6
(subtract 6, both sides)
(add 7, both sides)
x
4x = 12
4x 12
4=4
8=2
ax x 8 = 2 x 8
(divide by 4, both sides)
(multiply by 8, both sides)
x =3
x = 16
Solve the following problems using the Properties of Equality. Show each step. Write the
answer on the line.
1. 4x 7 = 9
2. 7x + 4 = 32
3. 5x + 3 = 18
9. (4) + 6 = 13
10.
11.
2
— 5 = 23
15 —
3 = -2
4. 2x — 6 = 14
12. (1-) + 12 = 7
5. 6x + 11 = 35
13. x + 3 = 19
6
6. -4x + 3 = 31
14. 12x + 9 = 12
4
7. -3x — 5 = 28
8.6 +
15 = 16
15.
9
—5 =8
16.x +
5 =
- 17
+5
3
-
v(5
Solving Equations—Variable on TWo Sides
0111.001.0, 4044•Muff, r.11.1PMC, ...10i , s.....1*
You may find complicated equations with variables on both sides. To solve these
equations, use the Addition Property of Equality that will remove a variable from one
side. Then solve for the variable.
Examples:
A.
2x + 3 = x — 3
2x+ 3 —x=x— 3 —x
(subtract x, both sides)
x + 3 = -3
x + 3 — 3 = -3 — 3
(subtract 3, both sides)
x = -6
B.
5x— 10 = 3x+ 4
5x-10 —3x= 3x+ 4 — 3x
(subtract 3x, both sides)
2x — 10 = 4
2x— 10 + 10 = 4 + 10
(add 10, both sides)
2x = 14
2x _ 14
2
2
(divide by 2, both sides)
x=7
Solve the following problems using the Properties of Equality. Show each step. Write the
answer on the line.
1.3x+ 2 = 2x+ 6
9. 6x + 7 = 8x-13
2.5x+ 3 = 4x+ 10
10. 1 + 2x= 5 —2x
3. 4x — 7 = 2x — 3
11. 13 + 3x= 13 + 2x
4. 10x
—
15 = 6x— 7
12. 5 — 2x=x— 7
S. 2x+ 25 =x+ 11
13. - 12 +x= 15 — 2x
6.13x+ 15 = 7x+ 27
14. -3 — x = 24 — 4x
7. 3x — 1 = 4x + 5
15. x + 3 = x + 6
8. 2x+ 9 =5x+ 3
16. -5x— 10 =-5x+2
ANSWER KEY
Page
7 +41 ei rade
1. -2 2. 42 3. -37 4. 17 5. 84 6. - 146 7. 87 8. 902 9. 237 10. -27
11. 14 12. -4,484 13. 1,377 14. -27 15. 34 16. -38
Page q
1. 16 2. 15 3. -7 4. - 21 5. -30 6. - 104 7. 56 8. 41 9. -20 10. 150
11. 503 12. - 104 13. - 336 14. -431 15. -8 16. 18
1
-
paqe I
1. 56/. 60 3. 6 4. 764 5. 76 6. 9,486 7. 145 8. 103 9. 211
10. 1,653 11. 289 12. 32
.t the
Page 10
1. - 63 2. - 60 3. 108 4. -77 5. -310 6. 9 7. -2 8. -5 9. -2 10. -40
11. 4 12. 21 13. -47 14. -8 15. 94 16. 82
PoT
l. 4 2. 9 3. 25 4.
16 5. 27 6. 100 7. 64 8. 216 9. 343 10. 10,000
11. 144 12. 225 13. 100,000 14. 72 15. 10,000,000
n
Page S
1. 17 2. 58 3. 1 4. 20 5. 14 6. 4 7. 72 8. 27 9. 552 10. 240
11. 37 12. 5 13. 17 14. 2 •
Page 11
1. - 14 2. -4 3. - 11 4. -21 5. 11 6. 25 7. 24 8. -51 9. 24 10. 119
11. 40 12. 18 13. -4 14. -90 15. 847 16. - 126 17. 1 18. 10
19. --20 20. - 10
Page 4-
1.2x 72.2 x 3 x 33.2 x 2 x 2 4. 3x 3 x 35.2 x 2 x 3 x 3
6.1 x 317.5 x 5 x58. 2x2x2x2x2x 39.2x 2x2 x 2 x 3
10. 3 x 3x 511.2 x 2 x 2 x 712.11 x
Page .
1. 5 / 8 2. 5/6 3. 1 1/6 (7/6) 4. 2 3/4 5. 1/12 6. 1/3 7. 2 1/12 (25/12)
8. 1 3/4 (7/4) 9. 7 1/12 10. 11 1/6 11. 3 7/12 12. 13/24
Page (C)
1. 3/8 2. 1/6 3. 9/20 4. 1 1/2 (3/2) 5. 1 7/9 (1619) 6. 1 3/5 (8/5)
7. 1 8. 4 9. 3/4 10. 20 11. 7 1/2 12. 7 13. 3 3/4 sq ft
14. 12 1/2 sq ft
Page
1. 12x 2. x/5 3. x + 16 4. x - 15 5. x4 6. x - 3 7. 2x
12
8. 3/4x + 25 Examples of answers: 9. a number plus 9 10. a
number decreased by 12 11. twice a number minus 5 12. half of a
number plus 18 13. a number squared plus 9 14. five times a
number decreased by 13
-
Page 13
1. 75 2. 110 3. 126 4. 176 5. 104 6. 243 7. 300 8. 12x 9. 20y
10. 3x + 15 11. 60 + 15x 12. 5x + Sy 13. 23x 14. lOy -y2
15. x2 + 9x 16. 2x2 - 3x
Page 14-
1. 3 2. 13 3. - 13 4. -40 5. 36 6. 40 7. -3 8. -5 9. 3 10. 7 11. -3
12. - 12 13. 40 14. 64 15. -96 16. 90 '
Page 15
1. 4 2. 4 3. 34. 10 5. 4 6. -7 7. - 11 8. 6 9. -28 10. 56 11. 15
12. 15 13. 48 14. 1 15. 39 16. 18
Page lip
1. 4 2. 7 3. 2 4. 2 5. - 14 6. 2 7. -6 8. 2 9. 10 10.
1 11. 0 12. 4 13. 9 14. 9 15. no solution 16. no solution
Page 7
1. -9, 3, 7, 15 2. -345, - 256, 241, 265 3. - 29, - 15, 38, 57
4. -49, -38, -23, 34 5. -321, -305, -289, 192 6. -18, -15, 21, 28
7. - 11, 5, 12 8. -4 1/27, 3/4, 5 1/2 9. -10 3/4, -8 1/4, 1/2, 5 1/4
10. - 12, -5 2/3, -3 1/3, 4 1/2, 11 11. -2 3/4, -2 1/4, 1 1/2, 3 1/4
12. -9 7/8, -8 1/8, -3 3/8, 1 7/8, 2 5/8 Writing Answers will vary.