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PRE-ALGEBRA
Summer Packet
VANDEBILT CATHOLIC HIGH SCHOOL
Incoming 8th Grade
EXAMPLES
Section I
Objective: Write an algebraic expression to represent unknown quantities with one unknown
and 1 or 2 operations
Examples:
The examples below show algebraic expressions written as mathematical
9 more than a number
the sum of 9 and a number
x+9
a number plus 9
a number increased by 9
the total of
4 subtracted
x and 9
from a number
a number minus 4
h-4
4 less than a number
a number decreased by 4
the difference of hand 4
6 multiplied by 9
6g
6 times a number
the product of 9 and 6
a number divided by 5
the quotient of
t and
divide a number by 5
5
t
5
expressions.
Section II
Objective: Simplify using given operations and by combining like terms
Examples:
The examples below show how to simply expressions by combining like terms and performing
indicated operations.
2x + 4x-7
Determine
6x-7
Combine like terms
2(x + 3) - 5x
Distribute
2x + 6 - 5x
Determine
-3x+
like terms
6
like terms
Combine like terms
Section III
Objective: Solving equations for missing variables
Examples:
The examples below show how to solve equations using addition, subtraction, multiplication,
and division.
2x
+5 = 7
Use inverse operations
-5 -5
Subtract 5 from both sides
2x
+2
=2
Isolate x
+2
Divide 2 on both sides
x=l
to isolate the variable
=
3 (2x - 1)
6x - 3
21
= 21
+3 +3
6x =
24
Distribute
Use inverse operations to isolate 6x
Add 3 to both sides
Isolate x
Divide 6 on both sides
x=4
Section IV
Objective: Solving proportions
Examples:
The examples below show how to solve proportions
x
24
12
3
12 x 24
288
Cross multiply to solve for the missing value
=3xx
= 3x
-;- 3 -;- 3
2
=
Multiplication
Isolate x
Divide 3 on both sides
= 96
x
X
by cross multiplication.
14
Cross multiply to solve for the missing value
7
2 x 14
28
= 7xx
= 7x
-;-7 -;-7
x=4
Multiplication
Isolate x
Divide 7 on both sides
Section V
Objective: Performing operations with negative integers
Examples:
The examples below show how to perform operations with negative integers. These are just
some of the possibilities.
-4x 6
=
= Negative
Negative -;-Positive
= Negative
-4
-3+5
=
Negative + Negative
-8
-24-;-.6
=
= Negative
-24
-3+ -5
=
Negative x Positive
Negative + Positive
2
= Takes
the sign of the integer with the larger
absolute value
Examples:
Rational Numbers:
Helpful processes and tips
Multiplying Fractions and Mixed Numbers
1.
Change any mixed numbers to improper fractions
2.
Cross cancel any numerator with any denominator by dividing each by a common factor
3.
Multiply numerators together
4.
Simplify ...put fraction in simplest form (keep as an improper fraction)
then multiply denominators together
Dividing Fractions and Mixed numbers
1.
Change any mixed number to an improper fraction
2.
Keep the first fraction, change the division sign to a multiplication
fraction
sign and flip the second
this means multiplying by the reciprocal
3.
Multiply numerators together
4.
Simplify ...put fraction in simplest form (keep as an improper fraction)
then multiply denominators together
Adding and Subtracting Fractions and Mixed Numbers
1.
Change any mixed number to an improper fraction
2.
Find common denominators
3.
Keep the denominator and add numerators
4.
Simplify ...put fraction in simplest form (keep as an improper fraction)
Section VI
Objective: Solving expressions using order of operations
Examples:
The examples below show how to solve expressions using order of operations.
First, let's recall Order of Operations.
Parenthesis
Exponents
Multiplication
Division
Addition
&
(left to right)
&
5 ubtraction
(3
+ 1) +
4
+ 24
4+6
= 10
(left to right)
(4
-i-
x 6) -;-4
4
Do what's inside of the parenthesis
Division
Add
Section VII
Objective:
Graph given coordinates
Examples:
The examples below show how graph ordered pairs onto a coordinate plane.
First, let's recall Quadrants.
--1-
I
Y
I
Quadrant II
(-, +)
-
Quadrant!
(+,+)
2
Graphing Ordered Pairs:
-s
s
0
x
Quadrant Ilf
-2
2.
Move along the y-axis second
3. Plot the point and label with the given variable
Quadrant IV
(+, -)
(-, -)
1. We move along the x-axis first
-s
r..
.•
Graph the following ordered pairs:
E
..
'.
:;.
~
A (1,3)
/J.
B
B
(3,1)
•
~
C (3,-3)
.,
-
D (-4,2)
'}
c
•
,.,f'
E (-1,5)
.;
c
-,
F (-3,-3)
Section VIII
Objective: Order rational numbers on a number line
Examples:
The examples below show how use a number line to help order rational numbers from least to
greatest.
First, let's recall negative and positive integers and the number line .
•• I I I I I I I I I I I , I I I I I I I I I ~
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
negative
0
1 2
3 4
zero
5 6
7 8 9 10
positive
14
20}
Order the rational numbers from least to greatest:
1.
6
3
5 } 10
Change fractions to decimals
2.
Use the number line to plot the points
3.
Rewrite using the original rational numbers
-1.2
-0.4
< I : I + I II I I I I t I I
-1
I /I
o
0.3
t I I I
0.7
+ I
II I I I I)
1
Section IX
Objective:
Find area and perimeter of shapes
Examples:
These are the formulas that must be used to find the area and perimeter of the geometric
figures.
Area
0'a polygon
= The amount of space inside the boundary of a flat (2-
dimensional) object
Perimeter
0'a polygon
the sum of the sides
Formulas
+ 2w
A
= lw
51
+ 52 + 53 + 54
A
= bh
=
51
+ 52 + 53
A
=
=
51
+ 52 + 53 + 54
Rectangle:
P = 21
Square:
P
= 45
Parallelogram:
P
=
Triangle:
P
Trapezoid:
P
Circle:
Circumference
=
ttd
!bh
2
VANDEBILT CATHOLIC mGH SCHOOL
PRE-ALGEBRA
©
2016
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reserve
d.
Incoming 8th grade
Summer Packet
Please show your work for all problems given. Make sure you BOX off your answers. Please
refer to the example problems attached to the packet if you have any questions or need any
guidance.
I. Write each as an algebraic expression when given as a verbal expression and as a verbal
expression when given an algebraic expression.
1) 29 decreased by 4
2) the product of 9 and x
3) 3 increased by 8
4) the sum of2 and n
5) 12Y
6) c - 16
7) 4 + v
8) n + 5
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Il. Simplify each expression by combining like terms where necessary.
9) 4 + 3r + 8
10) 2 + 5x - 3x + 7
ll)n+l-n
12) 1 - lOp - 4p
13) 4x+ 3x
14) 6a - 8 + lOa
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III. Solve each equation given.
15) v- 5 =-4
17) 4=-
16) 2 = a-7
x
a
18) -= -16
19
20
19) 36 = 17 + x
21) -3
=a-
20) 13=-4+b
19
x
22) 3 =12
23) 15 = -5n
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25) 5n = 70
26) 20 = 2x + 8x
27) -1 = 3 - 8k + 4
28) -7k- 4 + 6k= -12
29) -5
= 7 + 6x + 6x
30) -4=-3a+4a
31) -5(7m - 5) = -115
32) 6(1 - 4a) = 126
33) -7(3x - 6) = -84
34) 5(3b -
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IV. Solve each proportion.
9
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42) -35
41) 90
9
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VI. Order of operations. Use order of operations to simplify each expression.
55) 4 + 4 + 6 x 3 - 4
56) 5 - (1 + 1 + 3) 7 5
58) 3 - 3 + 4 x 3 x 3
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VII. Graph each ordered pair. Label with the corresponding letter.
61) G (1,0), R (-3, 1), V (-2, -3)
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VID. Order from least to greatest. Look at each rational number. Put them on a number line.
Order them from least to greatest using their original forms.
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IX. Find the area and perimeter of each.
69) A square with a side length of 4 inches.
70) A triangle with a height of 4 inches and sides of 5 inches, 9 inches and 7.5 inches.
71) A rectangle with a width of 12 centimeters and a length of21 centimeters.
72) A parallelogram with a side lengths of 20 centimeters top and bottom and 10 em on each side with a
height of 8 em.
73) A rectangle with a length of 3 yards and a width of 1.5 yards.
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