Download Pre-requisite Skills Review Sheets

Integer Addition, Subtraction,
Multiplication, Division
BASIC DEFINITIONS:
INTEGERS – Positive and Negative numbers (and zero) whose decimal digits are zeros.
ABSOLUTE VALUE – Distance from zero on a number line.
OPPOSITES – Two numbers the same distance from zero on a number line but on
different sides of zero.
INTEGER ADDITION:
- Do the integers have the same sign?
YES
NO
-ADD their absolute values
-SUBTRACT their absolute values
-Keep the common sign
-Keep the sign of the integer with the larger
absolute value.
INTEGER SUBTRACTION:
- Add the opposite. How?
Step 1:
Step 2:
Step 3:
Step 4:
Keep the first integer the same
Change the subtraction symbol to addition
Change the sign (to its opposite) of the sign that follows the subtraction symbol
Follow the rules of addition above.
INTEGER MULTIPLICATION and DIVISION:
- Do the integers have the same sign?
YES
NO
-MULTIPLY or DIVIDE their absolute values
-MULTIPLY or DIVIDE their absolute values
-Your answer will always be positive
-Your answer will always be negative
ALGEBRAIC PROPERTIES
These basic algebraic properties, which do not affect an expression’s outcome, are categorized by:
-Movement of terms
-Grouping of terms
-Special results from performing certain operations.
MOVEMENT OF TERMS:
COMMUTATIVE PROPERTY
(To ‘commute’ means ‘to move’)
of Addition:
of Multiplication:
a b b  a
a b  b a
5 4  45
 3  7  7  (3)
x 6 6 x
5 4  45
 3  7  7  (3)
x 6  6 x
GROUPING OF TERMS:
ASSOCIATIVE PROPERTY
(To ‘associate’ implies who is grouped together)
of Addition:
a  (b  c)  (a  b)  c
5  (4  3)  (5  4)  3
 3  [7  (1)]  (3  7)  (1)
x  (6  y )  ( x  6)  y
of Multiplication:
a  (b  c)  (a  b)  c
5  (4  3)  (5  4)  3
 3  [7(1)]  (3  7)  (1)
x  (6 y )  ( x  6) y
SPECIAL RESULTS FROM PERFORMING CERTAIN OPERATIONS:
DISTRIBUTIVE
PROPERTY
IDENTITY
PROPERTY
a(b  c)  ab  ac
of Addition:
of Multiplication:
a0a
a 1  a
50 5
 3  0  3
x0 x
5 1  5
 3 1  3
x 1  x
5(4  3)  5  4  5  3
 3[7  (1)]  3(7)  (3)( 1)
x (6  y )  x  6  x  y
*Distributive Property in reverse, ab  ac  a (b  c) , can still be called the Distributive Property,
but is more commonly known in algebra as ‘FACTORING’ (through GCF).
2
EXPRESSIONS and EQUATIONS
EXPRESSION – Collection of numbers, operations, and variables.
2x  7
Examples:
a4
5c
 8v  5
x
3
7
 12.4
EQUATION – Two expressions separated by an equals sign.
2x  7  12.4
Examples:
a47
5c  40
EVALUATING EXPRESSIONS:
To evaluate an expression, substitute/replace the variable with the number and simplify:
(7 )  4
5( 6 )
w
3
30
3
11
30
10
a  4 if
Examples:
a7
5c
if
c6
if
w  30
SOLVING EQUATIONS:
OPEN SENTENCE – Equation with at least one variable to be solved.
Equations can be True, False, or Open.
18  32  50
True
12  4  x
Open
45  15  25
False
REPLACEMENT SET – Collection of numbers that are substituted in for the variable(s).
Example:
Solve for
?
Solutions:
2(0) (0)  3
03
NO
2x  x  3
{ 0, 1, 2, 3 }
using replacement set
?
2(1) (1)  3
24
NO
?
2(2) (2)  3
45
NO
?
2(3) (3)  3
66
YES
Solution Set: {3}
3
To solve an equation without a replacement set provided, performing the inverse operation of the one(s) in
the original equation helps to isolate the variable.
Examples:
y  4  18
  4  4
y
 22
w  5  12
  5  5
w
 7
4 x  20
4x
 204
4
x5
f
3
f
3
 8
   38
3
f  24
WRITING EQUATIONS:
Addition Words
Subtraction Words
Multiplication Words
Division Words
Sum
Add
More than
Increased by
Difference
Subtracted from
Less than
Fewer than
Decreased by
Product of
Multiplied by
Times
Double
Triple
Quotient
Divided by
“Is” means ‘equal to’.
“Of” often means multiplication.
To write an equation, use your key words to translate the phrases into algebraic expressions/equations.
Examples:
Solutions:
Five more than g is 34
5  g  34
 5   5
g  29
4
(or also g  5  34 )
72 is one-sixth of y
72  16  y
672  616 y 
432  y
(or also 72 
y
6
)
EXPONENTS
Exponents show repeated multiplication in shorthand form. An exponent tells us how
many times to multiply the base by itself.
EXAMPLES:
53  5  5  5 In this problem, 5 is called the base and 3 is its exponent.
4a 3  4  a  a  a Note in this problem that 3 is the exponent of base a , not the 4.
(4a)3  4a  4a  4a However in this example the 4 is included in the repeated
multiplication. Both 4 and a are bases to exponent 3.
-Writing an expression as just a base with its exponent is called writing it in exponential
3
notation, such as 4  a  a  a becoming 4a ‘in exponential notation’.
ORDER OF OPERATIONS
An order has been agreed upon to which operations are performed before others. Several shortcut ways to
remember this ranking system have been developed, with most popular being PEMDAS, or “Please
Excuse My Dear Aunt Sally”. However, note that although there are six operations, two ranks of order
have two operations in them.
Step #1:
Step #2:
Step #3:
Step #4:
Parentheses – Compute within any grouping symbol first, if available.
Exponents – Compute powers next, if available.
Multiply or Divide – Compute in order from left to right.
Add or Subtract – Compute in order from left to right.
- If more than one grouping symbol exists within a problem, such as (parentheses), [brackets], or {braces},
work from the inside out.
EXAMPLES: Evaluate the following by using the Order of Operations:
x 4 if x  2
m2  (5)
(2) 4  2  2  2  2
16
(6) 2  6  6  (5)
36  (5)  31
(2) 3  (2)(2)(2)  2
 8  2  6
(3a)3 if a  2
3a 3 if a  2
16  4  32
[3(2)]3  [6]3
6  6  6  216
3  23  3  8
24
16  4  9
4  9  13
if
m6
n3  2
if
n  2
1  [4  (17  23)  23 ]
1  [4  40  23 ] 
1  [4  40  8] 
1  [4  5] 
1  [1]  0
5