Download INTRO PACKET

INTRO
PACKET
Algebra 1
Name _________________________________________________
Date Received: _________________ Date Due: ______________
Math Teacher: ___________________________________________
August 2012
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
2
Name _____________________________________Integer Operations: Add/ Subtract Alg. 1 Intro Packet
Simplify by performing the operation. Do not use a calculator:
1)
 4  ( 7 )
2)
 3  (1)
3)
69
1) _______ 2) ______
3) _______ 4) ______
4)
(2)  (3)
5)
3  11
6)
(6)  (10)
5) _______ 6) ______
7) _______ 8) ______
7)
 7  ( 7 )
8)
 6  ( 4)
9)
 13  (3)
10)
47
11)
3  (9)
12)
29
9) _______
10) ________ 11) ______
12) ________ 13) ______
13)
12  (2)
14)
 5  10
15)
6  (11)
14) ________ 15) ______
Simplify by performing the operation. Do not use a calculator:
1)
47
2)
 3  (1)
3)
69
1) _______ 2) ______
3) _______ 4) ______
4)
 23
5)
 13  (3)
6)
 6 10
5) _______ 6) ______
7) _______ 8) ______
7)
10)
77
47
8)
 6  (4)
11)
3  (9)
9)
3  11
12)  2  9
9) _______
10) ________ 11) ______
12) ________ 13) ______
13)
12  (2)
14)
 5 10
15)
6 11
14) ________ 15) ______
3
Name ______________________________________Integer Operations: Mult. & Divide
Alg. 1 Intro Packet
Simplify by performing the operation. Do not use a calculator:
1)
 5 2
2)
(7)( 3)
3)
(10)(5)
1) _______ 2) ______
3) _______ 4) ______
5) _______ 6) ______
4)
 312
5)
9 ( 2)
6)
( 1)( 3)
7) _______ 8) ______
9) _______
7)
10)
6(8)
 2 7
8)
11)
(10)( 5)
2(2)
9)
 5 8
12) (11)7
10) ________ 11) ______
12) ________ 13) ______
14) ________ 15) ______
13)
35  (5)
14)
42  (3)
15)  28  7
16) ________ 17) ______
18) ________
16)
19)
 14  2
48
8
17)
9  (3)
18)
 3 1
20)
 10
5
21)
 40
 20
19) ________ 20) ______
21) ________ 22) ______
23) ________ 24) ______
22)
 21  (7)
4
23) 2  ( 2)
24) 100  2
Name ______________________________________________________Alg. 1 Intro Packet
MIXED INTEGER REVIEW
Add, subtract, multiply or divide the following integers. Do not use a calculator:
1)
2  ( 7 )
2)
4  ( 3)
1) ________ 2) _______
3) ________ 4) _______
3)
15  3
4)
(9)( 7)
5) ________ 6) _______
7) ________ 8) _______
5)
 3  (6)
6)
 45
9) ________ 10) ______
11) _______ 12) ______
7)
73
8)
 2  ( 6 )
13) _______ 14) ______
15) _______ 16) ______
17) _______ 18) ______
9)
 4 5
10)
 12  (7)
11)
15  7
12)
11 5
13)
8  (21)
14)
(3)( 9)
15)
 24  4
16)
 9  (13)
17)
(2)( 25)
18)
 50  30
19)
 56
8
20)
32  ( 37)
19) _______ 20) ______
5
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).
6
Name _________________________________________Algebraic Properties
Alg. 1 Intro Packet
Match the expression change with the algebraic property that justifies it:
1)
033
2)
3 6  63
3)
(2  5)8  2(5  8)
4)
1 52  52
5)
1) ____
2) ____
[A] Commutative Property of Addition
3) ____
4) ____
[B] Commutative Property of Multiplication
5) ____
(1  2)  3  1  (2  3)
6) ____
6)
7 8  8 7
7)
5 1  1 5
8)
4(3  7)  (4  3)  7
9)
27  7 2
10)
( xy)  z  z  ( xy)
11)
4(3  7)  4  3  4  7
12)
27  7 2
*13)
[C] Associative Property of Addition
7) ____
8) ____
[D] Associative Property of Multiplication
9) ____
10)____
11) ___
[E] Distributive Property
12)____
13)____
[F] Additive Identity (‘Identity Property of Addition’)
18 x  9  9(2 x  1)
[G] Multiplicative Identity (‘Identity Property of Multiplication’)
Replace the question mark with the missing information:
14) Associative Property of Addition:
6  (7  d )  (? 7)  d
15) Associative Property of Multiplication:
16) Distributive Property:
5  3  (4)  5  ? (4)
14) ______
15) ______
16) ______
7( x  5)  ? x  ? 5
17) ______
17) Additive Identity (Identity Property of Addition):
18) Commutative Property of Addition:
? 0  m
18) ______
jk  mn  mn  ?
7
Name _________________________________________Distributive Property
Alg. 1 Intro Packet
Use the Distributive Property to simplify the following expressions:
1)
5( x  4)
2)  5( x  4)
1) _______________
2) _______________
3)
8( x  2)
3) _______________
4) 8(3x  4)
4) _______________
5)
2 (5  x )
5) _______________
6)  2(5  2b)
6) _______________
7) _______________
7)
4( 2 x  5)
8)  4(2b  5)
8) _______________
Simplify Like Terms:
9)
4 x  2 x  20
10) 11x  7 x  44  6
9) _______________
10) _______________
11) _______________
11)
3m  12  2m
12) 5h  4h  75  3h
12) _______________
13) _______________
14) _______________
13)
8 x  8  12 x  8
14) 21  18x  7
15) _______________
16) _______________
15)
4(b  5)  6
8
16)
 2 y  6( y  4)
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
45  15  25
False
12  4  x
Open
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}
9
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
10
(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
)
Name ___________________________________ One-Step Equations: Addition/Subtraction
Alg. 1 Intro Packet
Solve each equation by isolating the variable. Show all work:
1)
y  7  5
2)
t  8  1
3)
3  r  11
4)
6 p 9
5)
2  x6
6)
1 w7
7)
7  11  n
8)
 4  1  z
9)
3 9 j
Solve each equation by isolating the variable. Show all work:
10)
y  4  5
11)
t  5  1
12)
v  4  11
13)
13  x  7
14)
10  x  8
15)
 10  x  8
16)
p6  9
17)
 2  x 1
18)
5  a  (11)
11
Name _______________________________________ One-Step Equations: Mult./ Division
Alg. 1 Intro Packet
Solve each equation by isolating the variable. Show all work:
1)
3 y  15
2)
4t  16
3)
 6 x  12
4)
7 x  77
5)
 77  7 x
6)
 72  0.5x
7)
5 p  90
8)
 2b  10
9)
10  5w
Solve each equation by isolating the variable. Show all work:
10)
y  2
1
4
x5
13)
1
5
16)
1
8
12
p  6
11)
1
2
t  5
12)
1
3
14)
x
5
5
15)
x
5
18)
1
17)
x  ( 2 )  1
x  11
5
1
4
w
Name _______________________________________ One-Step Equations: Mixed Review 1 Alg. 1 Intro Packet
Solve each equation by isolating the variable. Show all work:
1)
 3 g  15
2)
24  y  61
3)
4)
 7  y  17
5)
c  ( 30)  12
6)
4b  20
 9
8)
t  8  10
9)
48  16 g
n  38  5
11)
4  6  y
12)
6  b  12
1
14)
x  3  3
7)
10)
13)
n
2
n
3
b
6
 5
15) The sum of a number
and 8 is negative 3.
13
Name _______________________________________ One-Step Equations: Mixed Review 2 Alg. 1 Intro Packet
Solve each equation by isolating the variable. Show all work:
 12
18)
20)
4b  60
21)
b  4  4
16  78  y
23)
x  25  44
24)
b 15  9
 30h  150
26)
c  ( 5)  28
27)
29)
3  x  3
16)
 3c  27
17)
19)
x  13  19
22)
25)
28)
1
6
14
x3
b
8
 7r  49
x  (4)  14
30) The difference of
a number and 1 is 3.
Name _______________________________________ Writing & Solving Equations Alg. 1 Intro Packet
A) Write and equation and B) Solve for the missing variable. Show all work:
1) Nine more than a number is four.
2) Ten less than a number is eight.
3) The product of negative five and a number .
is two-hundred fifteen.
4) One-fifth of a number is negative three.
5) A number decreased by sixteen is negative
twenty six.
6) Four less than a number is eight.
7) Four copies of a book cost $44. Find the price
of one book.
8) Jen added $150 to her savings account. Her
balance is now $525. How much was it before?
9) Terri is 60 inches tall. This is 24 inches more
than Kevin’s height. How tall is Kevin?
10) The perimeter of a square is 60 inches. Find
the length of each side.
15
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
3a3 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
1  [4  (17  23)  23 ]
1  [4  40  23 ] 
16
1  [4  40  8] 
1  [4  5] 
1  [1]  0
n  2
Name _________________________________________ Order of Operations, page 1
Alg. 1 Intro Packet
Find the answer by applying the Order of Operations. SHOW EACH STEP! Use example 1 as a guide:
1)
 15  7  2  1
 15  14  1
 29  1
 30
2)
6  2  3  (5)
3)
6  8  8  60
4)
32  8  5  6
5)
 20  5  4 8
6)
24  3  5  3  24
7)
10  12  12  (2)  6
8)
63 28  3
9)
18  3  4  2  8
10)
7  8 28  6
11)
6  (3  5) 2
12)
(14  3  4)  5
13)
(2  4)3  5  3
14)
(23  13)  6  18
15)
(8  6) 4  2  6
17
Name _________________________________________ Order of Operations, page 2
Alg. 1 Intro Packet
Find the answer by applying the Order of Operations. SHOW EACH STEP!
16)
(3  8)  11  23
19)
5  (6  8  18)  04  8
17)
 49  7  8  4(3  2)
18) 3  (5  8)  6  3  8
2
20) (4)( 9)  6  (3  4)
Substitute and simplify each expression
21) 3x
5
when x  2
24) b  5 when b  3
4
22) a  5 when a  5
23) 2  n
25) w  11 when w  5
26) x  y when x  2
2
3
3
when n  5
2
& y6
18