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Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Unit 1:
Operations
Unit 1: Operations - Lesson Topics:
Lesson 1: Integer operations, comparing and ordering (Text 1.3, 1.5, 1.6)
Lesson 2: Using rational numbers and absolute value (Text 1.2, pg.31)
Lesson 3: Properties (Text 1.4, 1.7)
Lesson 4: Irrational Numbers and Square Roots (Text 1.3, 10.2, 10.3)
Lesson 5: Order of Operations (with radicals, exponents and variables) (Text 1.2)
Lesson 6: Greatest Common Factor and Least Common Multiple (Text 8.2, 11.4)
Lesson 7 – Combine Like Terms (Text 1.7)
Lesson 8 – Distributive Property with Negatives (Text 1.3, 10.2, 10.3)
1
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 1: Integer operations, comparing and ordering
Objective:
(Text 1.3, 1.5, 1.6)
to add, subtract, multiply and divide real numbers
Vocabulary Reminders:
Opposites – two numbers that are the same distance from zero on the number line – also
called additive inverses – the sum of opposites is zero
Integers – whole numbers and their opposites
The Order of Operations always applies!
P
______________
E
______________
M/D
______________ or ______________ in order from left to right
A/S
______________ or ______________ in order from left to right
Addition Rules If signs are the same: add their absolute values; _________ the sign.
4+6=
-4 + (-6) =
If signs are different: subtract the smaller absolute value from the larger; keep the sign of the
______________ absolute value.
-4 + 6 = ____
4 + (-6) = ____
Subtraction Rules To subtract a number, add its opposite.
10 – 4 =
10 – (-4) =
10 + ____ = ____
10 + ____ = ____
4–6=
4 + ____ = ____
Multiplication/Division Rules If signs are the same: multiply/divide their absolute values; the sign is ______________.
4 • 6 = ____
-4 • (-6) = ____
24 ÷ 6 = ____
-24 ÷ (-6) = ____
If signs are different: multiply/divide their absolute values; the sign is ______________.
-4 • 6 = ____
4 • (-6) = ____
-24 ÷ 6 = ____
24 ÷ (-6) = ____
Multiplication Rules
(Pos)(Pos) = pos
(Neg)(Pos) = neg
(Pos)(Neg) = neg
(Neg)(Neg) = pos
Division Rules
Pos
=
Pos
pos
Neg
Pos
=
Pos
Neg
Neg
Neg
=
=
neg
2
neg
pos
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Operations with Fractions Review:
adding/subtracting fractions – 1) find common denominator 2) add/subtract numerators 3) keep the
denominator the same 4) simplify if needed
multiplying fractions/mixed numerals – 1) if mixed numeral, change to improper fraction 2) multiply
numerators 3) multiply denominators 4) simplify if needed
dividing fractions/mixed numerals – 1) if mixed numeral, change to improper fraction 2) change operation
to multiply AND use the reciprocal of the second fraction (flip it) 3) multiply numerators 4) multiply
denominators 5) simplify if needed
Class Practice:
1)
1  4

5)
 

7  7
 3
  
4  8
3
2)
6)
9) 7.08 + (-2.47) = ____
12) –[37 + (-15)] + (-8 + 14)
____ + ____ PEMDAS
____

 2

9  9
7
3)
 
1
2
7)
 3  6 
  
 10  8 

1
4
 3  6
   
 10    8 
10) -25.31 + 75.07 = ____
4)

7
5 10
1

8)  3  10 
 6  2 
 5  9 
11) -0.23 + (-0.51) = ____
13) 0.7 + (-2.4) + (-0.12) + 3.86 + (-0.59)
____ + ____
add all pos. and all neg.
____
3
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Evaluate for w = –6.7, x = 11.5, y = –4.9, z = 15.2,
F) [-x – (-z)] – z
G) -y – [w + (-z)]
[-(_____) – -(_____)] – ______
-(______) – [(______) + -(_____)]
[______ + (______)] – ______
(______) – [______]
______ – ______
______ + [______]
______
______
H)
K)
1  1
 5  3    
2  2
Evaluate
I)
J)
9(–7)
x
 2 yz for x = –20, y = 6 and z = –1
4
Evaluate  If a = -8 , b = -5 , c = -2 , and d = ½
 4c  8d 
d c
M)
N)
4
L)
–32 ÷ 8
 3  1 

 14    27 
 7  9 

Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Write an algebraic expression.
5
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
HW: p.7 #10-66 even
6
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 2: Using rational numbers and absolute value
(Text 1.2, pg.31)
Objective: To evaluate algebraic expressions for given value(s) or the variables and to write algebraic
expressions for word phrases.
To graph real numbers on a number line and to classify numbers in subsets
To compare and order real numbers and to use the concepts of opposites and absolute value.
Vocabulary Reminders:
Variable: A symbol used to represent one or more numbers, any letter but i may be used
Variable expression: An expression that contains one or more variables
Evaluate: Substitute a given number for each variable
Numerical expression: One or more #s connected by the following operations: +, -, x, 
Algebraic expression: Variable and numerical expression
Term – a number, variable, or the product of numbers and variables – a part of a variable
expression ex. n
6x – y
8x2 + 3x – 4
(1, 2, & 3 terms respectively)
Equations – a mathematical sentence that shows that two expressions have the same value
ex. – n = 5
6x – y = 7y
8x2 + 3x – 4 = 0
Simplify: Replace an expression with its simplest name
Reciprocal – the multiplicative inverse – for any nonzero a/b, the reciprocal is b/a – the
product of any nonzero number and its reciprocal is one (1) – zero does not have a reciprocal
Other Reminders:
words meaning addition: plus, increased, sum, more than
words meaning subtraction: minus, decreased, difference, less than
words meaning multiplication: times, product, of
words meaning division: divided, quotient, ratio
symbols meaning multiplication: 3  4, 3  n, 3  n, 3n, (3)n, 3(n), (3)(n)
symbols meaning division: 9  n, n 9 , 9/n
Examples:
Write an algebraic expression for each of the following word phrases.
1. The sum of 7 and some number
2. The difference between 18 and y
3
3. The product of 8 and some number
7
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Absolute Value
– the ____________ a number is from zero on the _________________
– the absolute value of a number n is written _____
|-4| = |4| because both 4 and -4 are ___ units away from the origin, 0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Practice:
1) |-7| = _____
4) |-7+9| = _____
2) |32| = _____
5) |5 ˗ 8| = _____
6) |5+(-5)| = _____
3) |5 ˗ 3| = _____
7) 4 ˗ |2 ˗ 4| = _____
8) How would you write "the absolute value of -5" using math symbols? _____
9) How would you write "The absolute value of 32 less 45" using math symbols?
___________________
Practice:
Simplify the following.
  3 
    
1.   4  =
2.  6  10
3. 5   4
8
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Comparing and Ordering Numbers:
Relationship between two numbers:

less than or equal to

5
5
_____ 
7
8
10)
Compare. 
11)
Compare.  

less than
5
8

5 3
3
_____  
7 8
7

12)
 2 
2 
1

1
 


Compare.
_____
 9 
9

13)

Order the following from least to greatest.
3 3 3
 , ,
5 4 8

HW: p.13 #10-50 even
9

not equal to
greater than
greater than or equal to
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 3: Properties
(Text 1.4, 1.7)
Objective: To identify properties for addition and multiplication
To simplify expressions using the properties
Properties of Real Numbers: Statements that are true for all real numbers
Identity Property for Addition:
a+0=a
3+0=3
0+a=a
0+3=3
Identity Property for Multiplication:
a*1=a
5*1=5
1*a=a
1*5=5
Zero Property for Multiplication:
a*0=0
5*0=0
0*a=0
0*5=0
Property of Negative One for Multiplication:
a * -1 = -a
5 * -1 = -5
Inverse Property for Addition:
a + (-a) = 0
7 + (-7) = 0
(-a) + a = 0
(-7) + 7 = 0
1
1
a
1
8 1
8
1
a 1
a
1
8  1
8
a
Inverse Property for Multiplication:
Commutative Property for Addition:
a+b=b+a
12 + 7 = 7 + 12
Commutative Property for Multiplication:
a*b=b*a
8*9=9*8
Associative Property for Addition:
Properties of Equality
Reflexive Property:
Symmetric Property:
Transitive Property:
Substitution Property:
a0
(a + b) + c = a + (b + c)
(-3 + 4) + 6 = -3 + (4 + 6)
Associative Property for Multiplication:
Distributive Property:
-1 * -a = a
-1 * -5 = 5
(a * b) * c = a * (b * c)
(2 * 10) * 12 = 2 * (10 * 12)
a(b + c) = ab + ac
7(4 + x) = 28 + 7x
a(b – c) = ab – ac
6(y – 3) = 6y - 18
a=a
6y = 6y
a = b, then b = a
24 = 6y, then 6y = 24
a = b, and b = c, then a = c
m = 6y, and 6y = 24, then m = 24
a = b, then a may replace b, or b may replace a in any statement
HW: p. 26 #9-45 mult. of 3; p.50 #21-24, 33-35
10
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 4: Irrational Numbers and Square Roots (Text 1.3, 10.2, 10.3)
Sets of Numbers: (Text 1.3) Objective: To find and use square roots.
Natural numbers: (Counting numbers)
Whole numbers:
Integers:
m
Rational numbers: A number that can be written in the form n ( a fraction)
when m and n are integers and n  0 (includes all
terminating and repeating decimals)
m
Irrational numbers: Cannot be written as n , when m and n are integers and
n  0 (includes all decimals that go on forever without a pattern)
Real numbers: The set of all rational and irrational numbers
Positive numbers: Numbers to the right of zero on a number line
Negative numbers: Numbers to the left of zero on a number line
Origin: The zero point on a number line or (0,0) in the coordinate plane
Real Numbers
Irrational
0.25
Integers
 5
Whole
17
-8
2
9
0

77
5.8761432…
Natural
42
0
11

½
1
3
Rational
Examples:
Complete the chart.
Natural
15
Whole
Integer
Rational
5
8
 5
HW: p. 26 #9-45 mult. of 3; p.50 #21-24, 33-35
3
11
Irrational
Real
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Objective: To find and use square roots.
Vocabulary:
16
Square root - If a2 = b, then ____ is a square root of ____.
radical symbol ↑ ↑ radicand
- radical symbol – indicates ___________ ___________
16 = ______
principal (positive) square root -

negative square root -  16 = ______

± - “plus or minus” - indicates ___________ the square roots -
16 = _______

perfect squares - the squares of ___________ - ex. (-2)2 = 4 or 112 = 121

Class Practice:
Simplify each expression a.
64 =
b.
d.
 0 =
e.
g.
 36 =
h.
 100 =
 16
9
=
16

c.
=
f.
49 =
 121 =
i.
1
=
25
Vocabulary Reminder:
Rational numbers – a number that can be written as a ___________ of two integers – as a decimal the digits
would ___________ or ___________
Irrational numbers - a number that can NOT be written as a ratio of two integers – as a decimal the digits
would NOT terminate or repeat
Class Practice:
Classify each as rational or irrational.
8
 225
 105 ?
Between what two consecutive integers is 12.34 ?
HW: p.20 #10-56 even
1
4
 75

83

12
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 4, Day 2 - Irrational Numbers and Square Roots (Text 1.3, 10.2, 10.3)
Objectives: To simplify radicals involving products and quotients; to solve problems involving radicals;
to rationalize the denominator of a fraction with radicals.
a  b  ab
Multiplying Square Roots
Simplify the following.
1.
25  4
2.
3  27
3.
8 8
4.
13  52
5.
2 2 3 2
6.
3 5 
7.
8  18
8.
3 8 2 2
9.
4 2 
A radical expression is in simplest form if
13.
5 300
2
ab  a  b
Factoring Square Roots
Simplify.
54
10.
2
* there are no perfect squares left under the radical
* there are no fractions left under the radical
* the denominator does not contain a radical
11.
20
12.
75
14.
192
15.
7 20
16.
x2
17.
x10
18.
x7
19.
8x 2
20.
16a 3
21.
x2 y5
HW: p.
13
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
a

b
Dividing Square Roots
a
b
or
a

b
a
b
22.
100
4
23.
4
9
24.
25
b4
25.
144
9
26.
96
12
27.
24
8
28.
25c3
b2
29.
11
49
30.
18
121
32.
27 x 2
144
33.
48n 6
31.
6n 3
125 x5
5 x3
When you have a square root in the denominator of a fraction that is not a perfect square, you should
rationalize the denominator. To rationalize the denominator, make the denominator a rational number
without changing the value of the expression. (square roots that can not be simplified are irrational)
Simplify.
1.
2
5
2.
3
3
3.
HW: p. 610 #12-68 mult. of 4
14
14
7
4.
9
10
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 4, Day 3 - Irrational Numbers and Square Roots (Text 1.3, 10.2, 10.3)
Objectives:
To simplify radicals involving addition and subtraction;
To solve problems involving sums and differences of radicals.
Simplifying Sums and Differences
You can simplify radical expressions by combining like terms. In expressions that contain radicals, the like
terms must have the same radical part.
like terms (radicals)
4 7
and
unlike terms (radicals)
 12 7
3 11
and
2 5
Identify the following pairs as like or unlike.
8 and 2 8
2 7 and  3 7
2 3 and 3 2
Simplify.
1.
4 x  12 x
2.
4 7  12 7
3.
4 7  12 6
4.
2 5 5
5.
3 45  2 5
6.
3 3  2 12
Multiplication with Addition/Subtraction. Use the distributive property.


7.
3 2 3
10.
3 5 2 2 6

8.


2 2 3
11.

3 2
HW: p.616 #9-25 -- or -- #10-28 even, 45-49
15
9.

24  2 6

simplify

2 1  2 10
24 first

Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 5: Order of Operations
(Text 1.2)
Objective: To simplify numerical and algebraic expressions by using the rules for order of operations
Order of Operations:
(with radicals, exponents and variables)
Parentheses
Exponents
Multiplication/Division*
Addition/Subtraction**
* Multiplication and Division have equal importance – do the operation that comes first
~ Work From Left to Right.
**Addition and Subtraction have equal importance – do the operation that comes first
~ Work From Left to Right.
 16 
8.    4(5)
2
7. 5 + 3(2)
9. 44(5) + 3(11)
3
3
10. 17(2)  42
 20 
11.    10(3)2
 5
 27  12 
12. 

 83 
13. (4(5))3
14. 25  42  22
 3(6) 
15. 

 17  5 
4
Evaluate each expression for s = 2 and t = 5.
19.
s4
4
 17
20. 3(t)3 + 10
22. –4(s) + t  5
2
3
 s2
23.  2 
 5t 
21. s3 + t2
2
 3s(3) 
24. 

 11  5(t ) 
HW: p. 13 #8-56 mult. of 4
16
2
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 6: Greatest Common Factor and Least Common Multiple (Text 8.2, 11.4)
Objective: To identify the GCF for a set of numbers (including variables)
To identify the LCM for a set of numbers (including variables)
Greatest Common Factor (GCF)
Least Common Multiple (LCM)
Find the GCF of 32 and 24.
Method 1 – “Rainbow Method”
List all factors of 32 and 24.
32 – 1, 2, 4, 8, 16, 32
Method 2 – Prime Factorization
List the prime factors of 32 and 24.
32 – 25
24 – 1, 2, 3, 4, 6, 8, 12, 24
24 – 23·3
Common factors: 1, 2, 4, 8
common prime factor is 2
GCF = 8
lesser power of that prime factor is 23
GCF = 23 = 8
Method 3 – Ladder Method
2
32
24
Is there a common factor?
2
16
12
yes
2
8
6
yes
4
3
no
↑
GCF = 2·2·2 = 8
for LCM, “use the “L”
LCM = 2·2·2·4·3 = 96
Find the GCF of 36m3 and 45m8.
Method 1
List all factors of 36m3 and 45m8.
36m3 – 1, 2, 3, 4, 6, 9, 18, 36 · m· m· m
Method 2
List the prime factors of 36m3 and 45m8.
36m3 – 22·32· m3
45m8 – 1, 3, 5, 9, 15, 45 · m· m· m· m· m· m· m· m 45m8 – 32·5· m8
Common factors: 1, 3, 9
GCF = 9m3
· m· m· m
common prime factor is 3 and m
lesser power of that prime factor is 32 and m3
GCF = 32 · m3 = 9m3
17
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Find the GCF of 36m3 and 45m8 using the Ladder method.
3
36m3 45m8
Is there a common factor?
3
12 m3 15 m8
yes
m
4 m3
5 m8
yes
m
4 m2
5 m7
yes
m
4m
5 m6
yes
4
5 m5
no
↑
GCF = 3·3·m·m·m = 9m3
for LCM, “use the “L”
LCM = 3·3·m·m·m·4· 5m5 = 180m8
Practice:
Find the GCF.
1. 60x4 and 17x2 _______________
2. 32y12 and 36y8 _______________
3. 16n3 , 28n2 and 32n5 _______________
4. 16m10 , 18m and 30m3 _______________
Find the LCM.
5. 60x4 and 17x2 _______________
6. 32y12 and 36y8 _______________
7. 16n3 , 28n2 and 32n5 _______________
8. 16m10 , 18m and 30m3 _______________
HW: p.482 #5-7, 15-20; p.675 #17-20 (LCM of denominators)
18
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Lesson 8 – Distributive Property
Objective:
(Text 1.7)
to use the distribute property to simplify expressions
Distributive Property:
For all real numbers a, b, and c:
a(b + c) = ab + ac
a(b – c) = ab – ac
(b + c)a = ba + ca
(b – c)a = ba – ca
Class Practice:
Simplify each expression.
1)
2x  3
2)
 x  4 2
3)
47 x  3
4)
2(n  6)
5)
 (7 x  2)
6)
 5(4 x  7)
7)
12 x  6 10  x 
8)
x 2  ( x 2  x)
10)
1
 x  2
2
11)
13)
15a  10
5
14)

9)
12) 1  {a  [a  (a  3)]}
18 x  12
3
15)
8n  6
4
The distributive property can help us with mental math. For example, I want to figure out how much I will
spend to buy four packages of candy for Halloween when each bag is $6.95, but I don’t have a calculator. A
quick way to do that mentally is to use the distributive property….
Coefficient – the number directly in front of the variable
Term – a number, variable, or the product of numbers and variables – a part of a variable expression
separated by + or – signs
Like terms – terms with exactly the same variable factors – same variable to the same power
19
Algebra 1
Mrs. Bondi
Unit 1 Notes: Operations
Class Practice:
Simplify each expression.
1)
2b  2  b  4
2)
 5  c  4  3c
4)
5a  2b  6  5b  9a
5)
4a  3  2 y  5a  7  4 y
7)
3x  4x  6  2
3)
3x  5 y  7 x  2 y
Solve each equation.
6)
b  5b  42
9)
Find three consecutive integers whose sum is -318.
10)
Three friends decide to order a box of egg rolls. Joe eats five egg rolls, Wanda eats two egg rolls,
and Jim eats one egg roll. There is a $1 delivery charge for any order. If the total cost is $7, how
much does each egg roll cost before the delivery charge is applied? Write an equation and solve.
HW: p.50 #18-40 even, 59-64, 69, 76-90 even
20
8)
7  4m  2m  1