Pre-Algebra
Tools for Algebra and Geometry
Order of Operations
When a numerical expression involves 2 or
more operations, there is a specific order in
which these operations must be performed.
In earlier grades you learned to solve
numerical expressions using “order of
operations” or as most of us say PEMDAS
which stands for
Parenthesis, Exponents,
(Multiplication/Division) then (Add/Subtract)
PEMDAS
Please Excuse (My Dear) (Aunt Sally)
Please Excuse (My Dear)(Aunt Sally)
The reason (multiplication & division) and (add &
subtract) are grouped is when those operations
are next to each other you do the math from left
to right. You do not necessarily do addition first
if it is written next to subtraction.
Variables and Expressions
Variable – a symbol used to represent a quantity
that can change.
Coefficient – the number that is multiplied by the
variable in an algebraic expression.
Numerical expression – an expression that contains
only numbers and operations.
Algebraic expression – an expression that contains
numbers, operations and at least one variable.
Constant – a value that does not change.
Evaluate – To find the value of a numerical or
algebraic expression.
Simplify – perform all possible operations including
combining like terms.
Numerical expression
8–6+2
Subtraction is first
Remember when those operations, (multiplication &
division) and (add & subtract) are grouped next to
each other you do the math from left to right.
When there are 2 or more operations, and we
use grouping symbols such as parenthesis or
brackets, you perform the inner most grouping
symbol first.
2 + 3[5 +(4-1)²]
2 +3[5 + (3)²]
2 + 3[5 + 9]
2 + 3[14]
2 + 42
44
Summary of Basic Steps
• Copy the problem.
• Simplify any grouping symbols (such as
parenthesis) first, starting with the inner most
group.
• Simplify any powers (exponents).
• Perform the multiplication & division in order
from left to right.
• Do the addition & subtraction last, from left to
right.
• Remember, if operations are written next to each
other, work from left to right.
It is very important to understand that it does
make a difference if the order is not
performed correctly!!!!!!!!!
70 – 2(5 + 3)
70 – 2(8)
68(8)
544 incorrect
(subtraction was done before multiplication)
70 – 2(5 + 3)
70 – 2(8)
70 – 16
54 correct
Order of Operations is not an
isolated skill.
This skill applies to almost every topic in Math.
Remember that “Aunt Sally” is used with
evaluating formulas, solving equations, evaluating
algebraic expressions, simplifying monomials &
polynomials, etc…
Order of Operations
Get your pencil and calculator ready and try
these problems.
1) 20 + 3(5 – 1) =
6) 48 / 3 + 5 =
2) 3 + 2²(1 + 8) =
7) 3(6 + 4)(5 – 3) =
3) (5 · 4)² =
8) 100 – 4(7 – 4)³ =
4) 2(3 + 5) – 9 =
9) 1² + 2³ + 3³ =
5) 2[13 – (1 + 6)] =
10) (24 – 6)/2 =
Here is an example of an algebraic expression.
4 is the coefficient.
X is the variable.
7 is the constant.
Algebraic Expression
4x + 7
4 = coefficient
x = variable
7 = constant
Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute a
given number for the variable, and find the value
of the resulting numerical expression.
X – 5 for x = 12
(12) – 5 = 7
2y + 1 for y = 4
2(4) + 1 = 9
6(n + 2) – 4 for n = 5, 6, 7
38, 44, 50
Evaluate each expression for
t = 0, x = 1.5, y = 6, z = 23
1) y + 5
6) 3(4 + y)
2) 3z – 3y
7) 3(6 + t) – 1
3) z – 2x
8) 2(y – 6) + 3
4) xy
9) y(4 + t) - 5
5) 4(y – x)
10) x + y + z
Properties of Addition
and Multiplication
Problem Solving
Commutative Property of Addition/Multiplication
• The order in which numbers are added does not
change the sum.
5+3=3+5
For any numbers a and b
a+b=b+a
• The order in which numbers are multiplied does
not change the product
2·4=4·2
For any numbers a and b
a·b=b·a
Commutative – switching places or
interchanging
• Think of the
commutative
property as
physically
changing places,
they commute or
substitute one
for the other.
Associative Properties of Addition/Multiplication
• The way in which addends are grouped does not
change the sum.
(2 + 4) + 6 = 2 + (4 + 6)
For any numbers a, b, and c.
(a + b) + c = a + (b + c)
• The way in which numbers are grouped does not
change the product.
(6 · 3) · 7 = 6 · (3 · 7)
For an numbers a, b, and c,
(a · b) · c = a · (b · c)
Associative – regroup the parts
• The associative
property can be thought
of as “friendships”
(associations). The
parentheses show the
grouping of two friends.
They don’t physically
move, they simply
change the one with
whom they are
associating.
Identity Properties of Addition/Multiplication
• The sum of a number and zero is the number.
6+0=6
For any number a,
a+0=a
• The product of a number and one is the number.
6·1=6
For any number a,
a·1=a
Identity – “I” remain the same or
“I” keep my identity.
• The identity element
here stays the same,
so if “I” add zero “I”
remain the same. If
“I” multiply by one, “I”
remain the same.
Multiplicative Property of Zero
• The product of a number and zero is zero.
5·0=0
For any number a,
a·0=0
Inverse property of addition/multiplication.
• The sum of a number and its opposite are equal to
zero.
5 + (-5) = 0
For any number a,
a + (-a) = 0
• The product of a number and its multiplicative
inverse equals one.
2·½=1
For any number a,
a · 1/a = 1
Inverse – what brings you back to
the identity element using that
operation?
• Think of the inverse
property as what
would you need to add
(multiply) to this
number to turn it into
an identity element?
The additive inverse
is the negative of the
number, and the
multiplicative inverse
is one divided by the
number.
Distributive Property
The sum of 2 addends (b + c) multiplied by a number
(a) is the sum of the product of each addend and
the number.
3(4 + 5) = 3(4) + 3(5)
For any number a, b, and c,
a(b + c) = ab + ac
or
(b + c)a = ab + bc
The expression a(b + c) is read “a times the quantity b plus c”
or
“a times the sum of b and c”
Distributive – multiply across the
parentheses.
• Using the distributive
property lets you multiply
each element inside the
parentheses by the element
outside the parentheses.
Consider the problem to
the left. The number in
front of the parentheses is
“looking” to distribute
(multiply) its value with all
of the terms inside the
parentheses.
Properties of Real Numbers
Properties of Real Numbers
Property
Example
1
Commutative Property of Addition
a+b=b+a
2+3=3+2
2
Commutative Property of Multiplication
a·b=b·a
2 · (3) = 3 · (2)
3
Associative Property of Addition
a + (b + c) = (a + b) + c
2 + (3 + 4) = 2 + (3 + 4)
4
Associative Property of Multiplication
a · (b · c) = (a · b) · c
2 · (3 · 4) = (2 · 3) · 4
5
Distributive Property
a · (b · c) = a · b + a · c
2 · (3 + 4) = 2 · 3 + 2 · 4
6
Identity Property of Addition
a+0=a
3+0=3
7
Identity Property of Multiplication
a·1=a
3·1=3
8
Additive Inverse Property
a + (-a) = 0
3 + (-3) = 0
9
Multiplicative Inverse Property
a · (1/a) = 1
3 · (1/3) = 1
10
Property of Zero
a·0=0
5·0=0
Variables and
Expressions
As people age, their blood
pressure rises. You can
approximate a person’s normal
systolic blood pressure by
dividing his/her age by 2 and
then adding 110.
How would you write this
problem in expression form to
solve mathematically.
The Language of Algebra
Algebra, like any language, is a language of symbols.
It is the language of math and must be learned as
any other language. You know the symbols of
division and addition, so you can write the bloodpressure relationship as:
age ÷ 2 + 110
In arithmetic, you could write:
□ ÷ 2 + 110
In algebra, we use variables, letters that represent
unknown values. In this case the letter x:
X ÷ 2 + 110
This is known as a algebraic expression.
Expressions like a ÷ 2 + 110 can be evaluated
by replacing the variables with numbers and
then finding the numerical value of the
expression.
If Samantha is 18 years old, she could estimate her
blood pressure by evaluating the expression,
18 ÷ 2 + 110
a ÷ 2 + 110 =
(18) ÷ 2 + 110 = substitute 18 for a
9 + 110 = order of operations, division first
119
When reading a verbal sentence and writing an
algebraic expression to represent it, there are
words and phrases that suggest the operations
to use.
Addition
Plus
Sum
More than
Increased by
Total
In all
Subtraction
Minus
Difference
Less than
Subtract
Decreased by
Multiplication
Times
Product
Multiplied
Each
Of
Division
Divided
quotent
Translating Word Phrases into
Math Expressions
While the table on the previous slide gives you an
idea about phrases that translate to math
operations, being able to identify the key words
that determine the operations (+, -, ·, ÷) that will
be used to solve problems takes practice.
Write an expression for each phrase.
1)
2)
3)
4)
5)
6)
7)
8)
a number n divided by 5
the sum of 4 and a number y
3 times the sum of a number b and 5
the product of a number n and 9
the sum of 11 times a number s and 3
7 minus the product of 2 and a number x
6 less than a number x
7 times the sum of x and 6
Write an algebraic expression to evaluate the
word problem:
1)
2)
Samantha purchased a 200-minute calling card
and called her father from college. After
talking with him for t minutes, how many
minutes did she have left on her card? Write
and solve an expression to represent the
number of minutes remaining on the calling card.
Jared worked for h hours at $5 per hour.
Write an expression to determine how much
money Jared earned. How much money will
Jared earn if he works a total of 18 hours?
Evaluating Expressions
Get your pencil and calculator ready and try
these problems.
Evaluate each expression if
m = 4, n = 11, p = 2, q = 5.
1) 8 + n
2) 10 – q
3) 3m
4) 24 – 4q
5) 5m ÷ 2
6)
7)
8)
9)
10)
n + 110 – 2p
mn + p
10m + m
2m – p
m+6÷p
Combining Like Terms
Using the language of algebra
Combining Like Terms
Have you ever heard the phrase “You are trying to
compare apples to oranges”.
Explain what you think this phrase means.
Apples and oranges cannot be compared because
they are unlike objects.
Term – the parts of an expression that are
added or subtracted.
(x + 2) (2x – 4)
Like terms – 2 or more terms that have the
same variable raised to the same power.
(in the expression 3a + 5b + 12a, 3a and 12a
are like terms.)
To simplify an expression – perform all
possible operations, including combining like
terms.
Add or Multiply?
x + x
x
x
1x + 1x = 2x
x
x
Add or Multiply?
x + y
x
1x + 1y = x + y
y
x
y
A procedure frequently used in algebra is the
process of combining like terms. This is a way
to “clean-up” an equation and make it easier to
solve. For example, in the algebraic
expression 4x + 3 + 7y, there are three terms:
4x, 3, and 7y.
Remember the 4 and 7 are coefficients.
Let’s say we are given the equation below. It looks
very complicated, but if we look carefully,
everything is either a constant (number), or the
variable x with a coefficient (4x).
Remember, a coefficient is the number by which a variable
is being multiplied (the 4 in 4x is the coefficient)
The “like terms” in the equation are ones that
have the same variable. All constants are like
terms as well.
This means 15, 10, 6, and -2 are all like terms,
and the other is 4x, -3x, 5x, and 3x. To
combine them is pretty easy, you just add
them together and make sure they are all on
the same side of the equation.
Since the 15 and 10 are both constants we
combine them to get 25. The 4x and -3x each
have the same variable (x), so we can add them
to get 1x. Doing the same on the other side
we arrive at 25 + 1x = 4 + 8x. The process is
still not finished.
There are still some like terms, but they are
on opposite sides of the equal sign. Since we
can do the same thing to both sides we just
subtract 4 from each side and subtract 1x
from each side.
What remains is 21 = 7x.
Now it’s just a simple process of dividing by
seven on each side and we arrive at our answer
of x = 3.
Combining like terms enables you to take that
huge mess of an equation and make it
something much more obvious to solve.
Combine the following:
1)
14a – 5a
2)
7x – 3x
3)
12g + 7g
4)
7y + 8 – 3y – 1 + y
5)
5t + 7p – 3p -2t
Simplify Algebraic Expressions by
combining like terms.
Simplify:
6(n + 5) – 2n =
6 (n) + 6(5) – 2n = Distributive Property
6n + 30 – 2n = 6n and 2n are like terms
4n + 30 Combine coefficients 6 – 2 = 4
Remember that a term like “x” has a coefficient of
1, so terms such as x, n, or y can be written as 1x,
1n, or 1y.
Example:
2a + 5b + 5 – a + 3
How many terms are in this expression?
What are the like terms?
Simplify by combining like terms.
a + 5b + 8
Example:
2 + 8(3y + 5) - y
What would be the first step in simplifying the
expression?
Use the Distributive Property to simplify 8(3y + 5),
8(3y) + 8(5),
24y + 40
Combine like terms.
2 + 24y + 40 –y
42 + 23y
Math Humor
Teacher: Sam, please
evaluate x + 4 if x = 7.
Sam: Is this a trick
question? Yesterday
you told us that x = 2!