Transcript Pre-Algebra

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
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?