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Order of Operations NOTES

[Algebra Map]  [Exponents]   [Expressions]  [Simplifying]  [Rules]  [Translations]

  Algebra Map

In the concept map below, various aspects of algebra are found, as well as how the aspects relate to different parts of "language" such as nouns, verbs, and pronouns.

Algebra concept map



An exponent is "shorthand" for the repeated multiplication of the same factor (base).  Once the pattern is learned, the writing out "longhand" of the exponent is called expanded notation.

b=base; e=exponent; n=number; the base is raised to the exponent value and multiplies out to a number

Said: Three raised to the fourth power for 34



In algebra, recognition of and use of patterns is very important.  In arithmetic, very short numerical expressions were presented and you were asked to "solve" or "find the answer".  Now, with algebra, the expressions can be longer and more complex, as well as contain not only numbers and operators, but variables (letters) inside the expression.  Numerical expressions are made of numbers and operators.  Algebraic expressions are made of numbers and variables with operators.

For example:
Numerical expression:  2 + 3(6) - 2
(just numbers and operators)

Algebraic expressions:  2 + x,  or  5y,  or  2x - 1 + z
(have variables involved)



Simplifying an expression basically is lessening (getting rid of) the operators as much as possible.  When asked to "simplify" a given expression, first it should be determined what type of expression (numerical or algebraic) is given.  If it is numerical, then the expression's simplification will follow the "rules" (see Order of Operation below) and  result in one number.  If the expression is algebraic, the simplification still follows the order of operations, but only with "like terms"... called "combining like terms". The simplification will have variable(s), a number and usually at least one operator in it.

  Order of Operation (the "rules")

Basically "4" steps:

  1. Do all operations within grouping symbols (parentheses, brackets)
  2. Evaluate the number value for any exponents
  3. Multiply or divide in order from left to right
  4. Add or subtract in order from left to right

A "visual" representation of these rules:

Visual representation of the order of operations: work what's listed from top to bottom.



Different Ways to Show Multiplication:

"X"  (times sign) 3 x 5
" .  " (raised dot) .  5
* (asterisk, used with computers) 3*5
(#)(#) parentheses (3)(5)
# next to letter 3x = (3)(x)
letter next to letter xyz = x (y)(z)
exponents 23 = 2 x 2 x 2

Different Ways to Show Division:

Division symbol (Division symbol: a horizontal line with a dot above and below) 2 Division symbol: a horizontal line with a dot above and below 5
Slash symbol ( / ) 2/5
Fraction bar 2


  • Division by zero is "undefined"
  • Zero divided by another number is zero

Translating scenarios into algebraic statements is important to be able to do when solving questions with algebra.  Recognizing patterns and being able to convert them into algebraic terms is one of the primary goals of learning algebra.  The following are some words commonly used for the four common operators of addition, subtraction, multiplication, and division.

Common words used in English:

Addition Subtraction Multiplication Division
Added to
More*, More than*
Increased by
Total, totaling
Subtracted from
Less than *
Decreased by
Twice, Double ( x 2)
Triple ( x 3 )
Quotient of
Divided by

*see "Tricky Translations" below

"Bump Ahead" road signTricky Translations

Translating English phrases into algebraic phrases is an important part of learning to use algebra as a tool to find solutions to everyday questions.  In English, there are sometimes several ways to express a situation.  In algebra, there is not multiple ways of expressing a situation and translating "English" to "algebraic terms" needs to be done with care.

  • Less... Less than... Is Less Than

Be cautious when you are translating to be exact, especially with the word phrases: "Less, Less than, and Is less than." Read the following English phrase and notice how the algebraic phrase is written.

Five less a number is two. 5 - n = 2
Five less than a number is two. n - 5 = 2
Five is less than a number. 5 < n


  • More...More than...Is More Than

Along with the subtraction phrases, care should also be taken when translating the word phrases: "More, More than, and Is more than." However, because of the commutative property of addition, it isn't as critical to get the order of the terms exact because the solution would yield identical results upon evaluation.

A number more five is seven. n + 5 = 7
A number more than five is seven. 5 + n = 7
A number is more than seven. n > 7