Key takeaway: The order of operations is a shared rule book for which calculation to do first, so that \(30 – 6 \times 4\) and every other expression has one correct value, remembered as PEMDAS.
Grade 66.EE.A.2cNumbers and Operations
Definition of Order of Operations
The order of operations exists because a string of numbers and symbols can be read in more than one way. Without a rule, \(30 – 6 \times 4\) might be 96 to one person and 6 to another, and a calculator, a spreadsheet and a student would disagree.
The memory aid PEMDAS lists the stages: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Multiplication and division share one level, as do addition and subtraction, so inside a level you work from left to right. Grouping symbols such as brackets and fraction bars count as parentheses. Everyone who follows the same order of operations lands on the same number.
Notation
The four stages, from first to last:
- P: parentheses and other grouping symbols, innermost pair first
- E: exponents
- MD: multiplication and division as they appear, left to right
- AS: addition and subtraction as they appear, left to right
The expression \(5 + 6 \times (9 – 3)^2 : 4\) uses all four stages.
Key facts
- Multiplication does not always come before division; whichever appears first from the left goes first.
- Addition and subtraction share a level in the same way.
- A fraction bar groups the top and the bottom, so simplify each before dividing.
- Parentheses can override the usual order of operations and force an addition to come first.
Order of Operations example
Evaluate \(5 + 6 \times (9 – 3)^2 : 4\).
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Parentheses: \(9 – 3 = 6\), giving \(5 + 6 \times 6^2 : 4\).
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Exponent: \(6^2 = 36\), giving \(5 + 6 \times 36 : 4\).
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Multiplication and division from the left: \(6 \times 36 = 216\), then \(216 : 4 = 54\).
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Addition: \(5 + 54 = 59\).
The value of the expression is 59.
Common mistake
Watch out
Wen evaluates \(30 – 6 \times 4\) from left to right: \(24 \times 4 = 96\).
Why it is wrong
Multiplication outranks subtraction. First \(6 \times 4 = 24\), then \(30 – 24 = 6\). Going straight left to right is only correct inside a single level, such as a string of additions and subtractions.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations