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Math101learn.math101.caOrder of Operations
Order of operations is a convention for reading an unparenthesized expression consistently.
The convention gives expressions a shared, unambiguous meaning and underlies substitution, formulas, calculators, and programming. Understanding hierarchy is more dependable than chanting an acronym.
Intuition and core definition
Order of operations is a convention for reading an unparenthesized expression consistently. Grouping symbols come first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not always precede division, and addition does not always precede subtraction; each pair shares a level.
Notation, language, and conditions
Parentheses, brackets, fraction bars, and radical bars create groups. A fraction bar groups the entire numerator and denominator. Juxtaposition such as $3x$ indicates multiplication. The acronym BEDMAS or PEMDAS is a memory aid, but the left-to-right rule within equal-priority operations is part of the convention.
Why this idea matters
Operation conventions make a written expression unambiguous, with grouping and exponent structure taking priority over left-to-right operations at the same level.
A dependable method
- Mark every explicit or implied group, including fraction numerators and denominators.
- Evaluate innermost groups and exponents.
- Perform multiplication and division in the order they appear from left to right.
- Perform addition and subtraction in the order they appear from left to right.
- Rewrite one stage per line and estimate to catch implausible results.
Worked example
Representations and interpretation
An expression tree makes hierarchy visible: low branches contain grouped subexpressions, and the root contains the last operation. In the example, the final root is addition combining two independently simplified branches.
Reasoning about variations
Compare $24\div6\cdot2=8$ with $24\div(6\cdot2)=2$. Parentheses create a different expression. A horizontal fraction $\frac{24}{6\cdot2}$ already groups the denominator and therefore also equals $2$.
Common mistakes
How to check your work
- Use a calculator with explicit parentheses only after predicting the structure.
- Draw or describe the expression tree and confirm the last operation.
- Estimate magnitude and evaluate a second time by grouping already-computed subexpressions.
Practice
- Evaluate $20-12\div3\cdot2$.
- Evaluate $3+2(5-1)^2$.
- Evaluate $16\div(2+2)$.
Answers and brief solutions
Show answers
- $12$ $12\div3=4$, then $4\cdot2=8$, and $20-8=12$.
- $35$ The group gives $4$, its square is $16$, then $2\cdot16+3=35$.
- $4$ Parentheses give $4$, so $16\div4=4$.
Synthesis and transfer
Compare a spreadsheet formula with and without parentheses around a subtotal; evaluating both exposes how grouping can change a financial calculation even when every numeral is identical.
An expression tree makes the two spreadsheet formulas visibly different: one root operation may add a fee after multiplication, while another multiplies the entire subtotal. Parentheses alter the tree by declaring which calculation produces an input for the next. Multiplication and division share one priority level and are processed left to right, as do addition and subtraction; the mnemonic should not incorrectly rank one over its partner. Evaluating a small numerical case can expose a misplaced group, but the formula's intended meaning decides which structure is right. In financial models, that distinction may determine whether a tax applies to one item, a subtotal, or a subtotal plus service charge.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Evaluate $20-12\div3\cdot2$.
- $12\div3=4$, then $4\cdot2=8$, and $20-8=12$.
End of lesson
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