Operations & Algebraic Thinking

Order of Operations

Evaluating expressions with parentheses and brackets.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Why Everyone Needs to Agree on an Order

Try this before reading on: what is 3 + 4 × 5? If you worked left to right, you got 35. If you multiplied first, you got 23. Both paths follow a reasonable-sounding rule, and they give two completely different answers to the exact same expression. What do you think that means for how math actually gets written down?

It means an expression like 3 + 4 × 5 is genuinely ambiguous without a shared agreement about which operation to do first. Mathematicians settled that ambiguity once, for everyone, with a fixed order of operations — so the same expression always means exactly one thing, no matter who's reading it.

Definition — Order of Operations

The agreed-upon sequence for evaluating an expression: first, anything inside parentheses or brackets; next, exponents; then multiplication and division, worked left to right; and finally addition and subtraction, worked left to right.

Why does multiplication happen before addition, rather than the other way around, or just left to right? Think of 4 × 5 as naming a single quantity — 4 groups of 5 — the same way a numeral like '20' names one quantity. Evaluating multiplication first turns 3 + 4 × 5 into 3 + 20, a simple addition of two quantities, which is exactly what the expression is actually describing.

Worked Example — Evaluating Without Grouping Symbols

Evaluate 3 + 4 × 5. Multiplication comes before addition, so first find 4 × 5 = 20. Then add: 3 + 20 = 23. Checking against the correct order confirms 23, not 35.

Worked Example — Using Parentheses to Change the Order

Evaluate (3 + 4) × 5. The parentheses override the usual order, forcing the addition to happen first: 3 + 4 = 7. Then multiply: 7 × 5 = 35. Notice this is the other answer from the very first question — parentheses are exactly how you'd write '35' if that's really what you meant.

Worked Example — Nested Grouping Symbols

Evaluate 2 × [(6 + 4) ÷ 5]. Work from the innermost grouping outward: first (6 + 4) = 10, then [10 ÷ 5] = 2, and finally 2 × 2 = 4.

3 + 4 \times 5 = 3 + 20 = 23 \qquad (3 + 4) \times 5 = 7 \times 5 = 35

Tip

A common memory aid is PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) — but remember multiplication and division are actually tied in priority and worked left to right, not multiplication always before division.

Common Mistakes

Treating multiplication as always coming before division, or addition always before subtraction, rather than working left to right within each tied pair.
Multiplication and division share equal priority, worked left to right; addition and subtraction share equal priority, worked left to right. For 20 ÷ 4 × 2, divide first (left to right): 5 × 2 = 10, not 20 ÷ 8.
Solving the entire expression strictly left to right, ignoring the order of operations completely.
Scan the whole expression for parentheses and multiplication/division first, before doing any addition or subtraction, even if the addition appears earlier in the expression.

Key Takeaways

Without an agreed order of operations, the same expression could mean different things to different people.
The order is: parentheses/brackets, then exponents, then multiplication/division left to right, then addition/subtraction left to right.
Parentheses can override the usual order whenever an expression needs addition or subtraction done first.

Summary

The order of operations exists so every expression has exactly one correct value. The next lesson works in the opposite direction — starting from a description in words and writing the numerical expression that captures it.

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