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Order of operations

Order of operations is a convention, not a discovery. Mathematicians agreed that multiplication and division bind tighter than addition and subtraction so that one line means one thing to everybody, and brackets exist to override that when a different meaning is wanted. A child who understands it as an agreement rather than a law tends to remember it better.

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Multiplication before addition

2 + 3 x 4

  1. There are two operations here, so the order matters.
  2. Multiplication binds tighter than addition, so 3 x 4 goes first. That is 12.
  3. Now the line reads 2 + 12.
  4. The answer is 14.

Worth knowing. Read left to right, this would be 5 x 4, or 20. That is the wrong answer, and it is the exact mistake the rule exists to prevent — which is why every problem here is one where the two readings disagree.

Brackets change the answer

(2 + 3) x 4

  1. Brackets come first, whatever is inside them.
  2. 2 + 3 is 5.
  3. Now the line reads 5 x 4.
  4. The answer is 20.

Worth knowing. Same three numbers, same two operations, different answer. Comparing this with 2 + 3 x 4 side by side is the clearest way to show what brackets are for.

Where it usually goes wrong

  • Working strictly left to right, so 2 + 3 x 4 comes out as 20.
  • Treating multiplication as always before division, rather than working those two left to right as they come: 12 / 3 x 2 is 8, not 2.
  • Doing the brackets last because they look like an afterthought.

The Professor underlines the part he is going to do first before he does anything. One line, one decision, no arguments with himself later.

Practise it

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Questions teachers and parents ask

Is it PEMDAS, BODMAS or BIDMAS?

Three names for the same convention, from different countries. All of them mean: brackets, then indices or exponents, then multiplication and division together working left to right, then addition and subtraction together working left to right. The pairs matter more than the letters — reading the letters strictly makes 12 / 3 x 2 come out as 2 instead of 8.

When is this taught?

Common Core introduces grouping symbols and evaluating expressions in grade 5 (5.OA.A.1), and expressions with exponents follow in grade 6. Before that, problems are written so that reading order does not change the answer.

Why do multiplication and division come before addition?

Because multiplication is repeated addition, so it names a quantity that is already whole. In 2 + 3 x 4 the 3 x 4 is one thing — twelve — and adding 2 to it is the second step. The convention is a way of writing that without needing brackets every time.

Do the two sides of a division sign follow the same rule?

Yes, and it catches people out. 12 / 3 x 2 is worked left to right through the pair: 12 / 3 is 4, then 4 x 2 is 8. Multiplication does not go first just because M comes before D in the mnemonic.

Where this shows up

Standards

  • 5.OA · Operations and algebraic thinking
  • 6.EE · Expressions and equations
  • CCSS.MATH.CONTENT.5.OA.A.1 — Use parentheses, brackets or braces in numerical expressions, and evaluate expressions with these symbols.
  • CCSS.MATH.CONTENT.6.EE.A.2.C — Evaluate expressions, including those arising from formulas, using the conventional order of operations.

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