Order of Operations (BIDMAS)

The method, step by step

1. 1

   Brackets first, working outward

   3 × (4 + 2)² − 8 ÷ 4 → 3 × 6² − 8 ÷ 4

   Anything inside brackets is resolved before it interacts with anything outside them. Where brackets are nested, the innermost pair goes first — the brackets are effectively saying 'treat this as one number', and you cannot use a number you have not worked out.

2. 2

   Indices next

   3 × 6² − 8 ÷ 4 → 3 × 36 − 8 ÷ 4

   Powers and roots are resolved before any multiplying or dividing. A common error is computing 3 × 6 first and then squaring, which gives 324 instead of 108 — the index belongs to the 6 alone, not to the product.

3. 3

   Division and multiplication together, left to right

   3 × 36 − 8 ÷ 4 → 108 − 2

   These two rank equally. The M appearing before the D in BIDMAS is an accident of the acronym, not a rule. In 12 ÷ 3 × 2 the correct answer is 8, not 2, because you work left to right — this is the single most common failure of the acronym.

4. 4

   Addition and subtraction together, left to right

   108 − 2 = 106

   Same principle again: A and S rank equally and are read left to right. In 10 − 3 + 2 the answer is 9, not 5. Students who do all addition before all subtraction get this wrong roughly half the time, depending on how the question is written.

5. 5

   Check by rewriting with brackets

   12 ÷ 3 × 2 = (12 ÷ 3) × 2 = 8

   Where the order is ambiguous to you, insert the brackets your reading implies and see whether the expression still says what it said. This converts a memory question into a reading question, which is far more reliable under pressure.

BIDMAS is four levels, not six

The acronym lists six letters and implies six ranked stages, which is wrong and is the source of nearly every error in the topic. There are four levels: brackets, indices, then multiplication-and-division as one level, then addition-and-subtraction as one level. Within a level you read left to right.

Written honestly it would be B, I, DM, AS. Because it is not, students confidently compute 12 ÷ 3 × 2 as 12 ÷ 6 = 2, following the acronym exactly and getting the wrong answer. The acronym is not a rule; it is a mnemonic for one.

• 1. Brackets, innermost first
• 2. Indices — powers and roots
• 3. Division and multiplication — equal rank, left to right
• 4. Addition and subtraction — equal rank, left to right

This is agreement, not discovery

Order of operations is a convention rather than a mathematical truth. There is nothing in arithmetic forcing multiplication to outrank addition; mathematicians agreed on it so that written expressions have exactly one meaning. Saying this to students helps, because the rule stops seeming arbitrary once its purpose is stated.

It also explains why brackets exist at all. Brackets are how you override the convention when you want a different meaning, which reframes them from decoration into the most powerful notation in the expression.

Where it bites: calculators and fraction bars

A fraction bar acts as a bracket over everything above and below it. Typing (3+5)/(2+2) into a calculator as 3+5/2+2 gives 7.5 rather than 2, and the calculator is right — it did what was typed. This is the most expensive keying error on calculator papers.

Scientific calculators apply the convention correctly, which means they will not rescue a student who typed an expression that means something other than what they intended. The check is to read the typed line back and ask what it says.

How we teach order of operations

We teach four levels rather than six letters, and we test the two equal-rank levels deliberately. Any student who gives 2 for 12 ÷ 3 × 2 has learned the acronym and not the rule, and that is a five-minute conversation that prevents a recurring error.

We also insist on brackets when typing into a calculator, well beyond the point where they are strictly necessary. Redundant brackets cost nothing and remove the single most common source of silently wrong calculator answers.