Order of operations
National Curriculum: this is in the Year 6 programme of study, part of key stage 2 (Years 3 to 6). Schools have to teach the key stage 2 programme by the end of Year 6, and can introduce content earlier or later within it.
Children in Year 6 are 10 or 11, and turn 11 during the school year.
What it is
The order of operations is the agreed order for working out a calculation with more than one operation in it: brackets first, then multiplication and division, then addition and subtraction. So 5 + 2 × 3 is 11, because the 2 × 3 is done first, while (5 + 2) × 3 is 21.
Most parents were taught BODMAS or BIDMAS, which gets brackets right and misleads about the rest. Multiplication and division rank equally, and addition and subtraction rank equally; operations of the same rank are worked from left to right. 20 − 4 + 2 is 18, because the subtraction comes first on the page. A child treating the mnemonic as a queue does the addition first and answers 14.
In Year 6 this decides whether a calculation needs brackets, and which of two written statements is the correct one. It carries into secondary school algebra: 3 + 4n cannot be collected into 7n, for the same reason that 5 + 2 × 3 is not 21.
When children learn it
The order of operations is a Year 6 requirement: using it to carry out calculations involving the four operations. It comes after those four are secure in writing, and after square and cube numbers in Year 5, so a Year 6 question can put a square inside a calculation with the four operations.
What it looks like when it goes wrong
These patterns come up again and again in children's answers on this skill. Each one points at a specific missing idea, not carelessness.
- The calculation worked straight across. 5 + 2 × 3 answered as 21. It was read like a sentence, left to right, so the addition happened first. This is what a child does before the rule is taught, and it is replaced by a habit rather than by an explanation: look at the whole calculation and decide what goes first, before writing anything.
- The brackets are passed over. (5 + 2) × 3 answered as 11. The multiplication was done first, out of a rule learned as multiply before you add. Brackets outrank everything else, and a bracket is on the page precisely because the usual order would give a different answer.
- A square read as a doubling. 4² + 3 answered as 11 rather than 19, because 4² was read as 4 × 2 rather than 4 × 4. The ordering cannot start until the square has a value, so this looks like an order-of-operations error and is not one.
How to help at home
- Before any working, ask them to point at the part that gets done first. Deciding that before the pencil moves is what the rule is for.
- Use one pair of calculations: 5 + 2 × 3 and (5 + 2) × 3. Same numbers, answers of 11 and 21, and the brackets are the only difference between them.
- Say the ranking out loud, because the mnemonic hides it: multiplication and division rank equally and are worked left to right, and so do addition and subtraction. Then ask for 20 − 4 + 2. An answer of 14 is the correction to make.
- Have them rewrite the calculation with brackets around the part that goes first: 5 + (2 × 3). It changes nothing about the answer, and it puts the decision on the page where it can be checked.