Lessons · Teaching · expressions, variables and the order of operations
The order of operations is a convention, and it has a reason
Brackets first, then exponents, then multiplication and division together from left to right, then addition and subtraction together from left to right. Multiplication comes before addition because multiplication is a compressed addition, and unpacking it first keeps the meaning.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
Every calculator, spreadsheet and programming language agrees on this order, so it is not a school rule. A teacher who presents it as arbitrary invites an argument they will lose; presenting it as a shared convention that makes 2 + 3 x 4 mean one thing rather than two is honest and ends the argument.
How to think about it
Work outward from the innermost bracket. Within a level, go strictly left to right: multiplication and division rank equally, so 12 divided by 3 times 2 is 8 and not 2. Substitute values into a variable only after the expression is understood.
Worked example
2 + 3 x 4 = 2 + 12 = 14, not 20Three lots of four, plus two. The multiplication is a package to open first.
12 / 3 x 2: equal rank, so left to right, giving 4 x 2 = 8Not 12 / 6. This is the one that divides a staffroom.
(5 + 3) x 2 = 16, because brackets say do this firstBrackets exist precisely to override the default order.
Evaluate 3n + 4 when n = 5: 3 x 5 + 4 = 19Substitute, then apply the order. Not 3 x 9.
Your turn
Name which of multiplication and division is done first when both appear.
They rank equally, so work from to right
Solve one, graded on the server
The trap
Believing addition always comes before subtraction because the mnemonic puts it first. They rank equally and run left to right: 10 - 4 + 2 is 8, not 4.