The acronyms side by side
Orders, indices, exponents and powers all mean the same thing: squares, cubes, square roots and so on. Brackets and parentheses are the same thing too.
| Acronym | Where it is used | Letters |
|---|---|---|
| BODMAS | UK, Australia, New Zealand, India | Brackets, Orders, Division, Multiplication, Addition, Subtraction |
| BIDMAS | UK | Brackets, Indices, Division, Multiplication, Addition, Subtraction |
| BEDMAS | Canada | Brackets, Exponents, Division, Multiplication, Addition, Subtraction |
| PEMDAS | United States | Parentheses, Exponents, Multiplication, Division, Addition, Subtraction |
All four acronyms describe exactly the same rule. Only the words differ.
The rule in four steps
Steps three and four are where the acronym misleads. BODMAS puts D before M and PEMDAS puts M before D, but neither is right on its own. Multiplication and division have equal rank, and so do addition and subtraction. When two operations of the same rank sit next to each other, you simply work left to right.
- Brackets first. Work out anything inside brackets, innermost pair first.
- Then powers and roots (orders, indices or exponents).
- Then multiplication and division, together, working from left to right.
- Then addition and subtraction, together, working from left to right.
Worked examples
Example 1: 8 minus 3 plus 2. Addition and subtraction have equal rank, so go left to right: 8 minus 3 is 5, then 5 plus 2 is 7. A child who does the addition first gets 8 minus 5, which is 3, and that is wrong.
Example 2: 12 divided by 3 times 2. Equal rank, left to right: 12 divided by 3 is 4, then 4 times 2 is 8. Doing the multiplication first gives 12 divided by 6, which is 2, and that is the classic PEMDAS mistake.
Example 3: 4 plus 3 times 5. Multiplication before addition: 3 times 5 is 15, then 4 plus 15 is 19. Not 35.
Example 4: (4 plus 3) times 5. The brackets change everything: 4 plus 3 is 7, then 7 times 5 is 35.
Example 5: 2 plus 3 squared. The power comes first: 3 squared is 9, then 2 plus 9 is 11. Not 25, which is what you get if you add first and then square.
Example 6: 20 minus 2 times (6 minus 2) squared. Brackets: 6 minus 2 is 4. Power: 4 squared is 16. Multiply: 2 times 16 is 32. Subtract: 20 minus 32 is negative 12.
How to teach it so it sticks
- Teach the ranks, not the letters. Draw four rows: brackets, powers, multiply and divide, add and subtract. Children see that the middle rows hold two operations of equal rank.
- Give pairs of near-identical questions, one with brackets and one without, and ask children to predict which will be bigger before they calculate.
- Use the equal-rank trap deliberately: a few questions like 10 minus 4 plus 1 in every set catch the children who are following letters instead of rules.
- Link it to real calculators. A basic four-function calculator ignores the rule; a scientific calculator follows it. Typing 4 + 3 x 5 into both and getting 35 and 19 is a memorable lesson.
Why the rule exists at all
The order of operations is a convention, not a law of nature, but it is a convention everyone agrees on so that a written expression means one thing. Without it, 4 + 3 x 5 could mean 19 or 35 depending on who read it, and no formula, spreadsheet or line of code would be reliable. That is the honest answer when a child asks why multiplication goes first: because we all agreed, and agreeing lets us write less.