Order of Operations
Use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies.
BODMAS is not a rule someone invented to catch you out. It is the agreed order that makes every calculation mean one thing.
Builds onYr 6 · Four operations with whole numbers
Why this works
Consider . Two people could read this two different ways:
Left to right: , then .
Multiplication first: , then .
Both are reasonable readings, but they disagree. A calculation that can mean two things is useless — so mathematicians agreed on one order, worldwide.
The agreed order is Brackets, Order (powers), Division and Multiplication, Addition and Subtraction.
So . Not because 14 is more natural, but because that is the convention everyone follows.
The part almost everyone gets wrong: D and M rank equally, and A and S rank equally. When operations tie, you work left to right. BODMAS is four levels, not six.
Worked example
e.g.
- Brackets first. Deal with everything inside them.
- Order — powers and roots.
- Division and Multiplication, working left to right. Here comes first.
- Still on the same level, keep going left to right.
- Addition and Subtraction last, again left to right.
Common mistakes
Wrong
Right
Treating multiplication as ranking above division. They are equal, so you go left to right — and division comes first here.
Wrong
Right
The same mistake with addition and subtraction. They tie, so work left to right: subtract first because it appears first.
Wrong
Right
Brackets come before powers. Add inside the bracket first, then square the result.
Wrong
Right
Without brackets, the power applies only to the 3, and the minus sign is applied afterwards. If you want , you must write .
Practice
FluencyGet quick and accurate at the method.
1
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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ReasoningExplain why. Say it in your own words.
1
Kai says because the M in BODMAS comes before the D. What is wrong with his reasoning?
AnswerD and M rank equally — the letters are only in that order because BODMAS has to be spelled somehow. When operations tie you work left to right, so first, then .
2
Where would you put brackets in to make the answer as large as possible?
Answer. Grouping the addition and the subtraction so both are done before multiplying gives the largest result — larger than the unbracketed 13.
3
Explain why and give different answers.
AnswerIn the power binds tightly to the 4 only, so you square 4 to get 16 and then apply the minus, giving . In the brackets mean the whole of is squared, giving .
4
Why does mathematics need an agreed order of operations at all?
AnswerWithout one, the same written expression would mean different things to different people, and no formula could be relied upon. The order is a convention, like driving on an agreed side of the road — the choice is arbitrary but agreement is essential.
AppliedThe same maths, inside a real question.
1
A cafe sells coffee for and cake for . Mia buys 3 coffees and 2 cakes. Write one expression for the total and evaluate it.
Answer. Both multiplications happen before the addition, which is exactly what the situation requires.
2
A taxi charges a flagfall plus per km. Write an expression for a 12 km trip and work out the cost.
Answer. The multiplication must come first, or you would wrongly charge the flagfall per kilometre.
3
A rectangular garden is 8 m by 5 m. A square shed of side 3 m sits inside it. Write an expression for the remaining garden area and evaluate it.
Answer m². The power and the multiplication both come before the subtraction.
4
Four friends share the cost of a pizza order and a delivery fee. Write an expression for each person's share.
Answer. The brackets are essential — without them you would divide only the delivery fee.