KS3 · Maths

Order of operations (BIDMAS)

What's 4 + 3 × 5? Left to right gives 35; the priority of operations gives 19. Same numbers, two answers — and only one is right.

Which operation first?

Sort each calculation by its first move

Pick a calculation, then pick the row of the priority diagram that its first operation comes from.

Still to sort

Row 1 · Brackets (written or hidden) (0)

Anything grouped together: numbers in brackets, or a sum sitting under a root sign or a long fraction line.

Where the line is: A root over one number, like √64, belongs in Row 2. A root stretching over a sum, like √(16 + 9), hides a bracket — so the sum underneath comes first.

Row 2 · Roots and powers (0)

Squares, cubes and square roots, such as 5², 2³ or √64.

Where the line is: Only comes first when there's no bracket, written or hidden, anywhere in the calculation.

Row 3 · Multiplication and division (0)

× and ÷ share this row. Neither one outranks the other.

Where the line is: Division is not lower than multiplication. They are equal partners on the same row.

Row 4 · Addition and subtraction (0)

+ and − share the bottom row, so they usually come last.

Where the line is: Only comes first when nothing in the calculation comes from a higher row.

8 of 8 still to sort.

Don't work anything out yet. The priority diagram has four rows, read from the top down. For each calculation, decide which row its FIRST operation comes from — then read why.

Watch out: A root sign or a long fraction line groups everything underneath it, even when no brackets are written. That's a hidden bracket, and it goes first.

Same row, same rank

What happens when two operations share a row?

Look at 24 ÷ 8 × 3 and 10 − 6 + 4. Each one uses two operations from the same row of the priority diagram.

Which of these is closest to what you think right now?

Multiplication comes before division, and addition comes before subtraction. So I'd do 8 × 3 and 6 + 4 first.You always just work from left to right.× and ÷ are equal partners, and so are + and −. I can write 24 ÷ 8 as 24/8, and − 6 as + (−6), so each number keeps its own sign.The order doesn't really matter. You get the same answer whichever way you go.

How sure are you?

I'm sureI think soI'm guessing

Multiplication comes before division, and addition comes before subtraction. So I'd do 8 × 3 and 6 + 4 first.

This is a common misconception

It's a tempting idea — but it gives 24 ÷ 24 = 1 and 10 − 10 = 0, and both are wrong. × and ÷ sit on the SAME row, and so do + and −. Neither outranks its partner. Here's the trick: write the division as a fraction. 24 ÷ 8 × 3 becomes 24/8 × 3 = 3 × 3 = 9. Now the 8 can't be borrowed by the multiplication, because it's clearly the number you divide by.

You always just work from left to right.

This is a common misconception

For these two, left to right happens to work, because each one only uses a single row. But it breaks the moment rows mix. Take 30 − 24 ÷ 8: left to right gives 6 ÷ 8, but division is on a higher row, so it's 30 − 3 = 27. The rows come first. Left to right is only safe inside one row.

× and ÷ are equal partners, and so are + and −. I can write 24 ÷ 8 as 24/8, and − 6 as + (−6), so each number keeps its own sign.

This matches the evidence

Spot on. 24/8 × 3 = 3 × 3 = 9. And 10 − 6 + 4 = 10 + (−6) + 4. Now every operation is an addition, so you can add in any order: 10 + 4 + (−6) = 8 as well. That's the commutative property of addition — you can swap the order of the numbers without changing the total. And −6 is the additive inverse of 6, because 6 + (−6) = 0.

The order doesn't really matter. You get the same answer whichever way you go.

This is a common misconception

Try it and see: 24 ÷ 8 × 3 gives 9 one way and 1 the other. The same numbers really can give different answers in different orders. That's exactly why everyone follows one agreed priority of operations — so we all reach the same answer.

Operations on the same row are equal. Write a division as a fraction and a subtraction as adding the additive inverse, and the same-row trap disappears.

One full pass down the diagram

Problem

Work out 12 + (4 + 6) × 5²

Spot the slip

Where does this answer go wrong?

Work out 50 − √(64 + 36) ÷ 2, where the root sign stretches over the whole of 64 + 36.

Jamie's working — which line goes wrong?

Hidden bracket first: the root sign covers 64 + 36, so that part comes before anything else.
√(64 + 36) = √64 + √36 = 8 + 6 = 14
Division next: 14 ÷ 2 = 7
Subtraction last: 50 − 7 = 43

Now with negatives

Fill in the missing steps

Work out 80 − (−12) ÷ 4 × 2

Scan the rows. The brackets round −12 just keep the negative sign clear — there's nothing inside them to work out. The highest row here is × and ÷.Writing negative numbers in brackets stops you mixing up a negative number with a subtraction.
missing step

Which line is step 2?

Subtract first: 80 − (−12) = 92, giving 92 ÷ 4 × 2× and ÷ share a row, so take them in turn from the left: (−12) ÷ 4 = −3, giving 80 − (−3) × 2Multiply first: 4 × 2 = 8, giving 80 − (−12) ÷ 8

Your turn

Decimals — and every row at once

Work out 8.2 − 3.4 × (−2) + 12/(2.5 + 7.5), where the fraction line runs under the whole of 2.5 + 7.5. Write down each stage of your working, then mark it. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Deals with the hidden bracket first: 2.5 + 7.5 = 10, so 12/10 = 1.2 (1 mark)
Does the multiplication before any adding or subtracting: 3.4 × (−2) = −6.8 (1 mark)
Rewrites subtracting −6.8 as adding its additive inverse: 8.2 − (−6.8) = 8.2 + 6.8 = 15 (1 mark)
Final answer: 16.2 (1 mark)

Model answer

Hidden bracket under the fraction line: 2.5 + 7.5 = 10, so 12/10 = 1.2. Multiplication next: 3.4 × (−2) = −6.8. Now the calculation reads 8.2 − (−6.8) + 1.2 = 8.2 + 6.8 + 1.2 = 15 + 1.2 = 16.2.

Decimals follow exactly the same rows as whole numbers. Use the column method when you add decimals, and check each stage before you move on.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same numbers, different order, different answer. So we all agree on one order — and it's easier than it looks.

What you need to know

The priority diagram, from the top down: brackets → roots and powers → multiplication and division → addition and subtraction.
Operations on the same row have equal priority: × does not beat ÷, and + does not beat −.
A root sign or fraction line that stretches over a calculation groups it like a hidden bracket: √(16 + 9) = √25 = 5, not √16 + √9.
Writing a division as a fraction makes the divisor clear: 24 ÷ 8 × 3 = 24/8 × 3 = 9.
Any subtraction can be rewritten as adding the additive inverse: 20 + 16 − 4 = 20 + 16 + (−4) = 32.
The priority of operations is the same for positive and negative numbers, whole numbers and decimals.

The big picture

An operation takes an input value to an output value — addition, subtraction, multiplication, division, roots and powers are all operations. When a calculation has more than one, the order matters, so everyone follows the same priority of operations, read from the top down: brackets first, then roots and powers, then multiplication and division (equal partners on one row), then addition and subtraction (also equal). A root sign or a long fraction line groups what's under it like a hidden bracket. The same order works for negative numbers and decimals: write negatives in brackets and rewrite subtractions as adding the additive inverse.

Key points

1An operation, such as +, −, ×, ÷, a root or a power, takes an input value to an output value.

2Following one agreed priority means everyone works a calculation the same way and gets the same answer.

3Brackets group numbers together and are always done first: 100 ÷ (4 + 6) = 100 ÷ 10 = 10.

4When two operations share a row, it doesn't matter which you do first, as long as each number stays with its own operation.

5Once every subtraction is an addition of a negative, the numbers can be added in any order — addition is commutative.

6Write negative numbers in brackets, such as (−12), so you don't confuse a negative number with a subtraction.

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Worked example

Problem

Work out 100 + (4² + 4) ÷ (−2) + √(20 − (−5)), where the √ covers all of 20 − (−5).

⚠ Watch out

Splitting a hidden bracket. √(49 + 36) is not √49 + √36, and 30 over 2 + 4 (with 2 + 4 all under the fraction line) is 30 ÷ 6 = 5, not 30 ÷ 2 + 4 = 19. Group what's under the sign first.

🧠

Memory hook

Start on the top floor and walk down: Brackets → Roots and powers → × and ÷ → + and −. Operations that share a floor share the rank — neither one pushes in front.

✓

Check yourself

Work out 50 − (−100 + 149) ÷ (−7) without a calculator. Which operation goes first? (Answer: 57 — bracket 49, then 49 ÷ (−7) = −7, then 50 + 7.)

Flashcards

(14)

What is an operation in maths?

Something that takes an input value to an output value. Addition, subtraction, multiplication, division, roots and powers are all operations.

Why do we need an agreed priority of operations?

So everyone approaches a calculation the same way and reaches the same answer. The same numbers can give different answers in different orders.

What are the four rows of the priority diagram, from the top?

1. Brackets 2. Roots and powers 3. Multiplication and division 4. Addition and subtraction.

Does multiplication come before division?

No. They share a row and have the same priority — neither one outranks the other.

Why write a division as a fraction, e.g. 24 ÷ 8 × 3 as 24/8 × 3?

It makes the divisor clear, so it can't be borrowed by the multiplication: 24/8 × 3 = 3 × 3 = 9, not 24 ÷ 24 = 1.

What is the additive inverse of a number?

The number you add to it to get 0. The additive inverse of 6 is −6, and the additive inverse of −6 is 6.

How can you rewrite a subtraction such as 20 − 4?

As adding the additive inverse: 20 − 4 = 20 + (−4).

What does it mean for an operation to be commutative?

The values can be written in either order without changing the result. Addition is commutative: 4 + (−6) = (−6) + 4.

What does a long fraction line do to the numbers above or below it?

It groups them like a hidden bracket, so work them out first: 25 over 4 + 1 is 25 ÷ 5 = 5.

Is √(16 + 9) the same as √16 + √9?

No. The root sign groups 16 + 9, so √(16 + 9) = √25 = 5. But √16 + √9 = 4 + 3 = 7.

12 ÷ 2 + 20 ÷ 4 has two divisions. What can you do with them?

Do both at the same time, because they're on the same row: 6 + 5 = 11.

Why write negative numbers in brackets, like (−3)?

So you don't confuse a negative number with a subtraction.

What is 100 − (−4)?

104. Subtracting −4 is the same as adding its additive inverse, 4.

How can you work out 12 + (−3) − 5 − (−10) quickly?

Rewrite it as all additions: 12 + (−3) + (−5) + 10. Positives: 22. Negatives: −8. Total: 22 + (−8) = 14.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Sort each calculation by its first move

What happens when two operations share a row?

Where does this answer go wrong?

Fill in the missing steps

Decimals — and every row at once

More KS3 Maths topics