Four Operations & Priority of Operations (BIDMAS) — AQA GCSE Mathematics

SPEC N2, N3 • NUMBER

⭐ HIGH PRIORITY

Four Operations & BIDMAS

The priority of operations. Non-calculator methods for Paper 1. Sign rules for negatives. The traps AQA uses every year — and how to avoid every one.

Why BIDMAS Matters

Consider this expression:

5 + 3 × 2 = ?

Left to right

5 + 3 = 8

8 × 2 = 16

BIDMAS

3 × 2 = 6

5 + 6 = 11

The Agreement

BIDMAS is the universally agreed priority order for operations. Without it, the same expression could mean two different things. Maths can't tolerate ambiguity — so mathematicians agreed on a single rule: 11 is correct.

The Four BIDMAS Levels

Work through the levels in order, top to bottom. But the middle two hold a pair of equal operations.

Brackets — always resolve innermost first

Indices — powers and roots (squaring, cubing, square roots)

D / M

Division & Multiplication — equal priority, left to right

A / S

Addition & Subtraction — equal priority, left to right

BODMAS vs BIDMAS

Some students learn BODMAS — O for "Orders" instead of I for "Indices." They mean exactly the same thing. AQA accepts either.

Left to Right — Within a Level

This is the single most common BIDMAS error. Division and multiplication have equal priority. Don't treat them as "D before M" — work left to right.

Worked example: 12 ÷ 4 × 3

✘ WRONG — "D before M"

12 ÷ 4 × 3

4 × 3 = 12 12 ÷ 12 = 1= 1 ✘

✔ RIGHT — Left to right

12 ÷ 4 × 3

12 ÷ 4 = 3 3 × 3 = 9= 9 ✔

Memory hook

The letters D and M appear in that order in BIDMAS for the acronym only. They share a level. Same for A and S. Within a level: always left to right.

01MCE Worked Example 2 MARKS

Calculate:

5 + 3² × (8 − 6) − 12 ÷ 4

•

MMethod. Apply BIDMAS: Brackets → Indices → ÷ & × (left to right) → + & − (left to right).

•

CCalculate.

Brackets: 8 − 6 = 2 → 5 + 3² × 2 − 12 ÷ 4

Indices: 3² = 9 → 5 + 9 × 2 − 12 ÷ 4

× and ÷ (L→R): 9 × 2 = 18; 12 ÷ 4 = 3 → 5 + 18 − 3

+ and − (L→R): 5 + 18 = 23; 23 − 3 = 20

•

EEvaluate.Answer: 20

Mark scheme note: M1 for a correct BIDMAS step, A1 for the final answer. A student who writes only "= 20" with no working can lose the method mark if their answer is wrong.

Sign Trap: (−3)² vs −3²

One of AQA's favourite traps. The presence or absence of brackets changes the answer — and its sign.

With brackets

(−3)²

= (−3) × (−3) = +9= 9 ✔

Without brackets

−3²

= −(3²) = −9

= −9 ✘ (for 9)

The Rule

Indices apply only to the number they sit on. In −3², the index applies to 3 (not −3), then the minus sign is applied after — giving −9.

Safer habit

When substituting a negative value, always put it in brackets. If asked to evaluate x² when x = −3, write (−3)², not −3².

Negative Arithmetic — Sign Rules

Four rules cover every negative-number calculation at GCSE.

RuleExampleWhy

Adding a negative = subtracting−3 + (−4) = −7Two minus signs become one subtraction

Subtracting a negative = adding5 − (−3) = 5 + 3 = 8Two minus signs combine into a plus

Like signs multiply = positive(−3) × (−4) = 12Same signs → positive result

Unlike signs multiply = negative(−3) × 4 = −12Different signs → negative result

Division follows the same sign rules

(−12) ÷ (−3) = 4 (like signs → positive). (−12) ÷ 3 = −4 (unlike → negative).

Subtracting a Negative — Why

Of all the sign rules, "subtract a negative = add" is the one students most often get wrong. Here's why it works.

Start at 5. "Subtract −3" means undo a subtraction of 3 — which is the same as adding 3. End at 8.

5 − (−3) = 5 + 3 = 8

Long Multiplication — Grid Method

Paper 1 is non-calculator. You need a reliable way to multiply two-digit numbers by hand. The grid method partitions each number into tens and units, then multiplies each pair.

Worked example: 47 × 23

×407

20800140

312021

Add all products: 800 + 140 + 120 + 21 = 1081

Mark scheme insight

AQA awards method marks for "valid long-multiplication method: grid, column, or partitioning with correct partial products." On a 2-mark question, showing the grid guarantees the method mark even if a partial product is slightly wrong.

Long Division — The Bus Stop

Divide digit by digit, left to right, carrying the remainder across. Still the most reliable Paper 1 method.

672 ÷ 14 = 48

Check by multiplying back

If your answer is correct, the answer × divisor = dividend. Here: 48 × 14 should equal 672. Quick check: 48 × 14 = 48 × 10 + 48 × 4 = 480 + 192 = 672. ✔

The D-before-M Trap

AQA examiner reports flag this every year. Students treat "D before M" as a strict order.

Calculate: 24 ÷ 6 × 2

Weak student

"D comes before M in BIDMAS, so I do the division, but with a twist — I see a multiplication right next to the division, so let me do that first."

6 × 2 = 12

24 ÷ 12 = 2

Result: 2 ✘ (loses accuracy mark)

Strong student

"D and M are on the same level. Work left to right. Division comes first because it's written first."

24 ÷ 6 = 4

4 × 2 = 8

Result: 8 ✔

Examiner Alert: D before M

What examiners report every year

Students treat D and M as a strict priority order, applying division before multiplication regardless of position. This is the single most common BIDMAS mistake at GCSE, flagged repeatedly in AQA examiner reports (WRE).

What to write instead

D and M share priority. Same for A and S. Within a level, always left to right — no exceptions.

24 ÷ 6 × 2 = 4 × 2 = 8

The one-line rule

Down the BIDMAS levels; left to right within a level.

Examiner Alert: Missing Brackets on Negatives

What examiners see

Students write −3² thinking it means 9. It doesn't — it means −9. The index applies only to the 3, then the minus sign is applied after.

What to write instead

When squaring a negative value, always use brackets: (−3)² = 9.

If a question gives you an expression like x² and says x = −3, substitute as (−3)², not −3².

Mark scheme specifics

AQA mark schemes explicitly state: accept "(−3)² = 9"; do not accept "−3² = 9" as equivalent — that notation is wrong even if the numerical answer happens to match the question's needs.

Common Errors

Students often writeCorrect versionWhy it matters

24 ÷ 6 × 2 = 2= 8 (left to right)BIDMAS D/M equal priority misapplied

−3² = 9−3² = −9; (−3)² = 9Sign/bracket priority error

5 − (−3) = 25 − (−3) = 8Double-negative error

Calculator-style answer to 47 × 23 with no method shownShow grid, column, or partitioningPaper 1 method marks lost

5 + 3 × 2 = 16 (left to right)= 11 (× before +)Priority order ignored

Practice Question 1

Work out:1 MARK

3 + 4 × 2 − 5

1

BIDMAS: no brackets, no indices. × before + and −.

2

4 × 2 = 8 → expression becomes 3 + 8 − 5

3

Left to right: 3 + 8 = 11, then 11 − 5 = 6.

Answer: 6

Practice Question 2

Work out:2 MARKS

(−3)² − 5

1

Indices first. (−3)² has brackets, so apply the square to all of −3.

2

(−3)² = (−3) × (−3) = +9 (like signs → positive).

3

Then subtract: 9 − 5 = 4.

Answer: 4

Common wrong answer: a student who reads the brackets as "just for decoration" writes −3² − 5 = −9 − 5 = −14. Mark scheme "do not accept."

Practice Question 3

Work out 47 × 23. Show your working.2 MARKS

(Paper 1 style — non-calculator)

1

Set up the grid. Partition 47 into 40 + 7 and 23 into 20 + 3.

2

Multiply each pair:

40 × 20 = 800 | 7 × 20 = 140

40 × 3 = 120 | 7 × 3 = 21

3

Sum all four products:

800 + 140 + 120 + 21 = 1081

Answer: 1081

Mark scheme: M1 for a valid method with correct partial products, A1 for the correct answer. A student who only writes "1081" with no working may score 1/2 if wrong, but 2/2 if correct — though showing working is always safer.

Summary

BIDMAS Priority Order

1

B rackets — innermost first

2

I ndices — powers & roots

3

D ivision & M ultiplication — equal priority, left to right

4

A ddition & S ubtraction — equal priority, left to right

Sign Rules for Negatives

(−3)² = 9 (brackets), but −3² = −9 (no brackets)

Subtracting a negative = adding ( 5 − (−3) = 8 )

Like signs multiply → positive ( (−3)(−4) = 12 )

Unlike signs multiply → negative ( (−3)(4) = −12 )

Paper 1 Non-Calculator Methods

Long multiplication: grid method — partition and multiply each pair, sum all products

Long division: bus stop — divide digit by digit, carry remainder across

Always show method — method marks are awarded even if the final answer is wrong

The #1 Trap — D before M

Students treat D as strictly before M in BIDMAS. They aren't — they share a level. Within a level, always left to right.

24 ÷ 6 × 2 = 4 × 2 = 8 (not 2)

Ready for 1.03

Next up: Primes, Factors & Multiples — HCF, LCM, and prime factor decomposition using factor trees and index notation.