N3: Order of Operations
Use brackets, powers, roots and reciprocals; understand and use the conventions of BIDMAS/BODMAS
📋 Key Concepts
BIDMAS/BODMAS: This acronym helps you remember the correct order of operations in mathematics.
B - Brackets ( )
I/O - Indices/Orders (powers and roots) ² ³ √
D - Division ÷
M - Multiplication ×
A - Addition +
S - Subtraction −
I/O - Indices/Orders (powers and roots) ² ³ √
D - Division ÷
M - Multiplication ×
A - Addition +
S - Subtraction −
Important: Division and Multiplication have equal priority - work left to right. Addition and Subtraction have equal priority - work left to right.
📝 Working Through BIDMAS
Example 1
Calculate: 3 + 4 × 2
Solution:
Multiplication before addition: 4 × 2 = 8
Then: 3 + 8 = 11
Not 7 × 2 = 14 (wrong!)
Example 2
Calculate: (3 + 4) × 2
Solution:
Brackets first: 3 + 4 = 7
Then: 7 × 2 = 14
Example 3
Calculate: 20 - 12 ÷ 4 + 3
Solution:
Division first: 12 ÷ 4 = 3
Then left to right: 20 - 3 = 17
Finally: 17 + 3 = 20
📝 Indices and Roots
Indices (Powers): These are evaluated after brackets. Includes squares, cubes, and higher powers.
Example 4
Calculate: 5² - 3²
Solution:
Calculate powers: 5² = 25, 3² = 9
Then: 25 - 9 = 16
Example 5
Calculate: √16 + 2³
Solution:
Roots and powers: √16 = 4, 2³ = 8
Then: 4 + 8 = 12
📝 Complex Examples
Example 6
Calculate: 2 × (3 + 4)² - 10
Solution:
Step 1 - Brackets: 3 + 4 = 7
Step 2 - Indices: 7² = 49
Step 3 - Multiplication: 2 × 49 = 98
Step 4 - Subtraction: 98 - 10 = 88
Example 7
Calculate: 48 ÷ 4 × 3 - 6
Solution:
Division and multiplication (left to right):
48 ÷ 4 = 12
12 × 3 = 36
Then subtraction: 36 - 6 = 30
Example 8
Calculate: (12 - 4) ÷ 2 + 3 × 2²
Solution:
Brackets: 12 - 4 = 8
Indices: 2² = 4
Division: 8 ÷ 2 = 4
Multiplication: 3 × 4 = 12
Addition: 4 + 12 = 16
📝 Negative Numbers
Be careful: -3² means -(3²) = -9, NOT (-3)² = 9. The negative sign is not inside brackets.
Example 9
Calculate: -3² + 4 × 2
Solution:
Indices: -9 (the negative is not squared)
Multiplication: 4 × 2 = 8
Addition: -9 + 8 = -1
🧮 Worked Examples — Exam Style
BIDMAS: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right). Equal-priority operations run LEFT to RIGHT.
Calculate: 12 ÷ 4 × 3
Left to right: 12 ÷ 4 = 3, then 3 × 3 = 9 — NOT 12 ÷ 12 = 1
B1 for 9. The ÷-then-× order trap is examined precisely here — same priority, so left first.
Insert brackets to make 3 + 4 × 2 = 14 true.
(3 + 4) × 2 = 14 ✓
B1. Bracket questions test whether you understand WHICH operation BIDMAS prioritises.
Exam wording to expect: "Work out the value of …" (BIDMAS silently assumed), "insert brackets".
❓ Practice Questions
Q1: Calculate: 8 + 4 × 3
Q2: Calculate: (15 - 6) ÷ 3
Q3: Calculate: 2³ + 4² - 10
Q4: Calculate: 50 ÷ 5 × 2 + 3
Q5: Calculate: (2 + 3) × (8 - 4)²
Q6: Calculate: 20 − 3 × 4 + 6 ÷ 2
Q7: Calculate: (2 + 3)² − 4 × (6 ÷ 3)
✅ Answers
- 4 × 3 = 12, then 8 + 12 = 20
- 15 - 6 = 9, then 9 ÷ 3 = 3
- 8 + 16 - 10 = 14
- 50 ÷ 5 = 10, 10 × 2 = 20, 20 + 3 = 23
- 5 × 16 = 80
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. × before +: 8 + 12 = 20
Q2. Brackets first: 9 ÷ 3 = 3
Q3. Powers first: 8 + 16 − 10 = 14
Q4. Left to right: 50 ÷ 5 = 10, × 2 = 20, + 3 = 23
Q5. Brackets: 5 × 16 = 80
Q6. 20 − 12 + 3 = 11 — the − 3×4 binds first, then ÷ before final +
Q7. 25 − 4 × 2 = 25 − 8 = 17
📌 Key Facts to Memorise
- Order: Brackets → Indices → Divide/Multiply (left→right) → Add/Subtract (left→right)
- Same priority → work left to right (12 ÷ 4 × 3 = 9)
- A minus sign applies to the WHOLE product after it: 10 − 2 × 3 = 4
- Fraction bars act as brackets: (a + b)/c does the top first
- Check calculator usage: type expressions exactly, or use bracket keys
🗺️ In the Exam
BIDMAS: 1–3 marks on Paper 1, and silently inside every multi-step question. Left-to-right equal priority is the examined subtlety.
Links to: four operations (N2), substitution (A2), negative numbers (N2).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always check for brackets first
- Remember division and multiplication are equal - work left to right
- Watch out for negative signs outside brackets: -3² ≠ (-3)²
- Show your working clearly, one step per line
- Double-check by entering the calculation on your calculator
🧠 Problem-Solving Strategies
Problem-Solving
For BIDMAS problems, work through one operation at a time and write each step on a new line. If an expression has both multiplication and division (or addition and subtraction), work left to right. When a question asks you to insert brackets to make a statement true, test each possibility systematically.
Multi-Step Problem
Insert brackets to make this calculation correct: 3 + 4 × 5 − 2 = 33
Solution: Try (3 + 4) × 5 − 2 = 7 × 5 − 2 = 35 − 2 = 33 ✓. The brackets need to go around 3 + 4.
⚠️ Common Errors
Watch Out!
1. Wrong: 3 + 4 × 2 = 14 (working left to right) Correct: Multiplication first: 4 × 2 = 8, then 3 + 8 = 11
2. Wrong: -3² = 9 (squaring the negative) Correct: -3² = −(3²) = −9; it is NOT (−3)² = 9
3. Wrong: 12 ÷ 3 × 2 = 2 (doing multiplication first) Correct: Division and multiplication are equal priority — work left to right: 12 ÷ 3 = 4, then 4 × 2 = 8
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Two students evaluate 24 − 18 ÷ 3 + 2 × 5. Alice gets 4. Bob gets 28. (a) Show which student is correct. (b) Explain the mistake the other student made. (c) Use brackets in two different ways to make the answer equal to (i) 4 and (ii) 28.
(a) Correct BIDMAS: 18 ÷ 3 = 6, 2 × 5 = 10, then 24 − 6 + 10 = 28. Bob is correct.
(b) Alice did left to right: 24 − 18 = 6, 6 ÷ 3 = 2, 2 + 2 = 4, 4 × 5 = 20 — or similar, ignoring BIDMAS.
(c)(i) (24 − 18) ÷ (3 + 2) × 5 = 6 ÷ 5 × 5 = 6. Or: 24 − (18 ÷ (3 + 2)) × 5 = 24 − (18 ÷ 5) × 5 ≈ 6. To get exactly 4: (24 − 18 ÷ 3 + 2) × 5 needs adjustment. One way: 24 − (18 ÷ 3 + 2) × 5 = 24 − (6 + 2) × 5 = 24 − 40 = −16. Try: (24 − 18) ÷ 3 + 2 × 5 = 6 ÷ 3 + 10 = 2 + 10 = 12. For 4: ((24 − 18) ÷ 3 + 2) × 5 doesn't work. Correct: (24 − (18 ÷ 3 + 2 × 5)) = 24 − (6 + 10) = 8. For 4: (24 − 18) ÷ (3 + 2 × 5) = 6/13 — not 4. To get 4: (24 − 18 ÷ 3 + 2) is not 4... Let's try: 24 − (18 ÷ (3 + 2) × 5) = 24 − (18 ÷ 5 × 5) = 24 − 18 = 6. For exactly 4: (24 − 18) ÷ 3 + 2 = 4. (c)(ii) For 28, the original expression already gives 28 without brackets, but 24 − (18 ÷ 3) + (2 × 5) = 24 − 6 + 10 = 28.
Mark scheme: 2 marks for correct evaluation showing 28, 2 marks for clear explanation of the mistake, 2 marks for correct bracket placements
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A teacher writes the expression 2 + 3² × 4 on the board. Two students disagree on the answer.
(a) Calculate the correct value.
(b) Student A says "The answer is 56 because I did 2 + 3 = 5, then 5² = 25, then 25 × 4 = 100." What mistake did Student A make?
(c) Where should brackets be placed to make Student A's answer of 100 correct? Justify your answer.
Answers: (a) Indices first: 3² = 9, then 9 × 4 = 36, then 2 + 36 = 38. (b) Student A added before applying the index — they treated (2 + 3)² instead of 3². (c) (2 + 3)² × 4 = 5² × 4 = 25 × 4 = 100. This justifies Student A's working if brackets were intended.
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