Chapter 1-17: Number Exam Technique

📝 Paper 1 Exam Technique (Non-Calculator - May 15, 2025)

Paper 1 is non-calculator - you must master mental arithmetic and written methods
Time is critical - aim for 1 minute per mark (80 marks in 90 minutes = 1.125 min/mark)
Show all working - even if wrong, method marks can be awarded
Check answers - use estimation to verify your answers make sense
BIDMAS/BODMAS - always appears in Paper 1, usually worth 2-3 marks
Standard form - frequently tested in Papers 2 & 3 (calculator allowed)
Surds (Higher) - Paper 1 favorites, often 3-4 mark questions
Bounds (Higher) - typically Paper 3, multi-step problem worth 4-6 marks

⏱️ Time Management Strategy

Foundation Tier: Aim for 1-1.5 minutes per mark. This gives you time to check answers.

Higher Tier: Harder questions may need 2 minutes per mark. Budget time carefully.

Paper 1 (90 minutes, 80 marks):

• 1-2 mark questions: 1-2 minutes each
• 3-4 mark questions: 3-5 minutes each
• 5-6 mark questions: 6-8 minutes each
• Leave 10 minutes at the end to check

Real Past Paper Questions

Question 1 (AQA 2023 Paper 1) 2 marks

Work out:

3 + 4 × 5

✓ Mark Scheme

Step 1: Multiply first (BIDMAS): 4 × 5 = 20 (1 mark for method)

Step 2: Add: 3 + 20 = 23 (1 mark for correct answer)

Answer: 23

⚠️ Common Mistakes (Examiner Reports)

Adding first: 3 + 4 = 7, then 7 × 5 = 35 (WRONG - ignores BIDMAS)
Not showing working - loses method mark if answer is wrong

Question 2 (Edexcel 2022 Paper 1) 3 marks

Work out the value of:

(2 + 3)² - 4 × 2

✓ Mark Scheme

Step 1: Brackets first: 2 + 3 = 5 (1 mark)

Step 2: Indices: 5² = 25 (1 mark)

Step 3: Multiply: 4 × 2 = 8

Step 4: Subtract: 25 - 8 = 17 (1 mark for correct answer)

Answer: 17

⚠️ Common Mistakes (Examiner Reports)

Doing 2 + 3² = 2 + 9 = 11 (forgetting to do brackets first)
Getting 5² = 10 (confusing squaring with multiplying by 2)
Subtracting before multiplying: 25 - 4 = 21, then 21 × 2 = 42

Question 3 (AQA 2023 Paper 2) 2 marks

Write 0.00034 in standard form.

✓ Mark Scheme

Step 1: Move decimal point 4 places to the right to get 3.4 (1 mark)

Step 2: Count moves: 4 places = 10⁻⁴ (1 mark for correct answer)

Answer: 3.4 × 10⁻⁴

⚠️ Common Mistakes (Examiner Reports)

Writing 34 × 10⁻⁵ (number must be between 1 and 10)
Using positive power: 3.4 × 10⁴ (for small numbers, power is negative)
Writing 3.4 × 10⁻⁵ (counting wrong number of decimal places)

Question 4 (Higher - OCR 2022 Paper 1) 3 marks

Simplify:

√50 + √18

✓ Mark Scheme

Step 1: Simplify √50 = √(25×2) = 5√2 (1 mark)

Step 2: Simplify √18 = √(9×2) = 3√2 (1 mark)

Step 3: Add: 5√2 + 3√2 = 8√2 (1 mark)

Answer: 8√2

⚠️ Common Mistakes (Examiner Reports)

Adding under the root: √(50 + 18) = √68 (WRONG)
Leaving answer as 5√2 + 3√2 (not fully simplified)
Incorrectly simplifying to √50 = 25 or √18 = 9

Question 5 (Higher - AQA 2023 Paper 1) 4 marks

Rationalize the denominator:

12/(3 + √3)

✓ Mark Scheme

Step 1: Multiply by conjugate: 12/(3 + √3) × (3 - √3)/(3 - √3) (1 mark)

Step 2: Numerator: 12(3 - √3) = 36 - 12√3 (1 mark)

Step 3: Denominator: (3 + √3)(3 - √3) = 9 - 3 = 6 (1 mark)

Step 4: Simplify: (36 - 12√3)/6 = 6 - 2√3 (1 mark)

Answer: 6 - 2√3

⚠️ Common Mistakes (Examiner Reports)

Not multiplying by the conjugate (3 - √3)
Getting (3 + √3)(3 - √3) = 9 - 3 wrong (should be 9 - 3 = 6)
Not simplifying final fraction correctly
Leaving answer as (36 - 12√3)/6 without simplifying

Question 6 (Foundation - Edexcel 2022 Paper 1) 2 marks

Write down all the factors of 24.

✓ Mark Scheme

Method: Find all pairs of numbers that multiply to give 24

1 × 24, 2 × 12, 3 × 8, 4 × 6

Answer: 1, 2, 3, 4, 6, 8, 12, 24 (2 marks for all correct, 1 mark for at least 5)

⚠️ Common Mistakes (Examiner Reports)

Missing 1 or 24 (common omission)
Listing only even factors: 2, 4, 6, 8, 12, 24
Including multiples instead: 24, 48, 72...

Question 7 (Both Tiers - AQA 2023 Paper 1) 3 marks

Find the HCF and LCM of 12 and 18.

✓ Mark Scheme

Prime factorization method:

12 = 2² × 3

18 = 2 × 3²

HCF: Take lowest powers: 2¹ × 3¹ = 6 (1 mark)

LCM: Take highest powers: 2² × 3² = 4 × 9 = 36 (2 marks)

⚠️ Common Mistakes (Examiner Reports)

Confusing HCF and LCM (getting them backwards)
Multiplying 12 × 18 = 216 for LCM (inefficient and often wrong)
Not using prime factorization for systematic approach

Question 8 (Higher - OCR 2022 Paper 3) 5 marks

A rectangle has length 8.5 cm (to 1 decimal place) and width 4.2 cm (to 1 decimal place).

Calculate the upper bound for the area of the rectangle.

✓ Mark Scheme

Step 1: Upper bound of length = 8.5 + 0.05 = 8.55 cm (1 mark)

Step 2: Upper bound of width = 4.2 + 0.05 = 4.25 cm (1 mark)

Step 3: For maximum area, use upper bounds for both (1 mark)

Step 4: Area = 8.55 × 4.25 = 36.3375 cm² (1 mark)

Step 5: Don't round the answer (1 mark for keeping all decimal places)

Answer: 36.3375 cm²

⚠️ Common Mistakes (Examiner Reports)

Using 8.6 and 4.3 as upper bounds (should add 0.05, not 0.1)
Rounding final answer to 36.34 or 36.3 (must give exact value)
Using lower bounds instead of upper bounds
Adding the measurements instead of multiplying

Question 9 (Higher - Edexcel 2023 Paper 1) 3 marks

Express 0.1̇8̇ as a fraction in its simplest form.
(The dots indicate recurring decimals: 0.181818...)

✓ Mark Scheme

Step 1: Let x = 0.181818... (1 mark)

Step 2: 100x = 18.181818... (multiply by 100 as 2 digits recur)

Step 3: 100x - x = 18, so 99x = 18 (1 mark)

Step 4: x = 18/99 = 2/11 (simplified) (1 mark)

Answer: 2/11

⚠️ Common Mistakes (Examiner Reports)

Multiplying by 10 instead of 100 (need to match number of recurring digits)
Not simplifying 18/99 to 2/11
Writing 18/100 (treating it as 0.18 not 0.1̇8̇)

Question 10 (Foundation - AQA 2022 Paper 1) 2 marks

Estimate the value of:

48.7 × 5.2

✓ Mark Scheme

Step 1: Round each number to 1 significant figure (1 mark)

48.7 ≈ 50

5.2 ≈ 5

Step 2: Multiply: 50 × 5 = 250 (1 mark)

Answer: 250

(Actual answer: 253.24, so estimate is very close)

⚠️ Common Mistakes (Examiner Reports)

Calculating exact answer 253.24 (question asks for estimate)
Rounding to 49 × 5 = 245 (should round to 1 s.f.)
Not showing the rounding step (loses method mark)

📊 What Different Grades Look Like

Grade 4-5 (Foundation/Higher):

• Can do basic BIDMAS questions (Questions 1-2)

• Understands factors and multiples (Question 6)

• Can write numbers in standard form (Question 3)

• Can estimate calculations (Question 10)

Grade 6-7 (Higher):

• Can simplify surds (Question 4)

• Understands HCF/LCM using prime factorization (Question 7)

• Can work with upper and lower bounds (Question 8)

Grade 8-9 (Higher):

• Can rationalize denominators with surds (Question 5)

• Can convert recurring decimals to fractions (Question 9)

• Applies bounds to complex multi-step problems

• Shows clear, logical working under exam pressure

📝 Key Command Words in Number Questions

Work out Calculate the answer and show all your working clearly

Simplify Write in simplest form (e.g., surds, fractions)

Estimate Round to 1 significant figure first, then calculate

Express Write in the specified form (e.g., standard form, fraction)

Find Calculate and show method (similar to "work out")

Calculate Work out the answer, showing all steps

⚡ Quick Revision Summary

BIDMAS/BODMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction
Standard Form: a × 10ⁿ where 1 ≤ a < 10
Surds (Higher): √(a×b) = √a × √b, simplify by finding square factors
Rationalize (Higher): Multiply by conjugate (a + √b)(a - √b) = a² - b
Bounds (Higher): For max area use max × max, for min use min × min
Recurring decimals (Higher): Multiply by 10ⁿ where n = number of recurring digits
Estimation: Round to 1 s.f. then calculate
HCF: Lowest powers in prime factorization
LCM: Highest powers in prime factorization
Time per mark: ~1-1.5 minutes, check answers with estimation