Lesson 1: Number Systems and Place Value

Every calculation you will ever do — from splitting a restaurant bill in Manchester to computing mortgage interest in London — rests on the number system. This lesson builds the foundations for all of GCSE mathematics: the types of numbers, place value, ordering, rounding, and estimation. These are not just 'easy' topics to rush past — examiners love testing them in context, and fluency here makes every later topic faster. We use pounds and pence throughout, because money is where number sense meets real life.

Types of numbers

Numbers come in families. Natural numbers (1, 2, 3, ...) count things; integers (..., −2, −1, 0, 1, 2, ...) include negatives and zero — think of temperatures in the Scottish Highlands dropping to −6°C, or a bank overdraft of −£250. Rational numbers can be written as fractions (3/4, −7/2); irrational numbers cannot (π, √2). Together, rationals and irrationals make the real numbers — every point on the number line.

Prime numbers — divisible only by 1 and themselves (2, 3, 5, 7, 11, ...) — are the atoms of arithmetic: every integer factors uniquely into primes (prime factorisation).

Worked example: express 360 as a product of prime factors. Divide by the smallest primes: 360 = 36 × 10 = (2² × 3²) × (2 × 5) = 2³ × 3² × 5. Check: 8 × 9 × 5 = 360. ✓ This skill underpins HCF and LCM, which GCSE tests regularly.

Multiples and factors appear constantly in UK contexts: bus timetables (the 24 and 36 buses from Leeds city centre — when do they coincide? Find the LCM of 24 and 36: 24 = 2³ × 3, 36 = 2² × 3², so LCM = 2³ × 3² = 72 minutes), or sharing £48 equally among groups (factors of 48). The HCF is found by taking the lowest powers of the common primes: HCF(24, 36) = 2² × 3 = 12.

Place value and ordering

Our decimal system is positional: each digit's value depends on its place. In £4,073.58, the 4 is worth £4,000, the 7 is worth £70, the 3 is worth £3, the 5 is worth 50 pence (£0.50), and the 8 is worth 8 pence (£0.08). Reading large numbers confidently matters: the UK national debt exceeds £2,700,000,000,000 — two point seven trillion pounds. GCSE problems often ask you to write numbers in words or figures, where a misplaced zero costs marks.

Ordering decimals is a classic exam trap: which is larger, £4.09 or £4.9? Line up the decimal points and pad with zeros: 4.90 is greater than 4.09. Similarly, 0.505 is less than 0.55. Compare digit by digit from the decimal point, padding 0.55 to 0.550: the tenths are equal (5), but the hundredths differ (0 < 5), so 0.505 < 0.550.

Negative numbers follow one rule: further left on the number line means smaller. So −3 > −8, and −0.75 < −0.5. In context: a temperature of −8°C in Braemar is colder than −3°C in London; an overdraft of −£1,200 is worse than −£400. When adding and subtracting negatives, use the number-line picture: 5 − 8 = −3 (move left), −4 − 6 = −10, −3 + 9 = 6. Two minuses make a plus only in multiplication and division: (−4) × (−3) = 12.

Rounding and significant figures

Rounding is everywhere in British life: shop prices (£2.99), petrol (142.9p per litre), and government statistics. The rule: look at the next digit — 5 or more rounds up. Round £47.68 to the nearest pound: £48. Round 0.0467 to two significant figures: the first significant digit is 4, the next digit is 6 (≥ 5), so 0.047. Significant figures count from the first non-zero digit: 0.004070 has four significant figures (4, 0, 7, 0).

GCSE distinguishes decimal places from significant figures — a common confusion. 3.14159 to 3 decimal places is 3.142; to 3 significant figures it is 3.14. Money is usually given to 2 decimal places (£7.50), while measurements use significant figures (a 2.5 km run). Exam tip: always check which one the question asks for, and show the unrounded value first if you use a calculator.

Worked example: a train ticket from Birmingham to London costs £68.40. Round to (a) the nearest £10, (b) 2 significant figures. (a) £68.40 → £70 (the units digit 8 ≥ 5, so the tens round up). (b) 2 s.f.: the first two digits are 6, 8; the next digit is 4 (< 5), so £68.

Bounds (Higher tier). A measurement rounded to 2 significant figures, such as 8.4 cm, is not exact: the true value lies between 8.35 cm and 8.45 cm. GCSE calls these the lower bound and upper bound, and Higher papers use them in error-interval questions: if x = 8.4 (to 1 decimal place), write 8.35 ≤ x < 8.45.

Estimation — the art of the sensible answer

Estimation means replacing awkward numbers with friendly ones before calculating: 48 × 21 ≈ 50 × 20 = 1000. This is a survival skill in exams (checking calculator work) and in life (is £1,950 a sensible quote for rewiring a flat? 10 sockets × £150 + 2 days' labour × £250 = £2,000 — yes, sensible). GCSE awards marks for estimation, usually one mark for the rounding and one for the answer.

The method: round each number to one significant figure, then compute. Worked example: estimate 4.87 × 19.3. Round: 5 × 20 = 100. (Actual: 93.99 — close.) For division: 412 ÷ 19 ≈ 400 ÷ 20 = 20. For expressions with both, estimate top and bottom separately: 297² ÷ 58 ≈ 300² ÷ 60 = 90,000 ÷ 60 = 1,500.

Worked example: a school prints 483 worksheets at 7.6p each. Estimate the cost. Round to 1 s.f.: 500 × 8p = 4,000p = £40. (Actual: £36.71 — the estimate is the right size, so a calculator answer of £367 or £3.67 would ring alarm bells.)

Estimation also catches calculator errors: if you compute 12% VAT on £240 and get £2,880, estimation (10% of £240 = £24, so 12% ≈ £29) tells you instantly that you misplaced the decimal. Examiners report that the most common cause of lost marks is not wrong methods but unchecked arithmetic — estimate first, calculate second, compare last.

Order of operations (BIDMAS)

BIDMAS — Brackets, Indices, Division/Multiplication, Addition/Subtraction — is the universal agreement that makes 3 + 4 × 5 equal 23, not 35. Multiplication comes before addition: 3 + (4 × 5) = 23. Brackets override: (3 + 4) × 5 = 35. Indices come before multiplication: 2 × 3² = 2 × 9 = 18, not 36.

Worked example: compute 48 ÷ (6 − 2)² + 3 × 7. Brackets first: 6 − 2 = 4. Indices: 4² = 16. Division: 48 ÷ 16 = 3. Multiplication: 3 × 7 = 21. Addition: 3 + 21 = 24. Work strictly left to right within each BIDMAS level, and show each step — method marks are awarded for correct intermediate steps even if the final answer slips.

Real context: a phone contract costs £12 per month plus 8p per minute. The bill for 150 minutes is 12 + 0.08 × 150 = 12 + 12 = £24 — not (12 + 0.08) × 150. BIDMAS is not an abstract rule; it is how every formula, from phone bills to physics, is read. Misapply it and you misread the world.

Problem solving — a trip budget, start to finish

Plan first, calculate second: a school trip needs (1) a coach cost, (2) ticket costs, (3) a per-pupil charge. A coach costs £420 flat. Entry tickets cost £7.25 per pupil for 47 pupils; 5 staff go free. Tickets: 47 × £7.25 = £340.75 (compute exactly: 47 × 7 = £329, 47 × 0.25 = £11.75). Total: 420 + 340.75 = £760.75. Per pupil: 760.75 ÷ 47 = £16.19 (2 d.p.). The school charges £16.50 — sensible rounding up, because charging £16.19 exactly is impractical and a small surplus covers surprises. Estimate check first: 47 × £7 ≈ £330, plus £420 ≈ £750 total, ≈ £16 per pupil — the exact answer sits comfortably beside it. Notice the workflow: estimate, exact calculation, then a rounding decision made for a stated reason.

Bounds in action (Higher)

Bounds turn one rounded measurement into a range of possibilities — and exam questions ask for best and worst cases. A hall is 12.4 m by 8.3 m, both to 1 decimal place. The true length lies in [12.35, 12.45) and the width in [8.25, 8.35). Maximum possible area = 12.45 × 8.35 = 103.9575 ≈ 104.0 m²; minimum = 12.35 × 8.25 = 101.8875 ≈ 101.9 m². So the area is between about 101.9 m² and 104.0 m² — a spread of 2 m² from measurements that 'look' exact. Flooring ordered from the middle value (12.4 × 8.3 = 102.9 m²) could leave you short: order from the upper bound. This is why builders add a wastage percentage — bounds mathematics in a hi-vis vest.

Common mistakes

Examiners' reports return to the same number errors every year:

• Decimal places vs significant figures. 0.00456 to 2 s.f. is 0.0046, but to 2 d.p. it is 0.00. Read which one is asked.
• Ordering decimals by length. 0.55 is bigger than 0.505 even though 505 'looks bigger' than 55. Pad with zeros and compare place by place.
• Rounding by chain. Round once, from the original number. Rounding 4.449 to 2 d.p. gives 4.45 directly (third digit 9 ≥ 5) — do not round 4.449 → 4.45 → 4.5 in stages.
• BIDMAS slips. 2 + 3 × 4 = 14, not 20; 2 × 3² = 18, not 36. Multiplication and indices outrank addition.
• Negative ordering. −8 is smaller than −3. 'Bigger debt' means a more negative number.
• Estimation with no rounding shown. The mark is for the rounded calculation: write 50 × 20 = 1000, not just the final estimate.

Exam Experience

GCSE papers test these skills in short, sharp questions. Try each one, then check the mark scheme.

Question 1. Estimate the value of (48.7 × 11.9) ÷ 5.8. [2 marks]

Question 2. (a) Write down the prime factorisation of 84. (b) Write down the prime factorisation of 126. (c) Hence show that the HCF of 84 and 126 is 42. [3 marks]

Question 3. A cinema sells 237 tickets at £8.50 each. (a) Estimate the total amount taken. (b) Is your estimate an overestimate or an underestimate? Give a reason. [3 marks]

Mark scheme.

• Q1: M1 for rounding all three numbers to 1 s.f. (50, 10 or 12, 5 or 6). A1 for an answer of 100 (accept 80–120 following your rounding, e.g. 50 × 12 ÷ 6 = 100).
• Q2: M1 for a correct prime factorisation of one number (84 = 2² × 3 × 7). M1 for both (126 = 2 × 3² × 7). A1 for selecting the lowest powers of the common primes: 2 × 3 × 7 = 42, so the HCF is 42 as required.
• Q3: M1 for rounding both values (e.g. 237 → 200 or 240, £8.50 → £9). A1 for a correct product of your rounded values (e.g. 240 × 9 = £2,160). B1 for 'overestimate' with the reason that both numbers were rounded up.