AQA GCSE Mathematics: Number (8300)

This guide covers Topic 1 Number, the first of six topic areas in AQA GCSE Mathematics (8300), first teaching September 2015. This content matches the Department for Education’s Mathematics GCSE subject content document and is common to all exam boards – the six topic areas and their weighting are set by Ofqual, not AQA independently. The qualification is tiered (Foundation and Higher), assessed across three written papers, with the same tier throughout.

Where this fits in 8300

Number carries roughly 25% of Foundation tier marks and 15% of Higher tier marks – joint largest at Foundation, level with ratio, proportion and rates of change (also 25%), the two together accounting for half of the Foundation marks. Because all content can be assessed on any of the three papers, number skills such as place value, ordering and rounding underpin questions across every other topic area, from algebra to statistics.

Syllabus coverage

AQA GCSE MATHEMATICS (8300) — TOPIC 1 NUMBER

• 3.1.1 Structure and calculation — ordering positive and negative integers, decimals and fractions; the four operations applied to integers, decimals and fractions; place value; relationships between operations including inverses; conventional notation for priority of operations; prime numbers, factors and multiples; and systematic listing strategies
• 3.1.2 Fractions, decimals and percentages — converting between fractions, decimals and percentages, and calculating with each
• 3.1.3 Measures and accuracy — using standard units of measure and related concepts including estimation, error intervals and (Higher tier only) upper and lower bounds

Four techniques in this topic sit in the specification’s Higher content only column and are labelled (Higher tier) below: calculating with fractional indices (N7), calculating exactly with surds and rationalising denominators (N8), changing recurring decimals into fractions (N10), and upper and lower bounds (N16). Everything else here is Foundation content, assessed at both tiers.

How to approach it

Structure and calculation (3.1.1) is the largest and most foundational sub-topic – fluency with the four operations and correct use of priority of operations (brackets, powers, roots) needs to be automatic before tackling later topics, since errors here compound into algebra and statistics questions. Fractions, decimals and percentages (3.1.2) is tested heavily in context (for example, household finance calculations such as VAT, profit and interest), so practise converting fluently between all three forms rather than memorising isolated procedures. Measures and accuracy (3.1.3) rewards precision – know the difference between rounding, truncation and bounds, since Higher tier questions frequently ask for the upper and lower bounds of a measured or rounded quantity.

Official syllabus

AQA GCSE Mathematics (8300) specification, for exams from May/June 2017 onwards — aqa.org.uk.

Types of number and place value

Integers, factors, multiples, primes, squares and cubes underpin the whole topic. A prime has exactly two factors, so 1 is not prime and 2 is the only even prime.

Every integer can be written as a product of prime factors, which is the fastest route to HCF and LCM:

60 = 2^2 x 3 x 5
72 = 2^3 x 3^2

HCF = lowest power of each shared prime = 2^2 x 3 = 12
LCM = highest power of every prime present = 2^3 x 3^2 x 5 = 360

Fractions, decimals and percentages

Converting fluently between the three forms is assumed throughout the paper. To add or subtract fractions use a common denominator; to divide, multiply by the reciprocal.

Percentage change problems appear constantly:

percentage change = (change / original) x 100
multiplier for a 15% increase = 1.15
multiplier for a 15% decrease = 0.85

For reverse percentage questions, divide by the multiplier rather than multiplying. Compound interest uses repeated multipliers: value after n years = P x (multiplier)^n.

Converting a recurring decimal to a fraction (Higher tier, N10) uses a standard algebraic technique. To convert 0.454545… (written 0.4̇5̇) to a fraction: let x = 0.454545…, so 100x = 45.454545… (multiplying by 100 shifts the decimal point past one full repeating block, since the block has two digits). Subtracting the first equation from the second eliminates the repeating part: 100x − x = 45.4545… − 0.4545…, giving 99x = 45, so x = 45/99, which simplifies to 5/11. The power of 10 used to multiply by must match the length of the repeating block — a two-digit block needs ×100, a one-digit block needs ×10 — or the subtraction will not cancel the recurring part correctly.

Indices, standard form and surds

Index laws: a^m x a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), a^0 = 1 and a^-n = 1/a^n (calculating with roots and integer indices is Foundation, N7). (Higher tier, N7) a^(1/n) is the nth root, and calculating with fractional indices is Higher only.

Standard form is A x 10^n where 1 <= A < 10. (Higher tier, N8) Surds are exact and should be simplified and rationalised rather than evaluated as decimals.

Rounding, bounds and estimation

Estimate by rounding each value to 1 significant figure. A length given as 8.4 cm to the nearest 0.1 cm lies in the error interval 8.35 <= length < 8.45, which is Foundation content (N15). (Higher tier, N16) Describing 8.35 and 8.45 as the lower and upper bounds and using bounds in a calculation is Higher only: when dividing, the largest result comes from the largest numerator with the smallest denominator.

Worked example

A jacket costs £68 after a 15% reduction. Find the original price.

The sale price is 85% of the original, so the multiplier is 0.85.

original = 68 / 0.85 = £80

Check by working forwards: 80 x 0.85 = 68. This check catches the most common error, which is finding 15% of 68 and adding it on — that gives £78.20 and is wrong, because the percentage applies to the original, not the sale price.

Common mistakes

Multiplying instead of dividing in reverse percentage questions. Treating 1 as a prime number. Adding fractions by adding numerators and denominators separately. Writing standard form with A outside the range 1 to 10, such as 12 x 10^3. Rounding partway through a calculation instead of at the end. (Higher tier) Taking the upper bound of a division as upper/upper rather than upper/lower.

Quick revision checklist

• Write any integer as a product of primes and use it to find HCF and LCM.
• Convert confidently between fractions, decimals and percentages.
• Use multipliers for increase, decrease, reverse percentage and compound interest.
• Apply the index laws, including negative powers; (Higher tier) fractional powers.
• Write and calculate with numbers in standard form.
• (Higher tier) Simplify and rationalise surds.
• (Higher tier) Convert recurring decimals to fractions.
• Write error intervals; (Higher tier) find upper and lower bounds and use them correctly in a calculation.