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 – the single largest weighting at Foundation. 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 and bounds

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 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, a^-n = 1/a^n, and a^(1/n) is the nth root.

Standard form is A x 10^n where 1 <= A < 10. 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. Bounds matter for accuracy questions: a length given as 8.4 cm to the nearest 0.1 cm has a lower bound of 8.35 and an upper bound of 8.45. 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. 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 all index laws, including negative and fractional powers.
• Write and calculate with numbers in standard form.
• Simplify and rationalise surds.
• Find upper and lower bounds and use them correctly in a calculation.