How do I find the LCM and HCF quickly?

Use prime factorisation: write each number as a product of primes. For the HCF, multiply the lowest power of common primes. For the LCM, multiply the highest power of all primes present. For example, for 12 (2^2 × 3) and 18 (2 × 3^2), HCF = 2 × 3 = 6, LCM = 2^2 × 3^2 = 36. Practice with numbers like 24 and 36 to get faster.

What is the difference between a factor and a multiple?

A factor of a number divides it exactly without leaving a remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4. A multiple is the result of multiplying a number by an integer. For example, 12 is a multiple of 3 because 3 × 4 = 12. So factors are smaller than or equal to the number, while multiples are larger than or equal to the number.

How do I convert a recurring decimal to a fraction?

Let x equal the recurring decimal. Multiply by a power of 10 to shift the decimal point so that the recurring part aligns. Subtract the original equation to eliminate the recurring part, then solve for x. For example, 0.3̅ (0.333...): let x = 0.333..., so 10x = 3.333..., subtract: 9x = 3, so x = 1/3. For more complex ones like 0.1̅4̅ (0.141414...), use 100x.

When do I use BIDMAS?

BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) tells you the order of operations. Use it whenever you have a calculation with more than one operation. For example, 3 + 4 × 2 = 3 + 8 = 11, not 14. Remember that division and multiplication have equal priority, so work left to right. Similarly for addition and subtraction.

How do I calculate percentage change?

Percentage change = (change in value ÷ original value) × 100%. For example, if a price increases from £50 to £60, the change is £10, so percentage increase = (10 ÷ 50) × 100% = 20%. If it decreases, the change is negative, giving a percentage decrease. Always use the original value as the denominator, not the new value.

What is standard form and how do I use it?

Standard form is a way of writing very large or very small numbers as A × 10^n, where 1 ≤ A < 10 and n is an integer. For example, 3000 = 3 × 10^3, and 0.0005 = 5 × 10^-4. To multiply numbers in standard form, multiply the A parts and add the exponents: (2 × 10^3) × (3 × 10^4) = 6 × 10^7. For division, divide the A parts and subtract exponents.

Number

AQA

GCSE

Algebra involves the use of symbols and notation to represent mathematical relationships, expressions, and functions. Students learn to manipulate algebraic expressions, solve various types of equations and inequalities, and interpret graphical representations of linear, quadratic, and other functions.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

Number is a foundational topic in AQA GCSE Mathematics, covering the properties and operations of integers, fractions, decimals, percentages, and surds. It includes understanding place value, the four operations (addition, subtraction, multiplication, division), factors, multiples, primes, powers, roots, and standard form. Mastery of Number is essential because it underpins all other areas of maths, from algebra to geometry, and is heavily tested across all three exam papers.

In the AQA specification, Number accounts for approximately 15-20% of the total marks, with questions ranging from basic arithmetic to problem-solving involving real-life contexts like finance and measurement. Students must be confident with both mental and written methods, as well as using calculators efficiently. Topics such as rounding, error intervals, and compound measures (e.g., speed, density) also fall under Number, making it a versatile and high-impact area to revise.

Why does Number matter? Beyond exams, numerical literacy is crucial for everyday life—managing budgets, interpreting data, and making informed decisions. In the wider curriculum, Number skills are prerequisites for topics like ratio, proportion, and rates of change. By mastering Number, students build the confidence and fluency needed to tackle more complex mathematical challenges.

Key Concepts

Core ideas you must understand for this topic

→Place value and the four operations: Understand the value of digits in decimals and large numbers; perform addition, subtraction, multiplication, and division accurately with integers and decimals.
→Factors, multiples, and primes: Know how to find the highest common factor (HCF) and lowest common multiple (LCM) using prime factorisation, and recognise prime numbers up to 100.
→Powers and roots: Work with square numbers, cube numbers, and their roots; understand index laws for multiplying and dividing powers (e.g., a^m × a^n = a^(m+n)).
→Fractions, decimals, and percentages: Convert between these forms fluently; perform calculations including finding a fraction of an amount, percentage increase/decrease, and reverse percentages.
→Standard form: Write very large or very small numbers as A × 10^n where 1 ≤ A < 10, and perform calculations with standard form using calculator and non-calculator methods.

What You Need to Demonstrate

Key skills and knowledge for this topic

Correct use and interpretation of algebraic notation
Accurate substitution of numerical values into formulae
Correct simplification of expressions by collecting like terms and using laws of indices
Correct expansion of brackets and factorisation of expressions
Accurate solution of linear and quadratic equations
Correct identification of gradients and intercepts from linear graphs
Accurate plotting of functions and interpretation of graphical features
Correct derivation of equations from word problems

Marking Points

Key points examiners look for in your answers

Correct use and interpretation of algebraic notation
Accurate substitution of numerical values into formulae
Correct simplification of expressions by collecting like terms and using laws of indices
Correct expansion of brackets and factorisation of expressions
Accurate solution of linear and quadratic equations
Correct identification of gradients and intercepts from linear graphs
Accurate plotting of functions and interpretation of graphical features
Correct derivation of equations from word problems

Examiner Tips

Expert advice for maximising your marks

💡Always show your working out, as method marks are awarded even if the final answer is incorrect
💡Check your answers by substituting values back into the original equation
💡Ensure you are familiar with the calculator functions for solving equations if permitted
💡Read the question carefully to see if an exact answer (e.g., in terms of pi or surds) is required
💡Use a ruler for drawing straight-line graphs and ensure axes are clearly labelled
💡Show all working: Even if you use a calculator, write down the steps (e.g., converting to common denominators). Marks are awarded for method, and if your final answer is wrong, you can still get method marks.
💡Check your answers for reasonableness: After a calculation, ask yourself if the answer makes sense. For example, 15% of £200 should be less than £200, and if you get £300, you've likely multiplied instead of divided.
💡Use the correct rounding: When a question says 'give your answer to 3 significant figures', don't round to 3 decimal places. Know the difference: significant figures start from the first non-zero digit, while decimal places count after the decimal point.

Common Mistakes

Pitfalls to avoid in your exam answers

Errors in sign when expanding brackets or solving equations
Confusing the rules for indices (e.g., adding instead of multiplying)
Incorrectly identifying the gradient or intercept from a linear equation
Failing to include all solutions for quadratic equations
Misinterpreting inequality signs on number lines or graphs
Errors in substitution, particularly with negative numbers
Misconception: Multiplying by 0.1 makes a number smaller, so it must be division. Correction: Multiplying by 0.1 is the same as dividing by 10, which does make the number smaller. Students often confuse the effect of multiplying by decimals less than 1.
Misconception: When adding fractions, you add the numerators and denominators separately. Correction: Only add numerators when denominators are the same; if not, find a common denominator first. For example, 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2, not 2/9.
Misconception: A negative number squared is negative. Correction: (-3)^2 = 9, because multiplying two negatives gives a positive. Students often forget the brackets and write -3^2 = -9, which is correct only if the negative is not squared.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Basic arithmetic: confident addition, subtraction, multiplication, and division of whole numbers.
•Understanding of place value up to millions and down to thousandths.
•Familiarity with simple fractions (e.g., halves, quarters) and percentages (e.g., 50%, 25%).

Study Guide Available

Comprehensive revision notes & examples

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Likely Command Words

How questions on this topic are typically asked

Solve

Simplify

Expand

Factorise

Plot

Sketch

Rearrange

Show that

Ready to test yourself?

Practice questions tailored to this topic

Number

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

Before You Start

Study Guide Available

Likely Command Words

Ready to test yourself?

Related Topics in AQA GCSE Mathematics

E2E stub concept

Algebra

Geometry and measures

Probability

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

Number

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

How do I find the LCM and HCF quickly?

What is the difference between a factor and a multiple?

How do I convert a recurring decimal to a fraction?

When do I use BIDMAS?

How do I calculate percentage change?

What is standard form and how do I use it?

Before You Start

Study Guide Available

Likely Command Words

Ready to test yourself?

Related Topics in AQA GCSE Mathematics

E2E stub concept

Algebra

Geometry and measures

Probability