Notation, vocabulary and manipulation — AQA GCSE Mathematics

This topic covers the essential language and notation of algebra, including index laws, expanding and factorising, rearranging formulas, and solving linear inequalities. It is fundamental to all other areas of GCSE Mathematics because it provides the tools to express mathematical ideas clearly and manipulate expressions accurately.

Mastering notation and manipulation is crucial for success in algebra, geometry, statistics and problem-solving questions. AQA examiners expect students to use correct mathematical vocabulary and notation, and to show clear, logical steps when manipulating expressions and equations.

• →Index laws: a^m × 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.
• →Expanding brackets: multiply each term in the first bracket by each term in the second, then simplify by collecting like terms.
• →Factorising: finding common factors or using the difference of two squares, and factorising quadratic expressions into double brackets.
• →Rearranging formulas: changing the subject by performing inverse operations to both sides, keeping the equation balanced.
• →Inequality notation: <, >, ≤, ≥ and their meanings, including solving and representing solutions on a number line.

• Recognise that ab is the product a × b, not a + b.
• Interpret 3y as y + y + y and as 3 × y.
• Interpret a² as a × a, a³ as a × a × a and a²b as a × a × b.
• Interpret a/b as a ÷ b.
• Write coefficients as fractions rather than decimals, for example (1/2)x not 0.5x.
• Use brackets to group terms so the grouped quantity is treated as a single unit.
• Replace each letter in the expression or formula with its given numerical value, using brackets for negative values.
• Apply the correct order of operations (BIDMAS/BODMAS), ensuring indices are calculated before multiplication and division.
• Evaluate scientific formulae such as F = ma or v = fλ accurately by substituting the given quantities.
• Keep units consistent throughout the calculation and interpret the final numerical result in its correct context.
• Check the computed value against the multiple-choice options, ensuring no sign errors were made.
• Identifies an expression as terms combined by operations with no equals or inequality sign, for example 3x² + 5y.
• Identifies an equation as a statement of equality between two expressions, for example 4x + 1 = 13, and recognises that it can be solved.
• Identifies a formula as a rule linking quantities, for example A = πr², and recognises that it is used by substitution or rearrangement.
• Identifies an inequality as a comparison using <, >, ≤ or ≥, for example 2x − 3 ≥ 7.
• Identifies a term as a part of an expression separated by + or − signs, for example 3x² and 5y in 3x² + 5y.
• Identifies a factor as a number or expression that divides another exactly, for example 3 and x as factors of 3x².
• Defines an identity as an algebraic statement that is true for all values of the variable.
• Recognises the identity symbol ≡ and distinguishes it from the equals sign =.
• Tests a candidate identity by substituting values, for example x = 0 and x = 1, and checking that both sides agree.
• Distinguishes an identity such as 2(x + 3) ≡ 2x + 6 from an equation such as 2x + 6 = 10, which has a single solution.
• Explains that an identity cannot be solved for one value of the variable because it holds for every value.
• Collect like terms by adding or subtracting coefficients of terms with identical variable parts, for example 7a + 4a - 3a = 8a.
• Multiply a single term over a bracket using the distributive law, for example 5(3x - 2) = 15x - 10.
• Take out the highest common factor from every term, for example 12x³ + 8x² = 4x²(3x + 2).
• Apply the laws of indices to products and powers of algebraic terms, for example a⁴ × a³ = a⁷ and (b²)⁵ = b¹⁰.
• Apply the laws of indices to quotients of algebraic terms, for example c⁶ ÷ c² = c⁴.
• Simplify expressions that combine sums, products and powers, for example 2x(3x² + 4x) = 6x³ + 8x².
• Expand the product of two binomials by multiplying every term in the first bracket by every term in the second, for example (x + 2)(x + 6) = x² + 8x + 12.
• Collect like terms after expansion, for example (2x + 1)(x + 4) = 2x² + 9x + 4.
• Factorise x² + bx + c by finding two numbers with product c and sum b, for example x² + 9x + 20 = (x + 4)(x + 5).
• Recognise and factorise the difference of two squares, for example x² − 36 = (x + 6)(x − 6).
• Expand and simplify products involving surds (Higher tier only), for example (3 + √2)(3 − √2) = 9 − 2 = 7.
• Check a factorisation by expanding the brackets and confirming the original expression is recovered.
• Expanding two binomials correctly: multiply each term in the first bracket by each term in the second, then collect like terms, e.g. (x + 4)(x − 2) = x² + 2x − 8.
• Expanding three or more binomials by multiplying two brackets first and then multiplying the resulting expression by the remaining binomial.
• Factorising ax² + bx + c by finding two numbers with product ac and sum b, splitting the middle term, and factorising in pairs, e.g. 6x² + 11x + 4 = (3x + 4)(2x + 1).
• Simplifying surds by extracting the largest square factor, e.g. √50 = 5√2, and rationalising denominators where required.
• Simplifying algebraic fractions by factorising numerator and denominator fully and cancelling common factors, stating any excluded values where relevant.
• Substituting numerical values into standard formulae correctly, including using the correct order of operations and units.
• Rearranging a formula by applying inverse operations to both sides to isolate the required variable.
• Handling formulae involving fractions by multiplying both sides by the denominator before further rearrangement.
• Handling formulae involving powers or roots by taking the appropriate root or power of both sides, e.g. making r the subject of A = πr² gives r = √(A/π).
• Checking the rearranged formula by substituting a known value or by reversing the rearrangement.
• Distinguishes an equation (true for specific values) from an identity (true for all values), using the ≡ symbol correctly.
• Expands brackets correctly, for example 3(x + 2) becomes 3x + 6.
• Collects like terms accurately to simplify expressions, for example 2x + 6 + 4x becomes 6x + 6.
• Constructs a general algebraic argument, such as representing consecutive integers as n and n + 1 and showing their sum is 2n + 1.
• Recognises that testing numerical values alone does not prove an identity for all values.
• Defines appropriate algebraic variables to represent the numbers in the statement.
• Constructs a logical sequence of algebraic steps without missing links.
• Accurately expands brackets, simplifies terms, or factorises expressions as required.
• Concludes with a clear statement linking the final algebraic result back to the original claim.
• Recognise that a function assigns exactly one output to each input.
• Interpret simple expressions as functions, identifying the rule applied.
• Substitute a numerical input into a simple expression and calculate the output.
• Interpret a function machine or mapping diagram as an expression with inputs and outputs.
• Explain the meaning of f(4) as the output when the input is 4.
• Find the inverse of a simple linear function by rearranging and swapping the variable.
• Use correct notation for inverse functions, such as f⁻¹(x).
• Process a composite function such as fg(x) by applying g first and then f.
• Recognise that fg(x) means f(g(x)) and that order matters.
• Show that applying a function and its inverse in either order returns the original input.

• 💡In an MCQ, test each option by expanding or rewriting it into the form given in the question before choosing.
• 💡Check indices by counting how many identical factors are multiplied, not by multiplying the base by the index.
• 💡When a coefficient is a decimal, convert it to a fraction and compare with the options.
• 💡Write the substitution step explicitly before calculating so you can check each value has been placed correctly.
• 💡Use brackets around every negative value you substitute, especially when it is raised to a power.
• 💡Re-read the formula to confirm which letter represents which quantity before substituting.
• 💡Underline the key sign in each option: an equals sign points to an equation or formula, and an inequality sign points to an inequality.
• 💡Say the definition quietly to yourself before selecting: expression, equation, formula, inequality, term or factor.
• 💡Eliminate options that use the wrong label for a correct example, because MCQs often test vocabulary rather than calculation.
• 💡Substitute x = 0 and x = 1 into both sides; if the results differ, the statement is not an identity.
• 💡Look for the ≡ symbol, which marks an identity, and do not confuse it with the = sign of an equation.
• 💡Try to simplify both sides; if they become identical expressions, the statement is an identity.
• 💡Check a simplification by substituting a simple value such as x = 2 into both the original and simplified expressions; the results must match.
• 💡When taking out a common factor, divide every term by the factor and confirm the bracket multiplies back to the original expression.
• 💡Write powers with clear raised indices and keep bases distinct, so a² × b³ is not combined into a single power.
• 💡For multiple-choice questions, test each option with a numerical substitution rather than choosing by appearance.
• 💡After expanding, count the terms before collecting to ensure no product has been missed.
• 💡For factorising, list factor pairs of the constant term and test which pair adds to the coefficient of x.
• 💡Check a factorisation by expanding it mentally or on paper and comparing with the original expression.
• 💡In multiple-choice questions, substitute a simple value such as x = 2 into the original and each option to identify the equivalent expression.
• 💡Always show your expansion or factorisation steps clearly so method marks can be awarded even if the final answer is wrong.
• 💡Check your factorisation by expanding it back to the original expression; this catches sign and coefficient errors quickly.
• 💡When simplifying surds, look for the largest square factor to reach the simplest form in one step, and rationalise denominators if the question requires it.
• 💡Write down each step of your rearrangement so method marks can be awarded even if the final answer is incorrect.
• 💡Check your rearranged formula by substituting simple numbers back into both the original and rearranged versions to see if they give the same result.
• 💡Pay attention to units and whether the context requires a positive or negative root when taking square roots.
• 💡In multiple-choice questions, test each option by substituting a simple value such as x = 0 or x = 1, but remember that a single counterexample can disprove an identity while a proof needs algebra.
• 💡When asked to show equivalence, start from one expression and manipulate it step by step to reach the other, showing each algebraic stage clearly.
• 💡Use the correct symbol: = for equations that are conditional, ≡ for identities that are always true.
• 💡If a question asks you to argue mathematically, write a short chain of reasoning using algebra rather than just numerical examples.
• 💡Write your proof as a continuous argument, ensuring each algebraic step follows logically from the previous one.
• 💡Clearly state what your variables represent at the start of your proof.
• 💡Check your final line against the original statement to ensure you have proved exactly what was asked.
• 💡When asked to process a function, write the substitution step clearly before calculating.
• 💡Check whether the question uses function notation or a function machine, and keep the input and output distinct.
• 💡For multiple-choice questions, test each option by substituting a simple input such as 2 or 3 to see which rule matches.
• 💡For inverse functions, show your rearrangement steps clearly so the method is visible.
• 💡For composite functions, write out the inner function first, then substitute into the outer function.
• 💡Check your answer by substituting a simple value into the original and composite functions to see if the results match.
• 💡Always show your working, especially when rearranging formulas or solving inequalities, as method marks are awarded for correct steps even if the final answer is wrong.
• 💡Use precise mathematical language: say 'expand' not 'multiply out', 'factorise' not 'take out', and 'subject' not 'letter on its own'.
• 💡Check your answer by substituting a value back into the original equation or inequality to verify it works.

• Reading ab as a + b; correct by treating juxtaposition as multiplication, so ab = a × b.
• Reading a² as a × 2; correct by treating the index as repeated multiplication, so a² = a × a.
• Writing a coefficient as a decimal such as 0.5x; correct by writing it as a fraction such as (1/2)x.
• Ignoring brackets and applying an operation to only part of a grouped expression; correct by treating the bracketed group as one quantity.
• Substituting a negative number without brackets, so x² with x = −4 is wrongly evaluated as −16; correct by writing (−4)² = 16.
• Applying operations in the wrong order, for example adding before multiplying; correct by strictly following the order of operations.
• Treating a coefficient as an added term, for example reading 3y as 3 + y; correct by treating 3y as 3 × y.
• Calling any algebraic string an equation: an equation must contain an equals sign, so 5x + 2 is an expression, not an equation.
• Treating a formula as just another equation: a formula is a general rule linking quantities, such as A = πr², and is used by substitution or rearrangement rather than solved for one unknown value.
• Counting terms by the number of symbols: terms are separated by + or − signs, so 3x² + 5y has two terms even though it contains several symbols.
• Confusing factors with terms: in 3x², the factors are 3 and x (with x used twice), while 3x² itself is one term of the expression.
• Treating an identity as an equation to solve: an identity is true for all values, so there is no single solution to find.
• Assuming every statement with an equals sign is an identity: only statements true for all values qualify, so 2x + 6 = 10 is an equation, not an identity.
• Ignoring the identity symbol ≡: it signals that the statement holds for all values, whereas = in an equation usually signals a condition to solve.
• Testing only one value and concluding too soon: one agreeing value is not enough to prove an identity, so check several values or rearrange both sides.
• Adding unlike terms such as 3x + 4y to get 7xy; the correction is that only terms with identical variable parts can be collected, so 3x + 4y stays as 3x + 4y.
• Multiplying only the first term inside a bracket, writing 3(x + 5) as 3x + 5; the correction is to multiply every term, giving 3x + 15.
• Multiplying indices instead of adding them when multiplying algebraic terms, writing x² × x³ as x⁶; the correction is that the laws of indices require adding the powers, giving x⁵.
• Failing to apply a power to the coefficient inside a bracket, writing (3x²)³ as 3x⁶; the correction is to cube the coefficient as well, giving 27x⁶.
• Expanding (x + 3)(x + 5) as x² + 15 by multiplying only the first and last terms; the correction is to multiply all four pairs, giving x² + 8x + 15.
• Factorising x² + 7x + 12 as (x + 7)(x + 12); the correction is to find numbers with product 12 and sum 7, namely 3 and 4, giving (x + 3)(x + 4).
• Treating x² − 25 as (x − 5)²; the correction is that the difference of two squares factorises as (x + 5)(x − 5).
• Incorrectly evaluating the product of surds, such as √3 × (−√3) = −9; the correction is that √3 × √3 = 3, so the product is −3, giving 4 − 3 = 1 for (2 + √3)(2 − √3) (Higher tier only).
• Sign errors when expanding brackets with negative terms: for example, incorrectly writing (x − 3)(x + 5) as x² + 8x − 15 instead of x² + 2x − 15; correct by checking each product carefully.
• Forgetting to multiply all terms when expanding three binomials, such as omitting the product of the first and third brackets; correct by expanding two brackets first and then multiplying every term.
• Incorrect factorisation of ax² + bx + c when a is not 1, often by assuming the factors are simply (x + p)(x + q); correct by using the ac method or checking by expansion.
• Simplifying algebraic fractions by cancelling terms rather than factors, e.g. cancelling x from (x + 2)/x incorrectly; correct by factorising first and cancelling only common factors.
• Applying an operation to only one side of the formula; correct by always performing the same operation to both sides.
• Incorrectly handling negative signs or division when rearranging, such as forgetting to divide all terms; correct by working step by step and checking each inverse operation.
• Forgetting to take the square root when isolating a squared variable, e.g. leaving r² instead of r; correct by taking the positive square root where the context requires a positive length.
• Mixing up the subject and another variable, or not fully isolating the required variable; correct by ensuring the subject appears alone on one side.
• Treating an identity as an equation and solving it for a single value; instead, recognise that an identity must be shown true for all values, often by algebraic simplification.
• Assuming two expressions are equivalent after checking only one or two values; instead, provide a general algebraic argument that covers every possible value.
• Expanding a bracket incorrectly, for example writing 3(x + 2) as 3x + 2; instead, multiply every term inside the bracket by the factor outside.
• Misreading the ≡ symbol as an ordinary equals sign and attempting to solve; instead, interpret ≡ as indicating an identity that holds for all values.
• Using numerical examples instead of a general algebraic proof; instead, use variables to represent all possible numbers.
• Defining variables incorrectly, such as using n and m for consecutive numbers instead of n and n + 1.
• Stopping before reaching a clear conclusion; instead, finish with a statement that explicitly matches the claim.
• Misunderstanding: thinking f(4) means f multiplied by 4. Correction: f(4) means substitute 4 for x in the expression for f(x).
• Misunderstanding: treating the input and output as interchangeable. Correction: the input is the value put into the function; the output is the result after applying the rule.
• Misunderstanding: applying operations in the wrong order. Correction: follow the order shown in the expression or function machine, using conventional priority of operations.
• Misunderstanding: thinking f⁻¹(x) means 1 ÷ f(x). Correction: f⁻¹(x) is the inverse function that reverses the effect of f.
• Misunderstanding: applying f before g in fg(x). Correction: fg(x) means apply g first, then f.
• Misunderstanding: assuming fg(x) always equals gf(x). Correction: composite functions are not generally commutative; check the order.
• Misunderstanding: forgetting to swap x and y when finding an inverse. Correction: after rearranging, replace y with x to express the inverse in terms of x.
• Students often think that when solving an inequality, the inequality sign never changes. Correction: The sign reverses when multiplying or dividing by a negative number.
• Students sometimes believe that (x + 3)^2 = x^2 + 9. Correction: (x + 3)^2 = x^2 + 6x + 9, as it means (x + 3)(x + 3).
• Students may incorrectly rearrange formulas by moving terms without applying inverse operations to both sides. Correction: Always perform the same operation to both sides to maintain equality.

1. 1Day 1-2: Revise index laws and practice simplifying expressions with positive, negative and zero indices.
2. 2Day 3-4: Practice expanding single and double brackets, including expressions with negative terms, and factorising linear and quadratic expressions.
3. 3Day 5-6: Focus on rearranging formulas, including those with fractions and powers, and solving linear inequalities.
4. 4Day 7-8: Complete mixed exam-style questions under timed conditions, checking your notation and working.
5. 5Day 9-10: Review mistakes, revisit weak areas, and create a summary sheet of key rules and common pitfalls.

• 📋Simplify expressions using index laws: often 1-2 mark questions testing multiplication, division and powers of indices.
• 📋Expand and factorise: typically 2-3 mark questions where you must expand double brackets or factorise a quadratic expression.
• 📋Rearrange formulas: 2-3 mark questions where you must change the subject, sometimes involving fractions or square roots.
• 📋Solve inequalities: 2-3 mark questions where you solve a linear inequality and represent the solution on a number line.