Notation, vocabulary and manipulation — AQA GCSE Mathematics

Notation, vocabulary and manipulation covers the precise mathematical language and algebraic skills needed to read, write and rearrange expressions, equations, formulas and inequalities correctly. In AQA GCSE Mathematics, you must use correct notation such as index laws, function notation, inequality symbols and standard algebraic forms, and manipulate them fluently to solve problems and communicate exact mathematical meaning.

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.

• Writing an expression in accepted algebraic form, with the coefficient first and multiplication signs omitted (e.g., `5pq`).
• Correctly using index notation for repeated multiplication (e.g., `a × a × b` becomes `a²b`).
• Interpreting a coefficient as representing repeated addition (e.g., `3y` means `y + y + y`) as well as multiplication.
• Using fractional notation for division and for coefficients that would otherwise be recurring decimals (e.g., `(x+5)/3` or `x/3`).
• a correct substitution shown, with each value in the right place, even if the arithmetic that follows is wrong
• evaluating brackets and powers before multiplying, dividing, adding or subtracting
• the final value, given to the accuracy asked for and with units where the question wants them
• a fully correct answer scores on its own, but when the answer is wrong the method marks depend on the substitution being visible
• naming the correct object when asked to classify, for example calling a statement with a subject and two or more quantities a formula
• a response that stays an expression when an expression is asked for, with no equals sign added and no value produced
• a correct factor or a correct number of terms identified, such as naming 3 and x + 2 as factors of 3x + 6
• when giving a reason, identify the deciding feature, such as the presence of a relation sign
• expanding and collecting the side that needs it, even if the comparison that follows goes wrong
• equating the coefficients of matching powers, for example matching the x terms on both sides
• each correct missing value in a completed identity
• give the correct label and explain whether the equality holds for all permissible values or only particular values
• collecting each set of like terms correctly, with the sign in front of a term carried along with it
• a correct expansion of a single bracket, with every term inside multiplied, even if the collecting afterwards is wrong
• the highest common factor taken outside, and one for the bracket that is left being correct
• a correct index law used, such as adding the indices for a product of powers of the same base
• four correct products when two brackets are multiplied out, before any collecting is done
• collecting to a correct three-term quadratic with the right signs
• a correct pair of numbers whose product is the constant and whose sum is the coefficient of x
• the fully factorised answer written as a product of two brackets
• correctly expanding a product involving surds, e.g. (2 + √3)(2 − √3) = 4 − 3 = 1
• a correct product of the first two brackets, even if the third multiplication then goes wrong
• a correct pair of numbers multiplying to ac and adding to b for factorising a quadratic
• splitting the middle term correctly and factorising in pairs to get the correct brackets
• correctly simplifying a surd or collecting like surds
• multiplying the numerator and denominator by the correct expression to rationalise the denominator
• substituting numerical values correctly into a standard formula and evaluating, including with π
• a correct first step that clears a term or denominator away from the side the new subject sits on
• each correct rearranging step, so an early slip can still leave later method marks available
• gathering the new subject into a single term and factorising when it appears twice
• the final formula written with the new subject alone on one side and nothing else attached
• a correct algebraic form for the numbers described, such as 2n for an even number or 2n + 1 for an odd number
• expanding and collecting correctly to reach a single simplified expression
• a factorised form that makes the required property visible, such as showing a factor of three
• a closing statement that links the algebra back to the claim, and a final line that restates the claim with no algebra in front of it earns nothing
• correct general expressions for the numbers in the claim, with the letter defined as an integer
• expanding a squared bracket correctly, middle term included
• a fully simplified expression that shows the structure asked for, such as a factor of two or a factor of three
• a conclusion that states the property in words and refers back to the original claim
• a correct substitution into the rule, even if the arithmetic that follows is wrong
• each correct output when a machine is followed forwards through its steps
• reversing the operations in the correct order when the output is given and the input is wanted
• an expression written in terms of x when the machine acts on a letter rather than on a number
• applying the functions in the correct order in a composite, with the inner function used first
• a correct substitution of one function into the other, before any expanding
• the expanded and simplified composite expression
• a correct rearrangement when finding an inverse, and one for the answer written in terms of x

• 💡Pay close attention to the position of indices. An index applies only to the single letter or bracket immediately before it. For example, `3y²` means `3 × y × y`, not `(3y) × (3y)`.
• 💡When a question asks you to 'write an expression', do not include an equals sign or try to solve it. An expression is a collection of terms without a relation like `=`, `>`, or `<`.
• 💡Put brackets round every value you substitute, negatives especially, and let the brackets carry the ordering for you.
• 💡Write the substitution line out in full before you touch the calculator; that line is what carries the method mark.
• 💡Check the value is sensible for the quantity: a length or an area cannot come out negative.
• 💡Look at the command word before you start: simplify wants an expression back, solve wants a value or a range.
• 💡If the statement uses the three bar identity sign rather than an equals sign, it is telling you the two sides agree for every value, which usually sets the method.
• 💡Underline the quantity you are asked for; a formula question often hands you more values than you need.
• 💡Expand fully before you compare; most of the marks here are lost in the expansion rather than in the comparison.
• 💡Match like with like: number terms on one side must equal the number terms on the other, and the same for each power of x.
• 💡To show a statement is not an identity, one counter example is enough, so try an easy value such as x = 1.
• 💡Attach the sign in front of each term to that term before you start collecting; it heads off most sign errors.
• 💡After factorising, multiply your bracket back out in your head and check it returns the original expression.
• 💡Before using an index law, check the bases are the same, because the laws only apply when they are.
• 💡Expand in the same fixed order every time, first term against both, then second term against both, so no product goes missing.
• 💡Check a factorised answer by multiplying it back out; the middle term is where an error shows up.
• 💡If a quadratic has no x term and a subtraction, try the difference of two squares before anything else.
• 💡With three brackets, multiply the pair that gives the easiest quadratic first; the answer is the same whichever pair you start with.
• 💡List the factor pairs of ac systematically rather than guessing brackets and testing them.
• 💡Never cancel across an addition or a subtraction in an algebraic fraction; only a factor of the whole top and the whole bottom can go.
• 💡Decide which letter you want alone, list what is done to it, then undo those steps in reverse order.
• 💡If the new subject sits inside a bracket or under a root, deal with everything outside it first.
• 💡Substitute a simple set of numbers into both the original and rearranged formula; they should agree.
• 💡Define your letters in words before you use them, for example writing that n is any integer.
• 💡Finish with a sentence: the algebra earns the method marks and the sentence earns the conclusion mark.
• 💡If you think a claim is false, one counter example settles it, so try small values including zero and a negative.
• 💡Read what kind of number the claim is about and set the letters up for it before writing anything else.
• 💡Aim your working at a visible factor: pull out a two for even, a three for a multiple of three.
• 💡Keep the letter an integer throughout and say so, because the word integer is often part of the mark.
• 💡If a machine is described in words, draw it as boxes with arrows; the order of the boxes is what most of the marks depend on.
• 💡To get an input from an output, work from right to left and replace each operation by its inverse.
• 💡Check your answer by putting it back through the machine forwards.
• 💡For a composite at a number, work the inner function out first and feed that single value into the outer one.
• 💡Check an inverse by putting a number through f and then through your answer; you should arrive back where you started.
• 💡Label each line with the function you are applying so the order of your working is clear.
• 💡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.

• Confusing repeated multiplication with repeated addition, for example writing `a × a × a` as `3a` instead of `a³`. Correction: `3a` means `a+a+a`, while `a³` means `a×a×a`.
• Misapplying an index to multiple terms instead of the single base it follows, for example writing `3y²` as `(3y)²`. Correction: The index in `3y²` applies only to the `y`, not the coefficient `3`, so it means `3 × y × y`.
• Losing accuracy by converting a fractional coefficient to a rounded decimal, for example writing `x/3` as `0.33x`. Correction: Always use the exact fraction unless instructed to give a decimal answer.
• Losing a sign when a negative is substituted, so (−3)² is worked out as −9 rather than as 9.
• Multiplying before dealing with an index, so 3x² with x = 4 is worked out as (3 × 4)² instead of 3 × 16.
• Putting a value into the wrong letter when a formula uses several, such as swapping u and v in a motion formula.
• Giving the answer without the units the question asked for, or in units that do not match the quantities used.
• Turning an expression into an equation by adding a zero on the right and solving it, when the question only asked you to simplify.
• Counting 3x² and 3x as like terms because both contain x, when a term is fixed by its power as well as by its letters.
• Calling any rule with letters a formula, when a formula needs a subject and links two or more quantities.
• Confusing a factor with a term, so saying that 3x and 6 are the factors of 3x + 6.
• Trying one number, seeing both sides agree, and calling the statement proved, when an identity has to hold for every value.
• Writing an equals sign where the identity sign is required, which loses the distinction the question is testing.
• Matching the number terms but forgetting the coefficients of x, so only half of the missing values are found.
• Treating an identity as something to solve, and producing a value of x from a statement that is true for all x.
• Adding indices when terms are added rather than multiplied, so x³ + x⁴ is written as x⁷.
• Multiplying only the first term inside a bracket, so 3x(2x − 5) is written as 6x² − 5.
• Taking out a factor that is not the highest, leaving a bracket that can still be factorised, such as 2(6y³ + 4y²).
• Losing a sign when the term outside is negative, so −2(x − 3) is written as −2x − 6.
• Multiplying only the first terms and the last terms, so (x + 5)(x − 3) becomes x² − 15. Correction: include the two cross products, −3x and +5x, before collecting.
• Getting the middle term wrong when the constant is negative, writing x² − 2x − 15 for (x + 5)(x − 3). Correction: the cross terms are −3x and +5x, which collect to +2x.
• Squaring term by term, so (x + 4)² is written as x² + 16. Correction: (x + 4)² means (x + 4)(x + 4), giving x² + 8x + 16.
• Factorising a difference of two squares as (x − 7)² instead of (x + 7)(x − 7). Correction: (x − 7)² expands to x² − 14x + 49, not x² − 49.
• cancelling individual terms in an algebraic fraction instead of common factors, eg cancelling x in (x+2)/(x+5)
• when factorising ax²+bx+c, finding numbers that multiply to c instead of ac
• adding surds incorrectly, eg √3 + √5 = √8
• when expanding brackets with surds, calculating √a × √a as √a instead of a
• Doing something to one side only, such as subtracting a term on the left and leaving the right untouched.
• Dividing after taking a square root rather than before, so from A = πr² writing r = √A/π instead of r = √(A/π).
• Stopping while the new subject is still multiplied by something, leaving an answer such as 2πr = C.
• Cancelling a letter that is only part of a sum, for example cancelling the a in (a + b)/a.
• Trying three or four numbers, finding they work, and offering that as an argument for all numbers.
• Using the same letter for two quantities that can differ, so two different even numbers both become 2n.
• Starting from the statement being shown and rearranging it, rather than starting from one side and reaching the other.
• Reaching 3n + 3 and stopping, without saying what that tells you about the original claim.
• Writing consecutive odd numbers as n and n + 1, which are one odd and one even.
• Expanding (n + 1)² as n² + 1 and losing the 2n term.
• Checking the claim for one value and then asserting that it therefore holds for every value.
• Reaching a correct expression but never saying why it answers the claim, so the last mark goes unearned.
• Reading f(3) as f multiplied by three rather than as the value of the function when the input is three.
• Doing the steps of a number machine in the wrong order, adding before multiplying when the machine multiplies first.
• Reversing a machine but keeping its operations in the original order.
• Changing an operation instead of inverting it when working backwards, such as dividing where the step was a subtraction.
• Reading fg(x) from left to right and applying f first.
• Treating f⁻¹(x) as 1/f(x), which is the reciprocal and a different function.
• Dropping the bracket instead of expanding it, so 3(x + 5) + 2 is written as 3x + 5 + 2 rather than 3x + 17.
• Swapping the letters when finding an inverse but never rearranging, so the answer is still written in terms of y.
• 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.