AQA GCSE Level 2 Further Mathematics

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Number

1.1 - Fractions, percentages, ratio and proportionUse fractions, decimals, percentage multipliers and ratios confidently, so the arithmetic supports your reasoning. You will choose an exact form, keep track of what a percentage refers to and distinguish direct from inverse proportion. These familiar number skills will carry much more interesting...
1.2 - The product rule for countingCount thousands of possibilities without writing a single list. You will learn why counting multiplies, how to fill restricted positions first and how to split a count into cases, so that you can answer AQA's three-mark counting questions quickly and check them a second way.
1.3 - Surds and rationalising denominatorsKeep square roots exact while simplifying sums, multiplying brackets and removing surds from denominators. You will see why these methods work, then use them to turn an awkward fraction into a surprisingly simple exact answer.

Algebra

Algebraic manipulation

2.1 - Basic algebra skillsMake algebra easier to read by choosing a useful order, collecting like terms and multiplying a whole bracket correctly. This short lesson refreshes the GCSE skills that support the longer expressions you will meet next, and explains why each change preserves the value.
2.6 - Expanding and simplifying bracketsExpand longer products without losing a term or a sign. You will multiply a quadratic by a linear expression, handle three brackets and combine expansions, choosing a useful order and collecting only like terms.
2.7 - Binomial expansion with Pascal's triangleUse Pascal's triangle to expand a positive integer power of a two-term expression. You will also work out a single coefficient without expanding everything, then use coefficients to form equations involving an unknown constant.
2.8 - Factorising fullyTurn algebraic sums into useful products, including expressions with bracketed squares and quadratics in two variables. You will learn to recognise a common factor, choose the right factorisation and keep going until every available factor has been taken out.
2.9 - Simplifying, multiplying and dividing algebraic fractionsAn algebraic fraction obeys the same rules as a numerical fraction, but its factors sometimes hide inside brackets. You will simplify, multiply and divide fractions with numeric, linear and quadratic denominators, and keep track of values that make the original calculation undefined.
2.9 - Adding and subtracting algebraic fractionsAdding algebraic fractions means making their pieces the same size before combining them. You will choose common denominators, handle subtraction brackets and use factorisation to combine fractions efficiently, including expressions with quadratic denominators.
2.10 - Using and rearranging formulaeA formula can calculate a value or describe a relationship in either direction. You will substitute carefully, make a chosen letter the subject and handle formulae in which that letter appears twice or is raised to a power.
2.12 - Completing the squareRewrite a quadratic as a squared bracket plus a constant, without changing its value. You will complete the square with positive, negative and fractional bracket values, handle a coefficient in front of the square and work out unknown constants in identities.

Equations and graphs

2.14 - Solving linear and quadratic equationsChoose a useful method for a linear or quadratic equation, keep exact answers where they are needed and check which solutions fit a problem. You will connect factorisation, completing the square, the quadratic formula and graphical solutions, rather than treating them as unrelated tricks.
2.11 - The factor theorem for polynomialsTest whether a linear expression is a factor of a polynomial by making one substitution. You will use rational values, work out unknown coefficients and reduce a cubic equation to a quadratic that you already know how to solve.
2.13 - Sketching linear and quadratic graphsTurn a line or quadratic rule into a clear sketch with its important coordinates labelled. You will use intercepts, the sign of a leading coefficient, symmetry and completed squares to show a graph's structure and interpret what it tells you.
2.15 - Solving simultaneous equationsWork out values that satisfy two equations at the same time, using algebra and graph intersections. You will solve two linear equations and systems containing a squared variable or a product, keeping the correct coordinates together in every solution.
2.16 - Linear equations in three unknownsSolve three linear equations by reducing them to a familiar pair in two unknowns. You will choose an economical elimination or substitution, keep the working organised and check the final triple in every original equation.
2.17 - Solving linear and quadratic inequalitiesDescribe whole sets of solutions using inequalities. You will solve linear inequalities without losing their direction, use quadratic roots as interval boundaries and combine the result with any restrictions in the question.
2.18 - Index laws and index equationsUse negative and fractional indices as exact instructions, simplify expressions to single powers and solve equations involving powers. You will also recognise when an index equation hides an ordinary quadratic, so that a familiar method becomes available.
2.13 - Exponential and piecewise graphsSketch exponential curves, work out their constants from points and interpret their outputs on a restricted domain. You will also draw functions with different rules on up to three parts of the domain, showing exactly which values belong at the joins.

Proof and sequences

2.21 - nth terms of linear sequencesAn nth-term rule lets you go straight to any position in a sequence. You will build rules for numerical and algebraic linear sequences, then use them to decide whether a value occurs and which term first passes a boundary.
2.19 - Algebraic proofAlgebra can establish a pattern for every allowed integer at once. You will turn expressions into visible multiples, choose between expansion and factorisation, and represent consecutive, even and odd integers accurately enough to prove general statements.
2.20 - Using nth terms and limiting valuesA given nth term tells you both individual values and how a sequence behaves far along its list. You will substitute and compare terms exactly, match values in two sequences and work out limiting values using ordinary algebra.
2.22 - nth terms of quadratic sequencesQuadratic sequences have a changing step but a constant change in that step. You will use second differences or known term positions to build their nth terms, then solve quadratic equations to work out which positions have a given value.

Coordinate geometry

3.1–3.3 - Gradients, perpendicular lines and distanceMeasure a line's direction using its gradient and its length using Pythagoras' theorem. In 3.1 you will calculate gradients, in 3.2 you will recognise parallel and perpendicular directions, and in 3.3 you will work out exact distances and use them to check coordinate arguments.
3.4 - Dividing a line in a ratioUse equal steps along a straight line to work out a midpoint, a point in a given ratio or a missing endpoint. You will distinguish a point inside a segment from one outside it, so that the ratio and the stated point order select the right answer.
3.5, 3.6 - Equations of straight linesConnect a line's equation to its gradient, intercepts and points. In 3.5 you will interpret and construct equations in different forms, including parallel, perpendicular and vertical lines; in 3.6 you will use points and a gradient to draw the required portion of a line accurately.
3.7, 3.8 - Equations of circlesUse a circle's centre and radius to describe its points, recover its equation and solve intersections with a line. In 3.7 you will connect circles centred at the origin with right angles, chords and tangents; in 3.8 you will extend the equation to any centre and use the same geometry in shifted c...
3.9 - Tangents to a circleWork out the equation of a tangent by using the radius to its point of contact. You will handle circles with any centre, recognise horizontal and vertical exceptions, and combine a tangent equation with intercepts and an exact area.

Calculus

4.1, 4.2 - The gradient function of a curveA curve can be steep at one point and almost flat at another, so one gradient cannot describe the whole curve. You will interpret its gradient function, use supplied gradient functions to work out local gradients and explain what those gradients mean as rates of change.
4.3 - Differentiating sums of powers of xObtain a gradient function directly from a curve's equation. You will differentiate integer powers and their sums, rewrite reciprocal terms as negative powers and simplify products or fractions before applying the same rule.
4.4 - Tangents and normals to curvesTurn a curve's local gradient into the equation of a straight line. You will differentiate to obtain tangent gradients, use perpendicular gradients for normals and handle the special case of a horizontal tangent with a vertical normal.
4.5 - Increasing and decreasing functionsUse a gradient function to describe where a curve rises or falls as you move to the right. You will solve derivative inequalities, prove that a function increases for every real input and handle a parameter without confusing a positive gradient with an isolated horizontal tangent.
4.6, 4.7 - Stationary points and the second derivativeLocate a curve's stationary points and decide whether each is a maximum, a minimum or neither. You will use the second derivative, which measures how the gradient changes, and check gradient signs when that test cannot decide.
4.8 - Maximum and minimum problemsUse calculus to choose the largest or smallest value that a problem actually allows. You will turn a geometric constraint into one expression, minimise positive reciprocal expressions and check domain boundaries as well as stationary points.
4.9 - Sketching curves with known turning pointsBuild a reliable curve sketch from its known maximum and minimum points, axis intercepts and far-end behaviour. You will distinguish crossings from touches and use the finished shape to interpret gradients and count solutions without inventing coordinates.

Matrix transformations

5.1, 5.2 - Multiplying matrices and the identity matrixUse a matrix to organise numbers and carry out a calculation in a fixed order. In 5.1 you will multiply by a scalar, multiply a matrix by a column and multiply two matrices; in 5.2 you will discover the special matrix that leaves every input unchanged.
5.3 - Transformations of the unit squareBuild a transformation matrix by following two corners of a unit square, then use it to map the remaining corners and other points. You will connect the columns to enlargements, reflections and rotations about the origin, so the standard matrices have a reason behind them.
5.4 - Combining transformations with matricesReplace successive transformations by one matrix product, keeping their order clear. You will compare reversed orders and use repeated multiplication to recognise cycles, so a long sequence of transformations can become a short calculation.

Geometry

6.1 - Perimeter, area and volumeChoose the right lengths and surfaces when calculating with flat shapes and solids. You will connect familiar area formulae to prisms, cylinders, cones, pyramids and spheres, keeping exact answers in terms of π where appropriate. A careful sketch often matters more than a long calculation.
6.1 - Angle facts and circle theoremsUse line, polygon and circle facts to work out angles with reasons. You will recognise the exact conditions behind each theorem, including which chord or arc an angle subtends. That turns a busy diagram into a short chain of deductions.
6.2 - Geometric proofBuild geometric arguments from stated facts, with a reason for every deduction. You will combine angle sums, parallel lines, circle theorems and congruent triangles to prove results for whole families of shapes. The diagram suggests a route; the argument establishes it.
6.4 - Pythagoras' theorem in 2D and 3DUse right-angled triangles to calculate exact lengths in flat diagrams and solids. You will recognise the specified Pythagorean triples, choose when to add or subtract squares and expose the two triangles behind a three-dimensional calculation. Keep roots exact until a question asks for rounding.
6.6, 6.7 - Trigonometry for any angleExtend sine, cosine and tangent beyond acute triangles using coordinates on a circle. You will use signs and reference angles, then sketch and read their graphs for angles of any size. All angles in this lesson are measured in degrees.
6.8 - Exact trigonometric valuesUse two special right-angled triangles to obtain exact lengths and trigonometric values without a calculator. You will keep surds exact and combine these values with reference angles to work throughout a full turn.
6.3 - Sine rule, cosine rule and triangle areaChoose a trigonometric rule from the information a triangle gives you. You will work out sides, angles and areas in triangles without a right angle, keeping exact values where possible and checking whether a sine calculation permits more than one triangle.
6.5 - Lengths and angles in 2D and 3DTurn a flat or spatial diagram into the triangle that contains the required length or angle. You will interpret bearings, distinguish line-to-plane from plane-to-plane angles and use the cosine rule when a triangle inside a solid is not right-angled.
6.9 - Trigonometric identitiesUse two identities to simplify trigonometric expressions, prove equalities and turn equations into familiar algebra. All angles in this lesson are measured in degrees.
6.10 - Solving trigonometric equationsSolve equations involving a single angle by combining algebra, reference angles and the graphs you already know. Every solution must satisfy both the original equation and its stated interval. All angles here are in degrees.