OCR GCSE Additional Mathematics (FSMQ)
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Algebra
AL1, AL3 - Algebraic vocabulary and polynomial operationsAlgebra lets you explain a whole pattern with one calculation. AL1 gives you the language for reading expressions, equations, identities and function notation; AL3 gives you the tools to add, subtract and multiply polynomials. You will build from familiar GCSE algebra to calculations with cubic e...
AL2 - Surds and exact valuesEvery decimal your calculator gives for √(2) is rounded, because the true decimal never ends and never repeats. This lesson shows you how to keep square roots exact: simplify them, collect them, expand brackets that contain them and clear them from the bottom of a fraction. OCR's "exact", "...
AL3, AL4 - Dividing polynomials and the factor theoremPolynomial division reverses the multiplication you already know. AL3 shows you how to find a quotient and remainder; AL4 turns a single substitution into a test for a linear factor. You will use the two ideas together to factorise cubics and find missing coefficients.
AL2 - Simplifying algebraic fractionsAlgebraic fractions obey the same rules as numerical fractions, but their factors can be hidden inside brackets and quadratics. You will simplify, combine, multiply and divide them while keeping track of values that would make the original expression undefined. The aim is an equal expression that...
AL5 - Completing the squareCompleting the square rewrites a quadratic as one squared bracket plus a constant. You will learn why the method works, handle positive and negative leading coefficients, and read off a quadratic's greatest or least value. The new form reveals information that is hard to see in an expanded expres...
AL6 - Setting up equations in one unknownAn equation turns a condition in a problem into something you can solve. You will choose one unknown, build linear, quadratic and cubic equations, and handle equations containing algebraic fractions. The final step is always to decide which mathematical solutions answer the original question.
AL6 - Simultaneous equations in two unknownsFind pairs of values that satisfy two equations at the same time. You will choose a sensible elimination or substitution, keep both answers when a quadratic appears, and turn a word problem into a complete pair of equations. A graph will explain what the algebra says about intersections.
AL7, AL8 - Linear and quadratic inequalitiesUse inequalities to describe whole ranges of possible values. AL7 supplies the rules for manipulating them; AL8 uses those rules, algebra and graphs to solve linear and quadratic inequalities, including ones you build from a problem. The important decisions are the direction of the sign and wheth...
AL9 - Graphing inequalities in two variablesShow all the coordinate pairs satisfying a linear inequality by drawing a boundary and choosing a side. You will distinguish strict from inclusive boundaries, shade unwanted regions and identify the points satisfying several inequalities at once. A picture can display infinitely many answers with...
AL10, AL11 - Recurrence relationships and modellingDescribe a sequence by telling each term how to make the next one. AL10 develops recurrence notation, including rules that use two earlier terms; AL11 turns repeated changes in balances and other quantities into mathematical models. You will see why the timing of a payment can matter as much as i...
Enumeration (counting and probability)
EN2 - Tree diagrams, tables and Venn diagramsA good picture keeps track of outcomes without missing any or counting one twice. You will construct and use trees for successive choices, two-way tables for paired outcomes and Venn diagrams for overlapping groups. The picture you choose depends on what the outcome actually records.
EN3 - Counting with the product ruleThe product rule counts a large collection of outcomes by breaking each outcome into small choices. You will decide how many choices are available at each stage, handle simple restrictions and explain why arranging distinct objects leads to factorials. The crucial step is choosing the stages well.
EN4 - Permutations (ordered selections)An ordered selection records both which objects are chosen and where they go. You will count permutations of distinct objects, derive the notation ⁿPr and handle restrictions on roles, positions and neighbouring objects. The starting question is always what makes two outcomes different.
EN5 - Combinations (unordered selections)Choosing a team records who is in it, rather than an order for its members. You will count these unordered selections with ⁿCr, explain why the formula divides out repeated orders and combine choices from separate groups. You will also distinguish “exactly”, “at least” and alternative case...
EN1 - The binomial expansionA power such as (2-3x)⁵ looks like a long multiplication, but counting tells you its terms directly. You will see why combinations appear in an expansion, expand a positive integer power, and find a requested coefficient or constant term without multiplying out every bracket.
EN2 - The binomial distributionWhen the same kind of chance experiment is repeated, you can count successes without drawing a huge tree. You will recognise when a binomial model fits, calculate exact probabilities and use a distribution to answer questions about several possible counts. The key is deciding what one trial and o...
EN6 - Counting and probability problemsA counting problem becomes manageable when you decide what counts as one outcome, whether order matters and which cases satisfy the restriction. You will combine combinations, ordered selections and binomial probabilities in problems where the method is yours to choose. Sometimes the shortest rou...
Coordinate geometry
CG1, CG2 - Distances and mid-pointsCoordinates let you measure a straight journey without drawing it to scale. CG1 turns horizontal and vertical changes into an exact distance; CG2 locates the point halfway along a segment. You will also work backwards when a length or a mid-point is known but an endpoint is missing.
CG3 - The equation of a circleA circle equation is a distance statement in disguise. You will write one from a centre, a point or a diameter, uncover the centre by completing the square, and find where a line meets a circle. Keeping squared lengths exact makes the geometry much simpler than measuring a drawing.
CG4 - Sketching and plotting polynomial graphsA graph turns an equation into a picture of all its solutions. You will plot straight lines and polynomial curves accurately, and make quick sketches from roots, intercepts and signs. You will also learn when a table of values can hide a bend, so that a neat drawing does not tell the wrong story.
CG5 - Gradient, intercept, tangent and normalA point fixes a location; a gradient fixes a direction. Combine them to find lines, perpendicular bisectors, and tangents and normals to circles and curves. You will use coordinate geometry and repeated intersections to justify the answers, keeping every line tied to the point it must pass through.
CG6–CG8 - Linear programmingLinear programming finds the best choice when several limits must hold at once. CG6 translates the situation into inequalities; CG7 uses their common region to maximise or minimise a quantity; CG8 identifies that quantity as an objective function. You will keep the meaning of each variable in vie...
Pythagoras' theorem and trigonometry
PT1 - Trigonometric ratios of any angleExtend sine, cosine and tangent beyond right-angled triangles by following a point around a circle. You will find signed and exact values, then sketch and plot the three trigonometric graphs in degrees.
PT2 - The sine and cosine rulesFind missing lengths, angles and areas in triangles without a right angle. You will choose a rule from the information given, match opposite sides and angles correctly, and keep enough accuracy for a final answer.
PT2 - Ambiguous case of the sine ruleRecognise when two given sides and a non-included angle describe two different triangles. You will find both possibilities, reject impossible angle combinations, and determine when there are no triangles or exactly one.
PT3, PT4 - Trigonometric identitiesUse the tangent ratio and the identity linking squared sine and cosine to find exact values and simplify expressions. You will also prove trigonometric identities by writing a clear chain of valid algebra, with care over any zero denominators.
PT5 - Solving trigonometric equationsFind every solution of a trigonometric equation in a stated interval, rather than stopping at the first calculator answer. You will use the symmetry and periods of sine, cosine and tangent, adjust intervals when the angle changes, and solve simple equations using the two trigonometric identities...
PT6 - Solving problems in two dimensionsTurn measurements and directions into a clear two-dimensional model, then choose the triangles that reveal an unknown height, distance or bearing. You will combine familiar trigonometry and Pythagoras, use two lines of sight when a distance is inaccessible, and build a height model for a point tr...
PT6 - Solving problems in three dimensionsFind lengths and angles in solids by identifying the two-dimensional triangles inside them. You will use horizontal projections to measure a line's inclination, handle observers at different heights, and explain why the line straight up a sloping plane is steeper than a diagonal route across it.
Calculus
CA1–CA3 - Differentiation and the gradient functionA straight line has one gradient; a curve can have a different gradient at every point. CA2 and CA3 explain what that changing gradient means, while CA1 gives you a quick way to calculate it for polynomials. You will find rates of change and tangent gradients without drawing tiny triangles on a g...
CA4 - Tangents and normals to curvesDifferentiation supplies the direction of a curve at a point. CA4 combines that direction with familiar straight-line geometry to find tangents and normals, including points where the tangent is horizontal. You will also work backwards from a line's direction to find where it can be tangent to a...
CA5, CA6 - Finding and classifying stationary pointsA horizontal tangent is a clue, not a verdict about a curve's shape. CA5 shows how to locate stationary points by differentiation; CA6 shows how to decide whether each is a maximum, a minimum or neither. You will give both coordinates and support every classification with a reason.
CA7 - Sketching curves using stationary pointsA sketch records a curve's important features without plotting a large table of points. CA7 uses stationary points, intercepts and the direction of travel to make the shape precise. You will also use a finished sketch to explain how many times a horizontal line meets a curve.
CA8–CA10 - Integration as the reverse of differentiationDifferentiation finds a gradient from a curve; integration can work back from the gradient to the curve. CA8 gives the polynomial integration rule, CA9 explains why it reverses differentiation, and the indefinite-integral part of CA10 introduces the constant of integration. You will use one known...
CA10–CA12 - Definite integrals and areas under curvesA definite integral turns an antiderivative into a number by giving it two limits. You will evaluate these integrals, use them to find areas bounded by a curve and recognise when a region must be split because the curve crosses the axis. Keeping the sign straight is what makes the same calculatio...
CA13 - Area between two curvesFind an area by integrating the vertical gap between its boundaries. You will choose which curve is above the other, find intersection limits and split a region when its upper or lower edge changes. A sketch decides the calculation before any integration begins.
CA14 - Kinematics with variable accelerationUse differentiation and integration to connect displacement, velocity and acceleration along a straight line. You will find instants of rest and greatest velocity, use initial conditions, and distinguish displacement from total distance when an object reverses direction. Signs tell the story of t...
CA15 - Constant acceleration formulaeRecognise when acceleration is constant and choose a formula using the quantities you know. You will calculate velocity, time and displacement, handle negative acceleration consistently, and find total distance when motion reverses. The formulae are short because the underlying velocity-time grap...
Exponentials and logarithms
EL1 - Exponential functions and their graphsExplore curves whose input appears in a power. You will calculate and plot exponential functions, recognise growth and decay from their graphs, and find the constants that control their height and rate of change.
EL2, EL3 - Logarithms and their lawsA logarithm answers a missing-power question. EL2 turns powers into logarithms and back; EL3 explains why products become sums and powers become multipliers. These laws will let you solve equations and reveal straight lines inside curved data.
EL4, EL5 - Straight-line graphs for exponential modelsTurn an exponential curve into a straight line using base-ten logarithms. EL4 supplies the rearrangement; EL5 lets you estimate the original constants from the line's gradient and intercept. You will use data, draw a line of best fit and judge what its predictions can reasonably claim.
EL4, EL5 - Straight-line graphs for power lawsDistinguish a power law from an exponential model, then use logarithms on both variables to make a straight line. EL4 gives the linear form of a power law; EL5 uses the fitted graph to estimate its constant and power. You will interpret those estimates and test the limits of predictions.
EL6 - Solving exponential equationsFind an unknown exponent by recognising a common base or using logarithms. You will solve equations with linear expressions in their exponents, including equations with the unknown on both sides, and show the analytical steps needed for detailed reasoning.
EL7 - Exponential growth and decayBuild exponential models for repeated percentage changes, determine their rates from observations and use logarithms to find times. You will distinguish a time within a continuous model from the first complete year or month when a threshold is reached, and interpret growth and decay in context.
Numerical methods
NM1 - Locating roots by change of signSome equations have awkward roots even when the equation itself is short. NM1 gives you a dependable way to trap a root between two numbers, narrow the gap and justify a rounded answer. The signs of the answers matter more than their size.
NM2, NM3 - Iteration and when methods failIteration feeds an answer back into a formula to obtain the next answer. NM2 uses this simple loop to approximate roots; NM3 explains why a sensible-looking loop, or a change-of-sign search, can sometimes fail. You will learn to show the numerical evidence and decide what it really establishes.
NM4, NM5 - Estimating gradients with chordsA table of points may describe a curve without giving you a formula to differentiate. NM4 uses a chord to estimate the gradient of a tangent; NM5 improves the estimate by choosing points more carefully. This gives a practical bridge between a measured change and an instantaneous rate.
NM6–NM8 - The trapezium rule and area estimatesA curve can enclose an area even when you have only a table of its heights. NM6 builds estimates from rectangles, NM7 replaces their flat tops with trapezia, and NM8 uses the curve's shape to judge the error and improve the estimate. The width of a strip and the quality of the data matter through...
NM9 - Numerical methods in contextA question about motion, a pump or a target time may hide a gradient, an area or a root. NM9 brings those numerical methods together: choose the mathematical quantity first, calculate with the available information, then interpret the answer with the right units and accuracy. The context tells yo...