Subjects / Leaving Certificate / Mathematics
Mapped to: Leaving Certificate Mathematics (current syllabus; check curriculumonline.ie for updates) · official specification (NCCA). Topic names are StudyRoom's own summaries.
Statistics and Probability
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Counting: arrangements and choicesUse the fundamental principle of counting, arrangements (n!) and combinations (nCr) to count outcomes.HLOL 16 q
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Probability concepts and relative frequencyDescribe chance on a 0 to 1 scale, find probabilities of equally likely outcomes and estimate them from experiments.HLOLFL 15 q
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Probability rules: union, independence and conditional probabilityApply the addition and multiplication rules, recognise independent and mutually exclusive events and find conditional probabilities.HLOL 15 q
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Expected value and fair gamesCalculate the expected value of a random variable and use it to judge whether a game or decision is fair.HLOL 15 q
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Bernoulli trials and the binomial distributionFind probabilities for repeated independent trials with two outcomes, including the first success and exactly r successes.HLOL 15 q
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Data collection, sampling and biasClassify data, choose sampling methods and recognise sources of bias in surveys and experiments.HLOLFL 15 q
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Representing data: charts, histograms and distributionsChoose and interpret charts, stem-and-leaf plots and histograms and describe the shape of a distribution.HLOLFL 15 q
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Measures of centre and spreadCalculate and interpret mean, median, mode, range, interquartile range and standard deviation.HLOLFL 15 q
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The normal distribution, empirical rule and z-scoresUse the empirical rule and, at Higher Level, z-scores and tables to find probabilities for normal data.HLOL 15 q
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Scatter plots and correlationInterpret scatter plots, the correlation coefficient and lines of best fit, and distinguish correlation from causation.HLOL 15 q
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Margin of error, confidence intervals and hypothesis testsUse the margin of error 1/√n and confidence intervals to test claims about population proportions.HL 15 q
Geometry and Trigonometry
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Coordinate geometry of the lineFind slopes, distances, midpoints, equations of lines, perpendicular distance and areas of triangles.HLOL 16 q
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Coordinate geometry of the circleWork with circle equations, centre and radius, points on circles, tangents and intersections with lines.HLOL 16 q
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Synthetic geometry: theorems and proofsUse and prove the prescribed geometry theorems on angles, triangles, parallelograms and circles.HLOL 15 q
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Geometric constructionsCarry out ruler-and-compass constructions such as bisectors, circumcircles and incircles.HLOL 15 q
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Enlargements and similar figuresFind scale factors and centres of enlargement and use the effect of enlargement on lengths and areas.HLOL 15 q
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Pythagoras and right-angled trigonometryUse Pythagoras' theorem and sin, cos and tan to solve right-angled triangles, including angles of elevation.HLOL 15 q
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The sine rule and cosine ruleSolve non-right-angled triangles with the sine rule and cosine rule, including the ambiguous case.HLOL 16 q
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Area of a triangle, arcs and sectorsUse area = ½ab sin C and find arc lengths and sector areas in degrees and radians.HLOL 15 q
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Radians, the unit circle and trigonometric graphsConvert between degrees and radians, use the unit circle for any angle and describe the period and range of trig graphs.HLOL 15 q
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Trigonometric identities and compound anglesUse and prove identities such as sin²A + cos²A = 1, compound angle and double angle formulae.HL 15 q
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Solving trigonometric equationsFind all solutions of trigonometric equations in a given interval, including general solutions.HL 15 q
Number
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Number systems, factors and primesClassify numbers as natural, integer, rational or real and work with factors, primes, HCF and LCM.HLOLFL 15 q
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Complex numbers and the Argand diagramAdd, multiply and divide complex numbers, find conjugates and moduli and show them on an Argand diagram.HLOL 16 q
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Polar form and De Moivre's theoremWrite complex numbers in polar form and use De Moivre's theorem for powers and roots.HL 15 q
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Arithmetic sequences and seriesFind the general term and the sum of an arithmetic sequence and solve problems with them.HLOL 15 q
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Geometric sequences and seriesFind the general term and the sum of n terms of a geometric sequence, including growth problems.HLOL 15 q
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Infinite geometric seriesFind the sum to infinity of a convergent geometric series and write recurring decimals as fractions.HL 15 q
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Indices and surdsApply the rules of indices, including fractional and negative indices, and simplify surds.HLOL 15 q
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Logarithms and exponential equationsUse the laws of logarithms and solve exponential and logarithmic equations.HL 15 q
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Approximation, scientific notation and errorRound to significant figures, use scientific notation and calculate percentage error and bounds.HLOLFL 15 q
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Everyday finance: percentages, VAT, tax and currencyCalculate percentage changes, VAT, income tax, unit prices and currency conversions.HLOLFL 15 q
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Compound interest and depreciationUse F = P(1 + i)^t for compound interest and reducing-balance depreciation, including AER.HLOL 16 q
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Present value, annuities and amortisationDiscount future sums to present value and calculate loan and mortgage repayments with the amortisation formula.HL 16 q
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Length, area and volumeFind perimeters, areas, surface areas and volumes of 2D shapes and solids such as cylinders, cones and spheres.HLOLFL 15 q
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The trapezoidal ruleEstimate the area of an irregular shape using the trapezoidal rule.HLOL 15 q
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Limits of sequences and functionsFind limits of sequences and functions by inspection and with the rules for limits, and connect them to sums to infinity and continuity.HL 15 q
Algebra
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Algebraic expressions and factorisingExpand, factorise and simplify algebraic expressions and fractions and change the subject of a formula.HLOL 16 q
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Linear equations and simultaneous equationsSolve linear equations and systems of two or three simultaneous linear equations, including word problems.HLOL 15 q
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Quadratic equationsSolve quadratics by factorising and the formula, and use the discriminant and the sum and product of roots.HLOL 15 q
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Simultaneous equations with one non-linearSolve a linear and a non-linear equation together by substitution and interpret the solutions graphically.HLOL 15 q
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Cubic equations and the factor theoremUse the factor theorem to find factors and roots of cubic polynomials.HL 15 q
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InequalitiesSolve linear, quadratic and modulus inequalities and show the solutions on a number line.HLOL 15 q
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Proof by inductionProve statements about series, divisibility and inequalities using mathematical induction.HL 15 q
Functions and Calculus
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Functions: domain, codomain and rangeInterpret functions as mappings, evaluate them and state their domain and range.HLOL 15 q
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Composite and inverse functionsForm composite functions and find inverse functions where they exist.HLOL 15 q
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Graphs of functions and transformationsSketch linear, quadratic, cubic, exponential and logarithmic graphs and describe transformations.HLOL 15 q
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Differentiation from first principlesUse the limit definition of the derivative to differentiate simple functions.HL 15 q
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Rules of differentiationDifferentiate using the power, product, quotient and chain rules, including trig, exponential and log functions.HLOL 15 q
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Maxima, minima and curve sketchingUse derivatives to find turning points, points of inflection and to solve optimisation problems.HLOL 15 q
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Rates of changeInterpret derivatives as rates of change, including velocity, acceleration and related rates.HLOL 16 q
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IntegrationIntegrate polynomial, exponential and trigonometric functions and evaluate definite integrals.HL 15 q
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Area under a curve and average valueUse definite integrals to find areas bounded by curves and the x-axis and the average value of a function.HL 15 q
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Exponential functions and modelsGraph and use functions of the form a·bˣ and eˣ to model growth and decay, including doubling and half-life.HLOL 19 q