# GCSE Maths

January 31, 2019

NumberStructure and calculation1. order positive and negative integers, decimals and fractions; use the symbols =, ≠,, ≤, ≥2. apply the four operations, including formal written methods, to integers, decimalsand simple fractions (proper and improper), and mixed numbers – all both positiveand negative; understand and use place value (e.g. when working with very largeor very small numbers, and when calculating with decimals)53. recognise and use relationships between operations, including inverse operations(e.g. cancellation to simplify calculations and expressions; use conventionalnotation for priority of operations, including brackets, powers, roots and reciprocals4. use the concepts and vocabulary of prime numbers, factors (divisors), multiples,common factors, common multiples, highest common factor, lowest commonmultiple, prime factorisation, including using product notation and the uniquefactorisation theorem5. apply systematic listing strategies including use of the product rule forcounting6. use positive integer powers and associated real roots (square, cube and higher),recognise powers of 2, 3, 4, 5; estimate powers and roots of any given positivenumber7. calculate with roots, and with integer and fractional indices8. calculate exactly with fractions, surds and multiples of π; simplify surdexpressions involving squares (e.g. 12 = 4× 3 = 4 × 3 = 2 3 ) andrationalise denominators9. calculate with and interpret standard form A x 10n, where 1 ≤ A 0 or a surd) and other sequences25.deduce expressions to calculate the nth term of linear and quadratic sequences.Ratio, proportion and rates of change1. change freely between related standard units (e.g. time, length, area,volume/capacity, mass) and compound units (e.g. speed, rates of pay,prices, density, pressure) in numerical and algebraic contexts2. use scale factors, scale diagrams and maps3. express one quantity as a fraction of another, where the fraction is less than 1 orgreater than 14. use ratio notation, including reduction to simplest form5. divide a given quantity into two parts in a given part:part or part:whole ratio;express the division of a quantity into two parts as a ratio; apply ratio to realcontexts and problems (such as those involving conversion, comparison, scaling,mixing, concentrations)6. express a multiplicative relationship between two quantities as a ratio or a fraction7. understand and use proportion as equality of ratios8. relate ratios to fractions and to linear functions9. define percentage as ‘number of parts per hundred’; interpret percentages andpercentage changes as a fraction or a decimal, and interpret these multiplicatively;express one quantity as a percentage of another; compare two quantities usingpercentages; work with percentages greater than 100%; solve problems involvingpercentage change, including percentage increase/decrease and original valueproblems, and simple interest including in financial mathematics10.solve problems involving direct and inverse proportion, including graphical andalgebraic representations11.use compound units such as speed, rates of pay, unit pricing, density andpressure12.compare lengths, areas and volumes using ratio notation; make links to similarity(including trigonometric ratios) and scale factors913.understand that X is inversely proportional to Y is equivalent to X is proportional to1Y ; construct and interpret equations that describe direct and inverse proportion14.interpret the gradient of a straight line graph as a rate of change; recognise andinterpret graphs that illustrate direct and inverse proportion15.interpret the gradient at a point on a curve as the instantaneous rate ofchange; apply the concepts of average and instantaneous rate of change(gradients of chords and tangents) in numerical, algebraic and graphicalcontexts16.set up, solve and interpret the answers in growth and decay problems, includingcompound interest and work with general iterative processes.Geometry and measuresProperties and constructions1. use conventional terms and notations: points, lines, vertices, edges, planes,parallel lines, perpendicular lines, right angles, polygons, regular polygons andpolygons with reflection and/or rotation symmetries; use the standard conventionsfor labelling and referring to the sides and angles of triangles; draw diagrams fromwritten description2. use the standard ruler and compass constructions (perpendicular bisector of a linesegment, constructing a perpendicular to a given line from/at a given point,bisecting a given angle); use these to construct given figures and solve lociproblems; know that the perpendicular distance from a point to a line is theshortest distance to the line3. apply the properties of angles at a point, angles at a point on a straight line,vertically opposite angles; understand and use alternate and corresponding angleson parallel lines; derive and use the sum of angles in a triangle (e.g. to deduce anduse the angle sum in any polygon, and to derive properties of regular polygons)4. derive and apply the properties and definitions of: special types of quadrilaterals,including square, rectangle, parallelogram, trapezium, kite and rhombus; andtriangles and other plane figures using appropriate language5. use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)6. apply angle facts, triangle congruence, similarity and properties of quadrilaterals toconjecture and derive results about angles and sides, including Pythagoras’Theorem and the fact that the base angles of an isosceles triangle are equal, anduse known results to obtain simple proofs7. identify, describe and construct congruent and similar shapes, including oncoordinate axes, by considering rotation, reflection, translation and enlargement(including fractional and negative scale factors) 108. describe the changes and invariance achieved by combinations of rotations,reflections and translations9. identify and apply circle definitions and properties, including: centre, radius, chord,diameter, circumference, tangent, arc, sector and segment10.apply and prove the standard circle theorems concerning angles, radii,tangents and chords, and use them to prove related results11.solve geometrical problems on coordinate axes12.identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids,prisms, cylinders, pyramids, cones and spheres13.construct and interpret plans and elevations of 3D shapes.Mensuration and calculation14.use standard units of measure and related concepts (length, area,volume/capacity, mass, time, money, etc.)15.measure line segments and angles in geometric figures, including interpretingmaps and scale drawings and use of bearings16.know and apply formulae to calculate: area of triangles, parallelograms, trapezia;volume of cuboids and other right prisms (including cylinders)17.know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr2;calculate: perimeters of 2D shapes, including circles; areas of circles andcomposite shapes; surface area and volume of spheres, pyramids, cones andcomposite solids18.calculate arc lengths, angles and areas of sectors of circles19.apply the concepts of congruence and similarity, including the relationshipsbetween lengths, areas and volumes in similar figures20.know the formulae for: Pythagoras’ theorem, a2 + b2 = c2, and the trigonometricratios, sinθ = oppositehypotenuse , cosθ = adjacenthypotenuseand tanθ = oppositeadjacent ; apply them to findangles and lengths in right-angled triangles and, where possible, generaltriangles in two and three dimensional figures21.know the exact values of sinθ and cosθ for θ = 00, 300, 450 , 600 and 900; know theexact value of tanθ for θ = 00, 300, 450 and 60022.know and apply the sine rule, sin sin sinabcABC = = , and cosine rule,2 22 a b c bc A =+− 2 cos , to find unknown lengths and angles23.know and apply 1 Area = sin2ab C to calculate the area, sides or angles of anytriangle.11Vectors24.describe translations as 2D vectors25.apply addition and subtraction of vectors, multiplication of vectors by a scalar, anddiagrammatic and column representations of vectors; use vectors to constructgeometric arguments and proofsProbability1. record describe and analyse the frequency of outcomes of probability experimentsusing tables and frequency trees2. apply ideas of randomness, fairness and equally likely events to calculateexpected outcomes of multiple future experiments3. relate relative expected frequencies to theoretical probability, using appropriatelanguage and the 0 - 1 probability scale4. apply the property that the probabilities of an exhaustive set of outcomes sum toone; apply the property that the probabilities of an exhaustive set of mutuallyexclusive events sum to one5. understand that empirical unbiased samples tend towards theoretical probabilitydistributions, with increasing sample size6. enumerate sets and combinations of sets systematically, using tables, grids, Venndiagrams and tree diagrams7. construct theoretical possibility spaces for single and combined experiments withequally likely outcomes and use these to calculate theoretical probabilities8. calculate the probability of independent and dependent combined events,including using tree diagrams and other representations, and know the underlyingassumptions9. calculate and interpret conditional probabilities through representationusing expected frequencies with two-way tables, tree diagrams and Venndiagrams.Statistics1. infer properties of populations or distributions from a sample, whilst knowing thelimitations of sampling2. interpret and construct tables, charts and diagrams, including frequency tables,bar charts, pie charts and pictograms for categorical data, vertical line charts forungrouped discrete numerical data, tables and line graphs for time series data andknow their appropriate use3. construct and interpret diagrams for grouped discrete data and continuousdata, i.e. histograms with equal and unequal class intervals and cumulativefrequency graphs, and know their appropriate use124. interpret, analyse and compare the distributions of data sets from univariateempirical distributions through: appropriate graphical representation involving discrete, continuous andgrouped data, including box plots appropriate measures of central tendency (median, mean, mode andmodal class) and spread (range, including consideration of outliers,quartiles and inter-quartile range)5. apply statistics to describe a population6. use and interpret scatter graphs of bivariate data; recognise correlation and knowthat it does not indicate causation; draw estimated lines of best fit; makepredictions; interpolate a

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Notting Hill and Ealing School Past Paper (Maths)

A maths past exam from 2008 for Notting Hill and Ealing School, part of the London Independent Girls' School Consortium