New GCSE Maths

Annotations:

• https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/254441/GCSE_mathematics_subject_content_and_assessment_objectives.pdf

1. Number
   1. Structure and calculation
      1. order positive and negative integers, decimals and fractions
      2. Add, Subtract, Multiply and Divide integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value
      3. Recognise and use relationships between operations, including inverse operations, use conventional notation for priority of operations, including brackets, powers, roots and reciprocals
      4. use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation,
      5. use positive integer powers and associated real roots (square, cube ...)
      6. apply systematic listing strategies including the product rule
      7. calculate with roots, and with integer and fractional indices
      8. calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares and rationalise denominators
      9. calculate with and interpret standard form A x 10n , where 1 ≤ A < 10 and n is an integer
   2. Fractions, decimals and percentages
      1. work interchangeably with terminating decimals and their corresponding fractions, change recurring decimals into their corresponding fractions and vice versa
      2. identify and work with fractions in ratio problems
      3. interpret fractions and percentages as operators.
   3. Measures and accuracy
      1. use standard units of mass, length, time, money and other measures
      2. estimate answers; check calculations using approximation
      3. round numbers and measures to an appropriate degree of accuracy ; use inequality notation to specify simple error intervals
      4. apply and interpret limits of accuracy, including upper and lower bounds
2. Algebra
   1. Notation, vocabulary and manipulation
      1. use and interpret algebraic notation
      2. substitute numerical values into formulae and expressions, including scientific formulae
      3. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
         1. taking out common factors
         2. multiplying a single term over a bracket
         3. collecting like terms
         4. expanding products of two or more binomials
         5. factorising quadratic expressions, including the difference of two squares;
         6. simplifying expressions involving sums, products and powers, including the laws of indices
      4. understand and use the concepts and vocabulary of expressions, equations, formulae, identities inequalities, terms and factors
      5. understand and use standard mathematical formulae; rearrange formulae to change the subject
      6. know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
      7. where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.
   2. Graphs
      1. plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient
      2. identify and interpret gradients and intercepts of linear functions graphically and algebraically
      3. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square
      4. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1 x with x ≠ 0, exponential functions =xyk for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size
      5. work with coordinates in all four quadrants
      6. plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
      7. recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.
      8. sketch translations and reflections of a given function
      9. calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
   3. Solving equations and inequalities
      1. solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph
      2. solve quadratic equations algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph
      3. solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically
      4. find approximate solutions to equations numerically using iteration
      5. translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution
      6. solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph
   4. Sequences
      1. generate terms of a sequence from either a term-to-term or a position-to-term rule
      2. recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions
      3. deduce expressions to calculate the nth term of linear and quadratic sequences.
3. Ratio, proportion and rates of change
   1. use scale factors, scale diagrams and maps
   2. express one quantity as a fraction of another,
   3. 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 contexts
   4. 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 real contexts and problems
   5. express a multiplicative relationship between two quantities as a ratio or a fraction
   6. use ratio notation, including reduction to simplest form
   7. define percentage ; interpret percentages and percentage changes as a fraction or a decimal, express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change
   8. understand and use proportion as equality of ratios
   9. solve problems involving direct and inverse proportion, including graphical and algebraic representations
   10. use compound units such as speed, rates of pay, unit pricing, density and pressure
   11. relate ratios to fractions and to linear functions
   12. interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
   13. compare lengths, areas and volumes using ratio notation; make links to similarity (including trigonometric ratios) and scale factors
   14. interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts
   15. set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes
   16. understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; construct and interpret equations that describe direct and inverse proportion
4. Geometry and measures
   1. Properties and constructions
      1. use conventional terms and notations; use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description
      2. use the standard ruler and compass constructions ; use these to construct given figures and solve loci problems
      3. 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 angles on parallel lines; derive and use the sum of angles in a triangle
      4. apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs
      5. derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language
      6. use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)
      7. identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional and negative scale factors)
      8. identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
      9. describe the changes and invariance achieved by combinations of rotations, reflections and translations
      10. apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
      11. identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
      12. solve geometrical problems on coordinate axes
      13. construct and interpret plans and elevations of 3D shapes.
   2. Mensuration and calculation
      1. use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)
      2. measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings
      3. know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)
      4. know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr2; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids
      5. calculate arc lengths, angles and areas of sectors of circles
      6. apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures
      7. know the formulae for: Pythagoras’ theorem and the trigonometric ratios, sinθ = opposite hypotenuse , cosθ = adjacent hypotenuse and tanθ = opposite adjacent ; apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures
      8. know the exact values of sinθ and cosθ for given values of θ (see attached)
      9. know and apply the sine rule and cosine rule, to find unknown lengths and angles
      10. know and apply Area = 1/2 ab SinC to calculate the area, sides or angles of any triangle.
   3. Vectors
      1. describe translations as 2D vectors
      2. apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs
5. Probability
   1. record describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
   2. apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
   3. relate relative expected frequencies to theoretical probability, using appropriate language and the 0 - 1 probability scale
   4. apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
   5. understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
   6. enumerate sets and combinations of sets, using tables, grids, Venn diagrams and tree diagrams
   7. construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
   8. calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
   9. calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.
6. Statistics
   1. construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs
   2. infer properties of populations or distributions from a sample, know the limitations
   3. use and interpret scatter graphs of bivariate data; recognise correlation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends
   4. interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
      1. appropriate graphical representation involving discrete, continuous and grouped data, including box plots
      2. appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including outliers, quartiles and inter-quartile range)
   5. interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, tables and line graphs for time series data
   6. apply statistics to describe a population