What mathematics do students study in A level ... - …

1 MEI 2009 Whatmathematicsdostudents studyin A level Mathematicscourses?Since the structure of A level mathematics (and Further mathematics ) was changed inSeptember 2004,studentswith only a single A level in Mathematicswill have studied only twoapplied modules (in addition to the four core modules, Core 1 to Core 4, which cover thecompulsory pure contentof the A level ).Please theofficial pure core for AS/A level mathematics , which must be covered by ALLAS/A combinations of modules studied for A level mathematics are:Core1, Core2, Core3,Core4+one of thecombinations of 2 applied modules shown belowStatistics1 Mechanics1 Mechanics1 Decision1 Decision1 Statistics1 Statistics1 Statistics2 Decision 1 Decision2 Mechanics1 Mechanics2 There are no prescribed appliedmodules that are required to be studied, hence students couldstudy anyoneof thesecombinations in order to gain an A level in followingpages summarise the approximate content of thecore modules in A levelMathematics, in AS/A level Further mathematics and in thefirst two modules ofeachappliedstrand.

2 However, there are differences between the content of such modules for the different Alevel specifications(and additionallya few other applied modules may be available fromspecificboards, by MEI).Those students who have studied an AS or A level in Further mathematics will have had theopportunity to study more applied mathematics modules thanthose with just the single A levelMathematics. This highlights the worth of the further mathematics qualification for those studentswho wish to study for mathematics -related degrees at find out more about Further is considerable scope for MEI to work with universities to help support the learning andteaching of undergraduates, for both first year mathematics courses and for mathematics servicecourses in mathematics -related more information, please to University underthe what we do you have any queries, please contact Charlie Stripp or StephenLee mathematics Algebra 9 Simultaneous equations 9 Solving quadratics, completion of square 9 Surds/indices 9 Logarithms 9 Inequalities (only involving linear and quadratic expressions, and the modulus function) 9 Polynomials (factor/remainder theorems)

3 9 Binomial expansion 9 Partial Fractions Trigonometry 9 Sine rule, cosine rule 9 Radians, arc length, sector area 9 Exact values of sin, cos, tan of standard angles 9 Sec, cosec, cot, arcsin, arccos, arctan 9 Compound/double angle formulae Sequences and Series 9 Arithmetic/geometric sequences/series 9 Sigma notation 9 Sequences defined recursively Curve Sketching 9 Graphs of quadratics, polynomials (from the factorised form) 9 Relationships between graphs of y = f(x), y = f(x + a), y = f(ax) Exponential and Log 9 Their graphs 9 Standard properties 9 Use in solving equations Coordinate Geometry 9 Equations of straight lines, gradient 9 Parallel and perpendicular lines 9 Equation of a circle 9 Circle theorems Functions 9 Composition 9 Inverses, calculating inverses 9 Even, odd, periodic functions 9 Modulus function 9 Inverse trig functions Parametric Equations 9 Finding gradients 9 Conversion from cartesian to parametric equations Calculus 9 Differentiation of powers of x, ex, ln x, sin x, cos x, tan x 9 Product rule, quotient rule 9 Chain rule 9 Integration by inspection 9 Integration by substitution (simple cases only) 9 Integration by parts 9 Differential equations (to include only variables which are separable)

4 9 Implicit differentiation 9 Volumes of revolution Vectors 9 Scalar product 9 Equations of lines 9 Intersection of lines Numerical Methods 9 Roots by sign change 9 Fixed point iteration Topics that students will not have met unless they have done AS/ A level Further mathematics include: Complex Numbers 9 Definitions, basic arithmetic 9 Argand diagram 9 Polynomial equations with complex roots 9 Polar form* 9 De Moivre s theorem* 9 Exponential notation* 9 nth roots of complex numbers* Matrices 9 Definitions, basic arithmetic 9 Matrices as linear transformation, composition 9 Determinant, inverse 9 Use in solving linear simultaneous equations, equations of planes and geometric interpretation 9 Characteristic polynomial* 9 Eigenvalues and eigenvectors* Proof 9 Language with proof 9 Proof by induction 9 Proving hyperbolic trig identities 9 Uniqueness of inverse matrix Curve Sketching 9 Graphs of rational functions 9 Conic sections* Series 9 Telescoping 9 Limits of series (beyond GPs)

5 9 Maclaurin/taylor series and approximations Coordinate Systems 9 Polar coordinates 9 Intrinsic coordinates* Calculus 9 Using inverse trig functions, hyperbolic trig functions 9 More advanced substitution 9 Integrating factor* Hyperbolic Trig Functions 9 Definition 9 Identities 9 Their use in calculus (*topics which are dependent on the material in the full A level Further mathematics having been studied, as opposed to AS level Further mathematics ) Depending on the specification followed and modules studied some students who have A level Further mathematics will also know some: Group Theory, Vector Spaces, Fields, Differential Geometry, Mutivariable Calculus, Differential Equations (including second order), Numerical Methods. Some students , particularly those who have not studied Further mathematics , may not have had the opportunity to have studied certain applied modules, M2, D2 to name but two, during their time in the sixth form.

6 The following pages give information on the approximate content of the first two modules of each applied strand. MECHANICS Newton s Laws Applied Along a Line 9 Motion in a horizontal plane 9 Motion in a vertical plane 9 Pulleys 9 Connected bodies Motion Graphs 9 Displacement-time, distance-time, velocity-time 9 Interpreting the graphs 9 Using differentiation and integration Constant Acceleration and SUVAT Equations 9 Introduction to the variables 9 Using the variables 9 Standard properties 9 Use in solving equations Vectors and Newton s Laws in 2 Dimensions 9 Resolving forces into components 9 Motion on a slope (excluding and including friction) Projectiles 9 Finding the maximum height, range and path of a projectile Collisions 9 Coefficient of restitution 9 Conservation of linear momentum 9 Impulse 9 Calculations involving a loss of energy Centre of Mass 9 Uniform bodies (symmetry) 9 Composite bodies 9 Centres of mass when suspended Equilibrium of a Rigid Body 9 Moment about a point 9 Coplanar forces 9 Toppling/sliding Variable Acceleration 9 Using differentiation in 1 and 2-D 9 Using integration in 1 and 2-D Energy, Work and Power 9 Work done 9 GPE, KE 9 Conservation of energy 9 Power (force does work)

7 Uniform Motion in a Circle 9 Angular speed 9 Acceleration 9 Horizontal circle, conical pendulum STATISTICS Correlation and Regression 9 Product moment correlation 9 spearman coefficient rank correlation 9 Independent and dependant variables 9 Least squares regression 9 Scatter diagrams Data Presentation 9 Bar charts, pie charts 9 Vertical line graphs, histograms 9 Cumulative frequency Discrete Random Variables 9 Expectation and variance of discrete random variables 9 Formulae extensions E(aX+b) The Binomial Distribution and Probability 9 Probability based on selecting or arranging 9 Probability based on binomial distribution 9 Expected value of a binomial distribution 9 Expected frequencies from a series of trials Probability 9 Measuring, estimating and expectation 9 Combined probability 9 Two trials 9 Conditional probability 9 Simple applications of laws Exploring Data 9 Types of data 9 Stem and Leaf diagrams 9 Measures of central tendency and of spread 9 Linear coding 9 Skewness and outliers Hypothesis Testing 9 Establishing the null and alternate hypothesis 9 Conducting the test and interpreting the results 9 Use of the binomial or normal distribution 9 Type 1 and type 2 errors Normal Distribution 9 Properties (including use of tables) 9 Mean and variance 9 Cumulative distribution function 9 Continuous random variables (probability density function and mean/variance)

8 9 As an approximation to binomial or Poisson distributions 9 t-distribution Poisson Distribution 9 Properties (including use of tables) 9 Mean and variance 9 Use as an approximation to binomial distribution Sampling/ Estimation 9 Randomness in choosing 9 Sample means and standard errors 9 Unbiased estimates of population means 9 Use of central limit theorem 9 Confidence intervals Chi-squared 9 Introduction 9 Conditions DECISION Graphs 9 Graphs Algorithms 9 Communicating 9 Sorting 9 Packing 9 Efficiency and complexity Networks 9 Prim 9 Kruskal 9 Dijkstra 9 TSP 9 Route inspection 9 Network flows Linear Programming 9 LP graphical 9 LP simplex 9 Two stage simplex Critical Path Analysis 9 Activity networks 9 Cascade charts Simulation 9 Monte Carlo methods 9 Uniformly distributed random variables 9 Non-uniformly distributed random variables 9 Simulation models Game Theory 9 Game theory 9 Using

9 Simplex Logic and Boolean Algebra 9 Logical propositions and truth tables 9 Laws of Boolean algebra 9 Combinatorial circuits and switching circuits Optimisation 9 Matchings 9 Hungarian algorithm 9 Transportation problem 9 Dynamic programming