1 A Level Maths Course Guide2 AS and A Level Maths Course Guide Welcome to A Level Maths. In this course you will develop your mathematical skills fro...

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Course Guide 2017-18

AS and A Level Maths Course Guide Welcome to A Level Maths. In this course you will develop your mathematical skills from GCSE, and will learn many new and powerful techniques that can be used in many other areas such as Science, Pharmacy, Finance, and Computer Programming to name but a few. This course also gives you the building blocks for any further study in a course that depends heavily on mathematical ideas, for example Physics, Engineering, and of course Maths!

Course Overview In Year 12 you will study two modules: Pure Maths and Applied Maths (Statistics and Mechanics). In Year 13 you will study the same again, but obviously to a greater depth. Your A Level grade will be determined from all of your modules studied, and will be assessed via two Pure Maths papers and one Applied Maths paper. A calculator is allowed for each paper. There is no coursework; all of the units are 100% exam. You will sit two Pure Maths exams and one Applied Maths exam at the end of Year 13. Each exam is two hours in length and each counts as 1/3 towards your overall grade. AS Maths In Year 12, you will be taught by two teachers; one teaching predominantly Pure Maths, and the other teaching predominantly Applied Maths (there is some “cross-over” between the units). For Pure Maths, you will be expected to provide your own folder. Your teacher will provide you with extensive class notes which will guide you through the course, and plenty of supporting exercises that will help you to practise the basics, and develop your problem-solving skills. Able students can expect to be stretched by some optional, very demanding exercises. You must bring your full folder to each lesson, as you will constantly need to refer to earlier topics. You will also be issued with an exercise book for starter activities, as it is essential that basics are practised throughout the course, and this will help to avoid “clutter” in your folder. It is essential that you bring a calculator to each lesson, as you will be expected to use it in each lesson. For Applied Maths, it would be a good idea to have a separate folder. The emphasis in this is on problem-solving. A greater proportion of time therefore will be spent on problem-solving, and you will not be provided with a set of class notes. You will also have access to a “large data set” provided by the exam board. You will not be expected to do coursework, but familiarity with this data set will be of benefit for the final exam. A Level Maths moves considerably more rapidly than GCSE, but we are sensitive to the differing needs of pupils, and recognise that not all students will progress through the course at the same speed. We do not therefore devote a set amount of time to each module; some modules will be completed very quickly, but some will need a greater amount of time for key ideas to sink in. All MCHS Maths staff are very supportive, and you should ask them for help whenever necessary. Your teacher will tell you when support is available.

At the end of each module you will be given an assessment. This will be marked with feedback on a PLC which will tell you which areas you were good at, and which need further improvement. You will receive a WWW and an EBI. The EBI will typically be further exercises on areas that you found difficult, or extension work for those who performed well in all areas. Longer modules may involve two assessments. There will be regular tests and exams which will be marked and graded according to A Level grade boundaries. These will form the basis of your reports. Pure Maths Year 12 This course offers some bridging material from GCSE to A Level study, and ensures that you are able use these ideas fluently before moving on to more advanced topics. Towards the end of the course you will meet calculus. This is a powerful technique that deals with how things change, and forms the basis of many “applied” topics, particularly the maths underlying Physics. The course is split into 9 units. Unit 1 (Proof): - The structure of mathematical proof and its associated language. Proof by deduction, proof by exhaustion, proof by counter-example, proof by contradiction. A lot of these ideas will be met throughout the A Level course. Unit 2 (Algebra and Functions): - Recaps and develops algebraic skills from GCSE such as simplifying, brackets, factorising, indices, surds. A lot of emphasis will be placed on self-study, with support if needed. - Recaps and develops quadratic equations (factorising, using the formula, completing the square) - linear and non-linear simultaneous equations, linear and quadratic inequalities, and using the discriminant to determine the number of solutions to a quadratic - Simplifying algebraic fractions, division of algebraic expressions, Factor-Remainder Theorem. Factorising cubics. Unit 3 ( Coordinate Geometry) - Gradient and intercept. Equation of a line in various forms. Parallel and perpendicular lines. Solving problems. - Equation of a circle, tangents, chords, diameter Unit 4 (Series): - The binomial theorem, understanding factorial notation, nCr, xpanding (a +b)n and (1 + x) n Unit 5 (Trigonometry): - Prove and use trigonometric identities to simplify expressions and solve equations. Determine all possible solutions using a CAST diagram or a graph. Unit 6 (Differentiation): - Understand what differentiation means, and understand the notation. Find gradients of curves, and the equation of the tangent and the normal. Find rates of change, and the

second differential. Determine criteria for a function to be increasing/ decreasing/ stationary. Classify stationary points. Solve problems involving maxima and minima Unit 7 (Integration): - Understand that integration is the inverse of differentiation. Find the indefinite integral of expressions involving indices. Find f(x) given f’(x) and a coordinate. Use integration to find the area bounded by a curve and the x axis, a curve and a line, or two curves. Understand the meaning of negative area. Unit 8 (Vectors): Unit 9 (Exponentials and logarithms): Assessment: 2 hour exam (non-calculator) in Summer Supporting text: Edexcel AS and A Level Mathematics Core 1

Terms 2 and 3 Core 2 (another third of AS Maths) Teachers: Mrs L Riggs, Mr C Arrowsmith This course develops the ideas from Core 1 and introduces some other important concepts such as radians and logarithms. It also develops your techniques of calculus and shows how they can be used to solve increasingly complex problems. You are allowed the use of a calculator for Core 2. Unit 1 (Algebra): Unit 2 (Geometry): - Sine rule, cosine rule etc. (Note: the emphasis will be on self-study here) Unit 3 (Logarithms): - Definition of a logarithm, logarithm laws, solving equations, change of base (Note: logs are very useful for Unit 7) Unit 4 (Coordinate Geometry): Unit 5 (Binomial Theorem): Unit 6 (Radian Measure): - Understand the definition of a radian and understand why they are used, Arc length, sector area, area of a segment. Use of calculator in RAD mode. Unit 7 (Geometric Progression): - Term to term rules, understand nth term formula, prove the formula for the sum to n terms and use to solve problems Unit 8 (Trigonometric graphs). Note: This may be combined with Unit 10 - Plot and sketch graphs of sin, cos, tan. Transform functions using eg f(x) + a. Find exact values for 30, 45, 60 degrees etc Unit 9 (Differentiation):

Unit 10 (Trigonometric identities and equations): Unit 11 (Integration): - Use the Trapezium Rule to estimate areas, and determine whether it is an under or over estimate. Assessment: 1 ½ hour exam in Summer (calculator allowed) Supporting text: Edexcel AS and A Level Mathematics Core 2

Terms 1,2 &3 Mechanics 1 Teacher: Miss G Goodwin This is very much an “applied” course, and shows how maths can be used to model and predict the behaviour of physical systems, for example the motion of a ball thrown off a cliff top. The emphasis is on applying Newton’s laws of motion and their resulting formulae, and these fundamental ideas will serve as a building block for more complicated ideas. A lot of the course complements ideas from Physics, but we treat it in a far more mathematical way. Topics studied are briefly described below. Again, there is no set time for each unit, as it sometimes takes longer for these fundamental ideas to sink in. Mathematical models in mechanics - Understand the assumptions made when using maths to model a mechanical system. Understand the physical meaning of words such as “smooth”, light”, “inextensible”, “particle” etc and how these affect the system. Know how to draw effective force diagrams Kinematics of a particle moving in a straight line - Memorise the equations of motion. Describe motion under gravity. Use distance-time and velocity-time graphs Statics of a particle - Identify all forces acting in a system, and draw them effectively on a diagram. Know how to resolve a force in a given direction, in particular perpendicular and parallel to an inclined slope. Understand how to model friction. Understand equilibrium. Dynamics of a particle - Use: Sum of forces acting on a particle = ma to solve problems. Solve connected particles problems, eg pulley systems. Thrust in a towbar/ tension in a tow rope, reaction force in a lift ( your weight doesn’t change, but your reaction force does) Momentum and impulse - Use momentum and impulse to find speeds of particles before and after collisions. Model particles connected by a string. Moments - Define moment/ turning force, and use to solve problems for systems in equilibrium Vectors - Use vectors to model dynamic and static systems.

Assessment: 1 ½ hour exam in Summer (calculator allowed) Supporting text: Edexcel AS and A Level Mathematics Mechanics 1

A Level Maths A Level Maths follows much the same format as AS Maths in terms of assessment and timings, but you can expect to meet a higher degree of challenge at this level In Core 3 and Core 4, you will further develop some of the ideas introduced in AS, such as differentiation, integration and trigonometry, and you will begin to look at developing precision, eg the construction of proofs and formal definitions of functions. For Mechanics 2, you will build on skills developed in Mechanics 1, and look in more detail at modelling more complex situations. When you have completed C0re 3 and 4, you will have learned all the trigonometry and differentiation you will ever need for a higher level course involving maths. If only the same were true of integration… In addition to this, you will be given Independent Learning Projects (ILPs). The majority of these are intended to recap information from AS Maths, and to help embed the basic ideas. You will be set one of these at the start of a new topic. The assessment is exactly the same as for AS Maths, but the timings of the courses are slightly different. Core 3 is started at the end of the AS exams in Year 12 (Term 3) and continued until Christmas the following year (Term 4). You will then study Core 4 during terms 5 and 6. Mechanics 2 is started in year 13, and continues throughout the year.

Terms 3 and 4 Core 3 Teachers in Year 13: Mrs L Riggs and Mr C Starr Unit 1 (Algebraic Fractions): - Simplifying algebraic fractions by cancelling common factors. Add, subtract, multiply and divide algebraic fractions. Use algebraic division to simplify fractions Unit 2 (Functions): - Function notation and definition including domain and range. Classifying functions as oneone, many-one etc. Composite functions, inverse functions and sketching the graph of the inverse. Unit 3 (The exponential function): - Define the exponential function ex. Find the inverse lnx. Sketch graphs and transform them, eg f(x) + 3, and solve equations Unit 4 (Numerical methods): - Determine intervals in which a solution lies, and use graphs to determine the number of solutions. Derive and use an iterative formula, and show that a solution is correct to a given number of decimal places Unit 5 (The modulus function): - Define the modulus function f(x) = │x│. Use the modulus function in transformations of graphs. Solve modulus equations. Unit 6 (Trigonometry): - Definition of sec, cosec, cot, and their graphs. Definition of arcsin, arccos, arctan and their graphs. Further Pythagorean identities, use to prove identities and solve equations. Find exact trigonometric ratios using CAST. Unit 7 (Further trigonometry): - Addition formulae, double and half angle formulae, Rcos(θ-α), sums of sines and cosines. Use all to prove identities and solve equations. Unit 8 (Differentiation): - Chain rule, product rule, quotient rule. Prove and use differential of ex, lnx, trig ratios etc

Supporting text: Edexcel AS and A Level Mathematics Core 3

Terms 5 and 6 Core 4 Teachers in Year 13: Mrs L Riggs and Mr C Starr Unit 1 (Partial Fractions): - Decompose a fraction into its partial fractions. Recognise the different methods required Unit 2 (Binomial Theorem): - Use binomial theorem when n is not a positive integer. Determine the range of values for which the series converges. Use the binomial theorem to estimate eg √2. Use partial fractions to simplify Unit 3 (Parametric equations): - Express a Cartesian equation in terms of parametric equations and vice versa. Solve problems eg intersections of curves with axes etc. Unit 4 (Further Differentiation): - Implicit and parametric differentiation. Prove the rules for ax and logax Unit 5 (Integration): - Integration as the inverse of differentiation, integration of f(ax+b) . Using trigonometric identities, especially integrals of sin2x and cos2x . Using partial fractions, particularly with limits. Integration by substitution. Integration by parts. Comparing exact and approximate answers using the trapezium rule. Parametric integration - Volume of revolution using Cartesian and parametric equations - Shortcuts: -Integrating f’(x)fn(x) and f’(x)/f(x). Unit 6 (Vectors): - Recap basic definitions of vectors and use to prove geometric theorems. Use the scalar product. Find the vector equation of a line in 3D space. Determine whether or not a pair of lines intersect, and find the angle between two lines.

Supporting text: Edexcel AS and A Level Mathematics Core 4

Terms 4,5 and 6 Mechanics 2 Teacher: Mr C Starr Topics covered: Kinematics of a particle moving in a straight line or plane: - Projectiles - Velocity and acceleration when displacement is a function of time - Differentiating and integrating vectors, and finding the motion of a particle in a plane Centre of Mass: - Find the centre of mass of a system of particles, or a combination of 2D shapes (plane lamina) - Determine equilibrium conditions for a lamina on an inclined plane, or hanging from a fixed point Work, Energy, Power: - Definitions of work, energy and power. Types of energy: potential, kinetic, and energy lost due to friction. Conservation of energy, and change in energy for a particle. Power: moving vehicles. Collisions: - Elastic collisions. Inelastic collisions and the coefficient of restitution. Impact, or successive impacts of a particle with a fixed surface. Energy lost in an inelastic collision Statics of rigid bodies. - Moment of a force, and conditions for equilibrium. Lots and lots of problems involving ladders…

Supporting text: Edexcel AS and A Level Mathematics Mechanics 2

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