A Level Further Mathematics

Transcription

1 A Level Further Mathematics Contents For courses in Year 13 starting from September 2014 Course Overview... 2 Schemes of Work... 3 Further Pure Further Pure Further Pure Decision Mechanics Mechanics Assessment Schedule Key Assessment points General Expectations Preparing for Lessons and Homework Assignments Tasks that we will set Independent study Community Service Resources Who to contact about the course... 14

2 Course Overview Examination Board: AQA ( Subject code: 6360 (5371 for AS, 6371 for A2) The Course In studying Further Mathematics A level you will cover six modules: Further Pure 1 (MFP1) Further Pure 2 (MFP2) Further Pure 3 (MFP3) Decision 2 (MD02) Mechanics 1B (MM1B without coursework) Mechanics 2B (MM2B without coursework) The six modules are equally weighted, i.e. each count for 16 2 / 3 % of the final mark. The modules are examined as follows: Module Exam Length of Paper Number of marks Total % Further Pure 1 1 written paper 1 hr 30 minutes / 3 Further Pure 2 1 written paper 1 hr 30 minutes / 3 Further Pure 3 1 written paper 1 hr 30 minutes / 3 Decision 2 1 written paper 1 hr 30 minutes / 3 Mechanics 1B 1 written paper 1 hr 30 minutes / 3 Mechanics 2B 1 written paper 1 hr 30 minutes / 3 All 6 modules are sat in the Summer of Year 13. There is no coursework requirement for A Level Further Mathematics. There are 5 Maths groups, and 2 Further Maths groups in Year 13. Syllabus Content The syllabus content for each of the modules can be found on the AQA website ( Notice at the start of each syllabus there is a list of formulae that you will be expected to KNOW for the exam. There are no formula sheets but you will be given a formulae booklet in each exam. You can download the formulae booklet from the AQA website. Page 2 of 15

3 Schemes of Work Further Pure 1 Chapter Section Lessons Syllabus reference 1. Roots of quadratic equations 1.1 Roots and coefficients 1.2 Expressions involving α and β 1.3 Forming new equations with related roots 1.4 Further examples 1.5 New equations by means of a substitution Complex numbers 2.1 Historical background 2.2 The imaginary number i 2.3 Complex numbers and complex conjugates 2.4 Combining complex numbers 2.5 Complex roots of quadratic equations 2.6 Equating real and imaginary parts 3. Inequalities 3.1 Introduction 3.2 Inequalities involving rational expressions 3.3 Multiplying both sides by the square of the denominator 3.4 Combining terms into a single fraction 4. Matrices 4.1 Introduction 4.2 The order of a matrix 4.3 Adding and subtracting matrices 4.4 Multiples of matrices 4.5 Multiplying two matrices 4.6 Special matrices (16.1) 5. Trigonometry 5.1 Exact sine, cosine tangent for multiples of 30 and General solutions of sinx=k and cosx=k, where 1 k General solution of tanx=k Matrix transformations 6.1 Introduction 6.2 Transformation matrices 6.3 Matrices associated with common transformations 6.4 Stretches and enlargements 6.5 Rotations about the origin 6.6 Reflections in a line through the origin 6.7 Composite transformations 7. Linear laws 7.1 Review of straight line graphs 7.2 Reducing a relation to a linear law 7.3 Use of logarithms to reduce equations of the form y=ax n to a linear law 7.4 Use of logarithms to reduce equations of the form y=ab x to a linear law Page 3 of 15

4 8. Calculus 8.1 Gradient of a chord and tangent 8.2 Gradient of curve at a point as the limit of the gradient of the chord 8.3 Use of binomial theorem 8.4 Improper integrals with limits involving infinity 8.5 Further improper integrals 9. Series 9.1 Sigma notation 9.2 Sum of first n natural numbers 9.3 Sums of squares and cubes 9.4 Questions involving algebra 10. Numerical methods 10.1 Change of sign to find roots of equations 10.2 Bisection method 10.3 Linear interpolation 10.4 Newton-Raphson iterative formula 11. Asymptotes and rational functions 12. Further rational functions and maximum and minimum points 10.5 Numerical method to find a point on a curve 10.6 Euler s step-by-step method 11.1 Asymptotes 11.2 Vertical asymptotes 11.3 Curves of the form y=(ax+b)/(cx+d) 11.4 Intersection of graphs of rational functions and straight lines 11.5 Use of graphs to solve inequalities 12.1 Rational functions with quadratic denominators 12.2 Rational functions of the form (px+q)/(ax 2 +bx+c) 12.3 Use of discriminant to find regions for which a curve is defined 12.4 Finding stationary points without calculus Parabolas, ellipses and hyperbolas 13.1 Parabolas and their vertices at the origin 13.2 Ellipses with their centres at the origin 13.3 Hyperbolas 13.4 Translations of curves 13.5 Intersections with straight lines Page 4 of 15

5 Further Pure 2 Chapter Section Lessons Syllabus reference 1. Complex numbers 1.1 Introduction 1.2 Complex numbers 1.3 Modulus and argument 1.4 Polar form of complex number 1.5 Addition, subtraction and multiplication of complex numbers 1.6 Complex conjugate and division of complex numbers 1.7 Products and quotients of complex numbers in polar form 1.8 Equating real and imaginary parts 1.9 Further consideration of modulus and argument Roots of polynomial equations 3. Summation of finite series 1.10 Loci on Argand diagrams 2.1 Introduction 2.2 Quadratic equations 2.3 Cubic equations 2.4 Relationship between roots of a cubic equation and its coefficients 2.5 Cubic equations with related roots 2.6 An important result 2.7 Polynomial equations of degree n 2.8 Complex roots of polynomial equations with real coefficients 3.1 Introduction 3.2 Summation of series by the method of differences 3.3 Summation of series by method of induction 3.4 Proof by induction De Moivre s Theorem 5. Inverse trigonometric functions 4.1 De Moivre s theorem 4.2 Using De Moivre to evaluate powers of complex numbers 4.3 Trigonometric identities using De Moivre 4.4 Exponential form of a complex number 4.5 Cube roots of unity 4.6 nth roots of unity 4.7 Roots of z n = a, where a is non-real. 5.1 Introduction and revision 5.2 Derivative of standard inverse trig. functions 5.3 Application to more complex differentiation 5.4 Standard integrals integrating to inv. trig. functions. 5.5 Applications to more complex integrals 6. Hyperbolic functions 6.1 Definitions of hyperbolic functions 6.2 Numerical values of hyperbolic functions 6.3 Graphs of hyperbolic functions 6.4 Hyperbolic identities 6.5 Osborne s Rule PTO PTO Page 5 of 15

6 6. Hyperbolic functions (ctd.) 7. Arc length and area of surface of revolution 6.6 Differentiation of hyperbolic functions 6.7 Integration of hyperbolic functions 6.8 Inverse hyperbolic functions 6.9 Logarithmic form of inverse hyp. fns Derivatives of inverse hyp. fns Integrals which integrate to inv. hyp. fns Solving equations 7.1 Introduction 7.2 Arc Length 7.3 Area of surface of revolution Page 6 of 15

7 Further Pure 3 Chapter Section Lessons Syllabus reference 1. Series and limits 1.1 The concept of a limit 1.2 Finding limits in simple cases 1.3 Maclaurin s series expansion 1.4 Range of validity of a series expansion 1.5 The basic series expansions 1.6 Use of series expansions to find limits 1.7 Two important limits 1.8 Improper integrals 2. Polar coordinates 2.1 Cartesian and polar frames of reference 2.2 Restrictions on the value of θ 2.3 Relationship between Cartesian and polar coordinates 2.4 Representing curves in polar form 2.5 Curve sketching 2.6 Area bounded by a polar curve Introduction to differential equations 4. Numerical methods for the solution of first order differential equations 5. Second order differential equations 3.1 Order and linearity 3.2 Families of solutions, general solutions and particular solutions 3.3 Analytic solution of first order linear differential equation: integrating factors 3.4 Complementary functions and particular integrals 3.5 Transformations of non-linear differential equations to linear form 4.1 Introduction 4.2 Euler s formula 4.3 The mid-point formula 4.4 The improved Euler formula 4.5 Error analysis: some practical considerations 5.1 Introduction to complex numbers 5.2 Working with complex numbers 5.3 Euler s identity 5.4 Formation of second order differential equations 5.5 Differential equations of the form ay +by +cy=0 5.6 Differential equations of the form ay +by +cy=f(x) 5.7 Second order linear differential equations with variable coefficients Page 7 of 15

8 Decision 2 Chapter Section Lessons Syllabus reference 1 Allocation Introduction 1.2 The Hungarian Algorithm 1.3 Non-square arrays 2 Network flows 2.1 Some important terms 2.2 Max flow / min cut theorem 2.3 The labelling process 2.4 Extensions 2.5 Minimum capacities 3 Critical Path Analysis 3.1 Activity networks 3.2 Earliest and latest starting times 3.3 Critical activities 3.4 Cascade diagrams 3.5 Resource levelling 4 Dynamic Programming 4.1 Negative edge weights 4.2 Various optimisation problems 4.3 Terminology 4.4 Min and max problems 4.5 Maximin and minimax problems 5 Simplex Algorithm 5.1 The simplex method 5.2 The tableau format 5.3 The simplex algorithm 5.4 Network problems 6 Game Theory 6.1 Zero-sum games 6.2 Play-safe strategies 6.3 Stable solutions 6.4 Mixed strategies x n games [6.6 m x n games not on syllabus] Page 8 of 15

9 Mechanics 1 Chapter Section Number of Lessons AQA syllabus reference 1 Kinematics in one dimension 2 Kinematics in two dimensions A Velocity and Displacement B Graphs of motion C Area under a velocity-time graph D Motion with constant acceleration E Constant acceleration equations A Displacement B Resultant displacement C Position vector D Velocity E Resultant velocity F Resultant velocity problems G Acceleration H Constant acceleration equations in two dimensions 3 Forces A Forces as vectors B Resolving a force C Resolving coplanar forces in equilibrium D Weight, tension and thrust E Friction 4 Momentum A Mass and momentum B Conservation of momentum C Conservation of momentum in two dimensions 5 Newton s laws of motion 6 Newton s laws of motion 2 A Force and momentum B Force, mass and acceleration C Solving problems in one dimension D Vertical motion E Resolving forces F Friction G Smooth inclined surfaces H Rough inclined surfaces I Motion in two dimensions A Modelling B Newton s third law of motion C Pulleys and pegs 7 Projectiles A Vertical motion under gravity B Motion of a projectile C Projectile problems D Release from a given height Page 9 of 15

10 Mechanics 2 Chapter Section Lessons Syllabus reference 1 Moments A Moments of a force B Equilibrium of a rigid body C Tilting D Non-parallel forces 2 Centre of mass A Centre of mass of a system of particles B A system of particles in a plane C Centre of mass by symmetry D Centre of mass of a composite body E Suspended objects 3 Variable acceleration A Motion in one dimension B Motion in two dimensions 1 C Motion in two dimensions 2 D Motion in three dimensions E Using Newton s laws in one dimension F Using Newton s laws in two or three dimensions Differential equations A Forming and solving a differential equation Uniform circular motion 6 Work, energy and power A Angular speed B Velocity and acceleration C Forces in circular motion D Problems needing resolving of forces A Work B Kinetic energy C Potential energy D Conservation of energy E Power 7 Hooke s Law A Elastic springs and strings 8 Motion in a vertical circle B Work done by a variable force C Mechanical energy A Circular motion with variable speed Page 10 of 15

11 Assessment Schedule In addition to homework which will be set and marked regularly, you will sit assessments throughout the year. These are very important milestones in the course. They allow you and your teachers to see where your areas of strength and weakness are. We also monitor your performance in these assessments and will use this information to highlight students who are not achieving well in the course. Where this is the case, we will inform you, your parents, and your Head of Year. To make best use of these assessments you must revise for them, and you must review your papers when they are returned to you so that you can see where you need improvements ahead of the final exams. Failure to achieve well in assessments consistently throughout the year will result in us advising your parents and your Head of Year that you should not continue with the course. Key Assessment points November Assessment Week Decision 2 assessment (on content covered up to October half term) Further Pure 1 assessments (on whole module) Mechanics 1 assessment (on content covered up to October half term) January First week back, Mechanics 1 assessment (on whole module) February Assessment Week Decision 2 assessment (on whole module) Further Pure 1 and Further Pure 3 assessment (on content so far for FP3) Further Pure 2 assessment (on content so far) Mechanics 2 assessment (on content so far) April (after Easter holidays) Further Pure 3 assessment (on whole module) Mechanics 2 assessment (on whole module) May/June May/June DECISION 2 final exam FURTHER PURE 1 final exam FURTHER PURE 2 final exam FURTHER PURE 3 final exam MECHANICS 1 final exam MECHANICS 2 final exam Page 11 of 15

12 General Expectations Students who have taken A Level Further Mathematics have achieved highly in recent years in this school. It is noticeable that the students who work in a committed and enthusiastic way all year achieve the higher grades, and this applies to all students irrespective of which set they were in at GCSE. You are expected to work hard. You are expected to ask questions and seek help when you need it. You are expected to be prepared for your lessons, and complete all homework and lesson preparation tasks with full effort. We also expect you to take responsibility for your own learning, in preparation for Higher Education in any subject. Homework This will be set regularly and should be completed as soon as it is set. Use your study periods to discuss problems with a friend or to speak to a member of staff. It is worth making a note of your teacher s non-teaching periods. Homework set by your teachers is the minimum you should do. Reading around the subject, working on further questions in order to ensure understanding and improving your notes all form a part of A Level study. It is advisable to spend at least two hours a day on this subject. The next section of this booklet gives more details about the types of homework you should expect to be set. Not completing a piece of homework or lesson preparation on time is taken very seriously as you need to be showing you are committed to achieving well in this course. If you do not complete tasks on time then we cannot guarantee that it will be marked. You may be asked to leave the lesson to go to complete the homework task and you will be expected to bring to this to your teacher and to catch up on the classwork that you have missed. If you are absent from class when a piece of homework is set, it is your responsibility to find out the work that you have missed, and to complete the homework on time. Attendance You have chosen a difficult A Level. It is essential that you attend all lessons. There has been a direct correlation between students underachieving or failing and those who miss lessons. If your attendance is not of the requisite standard you may be asked to leave the course. Calculators For all modules other than Core 1 a scientific or graphical calculator may be used. Scientific calculators can be bought through the school. Your teachers Your Maths teachers are usually available on the first floor of A Block, and you are expected to find us if you have any problems or questions about the work. You may also find it useful to have your teachers addresses. All teachers have addresses of the same form: for example, Mr M Arthur has the address marthur@twyford.ealing.sch.uk Good luck, and work hard! Page 12 of 15

13 Preparing for Lessons and Homework Assignments As a general rule, you should expect to work for two extra hours for Further Maths per day. This will include both tasks that we set specifically, and your own independent study. Tasks that we will set We will ask you to complete some work after every lesson. This may be lesson preparation for the next lesson or a longer piece of homework to be handed in to be marked. Lesson preparation will typically include some of the following: Completing questions from a short exercise or the support materials Reading ahead about the next topic Investigating a piece of mathematics Researching a topic in mathematics Learning or memorising important formulae, identities and techniques. This must be completed for the next lesson, and your teacher will ask you to show evidence of having completed it. Homework that is handed in will include: An assessed piece of work on each chapter Answering past exam questions either on a particular topic, or a variety of topics This will usually be set at end of a topic but may be more often for larger topics. You would need to allow at least an hour to complete a piece of homework and some may take significantly longer. You will be told when you need to hand this in to your teacher use your planner to keep organised with this. If you have difficulty completing the homework it if your responsibility to see your teacher well before the deadline for help to enable you to submit the homework on time. Independent study In addition to the work that we set, you will also need to spend time ensuring that you are confident with the topics that you have studied. Your independent study should include the following: Re-reading notes and examples from class Completing additional questions from the exercises (even if they have not been set specifically as homework) Reading ahead about the next topics in the course Completing practice exam questions (exam papers are available on the AQA website) Using the A Level resources on the MyMaths website - see the next page of this booklet. (Although there are some resources for Decision and Mechanics, Further Pure is not dealt with on MyMaths.) Learning or memorising important formulae, identities and techniques Page 13 of 15

14 Community Service Each year students volunteer to spend one of their study periods working as a classroom assistant in a lower school class. This benefits you as a student as it gives you a chance to deepen your own understanding of the subject by explaining it to other students. It also helps the younger students in the school to see a positive role model helping them with their studies. It is often good to support the lessons of one of your A Level teachers as this can help to build the working relationship you have with them. We will ask for volunteers at the start of the year, and will check your attendance and helpfulness throughout the year with the teacher you are supporting. Resources Each module has an accompanying textbook. You borrow these from the department rather than Student Services this year and return them at the end of the year. (Please ensure that your Y12 textbooks have already been returned.) You will also receive a set of support materials consisting of exam questions from each topic in the modules you are studying. We may set these as homework, and you can also use them in your independent study. We also offer support to students in Year 13 who are completing STEP papers as part of their university offer. AQA Examination Board: The current specification document shows the content of each module in the course. You can download past papers for each module, together with mark schemes. These won t be needed yet, but you will as you prepare for your exams. You can also download a copy of the formula booklet that you will be able to use in the exams. MyMaths: MyMaths has a library of A Level materials. This year the school s login details are: Login: twyford Password: factor Ask one of your teachers if you need your individual login and password details. Wider learning and stretch opportunities Independent learning is clearly important at university and Sixth Form study needs to include elements of wider reading and listening, especially in a subject that you intend to study beyond A Level. On Copia can be found a list of books that we would recommend for this purpose. In addition, on Copia, the department has produced a resource which lists websites of mathematical videos and podcasts, as well as other useful websites including those of relevant mathematical organisations, or, for example, Who to contact about the course Each A Level Further Maths group has three teachers. You should talk to your teachers in the first instance if you have any questions or concerns about your progress or the organisation of the course. Mr Palfreyman is Head of Key Stage 5 Maths, and Mr Arthur is Head of Maths. You should talk to them if you have any further questions or concerns. Mr Palfreyman often works in the Maths Office (Office 06), and Mr Arthur has an office (Office 05) next door to this on the Maths corridor. All teachers have addresses of the same form. As an example, Mr M. Arthur s is marthur@twyford.ealing.sch.uk Page 14 of 15

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