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1 Support for UCL Mathematics offer holders with the Sixth Term Examination Paper and the Advanced Extension Award Sixth Term Examination Paper (STEP) and Advanced Extension Award (AEA) examinations test advanced mathematical thinking and problem solving. Both examinations encourage you to: xplore different possible strategies to ›e analyse and answer questions se the techniques you know in unusual ›u and interesting ways evelop confidence, stamina and fluency ›d in working through unstructured problems Preparing for STEP and AEA examinations will reward you with stronger problem solving skills. These are essential in many disciplines, and invaluable for studying mathematics at university. 2 niversity College London provides ›U Mathematics offer holders with a comprehensive package to help with preparation for STEP and AEA examinations. During 2015/16, all offer holders have free access to: •Online Resources – extensive notes, questions and solutions to support independent study •Live Online Lectures – a series of seventeen live online lectures giving detailed advice on how to tackle STEP and AEA questions; to attend these all you need is a computer or tablet connected to the internet •Study Days – two study days at University College London giving you a chance to work face-to-face with expert tutors and other students on STEP and AEA questions. Here is a selection of feedback on similar support provided to previous applicants: Thank you so much for the online lectures, I found them invaluable and they were no doubt a vital part of my preparation for STEP. I found the whole package incredibly useful. I found the resources such as notes and solutions really helpful. The study day that I attended was brilliant. I took away a lot from the day. Online resources to support your independent study s a UCL offer holder you get ›A access to a set of online resources to support your independent study for STEP and AEA and to complement the live online lectures and study days. These are provided by Mathematics in Education and Industry (MEI), a charity that supports mathematics teaching and learning. MEI also provides highly-acclaimed online resources to support A level Mathematics and Further Mathematics. Arranged by mathematical theme these include: • full worked solutions to all STEP and AEA papers since 2005 • additional notes • additional questions and solutions (not from STEP and AEA past papers) • video worked solutions. 3 The Mathematics Department at UCL will send your username and password for the resources to you by email. You log in to the resources at www.integralmaths.org Live online lectures 4 opic-by-topic, this series of ›T live online lectures provides comprehensive coverage of the pure mathematics ideas, skills and techniques useful in STEP and AEA. Throughout the series there is an emphasis on: •providing insight into the type of thinking needed to solve problems successfully •building a knowledge-base of useful techniques, facts and ‘tricks’ •producing high-quality written explanations of solutions. STEP and AEA past paper questions will be considered in every session. The Mathematics Department at UCL will send your personal hyperlink to attend the online lectures to you by e-mail. Similar online sessions were provided to UCL mathematics applicants during 2014. Feedback comments included: Essential I really enjoyed the online lectures. It was helpful to see how to make connections between different areas of maths and how to produce elegant solutions. Everything was perfectly clear and very informative, thank you. Great sessions and really helpful! Live online lectures 5 Session title 2016 dates 1 What are STEP and AEA? The problem-solving mindset 28 January 2 Using algebraic identities and solving equations 4 February 3 Digits, divisibility and logarithms 11 February 4 Counting and placement 25 February 5 Quadratics, cubics and other polynomials 3 March 6 Handling inequalities 10 March 7 Geometrical problems 17 March 8 Factors, primes and irrationals 24 March 9 Summing series 14 April 10 The binomial expansion and approximations 21 April 11 Sequences and induction 28 April 12 Advanced techniques in trigonometry 5 May 13 Analysing the behaviour of functions – limits and curve sketching 12 May 14 Advanced techniques in integration 19 May 15 Differential equations 26 May 16 Vectors and coordinate geometry 2 June 17 Examination technique and advice 9 June Live online lectures 6 Live online lecture 1 What are STEP and AEA? The problem-solving mindset Live online lecture 2 Using algebraic identities and solving equations To find an effective way to approach a problem often requires a willingness to experiment and play around with the information you have been given: In this session, you’ll look at the importance of precision and care in algebra, particularly when finding roots of equations. This will include suggestions about how to set out working to minimise the chance of mistakes. An example which illustrates the need for care is the following, without care it’s easy to miss some of the roots: • looking at a special case of a problem might be helpful • keeping a track of your ideas and organising your thoughts is important • you should make good use of notation and diagrams. In this session examples will be used to illustrate these ideas. You’ll also see how problems can have many different solutions and discuss the notion of an elegant solution. Describe fully the roots of sin2x = sinx Advice will be provided about algebraic techniques including spotting how to factorise expressions. Can you see how the following factorises? a2b + abc + a2c +ac 2 + b 2a +b2c + abc + bc 2 Some standard but perhaps less well-known algebraic relationships will be considered. The following identity, for example, can be very useful: Live online lecture 3 Digits, divisibility and logarithms This session will look at how to deduce properties of a number from its digits; how to spot factors of a number; and techniques involving logarithms, including how logs can help you to find the first digit of a number. You need to remember that the digits of a number are not the same as the number itself. The two-digit number whose digits are a then b which you would write down as ‘ab’ is actually 10a + b. The three digit number ‘abc ’ is actually 100a + 10a + b. The use of indices and logarithms will be considered. How would you decide which of the following is correct? log2 3 < log4 8;log2 3 = log4 8;log2 3 > log4 8 a3 – b3 ≡ (a – b)(a2 + ab + b 2) › Example problem › Example problem › Example problem What is the area of the square in this diagram? For which positive integers n do there exists non-zero integers a and b such that a2 – b 2 = n Find all the positive integers solutions m and n of m! + 3 = n2? Live online lectures 7 Live online lecture 4 Counting and Placement Live online lecture 5 Quadratics, cubics and other polynomials Combinatorics is a branch of mathematics partly concerning the study of finite collections of objects. Aspects of combinatorics include counting the number of objects of a given kind or size, deciding when the objects can be arranged so that certain criteria can be met; and finding the “largest”, “smallest”, or “optimal” objects Examples of this are: Factorising and expanding polynomials; using techniques such as ‘completing the square’; using the factor and remainder theorem and employing substitutions to allow the use of polynomial techniques will all be considered in this session. How many three digit numbers are there whose digits sum to 25? Arrange the numbers 0 – 20 (inclusive) into seven separate groups each of three numbers so that when the numbers in each are added together, they make seven consecutive numbers? Confidence with problems like this is important - not only for problems which are directly of a combinatorial nature but also to keep track of the various cases that might develop in a piece of mathematical reasoning. The fundamental theorem of algebra will be discussed as well as ways to identify the number of real roots a polynomial equation has. It’s also useful to know the relationship between roots and coefficients of polynomials. For example: If ∝, β, γ are roots of the polynomial x3 + ax 2 + bx + c = 0 then a = –(∝ + β + γ ), b = ∝β + βγ + γ∝, c = – ∝βγ It’s useful to use curve sketching alongside algebraic techniques when dealing with polynomials. These ideas will be covered in this lecture. › Example problem › Example problem Imagine a three dimensional version of noughts and crosses played in a 3 × 3 × 3 cell cube. How many different winning lines are there? Calculate the coefficient of x40 in the expansion of (x 2 –1)14 (x4 + x 2 + 1)4 Live online lectures 8 Live online lecture 6 Handling inequalities Live online lecture 7 Geometrical problems Live online lecture 8 Factors, primes and irrationals Why is it that there are some inequalities that you would look to solve and other inequalities that you would look to prove? Being able to use geometrical properties to deduce information about a figure is important. Solving geometrical problems can also require: It’s clear that the statement • good diagrams Knowing its prime factorisation reveals a lot of information about a number. In particular it’s a useful way to understand how many factors a number has, what they are and the number’s relationship to other numbers via least common factors and highest common multiples. It will help you to answer questions like: 2x + 4 > 10 is true for some values of x and false for other values of x and so you would look to solve it. However the statement x 2 + y 2 ≥ 2xy is true for all real values of x and y and you would look to prove that that is the case. In this session a variety of techniques for solving and proving inequalities will be considered. • experimenting by adding elements such as lines or circles to diagrams to reveal relationships • using algebra alongside geometry where necessary. In this diagram what is the sum of the four shaded angles? Find the number of factors of 180. How many of these factors are divisible by 10? The classic proof that the square root of 2 is irrational uses prime factorisation. In this session you’ll look at questions involving these ideas as well as discussing what it means to say that a number is irrational. Mathematical reasoning involving irrational numbers will be considered in detail. This problem is reasonably straightforward but in producing a solution it would be important to be clear about which geometric properties you are using at all times. › Example problem › Example problem › Example problem Solve the inequality Show that the shortest distance from the point (∝, β ) to the line |m∝ + c – β | y = mx + c has the value √m2 + 1 If a, b, c, d are rational numbers and if p is irrational prove that a + bp = c + d p implies that a = c and b = d cos∅ +1 ≤1 where 0 ≤ ∅ < 2π and sin∅ ≠ 0 sin∅ Live online lectures 9 Live online lecture 9 Summing series Live online lecture 10 The binomial expansion and approximations Live online lecture 11 Sequences and induction General knowledge about summing series such as the formula for the sum of the first n whole numbers; the formula for the sum of the squares of the first n whole numbers; arithmetic and geometric series can be useful in STEP and AEA problems. The binomial expansion provides you with a quick way to expand brackets like: Can you identify all of these sequences? This knowledge combines with topics already covered to enable you to answer questions like: Prove that 1 1 1 1 + + + ... < n + 1 (n + 1)(n + 2) (n + 1)(n + 2)(n + 3) n Sigma notation will also be covered – can you show that 3 i ∑ ∑j i=1 j=1 2 = 20? (a + b) (a + b) (a + b) (a + b) (a + b) (a + b) It also provides you with series with finitely many terms which approximate functions like: f (x ) = 1 3 (1+2x ) 2 Computers use series such as those provided by the binomial expansion to evaluate certain functions. This lecture will take a look at how the binomial expansion can be used to produce approximations and how judgements about their accuracy can be made. A pocket calculator may evaluate the following as 1: 1 √1–1.5x10 –12 1, 8, 27, 64, 125, ... 1, 3, 6, 10, 15, 21, ... 1, 1, 2, 3, 5, 8, 13, 21, 34, ... 1, 2, 6, 24, 120, 720, ... As well as discussing sequences in general, this session will cover situations where sequences are defined or properties of sequences are deduced recursively (from one term to the next). This will lead naturally to the idea of proof by induction. Proof by induction is one way to prove a statement like “11n – 6 is divisible by 5 for all positive integers n.” The binomial expansion can be used to estimate how close to 1 this value really is. › Example problem › Example problem › Example problem The first and second terms of an arithmetic series are 100 and 95 respectively. The sum to n terms of the series is Sn Use the binomial expansion of √9 – x as far as the term in x 2 with x= 1 Prove, by induction, that for all positive integers n greater than or equal to 8 there exist non-negative integers x, y such that n = 3x + 5y Find the largest positive value of Sn 64 to find a rational number which approximates √23 Live online lectures Live online lecture 12 Advanced techniques in trigonometry Recalling the graphs of trigonometric functions and being able to use trigonometric identities fluently is very important in mathematical problem-solving. Can you see why: cos(2x) = cos4 x – sin4 x ? Familiarity with key facts is also important – it’s useful to be able to recall values such as: ( 54π (, sin(11π), tan(– 34π ( cos quickly. Trigonometry in integration and differentiation and in geometrical problems will also be considered. 10 Live online lecture 13 Analysing the behaviour of functions – limits and curve sketching Curve sketching is a very important skill in all mathematical subjects – from Economics to Engineering. Algebraic techniques such as factorising, long division and the binomial expansion can be very useful to determine the behaviour of functions. Of particular interest in this session will be limiting values of functions and their behaviour around asymptotes. For example it’s clear that the function: f (x) = x2 – 1 x–1 is not defined at x = 1, but there is such a thing as: lim = x⟶1 x –1 ? x–1 2 This session will cover techniques for analysing questions like this as well as for general curve sketching. Live online lecture 14 Advanced techniques in integration Proficiency in integration involves being able to choose correctly from a range of possible techniques to solve a given problem. It’s important to gain as much experience as you can with methods like integration by substitution and integration by parts because this will give you a much better instinct for what will and what won’t work. Familiarity with trigonometric and other relationships can also be very important in problems with integration. For example it’s quite easy to see that: π 2 π 2 0 0 ∫cos xdx = ∫sin xdx and that π ( π ∫cos x – π 2 2 π 2 ( dx = ∫cos xdx 0 Being able to recall relationships like these and spot when to use them can be important in questions involving integration. › Example problem › Example problem › Example problem Solve the equation What is tan lim = Calculate 1 x4 ∫ 2 dx 0 1 + x ( ∅2 ( = tan2 ∅ x⟶27 x – 27 1 x3 – 3 ? Explain why this is the case. Live online lectures 11 Live online lecture 15 Differential equations Live online lecture 16 Vectors and coordinate geometry Live online lecture 17 Examination technique and advice Differential equations describe how mathematical systems change. There are a variety of techniques for solving differential equations and examples of these will be given in this session. Practising vectors and coordinate geometry questions is very worthwhile. Confidence can be built quite quickly. A theme of this session will be situations which require working with general coordinates or vectors rather than with coordinates and vectors given explicitly. For example: In the final session we’ll look at some examination questions alongside examiners’ comments about them. This will give a clearer understanding of what is required to produce an answer to a STEP or AEA question that will attract as much credit as possible. Final pointers on examination technique will also be provided. Techniques such as substitution that can be used to reduce some differential equations to a form that can be dealt with using standard methods will be discussed. Sometimes problems will require a real life situation to be modelled by appropriate differential equations. Doing this is a skill in its own right and examples of this will be given in this lecture. Being able to interpret the solution of a differential equation is also very important. Sketching solutions to differential equations will be considered. A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d ), respectively. In terms of a, b, c and d, what are the coordinates of the other two vertices? The session will also cover a number skills that it is useful to have to hand when attempting vectors questions. For example it’s useful to be able to quickly deal with situations like this: If point A has position vector a and point B has position vector b relative to the origin find the position vector of the point P on the line AB which is 2 of the way from 3 A to B, in terms of a and b. › Example problem › Example problem Find two solutions of the differential equation dy 2 dy 3 + 2x =1 dx dx If A is the point (a, 0, 0), B is the point (0, b, 0) and C is the point (0, 0, c ) show that ( ( cos (ACB ) = c2 √(a 2 + c 2) (b 2 + c 2) Study days at UCL Two STEP & AEA study days designed to complement the online support will take place at UCL in 2015. They provide a great opportunity to come to UCL to learn and practise mathematics. The emphasis will be on gaining experience in answering past paper questions but there will also be teaching sessions on some selected topics. 12 1st Study Day 8 April 2016 all UCL offer holders are welcome to attend The focus of the day will be curve sketching and trigonometry. We will cover the curve sketching techniques needed in the STEP/AEA papers but not necessarily covered in A-level Maths and Further Maths, general solutions of trigonometric equations, and inverse trigonometric functions. 2nd Study Day 2 June 2016 for offer holders who make UCL their firm choice on UCAS The focus of the day will be calculus. After an extensive revision of integration techniques, we will examine different types of exam questions on integration. We will also look at differentiation and differential equations. To book a place at a study day email: admissions@math.ucl.ac.uk I really enjoyed the trigonometry and the graph questions and once again found that my confidence had improved. I enjoyed meeting people and working through the questions at the study day and I thought that the tutor was very good.