1 Support for UCL Mathematics offer holders with the Sixth Term Examination Paper The Sixth Term Examination Paper (STEP) examination tests advanced mathematical thinking and problem solving. The examination encourage you to: explore different possible strategies to analyse and answer questions use the techniques you know in unusual and interesting ways develop confidence, stamina and fluency in working through unstructured problems Preparing for the STEP examination will reward you with stronger problem solving skills. These are essential in many disciplines, and invaluable for studying mathematics at university.
2 University College London provides Mathematics offer holders with a comprehensive package to help with preparation for the STEP examination. During 2017/18, 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 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 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 3 As a UCL offer holder you get access to a set of online resources to support your independent study for STEP and to complement the live online lectures and study days. These are provided by 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 papers since 2005 additional notes additional questions and solutions (not from STEP past papers) video worked solutions. 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 Offer holders who make UCL their firm or insurance choice on UCAS will continue to have access to this material until the end of the examination period.
4 Topic-by-topic, this series of live online lectures provides comprehensive coverage of the pure mathematics ideas, skills and techniques useful in STEP. 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 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 have been provided to UCL mathematics applicants since 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!
5 Session title 2018 dates (all lectures start at 1800 UK time) 1 What is STEP? The problem-solving mindset 18 January 2 Using algebraic identities and solving equations 25 January 3 Digits, divisibility and logarithms 1 February 4 Counting and placement 8 February 5 Quadratics, cubics and other polynomials 15 February 6 Handling inequalities 1 March 7 Geometrical problems 8 March 8 Factors, primes and irrationals 15 March 9 Summing series 22 March 10 The binomial expansion and approximations 29 March 11 Sequences and induction 12 April 12 Advanced techniques in trigonometry 19 April 13 Analysing the behaviour of functions limits and curve sketching 26 April 14 Advanced techniques in integration 3 May 15 Differential equations 10 May 16 Vectors and coordinate geometry 17 May 17 Examination technique and advice 24 May
6 Live online lecture 1 What is STEP? The problem-solving mindset 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: 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. Live online lecture 2 Using algebraic identities and solving equations 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: 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? a 2 b + abc + a 2 c +ac 2 + b 2 a +b 2 c + abc + bc 2 Some standard but perhaps less well-known algebraic relationships will be considered. The following identity, for example, can be very useful: a 3 b 3 (a b)(a 2 + ab + b 2 ) 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? log 2 3 < log 4 8; log 2 3 = log 4 8; log 2 3 > log 4 8 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 a 2 b 2 = n Find all the positive integers solutions m and n of m! + 3 = n 2?
7 Live online lecture 4 Counting and Placement 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: 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. Live online lecture 5 Quadratics, cubics and other polynomials 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. 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 x 3 + 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. 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 x 40 in the expansion of (x 2 1) 14 (x 4 + x 2 + 1) 4
8 Live online lecture 6 Handling inequalities Why is it that there are some inequalities that you would look to solve and other inequalities that you would look to prove? It s clear that the statement 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. Live online lecture 7 Geometrical problems Being able to use geometrical properties to deduce information about a figure is important. Solving geometrical problems can also require: good diagrams 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? 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. Live online lecture 8 Factors, primes and irrationals 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: 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. Solve the inequality cos +1 sin 1 where 0 < 2π and sin 0 Show that the shortest distance from the point (, β) to the line m + c β y = mx + c has the value m 2 + 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
9 Live online lecture 9 Summing series 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 problems. This knowledge combines with topics already covered to enable you to answer questions like: Prove that 1 n + 1 + 1 (n + 1)(n + 2) + 1 (n + 1)(n + 2)(n + 3) +... < 1 n Sigma notation will also be covered can you show that 3 i = 1 i j = 1 j 2 = 20? The first and second terms of an arithmetic series are 100 and 95 respectively. The sum to n terms of the series is S n Find the largest positive value of S n Live online lecture 10 The binomial expansion and approximations The binomial expansion provides you with a quick way to expand brackets like: (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: 1 f(x) = (1+2x) 3 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 The binomial expansion can be used to estimate how close to 1 this value really is. Use the binomial expansion of 9 x as far as the term in x 2 with x = 1 64 to find a rational number which approximates 23 Live online lecture 11 Sequences and induction Can you identify all of these sequences? 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 11 n 6 is divisible by 5 for all positive integers n. 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
10 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) = cos 4 x sin 4 x? Familiarity with key facts is also important it s useful to be able to recall values such as: cos( 5π, sin(11π), tan 3π 4 ( 4 quickly. ( Trigonometry in integration and differentiation and in geometrical problems will also be considered. ( 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 = x2 1 x 1? x 1 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 cos xdx = sin xdx and that cos ( π 2 0 0 π π 2 ( π 2 π x dx = cos xdx 2 0 Being able to recall relationships like these and spot when to use them can be important in questions involving integration. Solve the equation tan( = tan 2 2 ( What is lim = x 27? x 27 x 1 3 3 Explain why this is the case. Calculate 1 x 4 1 + x dx 2 0
11 Live online lecture 15 Differential equations 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. 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. Live online lecture 16 Vectors and coordinate geometry 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: 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. Live online lecture 17 Examination technique and advice 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 question that will attract as much credit as possible. Final pointers on examination technique will also be provided. Find two solutions of the differential equation 3( dy 2 + 2x dy dx dx = 1 ( 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) = c 2 (a 2 + c 2 ) (b 2 + c 2 )
Study days at UCL 12 Two STEP study days will take place at UCL in 2018. 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. I really enjoyed the trigonometry and the graph questions and once again found that my confidence had improved. Study Day 1 Friday 13 April 2018 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 paper but not necessarily covered in A-level Maths and Further Maths, general solutions of trigonometric equations, and inverse trigonometric functions. Study Day 2 Friday 1 June 2018 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 enjoyed meeting people and working through the questions at the study day and I thought that the tutor was very good.