Mathematics (FM or IB HL) – pre-course information

Mathematics (FM or IB HL) – pre-course information

In the Further Mathematics A level or the IB HL Mathematics course you will meet a lot of exciting, interesting and beautiful mathematics. Both courses can also be demanding, and therefore we ask you to do some preparation based on topics that most students choosing them have covered in years 10 and 11. We have found that it is very important for students to be able to work quickly and accurately on these topics, so that when new ideas are introduced they can focus on understanding them properly.

It is therefore essential that you have worked on and understood all of the topics listed in this handout before lessons begin in September of Y12: some are from the GCSE/IGCSE syllabus, others form part of the Additional Mathematics course that most Further Maths students have followed in Year 11.

Please check the list overleaf and make sure that you are able to answer questions on these topics. The list is not as long as it looks! Most of the topics have been broken down into several smaller headings: you should be able to tick off many of them straight away.

You will find explanations and exercises to help you in the Additional Mathematics textbook and on the reference sheets attached to this handout.

You will also receive a question sheet covering all these topics. You must complete all of the questions during the summer holidays (this will help you to tell how well you understand the topics) and hand in your answers to your Maths teacher in your first Maths lesson in the Autumn term.

The Maths Department

TopicsFor some topics you will have to go back to GCSE notes if you need to review them. Where appropriate, references to chapters in the Additional Mathematics (AM) textbook have been given, otherwise please see the sheets at the end of this document for reference material.(Italicised topics are those which students often find need revision!)

ALGEBRAThe basic language of algebra [AM Ch 1]

Know and use accurately vocabulary and notation appropriate to the subject at this level.

Algebraic manipulation, eg simplifying algebraic fractions. Be able to change the subject of a formula.

Solution of equations [AM Ch1] Be able to solve linear equations in one unknown. Know how to solve an equation graphically. Be able to solve quadratic equations using factorising, the formula, and

completing the square. Be able to find the discriminant of a quadratic function and understand its

significance. Know how to use the method of completing the square to find the line of

symmetry and turning point of the graph of a quadratic function. Be able to solve linear simultaneous equations in two unknowns. Be able to solve simultaneous equations in two unknowns where one equation is

linear and one is of 2nd order (quadratic). Know the significance of points of intersection of two graphs with relation to the

solution of simultaneous equations.Inequalities [AM Ch2]

Be able to solve linear inequalities. Be able to solve quadratic inequalities.

Indices [GCSE] Understand and be able to use the laws of indices for all rational exponents. Understand the meaning of and be able to evaluate negative, fractional and zero

indices, both with and without a calculator.

CO-ORDINATE GEOMETRY Know and understand the equation y = mx+ c [GCSE]

The co-ordinate geometry of straight lines [AM Ch5] Know how to specify a point in Cartesian co-ordinates in two dimensions. Know the relationship between the gradients of parallel lines and perpendicular

lines. Be able to calculate the distance between two points. Be able to find the co-ordinates of the midpoint of a line segment joining two

points. Be able to form the equation of a straight line. Be able to draw a line when given its equation. Be able to find the point of intersection of two lines. Know how to plot a curve given its equation. Know how to find the point of intersection of a line and a curve.

The co-ordinate geometry of curves [AM Ch 5] Know how the find the point(s) of intersection of two curves. Understand that the equation of a circle, centre (0, 0), radius r is x2+ y2 = r2

Understand that (x – a)2 + ( y – b)2 = r2 is the equation of a circle with center (a,b) and radius r .

Know that:- the angle in a semicircle is a right angle;

- the perpendicular from the centre of a circle to a chord bisects the chord;- the tangent to a circle at a point is perpendicular to the radius through

that point.

POLYNOMIALS

Basic operations on polynomials [AM Ch 3] Know how to add, subtract and multiply polynomials. Know how to divide polynomials to find quotients and remainders.

The factor theorem [AM Ch 3] Understand the factor theorem and know how to use it to factorise a polynomial. Know how to use the factor theorem to solve a polynomial equation.

Know how to use the factor theorem to find an unknown coefficient.

The remainder theorem [AM Ch 3] Understand the remainder theorem and know how to use it.

Binomial expansions [AM Ch 4] Know how to use Pascal's triangle in the binomial expansion of (a + b)n where n is

a positive integer.

Know the notations nCr and(nr )

, and their relationship to Pascal's triangle. Know how to use nCr in the binomial expansion of (a + b)n where n is a positive

integer.

CURVE SKETCHINGVocabulary

Understand the difference between sketching and plotting a curve. [A sketch shows key features of a graph but need not be absolutely accurate, unlike a plotted curve.]

Quadratic curves [AM Ch1] Know how to sketch a quadratic curve with its equation in completed square form.

Polynomial curves [sheet] Know how to sketch graphs of polynomial functions.

Transformations [sheet] Know how to sketch curves of the forms y = f (x) + a and y = f (x - b) given the

curve of y = f (x )

TRIGONOMETRYBasic trigonometry [GCSE & AM Ch 7]

Know how to solve right-angled triangles using trigonometry. Be able to use the definitions of sin θ and cos θ for any angle. The sine, cosine

and tangent functions. Know the graphs of sin θ, cos θ and tan θ for all values of θ , their symmetries and

periodicities. Know the exact values of sin θ , cos θ and tan θ when θ is 0°, 30°, 45°, 60°, 90°

and 180°.Identities [AM Ch 7]

Be able to use sin θ/cos θ = tan θ (for any angle). Be able to use the identity sin2 θ+ cos2 θ = 1. Be able to solve simple trigonometric equations in given intervals.

Area of a triangle [GCSE] Know and be able to use the fact that the area of a triangle is given by ½ ab sin C

.The sine and cosine rules [GCSE & AM Ch 7]

Know and be able to use the sine and cosine rules.Radians [sheet]

Understand the definition of a radian and be able to convert between radians and degrees.

Know and be able to find the arc length and area of a sector of a circle, when the angle is given in radians.

CALCULUSThe basic process of differentiation [AM Ch 9]

Know that the gradient of a curve at a point is given by the gradient of the tangent at the point.

Know that the gradient of the tangent is given by the limit of the gradient of a chord.

Know that the gradient function dy/dx gives the gradient of the curve and measures the rate of change of y with respect to x.

Be able to differentiate y= kxn where k is a constant and n is any integer or fraction ≠ 1, and the sum of such functions.

Applications of differentiation [AM Ch 9 + sheet] Be able to use differentiation to find stationary points on a curve: maxima,

minima and points of inflection. Be able to use the second derivative, d2y/dx2 to determine the nature of

stationary points on a curve. Understand the terms increasing function and decreasing function. Be able to find the equation of a tangent and normal at any point on a curve.

Integration [AM Ch 10] Know that integration is the inverse of differentiation. Be able to integrate functions of the form kxn where k is a constant and n ≠ 1, and

the sum of such functions. Know what are meant by indefinite and definite integrals. Be able to evaluate definite integrals. Be able to find a constant of integration given relevant information.

Integration to find the area under a curve [AM Ch 10 + sheet] Know that the area under a graph can be found as the limit of a sum of areas of

rectangles. Be able to use integration to find the area between a graph and the x-axis. Be able to find an approximate value of a definite integral using the trapezium

rule, and comment sensibly on its accuracy.

Applications of calculus to kinematics [AM Ch 11] Use calculus to find formulae relating displacement, velocity or acceleration to

time Use calculus to solve problems with curved velocity-time graphs

FUNCTIONS & SETS [sheets]The language of functions

Understand the definition of a function, and the associated language. Know how to find a composite function, g○f(x ). [This notation means the same as

gf(x)] Know how to find the inverse of a function, f-1(x), and the circumstances in which

the domain of f(x) may need to be restricted.Set Notation

Know, use and interpret the notation and symbols: {x, y, …}, {x│ }, n(A), ∩, U, U

(means the same as E )

SHAPE & SPACE [GCSE] Know and use Pythagoras’ Theorem in 2-D and 3-D. Be able to calculate areas, perimeters, volumes, etc for standard shapes and

solids. Recognise and describe accurately translations, rotations, reflections and

enlargements. Use properties of similar shapes to calculate lengths, areas and volumes.

NUMBERCalculations [GCSE]

Use standard form Be able to approximate to given numbers of decimal places or significant figures Use fractions, decimals and percentages in ratio and proportion problems

Types of number [GCSE]

know the vocabulary of whole numbers: primes, multiples, factors, etc appreciate the distinction between integers, rational numbers, and irrational

numbersSurds (must be done without using a calculator) [AM Ch 2 + GCSE]

Be able to use and manipulate surds. Be able to rationalise the denominator of a surd.

STATISTICS & PROBABILITYStatistical calculations [GCSE]

find mean, mode and median values for raw data, from frequency tables, and from grouped frequency tables

Statistical diagrams [GCSE] draw and interpret cumulative frequency diagrams and histograms

Probability [GCSE] draw and use tree diagrams to solve probability problems

The Binomial distribution [AM Ch 4] understand when the Binomial distribution may be used and be able to calculate

probabilities in these cases

REFERENCE PAGES

Transformations of Graphs (using f(x) = x 2 + 2x – 3) type A Graphs of: y = f(x) (red)

y = f(x – 1) (blue)y = f(x + 3) (pink)y= f( x – 2) (green)

You can see that all the graphs are horizontal translations of the original (red) graph.You can work out what the translation will be using the number that is added/subtracted from x inside the function.

eg y = f(x – 2) is a translation of (20)y = f(x +3) is a translation of (−3

0 )

type BGraphs ofy = f(x) (red) y = f(3x) (blue) y = f(½ x) (pink) y = f(-x) (green) y = f (2x)(orange)

You can see that all the graphs are horizontal squashes or stretches of the original (red) graph. You can work out what the squash or stretch will be using the number that is multiplying x inside the function.eg y = f(3x) is squashed parallel to the x axis so it is 3 times thinner

y = f (½ x) is stretched parallel to the x axis so it is 2 times widery = f (-x) is reflected in the y axis but is not stretched or squashed.

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Differentiating powers of x with negative or fractional indices

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Using the second derivative (the differential of dy/dx)

Example

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strips to be able to decide.

In this example the convergence is unusually slow because of the high curvature of the curve.

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Extracts fromLongman Mathematics for IGCSE Book 2 (Turner, Potts, Waite, Hony)MEI structured Mathematics P1 (Hanrahan, Porkess, Secker)Modular Mathematics Module A Pure Maths 1 (Bostock, Chandler)

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