Number and algebra

The ordinary processes of number manipulation The ‘four operations’ and combination of them by use of brackets

Prime numbers, factors, multiples To include finding HCF and LCM in simple cases

Indices, powers and roots Use index notation and index laws for multiplication and division involving integer, fractional and negative powers

Simple manipulation of surds Students should understand what surds represent and their use for exact answers Manipulation will be simple For example: 5√3 + 2√3 = 7√3; √48 = 4√3; 10 × 1/√5 = 2√5

Rationalising the denominator For example: 15/(√7 − 2)

Natural numbers, integers and rational and irrational numbers Recognitions of these sets Proofs of irrationality will not be required

Weights, measures and money Carry out calculations using standard units of mass, length, area, volume and capacity, time and average speed Metric and SI units only Carry out calculations using money, including converting between currencies (where conversion is required, the rate of conversion will always be given)

Fractions, decimals, ratio, proportion and percentage Students will be expected to interchange any of these methods of fractional representation and to select the most appropriate to given situations Ratios and proportions are required in, at most, three proportions, i.e. a : b or a : b : c Students will be expected to use the four operations with fractions and decimals, and use percentages, ratio and/or proportion in problems

Expressing numbers to a given degree of accuracy Correction to a given number of decimal places or significant figures

Solve problems using upper and lower bounds where values are given to a degree of accuracy

Numbers in standard form a × 10^n, where n is an integer and 1 ≤ a < 10 Solve problems involving standard form Questions may involve the application of any of the techniques listed in 1 to problems of everyday personal, domestic or community life

The idea of a set

Set language and notation Questions may be set involving these ideas in the abstract or derived from practical situations

Union and intersection of sets Understand sets defined in algebraic terms

Number of elements in a set Use the notation n(A)

Complementary sets Use the notation A′

Subsets

Universal set, null set

Venn diagrams and their use in simple logical problems

Use of symbols to represent sets

The basic processes of algebra Collecting like terms, using the four operations, the rules of indices, with integers and fractional powers

The construction, interpretation and use of formulae and their manipulation To include change of subject of a formula and substitution

The factorisation of simple algebraic expressions

Use of the factor theorem Including application to cubics and factors of the form (ax + b) or (ax – b)

Algebraic division of a cubic by a linear factor

The manipulation of simple algebraic fractions, the denominators being numerical, linear or quadratic Simple cases involving sum, difference, product and quotient of algebraic fractions

Solution of equations of 1st, 2nd and 3rd degree containing one unknown quantity Solution of quadratics to include solution by factorisation, by graph, by completing the square or by formula Problems that result in the solution of such equations may also be set

Solution of linear simultaneous equations in two unknowns Simple questions may be set requiring the graphical solution of simultaneous linear equations

Solve simultaneous equations in two unknowns, one equation being linear and the other being quadratic

Solution of linear inequalities, and the representations of solutions on the number line and two-dimensional space Simple questions may be set requiring the graphical solution of simultaneous linear inequalities No questions will be set on linear programming

Solve quadratic inequalities in one unknown and represent the solution set on a number line

The idea of a sequence Being able to recognise sequences with a common difference or common integer sequences, and to continue a given sequence

The idea of a function of a variable

Function as a mapping or as a correspondence between the elements of two sets

Use function notation in the forms f(x) = … and f: x ↦ … .

Domain and range of a function Questions will not be set on continuity, but students will be expected to recognise when parts of the domain need to be excluded (e.g. x = 0 must be excluded from the domain of the function f where f(x) = 1/x)

Composite functions ‘fg’ will mean ‘do g first then f’

Inverse functions Finding the inverse of a function

Variation, direct and indirect proportion To include only the following: y ∝ x; y ∝ 1/x; y ∝ x²; y ∝ 1/x²; y ∝ x³; y ∝ 1/x³; y ∝ √x; y ∝ 1/√x

Rectangular Cartesian co-ordinates

Recognise that equations of the form y = mx + c are straight–line graphs with gradient m and intercept on the y-axis at the point (0, c)

Graphs and graphical treatment of the equation: y = Ax³ + Bx² + Cx + D + E/x + F/x², in which the constants are numerical and at least three of them are zero Students will be expected to draw and interpret graphs from given equations Use of the intersection of two curves (graphs) to solve equations

The gradients of graphs above by drawing Students will be expected to draw a reasonable tangent to the graph at a named point and to construct an appropriate right-angled triangle from which to calculate the gradient

Differentiation of integer powers of x Use of dy/dx notation

Determination of gradients, rates of change, maxima and minima, stationary points and turning points Students will either be required to differentiate or use graphical methods to arrive at solutions and relate their calculations to their graphs and vice versa

Applications to linear kinematics and to other simple practical problems This includes the drawing and interpretation of distance/time and speed/time graphs, and other graphs of a similar nature Students need to be able to understand the relationship between displacement or distance, velocity and speed, and acceleration, for example: ds/dt = v and dv/dt = a

Representation of data by a matrix

Add and multiply matrices of order no greater than 3 × 3, using correct row-by-column multiplication.

Multiplication of a matrix by a scalar

Use identity and zero matrices of order no greater than 3 × 3.

Determinants and inverses of non-singular 2 x 2 matrices Knowledge of singular matrices is not required

Transformations of the plane associated with 2 × 2 matrices Transformations include: Reflections in x = 0, y = 0 and y = ±x Rotations about the origin Enlargements with centre at the origin

Combination of transformations The matrix AB represents the transformation represented by B followed by the transformation represented by A