Maths – Key Stage 5

Year 12 – Core 1

Half TermTopics to be coveredKey skills covered
Autumn HT 1Algebra & Functions1. Laws of indices for all rational exponents.
2. Use and manipulation of surds.
3. Quadratic functions and their graphs.
4. The discriminant of a quadratic function.
5. Completing the square. Solution of quadratic equations.
6. Simultaneous equations: analytical solution by substitution.
7. Solution of linear and quadratic inequalities.
8. Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation.
9. Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.
10. Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
Autumn HT 21. Coordinate Geometry
2. Differentiation
1. Equation of a straight line, including the forms y – y1 = m(x – x1) and ax + by + c = 0.
2. Conditions for two straight lines to be parallel or perpendicular to each other.
3. The derivative of f(x) as the gradient of the tangent to the graph of y = f (x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives.
4. Differentiation of xn, and related sums and differences.
5. Applications of differentiation to gradients, tangents and normals.
Spring HT 11. Integration
2. Sketching Curves
1. Indefinite integration as the reverse of differentiation.
2. Integration of xn.
3. Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.
4. Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y= f(ax).
Spring HT 21. Arithmetic Sequences
2. Revision programme
1. Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form xn+1 = f(xn).
2. Arithmetic series, including the formula for the sum of the first n natural numbers
Summer HT 1
Summer HT 2

Year 12 – Core 2

Half TermTopics to be coveredKey skills covered
Autumn HT 11. Algebra & Functions
2. Logarithms & Exponentials
1. Simple algebraic division
2. Use of the Factor Theorem and the Remainder Theorem.
3. Laws of logarithms
Autumn HT 2Trigonometry1. The sine and cosine rules, and the area of a triangle in the form 0.5 ab sin C.
2. Radian measure, including use for arc length and area of sector.
3. Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.
4. Solution of simple trigonometric equations in a given interval.
Spring HT 11. Differentiation
2. Integration
1. Applications of differentiation to maxima and minima and stationary points, increasing and decreasing functions.
2. Evaluation of definite integrals.
3. Approximation of area under a curve using the trapezium rule.
4. Interpretation of the definite integral as the area under a curve.
Spring HT 21. Sequences & Series
2. Coordinate Geometry
1. Binomial expansion of (1 + x)n for positive integer n.
Summer HT 1Revision Programme begins1. Coordinate geometry of the circle using the equation of a circle in the form (x – a)2 + (y – b)2 = r2 and including use of the following circle properties:
i. The angle in a semicircle is a right angle;
ii. The perpendicular from the centre to a chord bisects the chord;
iii. The perpendicularity of radius and tangent.
Summer HT 2

Year 12 – Decision 1

Half TermTopics to be coveredKey skills covered
Autumn HT 11. Algorithms
2. Graphs
3. Route Inspection
1. The general ideas of algorithms and the implementation of an algorithm given by a flow chart or text.
2. Candidates should be familiar with bin packing, bubble sort, quick sort, binary search.
3. The minimum spanning tree (minimum connector) problem. Prim’s and Kruskal’s (greedy) algorithm.
4. Dijkstra’s algorithm for finding the shortest path.
5. Algorithm for finding the shortest route around a network, travelling along every edge at least once and ending at the start vertex. The network will have up to four odd nodes.
Autumn HT 21. Matching
2. Critical Path Analysis
1. Use of bipartite graphs for modelling matchings. Complete matchings and maximal matchings.
2. Algorithm for obtaining a maximum matching.
3. Algorithm for finding the shortest route around a network, travelling along every edge at least once and ending at the start vertex. The network will have up to four odd nodes.
Spring HT 11. Linear Programming
2. Revision Programme Begins
1. Formulation of problems as linear programs.
2. Graphical solution of two variable problems using ruler and vertex methods
3. Consideration of problems where solutions must have integer values.
Spring HT 2
Summer HT 1
Summer HT 1

Year 13 – Core 3

Half TermTopics to be coveredKey skills covered
Autumn HT 1TrigonometryKnowledge of secant, cosecant and cotangent and of arcsin, arccos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains.
Autumn HT 21. Algebraic fractions
2. Functions
1. Simplification of rational expressions including factorising and cancelling, and algebraic division.
2. Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs.
Spring HT 1Differentiation1. Differentiation of ex, ln x, sin x, cos x, tan x and their sums and differences.
2. Differentiation using the product rule, the quotient rule and the chain rule.
Spring HT 21. Exponentials
2. Transformation of graphs
3. Numerical Methods
1. The function ex and its graph.
2. The function ln x and its graph; ln x as the inverse function of ex.
3. The modulus function.
4. Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
5. Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous.
6. Approximate solution of equations using simple iterative methods, including recurrence relations of the form xn+1 = f(xn).
Summer HT 1Revision Programme begins

Year 13 – Core 4

Half TermTopics to be coveredKey skills covered
Autumn HT 11. Partial Fractions
2. The Binomial Expansion
1. Rational functions. Partial fractions (denominators not more complicated than repeated linear terms).
2. Binomial series for any rational n.
Autumn HT 21. Parametric Equations
2. Vectors
1. Parametric equations of curves and conversion between Cartesian and parametric forms
2. Vectors in two and three dimensions.
3. Magnitude of a vector.
4. Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations.
5. Position vectors.
6. The distance between two points.
7. Vector equations of lines.
8. The scalar product. Its use for calculating the angle between two lines.
Spring HT 1Differentiation1. Differentiation of simple functions defined implicitly or parametrically.
2. Exponential growth and decay.
3. Formation of simple differential equations.
Spring HT 2Integration1. Evaluation of volume of revolution.
2. Simple cases of integration by substitution and integration by parts. These methods as the reverse processes of the chain and product rules respectively.
3. Simple cases of integration using partial fractions.
4. Analytical solution of simple first order differential equations with separable variables.
5. Numerical integration of functions.
Summer HT 1Revision Programme

Year 13 – Statistics 1

Half TermTopics to be coveredKey skills covered
Autumn HT 1Probability1. Elementary probability.
2. Sample space. Exclusive and complementary events. Conditional probability.
3. Independence of two events.
4. Sum and product laws.
Autumn HT 21. Normal Distribution
2. Discrete Random Variables
1. The Normal distribution including the mean, variance and use of tables of the cumulative distribution function.
2. The concept of a discrete random variable.
3. The probability function and the cumulative distribution function for a discrete random variable.
4. Mean and variance of a discrete random variable.
5. The discrete uniform distribution.
Spring HT 11. Representation of Data (Location)
2. Representation of Data (Dispersion)
1. Measures of location mean, median, mode.
2. Measures of dispersion - variance, standard deviation, range and interpercentile ranges.
3. Skewness. Concepts of outliers.
Spring HT 2Representation of Data1. Histograms, stem and leaf diagrams, box plots.
2. Skewness. Concepts of outliers.
Summer HT 1CorrelationThe product moment correlation coefficient, its use, interpretation and limitations.
Summer HT 21. Regression
2. Revision Programme begins
1. Scatter diagrams. Linear regression.
2. Explanatory (independent) and response (dependent) variables. Applications and interpretations.