MEI Conference 2012
Extending GCSE
mathematics and making
links with A level
Kevin Lord
Kevin.lord@mei.org.uk
Options for extending the more able at GCSE
A comparison of GCSE Mathematics Higher tier A* -B content with the FMSQ Additional
Mathematics (OCR) and Level 2 Certificate in Further Mathematics (AQA) Specifications
The content from the specifications of these course is listed in the following table alongside the GCSE
Higher tier specification (exclusive of Foundation Tier) to show the overlap between the qualifications.
Topics
Higher
Tier Maths
Further
Maths
Additional
Maths
Calculations involving upper and lower bounds
Interpret, order and calculate with numbers in standard index form
Repeated Proportional Change, Direct and Indirect Proportion
Change the subject of a formula including cases where the subject is on
both sides of the original formula, or where a power of the subject appears
Use index laws for multiplication and division of integer, fractional and
negative powers
Use of Surds, rationalising denominator
Transformations of graphs and functions
Equations of straight lines: gradient, intercept, parallel and perpendicular
lines
Graphs of functions, linear, quadratic, cubic, reciprocal, exponential,
trigonometric
Algebra
Graphs of inequalities, showing required regions
Use graphs of linear inequalities to solve 2D max/min problems, using
objective function
Quadratic equations: factorising, solving, completing the square, difference
of two squares
Solve quadratic inequalities graphically
Simplify rational expressions
Solve simultaneous equations, algebraically and graphical, including ones
with linear and quadratic unknowns
Domain and range of a function
Factor Theorem
Algebraic proof
Sequences : nth term of quadratic sequences, limiting values
Construct simple loci including equation of a circle
Equation of a circle for any centre and radius
inc. remainder
theorem
Topics
Higher
Tier Maths
Further
Maths
Additional
Maths
Calculate length of arcs, area of sectors and segments
Volume and surface areas of complex shapes
Vectors : notation and use vectors, applied to simple geometrical proofs
Linear, area and volume scale factors of similar shapes and solids
Geometry
Circle theorem, applied to simple geometrical proofs
Area of triangle using 1/2 absinC
Pythagoras and Trigonometry applied to 3D problems, including angle
between line and plane
inc. angle bet. 2
planes
Sine and cosine rules to solve 2D and 3D problems
Use of ratios for sin, cos and tan of angles 30, 60 and 45
Use of trigonometric identities
Probability and
Statistics
Solving trigonometric equations in given intervals
Higher tier statistical methods
Probability trees for independent and dependent events, conditional
probability
Binomial expansion for positive integer powers
Calculate probabilities using Binomial distribution
Differentiation of polynomials
Equation of tangents and normals at any point on a curve
Calculus
Curve sketching using stationary points
Use differentiation to find stationary points
Determine nature of stationary points
Integration of polynomials : definite, indefinite integrals, area under a curve
Matrices
Applications of calculus to kinematics
Matrix multiplication for 2x2 or 2x1 matrices
Matrix representations of transformations, including combinations
A tick indicates that this topic is specifically listed in the course specification and is likely to be assessed. Cells
that are shaded represent the GCSE content that is assumed as prior knowledge in the extension qualifications
and may be required to solve problems.
Introductions to these specifications from the examination board
materials
AQA LEVEL 2 CERTIFICATE IN FURTHER MATHEMATICS
This qualification fills the gap for high achieving students by assessing their higher order mathematica
skills, particularly in algebraic reasoning, in greater depth without infringing upon AS Level
mathematics, thus preparing them fully to maximise their potential in further studies at Level 3. It
offers the opportunity for stretch and challenge that builds on the Key Stage 4 curriculum and is
intended as an additional qualification to the GCSE Mathematics, rather than as a replacement. The
content assumes prior knowledge of the Key Stage 4 Programme of Study and covers the areas of
algebra and geometry, which are crucial to further study in the subject, in greater depth and breadth.
This new qualification places an emphasis on higher order technical proficiency, rigorous argument
and problem solving skills.
It also gives an introduction to calculus and matrices and develops further skills in trigonometry,
functions and graphs.
The AQA Level 2 Certificate in Further Mathematics is an untiered Level 2 linear qualification for
learners who
 either already have, or are expected to achieve, grades A and A* in GCSE mathematics
 are likely to progress to A-Level study in mathematics and possibly further mathematics.
http://store.aqa.org.uk/qual/igcse/pdf/AQA-8360-W-SP.PDF
OCR FREE STANDING MATHEMATICS QUALIFICATION (ADVANCED): ADDITIONAL
MATHEMATICS
The target students for this course are those who have taken Higher GCSE either at the end of Year
10 or in January of Year 11, or who will take GCSE at the same time as Additional Mathematics.
Many of the students will go on to study AS and A level modules in Year 12, and for these the course
provides an introduction to the subject at that level, with the possibility of subsequent accelerated
progress into Further Mathematics.
There are others who will not continue with Mathematics beyond Year 11, and for these students this
unit provides a worthwhile and enriching course in its own right.
It is also expected that other students will follow this course and will benefit from it according to their
circumstances.
http://www.ocr.org.uk/download/kd/ocr_10611_kd_fsmq_spec.pdf
FMSP - GCSE Extension Materials
Number/Algebra
GCSE topic
Graphs of quadratic equations
NA 1 Surds
NA 2 Interpreting graphs
NA 3 Trial and improvement
NA 4 Functions
NA 5 Indices
NA 6 Simultaneous equations
NA 7 Solving quadratic equations
NA 8 Adding fractions
NA 9 NA 10 Inequalities
NA 11 Plotting curves
Shape, Space and Measure
SSM 1 SSM 2 SSM 3 SSM 4† SSM 5 SSM 6† GCSE topic
Trigonometry 1
Trigonometry 2
Circles
Pythagoras’ theorem
Loci
Vectors
Handling Data
HD1 HD2 HD3 GCSE topic
The mean from a frequency table
Interquartile range
Tree diagrams
http://www.furthermaths.org.uk/gcseextension.php
Problems from the Nrich Website
http://nrich.maths.org
Problems from the UKMT Maths Challenges
http://www.mathcomp.leeds.ac.uk/individual-competitions/