Revision of GCSE Specifications Draft Proposals Further Mathematics Draft Proposals for Consultation 2016
Content Page Introduction... 3 A. Specification at a Glance... 4 B. Subject Content for each Unit... 5 C. Summary of Changes... 9 New Content... 9 Content Remaining... 9 D. External Assessment... 10 E. Progression from Key Stage 3... 10 F. Progression to GCE... 12 G. Additional Comments... 12 H. Support... 13 2
Introduction Awarding Bodies are revising their GCSE and GCE specifications to ensure that both content and assessment continue to reflect the needs of learners and the society, economy and environment in which they live and work. The revision programme is now underway to review our GCSE and produce revised specifications for first teaching from September 2017. The new specification should provide opportunities for students to build upon the knowledge, understanding and skills developed at Key Stage 3, and the relevant requirements of the Northern Ireland Curriculum at Key Stage 4. This document has been designed to provide you with an outline of our draft proposals for the revised GCSE specification. For further information on the revision of GCSE Specifications go to: http://www.ccea.org.uk/the-revision/ 3
A. Specification at a Glance The table below summarises the structure of this GCSE course: Relevant information in relation to course content, assessment, weighting and availability is inserted below. Content Assessment Weighting Availability Unit 1 Pure Maths External exam with calculator 2 hours 15 minutes 55% January and Summer sittings starting Summer 2018 Unit 2 Applications (Choice of 2 sections out of 4) External exam with calculator 1 hour 45 minutes 45% January and Summer sittings starting Summer 2018 At least 40% of the assessment (based on unit weightings) must be taken at the end of the course as terminal assessment. 4
B. Subject Content for each Unit We have divided the course into two units. A brief description of each unit is provided below. Unit 1 Pure Maths Content Description PURE MATHS Algebraic Fractions Algebraic Manipulation Completing the square Simultaneous Equations Equation of a circle Add, subtract, multiply and divide rational algebraic fractions with linear and quadratic numerators and/or denominators; Manipulation of algebraic expressions and including expansion of 3 linear brackets; The coefficient of x 2 will always be 1. Applying this to solving quadratic equations, identifying minimum turning point Formation and solution of three equations in three unknowns Centred on the origin. Including finding the equation of a tangent at a given point on the circle; Transformation of functions Quadratic Inequalities Trigonometric Equations Differentiation f(x)±a, -f(x), f(x±a), f(-x) Restricted to quadratic expressions that factorise. Including using set notation to represent the solution set. Solve simple trigonometric equations leading to a maximum of 2 solutions in a given range Restricted to integer powers of x and includes the application of differentiation to gradient; finding equations of tangents and normal to points on a curve; simple optimisation problems and elementary curve sketching of a quadratic or cubic function Integration Logarithms Restricted to integer powers of x and including definite integration and finding the area under a curve; Laws of logs and including the use of log/log graphs in context. The solution of indicial equations. 5
Vectors Matrices Vector concept, sum of 2 vectors and scalar multiples of a vector, simple geometrical problems. (Proofs using vector geometry included); Performing addition, subtraction and multiplication on matrices, finding inverses and solving matrix equations and using matrices to solve 2x2 simultaneous equations; Unit 2 Applications Content Description Section 1 Mechanics Kinematics Displacement/time graphs and velocity/time graphs and their applications (excluding interception problems); Use of constant acceleration formulae; Vectors Use of i/j notation including the magnitude and direction of a vector and the application of i/j vectors in calculations Forces Newton s Laws of Motion Moments Resolving forces into components and finding the resultant of a set of forces. Apply the concept of equilibrium; Apply F = ma including the scenarios: Inclined plane (applied forces parallel to the plane) Two connected particles in rectilinear motion; Friction will be given as a value or ratio to mass; (F = µr will not be tested) Restricted to a horizontal uniform rod supported by 1 or 2 pivots Section 2 Statistics Measures of Central Tendency and Dispersion Calculation of mean and standard deviation. Including combining sets of data. Transformation of data sets; (Excluding the calculation of median.) 6
Probability Calculation of combined probabilities using the addition rule. Calculate and interpret conditional probabilities through representation using expected frequencies, two-way tables, tree diagrams and Venn diagrams; Including construction of Venn diagrams; Binomial Probabilities Normal Probabilities Bivariate Analysis Calculate binomial probabilities in context using Pascal s triangle and the expansion of (p+q) n where n<=6 Understand that the distribution of many real world variables takes the shape of a bell curve. Calculate a single probability from the Normal distribution using tables and z = (x-µ)/σ where the mean and standard deviation will always be given Including calculation of Spearman s Rank Correlation coefficient and calculation of equation of line of best fit; Section 3 Decision Maths Algorithms Linear Programming Time Series Critical Path Analysis Concept of an algorithm. List of steps with an ending condition. Applications. Modelling real life scenarios as linear programming problems Graphical solution of at most 5 inequalities in two variables Solutions may be real numbers or restricted to integers. Understand and use seasonal variation, cyclic variation, random variation and secular trend. Calculate appropriate moving averages (using only 3,4 or 5 points) Draw a trend line and use it to extrapolate. (The plotting of original data will be given.) Modelling of a project in context by an activity network. (Using activities on arc). Algorithm for finding the critical path. Nodes will show earliest and latest event times. Calculate float times. Identify critical path. Interpret in context. 7
Section 4 Informatics Representation of Numbers and Arithmetic Binary and Hexadecimal Number Systems Integer Representation - in one byte or two bytes. Representation of signed integers in one byte. Adding integers. Subtracting integers by adding complements. Floating point representation Appreciation of overflow and underflow in computer representation of real numbers. Counting Logic Number of permutations of r objects from n Number of Combinations of r objects from n Multiplicative Principle Boolean variables Boolean expressions: AND, OR and NOT Truth Tables (up to two variables) Proof of de Moivre s theorems. Algorithms Flowcharts (tracing and construction) Searching (using a linear search or binary search) Sorting ( using a bubble-sort algorithm or quicksort) Efficiency of algorithms Graphs Graphs (a collection of vertices and edges) Basic graphs: complete and bipartite graphs Eulerian Paths and Hamiltonian Circuits Trees and Spanning Trees. Kruskal s and Dijkstra s algorithms 8
C. Summary of Changes New Content What s new at a Glance Unit 1 Expanding 3 brackets Transformation of functions Equation of a circle Quadratic inequalities Log/log graphs Unit 2 Section 1 Mechanics Nothing Section 2 Statistics Binomial probabilities Normal probabilities Section 3 Decision Maths All new Section 4 Informatics All new Content Remaining What is remaining from the current Specification Algebraic fractions Completing the Square 3x3 Simultaneous equations Trigonometric equations Differentiation and its applications Integration and its application Laws of logarithms Matrices Vector geometry Kinematics 9
Forces Vectors Newton s Laws Moments Measures of central tendency and dispersion Probability Bivariate Analysis D. External Assessment Number of Papers 2 test papers Types of Questions Some short and some more testing extended questions. Length / Time Unit 1: 2 hours 15 minutes Unit 2: 1 hour 45 minutes Weighting Unit 1 55% Unit 2 45% Additional Information / Description Unit 2 candidates will choose 2 sections E. Progression from Key Stage 3 Cross Curricular Skills at Key stage 4 Communication Communicate effectively in oral, visual, written, mathematical and ICT formats, showing clear awareness of audience and purpose. Read contextualised questions for mathematical information, communicate reasoning in clear and organised ways and present ideas, findings and answers in a variety of appropriate formats for different audiences and purposes 10
Using Mathematics Demonstrate mental mathematical capability with simplification of expressions e.g. in algebra; Demonstrate financial capability in a range of relevant everyday contexts and tested using Moving Averages for example. Be aware of how the knowledge and skills developed in Further Mathematics can be applied to a range of situations/contexts e.g. how the study of Newton s Laws can be applied to solving real-life problems relating to dynamics Know which mathematics to choose for a particular situation/context and apply appropriate mathematical techniques accurately e.g. use trigonometric equations to solve further mathematics problems involving angles Using ICT Show deeper mathematical understanding by thinking critically and flexibly, solving problems and making informed decisions, demonstrating Using ICT where appropriate Thinking Skills and Personal Capabilities at key stage 4 Self-Management Demonstrate self-management by working systematically, persisting with tasks, evaluating and improving own performance; Working with Others Work effectively with others; Problem Solving Decide on the appropriate method and equipment to solve problems mental, written, calculator, mathematical instruments or a combination of these; Make links between cause and effects by considering, for example, the rates of change of a function with respect to one of its variables through the study of calculus Managing Information Research and manage information effectively to investigate and solve mathematical problems, including Using ICT where appropriate; Being Creative Demonstrate creativity and initiative when developing ideas and following them through; 11
Progression from relevant Area of Learning The strands of Number, Algebra, Shape Space and Measures and Handling Data studied at Key Stage 3 continue to be reflected in the nature of the content of GCSE Further Maths; Relevance of Learning to Everyday Life and Work, Independent and Lifelong Learning Real life scenarios are used in the teaching and assessment of the current specification and this practice will be carried forward into the new specification; The new Further Mathematics specification will help develop awareness and understanding of the skills required to be successful in employment and business and help develop an understanding of how these skills are transferable to the world of work. It will offer a broad range of different branches of mathematics to facilitate progression to advanced, further/higher education and beyond into a pathway of lifelong learning. F. Progression to GCE Opportunities for progression to GCE This specification will offer opportunities for students to develop skills needed to study A level Maths, Biology, Physics, Chemistry, Geography, Technology, Computing and Engineering; G. Additional Comments Additional comments The weighting of Units 1 and 2 has changed. Currently each unit is worth 50% of the overall award, but going forward the Unit 1 on Pure Mathematics will form 55% of the overall award and Unit 2 will therefore contribute 45%. This change has been made to incorporate more topics in Pure Mathematics and provide more opportunity for strengthening skills in algebraic manipulation e.g. the introduction of topics such as expanding 3 brackets, transforming functions and solving problems involving quadratic inequalities Choices have been introduced into Unit 2. Currently, Unit 2 comprises two equally weighted sections in Mechanics and Statistics and candidates are required to take both. The new specification will offer, within Unit 2, a choice of two sections from a selection of four, thus retaining sections in Mechanics and Statistics and providing two new sections in Decision Mathematics and Informatics. Each section will be 12
equally weighted. By offering more choice through the introduction of new areas of mathematics, it is CCEA s intention to make the Further Maths specification more appealing to young people and therefore increase uptake at both GCSE and GCE levels. H. Support The range of support provided by CCEA includes: Past papers; Mark schemes; Chief Examiner s report; Guidance on progression from Key Stage 3; Schemes of work; Centre support visits; Support days for teachers; Resource list; and Exemplification of standards. Additional support may be required as suggested below Possible additional support Additional support materials to be produced for teaching outlining the type of assessment on the new topics to be covered in the specification; Staff Development days on new topics; Centre support visits covering the implementation of new topics. 13