Further Mathematics
Encyclopedia
Further Mathematics is the title given to a number of advanced secondary
mathematics
courses. Higher and Further Mathematics may also refer to any of several advanced mathematics courses at many institutions.
In the United Kingdom, Further Mathematics describes a course studied in addition to the standard mathematics AS-Level and A-Level courses. In Victoria
, Australia it describes a course delivered as part of the Victorian Certificate of Education
. See the section on Victoria for a more detailed explanation. Globally, it describes a course studied in addition to GCE AS-Level and A-Level Mathematics, or one delivered as part of the International Baccalaureate Diploma.
and applied
modules. Whilst the pure modules - formerly known as Pure 4-6 (or Core 4-6), now known as Further Pure 1-3 (4 exists for the AQA
board http://www.aqa.org.uk/qual/pdf/AQA-5361-6371-W-SP-08.PDF) - are of a higher standard than those in the standard course, the applied modules need not be. The topics covered by Further Mathematics are more sophisticated and conceptually advanced compared to the single A-level Mathematics.
To achieve an A level in Further Maths, candidates must study six modules which have not already been used for their Maths A level. These six modules must consist of FP2 or FP3, and FP1 along with 4 other modules.
Some schools and colleges in places such as Pakistan, Hong Kong and India take examinations set by British boards and consequently the subject is offered internationally.
Because smaller schools and colleges may not be able to offer Further Mathematics (as it is a very low-intake course requiring well-trained teachers), most universities do not require the course, and may offer "catch-up" classes covering the additional content. Exceptions are the University of Warwick
, the University of Cambridge
where you must have Further Mathematics to at least AS level to study for a degree in mathematics; also University College London requires an A2 in Further Maths for its maths courses and is recommended for the Statistics courses; Imperial College also requires an A* in A2 Further Maths while other top end universities recommend it or promise lower offers in return.
Further Maths is currently the fastest-growing subject at A level, with the number of students increasing by 23% in 2006, and a network has been set up to offer the subject to pupils at schools that cannot provide it. http://www.economist.com/world/britain/displaystory.cfm?story_id=7950110
Further Maths is commonly expressed as the most difficult A-level currently offered in the U.K., this is mainly because it is the only subject to further the study (as an extra AS or full A-level) of one particular subject. Although the subject has about 60% of its cohort obtaining "A" grades, students taking the subject tend to be more able, those less likely to achieve top grades are usually discouraged from the course as it can disrupt their performance in other studies.
is the easiest level of mathematics. Any student wishing to undertake tertiary studies in areas such as Science, Engineering, Commerce, Economics, and some Information Technology courses, must undertake one or both of the other two VCE maths subjects- Mathematical Methods or Specialist Mathematics. The Further Mathematics syllabus in VCE consists of three core modules, which all students undertake, plus three modules chosen by the student (or usually by the school or teacher) from a list of six. The core modules are Univariate Data, Bivariate Data and Time Series. The optional modules are Number Patterns, Geometry and Trigonometry, Graphs and Relations, Business-Related Mathematics, Networks and Decision Mathematics, or Matrices.
(HL) Mathematics. It assumes knowledge of the core syllabus of the HL course, and consists of studying all four of the options studied at Higher Level, plus an extra geometry
unit.
The syllabus consists of:
Secondary education
Secondary education is the stage of education following primary education. Secondary education includes the final stage of compulsory education and in many countries it is entirely compulsory. The next stage of education is usually college or university...
mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
courses. Higher and Further Mathematics may also refer to any of several advanced mathematics courses at many institutions.
In the United Kingdom, Further Mathematics describes a course studied in addition to the standard mathematics AS-Level and A-Level courses. In Victoria
Victoria (Australia)
Victoria is the second most populous state in Australia. Geographically the smallest mainland state, Victoria is bordered by New South Wales, South Australia, and Tasmania on Boundary Islet to the north, west and south respectively....
, Australia it describes a course delivered as part of the Victorian Certificate of Education
Victorian Certificate of Education
The Victorian Certificate of Education or VCE is the credential awarded to secondary school students who successfully complete high school level studies in the state of Victoria, Australia. Study for the VCE is usually completed over two years, but it can be spread over a longer period in some cases...
. See the section on Victoria for a more detailed explanation. Globally, it describes a course studied in addition to GCE AS-Level and A-Level Mathematics, or one delivered as part of the International Baccalaureate Diploma.
Background
A qualification in Further Mathematics involves studying both purePure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
and applied
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
modules. Whilst the pure modules - formerly known as Pure 4-6 (or Core 4-6), now known as Further Pure 1-3 (4 exists for the AQA
Assessment and Qualifications Alliance
AQA is an Awarding Body in England, Wales and Northern Ireland. It compiles specifications and holds examinations in various subjects at GCSE, AS and A Level and offers vocational qualifications. AQA is a registered charity and independent of the Government...
board http://www.aqa.org.uk/qual/pdf/AQA-5361-6371-W-SP-08.PDF) - are of a higher standard than those in the standard course, the applied modules need not be. The topics covered by Further Mathematics are more sophisticated and conceptually advanced compared to the single A-level Mathematics.
To achieve an A level in Further Maths, candidates must study six modules which have not already been used for their Maths A level. These six modules must consist of FP2 or FP3, and FP1 along with 4 other modules.
Some schools and colleges in places such as Pakistan, Hong Kong and India take examinations set by British boards and consequently the subject is offered internationally.
Because smaller schools and colleges may not be able to offer Further Mathematics (as it is a very low-intake course requiring well-trained teachers), most universities do not require the course, and may offer "catch-up" classes covering the additional content. Exceptions are the University of Warwick
University of Warwick
The University of Warwick is a public research university located in Coventry, United Kingdom...
, the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
where you must have Further Mathematics to at least AS level to study for a degree in mathematics; also University College London requires an A2 in Further Maths for its maths courses and is recommended for the Statistics courses; Imperial College also requires an A* in A2 Further Maths while other top end universities recommend it or promise lower offers in return.
Further Maths is currently the fastest-growing subject at A level, with the number of students increasing by 23% in 2006, and a network has been set up to offer the subject to pupils at schools that cannot provide it. http://www.economist.com/world/britain/displaystory.cfm?story_id=7950110
Further Maths is commonly expressed as the most difficult A-level currently offered in the U.K., this is mainly because it is the only subject to further the study (as an extra AS or full A-level) of one particular subject. Although the subject has about 60% of its cohort obtaining "A" grades, students taking the subject tend to be more able, those less likely to achieve top grades are usually discouraged from the course as it can disrupt their performance in other studies.
List of the areas of study on the syllabus
Study areas vary with the examination board and the specification they set, although Edexcel's is the most popular and is given below..- Further Pure 1
- Complex Numbers
- Use of iterative methods to solve equations including Newton-Raphson methodNewton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
- ParabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
s and Rectangular Hyperbola - MatricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
- Summation of seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
using standard results - Proof by inductionMathematical inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
- Further Pure 2
- Inequalities
- Summation of seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
by the method of differences - Further Complex Numbers
- Linear, Ordinary Differential Equations, of the first and second order
- Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
- Polar CoordinatesPolar coordinate systemIn mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
- Further Pure 3
- Hyperbolic functions, including their differentiation and integration
- Conic Sections
- CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, including reduction formulaeIntegration by reduction formulaeIntegration by reduction formulae can be used when we want to integrate a function raised to the power n. If we have such an integral we can establish a reduction formula which can be used to calculate the integral for any value of n....
, surfaces of revolution and the inverse trigonometric functions. - Vectors, including the cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
and the triple scalar product - Further MatricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
Australia (Victoria)
In contrast with other Further Mathematics courses, Further Maths as part of the VCEVCE
VCE could refer to:* Victorian Certificate of Education, Year 11 and 12 qualification in Victoria, Australia* Vocational Certificate of Education, a Further Education qualification in the United Kingdom...
is the easiest level of mathematics. Any student wishing to undertake tertiary studies in areas such as Science, Engineering, Commerce, Economics, and some Information Technology courses, must undertake one or both of the other two VCE maths subjects- Mathematical Methods or Specialist Mathematics. The Further Mathematics syllabus in VCE consists of three core modules, which all students undertake, plus three modules chosen by the student (or usually by the school or teacher) from a list of six. The core modules are Univariate Data, Bivariate Data and Time Series. The optional modules are Number Patterns, Geometry and Trigonometry, Graphs and Relations, Business-Related Mathematics, Networks and Decision Mathematics, or Matrices.
International Baccalaureate
Further Mathematics, as studied within the International Baccalaureate Diploma is a Standard Level course that can only be taken in conjunction with Higher LevelIB Group 5 subjects
The Group 5 subjects of the IB Diploma Programme consist of four different mathematics courses and two computer science courses. To earn an IB Diploma, a candidate must pass one of the following four mathematics courses: Mathematical Studies SL , Mathematics SL, Mathematics HL , or Further...
(HL) Mathematics. It assumes knowledge of the core syllabus of the HL course, and consists of studying all four of the options studied at Higher Level, plus an extra geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
unit.
The syllabus consists of:
- Topic 1 - GeometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
- Topic 2 - StatisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and ProbabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The... - Topic 3 - Sets, RelationsRelation (mathematics)In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
, and GroupsGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity... - Topic 4 - SeriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
and Differential Equations - Topic 5 - Discrete MathematicsDiscrete mathematicsDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...