# Additional Mathematics

Additional Mathematics is a qualification in mathematics, commonly taken by students in high-school (or GCSE exam takers in the United Kingdom). It features a range of problems set out in a different format and wider content to the standard Mathematics at the same level.

## Additional Mathematics in Singapore

In Singapore, Additional Mathematics is an elective subject offered to pupils in secondary school—specifically those who have an aptitude in Mathematics and are in the Normal (Academic) stream or Express stream. The syllabus covered is more in-depth as compared to Elementary Mathematics, with additional topics including Algebra binomial expansion, proofs in plane geometry, differential calculus and integral calculus. Additional Mathematics is also a prerequisite for students who are intending to offer H2 Mathematics and H2 Further Mathematics at A-level (if they choose to enter a Junior College after secondary school). Students without Additional Mathematics at the 'O' level will usually be offered H1 Mathematics instead.
| Chapters | Subtopics |
| Algebra | A1 Quadratic functions A2 Equations and inequalities A3 Surds A4 Polynomials and partial fractions A5 Binomial expansions (Not tested in N(A)) A6 Exponential and logarithmic functions (Not tested in N(A)) |
| Geometry and Trigonometry | G1 Trigonometric functions, identities and equations G2 Coordinate geometry in two dimensions G3 Proofs in plane geometry (Not tested in N(A)) |
| Calculus | C1 Differentiation and integration |

### Examination Format

The syllabus was updated starting with the 2021 batch of candidates. There are two written papers, each comprising half of the weightage towards the subject. Each paper is 2 hours 15 minutes long and worth 90 marks. Paper 1 has 12 to 14 questions, while Paper 2 has 9 to 11 questions. Generally, Paper 2 would have a graph plotting question based on linear law. It was originated in the year 2003

## GCSE Additional Mathematics in Northern Ireland

In Northern Ireland, Additional Mathematics was offered as a GCSE subject by the local examination board, CCEA. There were two examination papers: one which tested topics in Pure Mathematics, and one which tested topics in Mechanics and Statistics. It was discontinued in 2014 and replaced with GCSE Further Mathematics—a new qualification whose level exceeds both those offered by GCSE Mathematics, and the analogous qualifications offered in England.

## Further Maths IGCSE and Additional Maths FSMQ in England

Starting from 2012, Edexcel and AQA have started a new course which is an IGCSE in Further Maths. Edexcel and AQA both offer completely different courses, with Edexcel including the calculation of solids formed through integration, and AQA not including integration.
AQA's syllabus mainly offers further algebra, with the factor theorem and the more complex algebra such as algebraic fractions. It also offers differentiation up to—and including—the calculation of normals to a curve. AQA's syllabus also includes a wide selection of matrices work, which is an AS Further Mathematics topic.
AQA's syllabus is much more famous than Edexcel's, mainly for its controversial decision to award an A* with Distinction (A^), a grade higher than the maximum possible grade in any Level 2 qualification; it is known colloquially as a Super A* or A**.
A new Additional Maths course from 2018 is OCR Level 3 FSMQ: Additional Maths (6993). In addition to algebra, coordinate geometry, Pythagorean theorem, trigonometry and calculus, which were on the previous specification, this course also includes:
* 'Enumeration' content, which expands the topic of the binomial distribution to include permutations and combinations
* 'Numerical methods’ content, which expands upon the informal graphical approximations in GCSE
* 'Exponentials and Logarithms’ content, which develops the growth and decay content and the graphs section of GCSE
* 'Sequences' content, which uses subscript notation to support the iterative work on numerical methods.

## Additional Mathematics in Malaysia

In Malaysia, Additional Mathematics is offered as an elective to upper secondary students within the public education system. This subject is included in the Sijil Pelajaran Malaysia examination.
Science stream students are required to apply for Additional Mathematics as one of the subjects in the Sijil Pelajaran Malaysia examination, while Additional Mathematics is an optional subject for students who are from arts or commerce streams.
Additional Mathematics in Malaysia—also commonly known as Add Maths—can be organized into two learning packages: the Core Package, which includes geometry, algebra, calculus, trigonometry and statistics, and the Elective Package, which includes science and technology application and social science application. It covers various topics including:
| | Chapters | Subtopics |
| Form 4 | 1) Functions | 1.1 Functions 1.2 Composite Functions 1.3 Inverse Functions |
| Form 4 | 2) Quadratic Functions | 2.1 Quadratic Equations and Inequalities 2.2 Types of Roots of Quadratic Equations 2.3 Quadratic Functions |
| Form 4 | 3) Systems of Equations | 3.1 Systems of Linear Equations in Three Variables 3.2 Simultaneous Equations involving One Linear Equation and One Non-Linear Equations |
| Form 4 | 4) Indices, Surds and Logarithms | 4.1 Law of Indices 4.2 Laws of Surds 4.3 Laws of Logarithms 4.4 Applications of Indices, Surds and Logarithms |
| Form 4 | 5) Progressions | 5.1 Arithmetic Progressions 5.2 Geometric Progressions |
| Form 4 | 6) Linear Law | 6.1 Linear and Non-Linear Relations 6.2 Linear Law and Non-Linear Relations 6.3 Applications of Linear Law |
| Form 4 | 7) Coordinate Geometry | 7.1 Divisor of a Line Segment 7.2 Parallel Lines and Perpendicular Lines 7.3 Areas of Polygons 7.4 Equations of Loci |
| Form 4 | 8) Vectors | 8.1 Vectors 8.2 Addition and Subtraction of Vectors 8.3 Vectors in a Cartesian Plane |
| Form 4 | 9) Solution of Triangles | 9.1 Sine Rule 9.2 Cosine Rule 9.3 Area of Triangles 9.4 Application of Sine Rule, Cosine Rule and Area of a Triangles |
| Form 4 | 10) Index Numbers | 10.1 Index Number 10.2 Composite Index |
| Form 5 | 1) Circular Measure | 1.1 Radian 1.2 Arc Length of a Circle 1.3 Area of Sector of a Circle 1.4 Application of Circular Measure |
| Form 5 | 2) Differentiation | 2.1 Limit and its Relation to Differentiation 2.2 The First Derivative 2.3 The Second Derivative 2.4 Application of Differentiation |
| Form 5 | 3) Integration | 3.1 Integration as the Inverse of Differentiation 3.2 Indefinite Integral 3.3 Definite Integral 3.4 Application of Integration |
| Form 5 | 4) Permutation and Combination | 4.1 Permutation 4.2 Combination |
| Form 5 | 5) Probability Distributions | 5.1 Random Variable 5.2 Binominal Distribution 5.3 Normal Distribution |
| Form 5 | 6) Trigonometric Functions | 6.1 Positive Angles and Negative Angles 6.2 Trigonometric Ratios of any Angle 6.3 Graphs of Sine, Cosine and Tangent Functions 6.4 Basic Identities 6.5 Addition Formulae and Double Angle Formulae 6.6 Application of Trigonometric Functions |
| Form 5 | 7) Linear Programming | 7.1 Linear Programming Model 7.2 Application of Linear Programming |
| Form 5 | 8) Kinematics of Linear Motion | 8.1 Displacement, Velocity and Accelerations as a Function of Time 8.2 Differentiation in Kinematics of Linear Motion 8.3 Integration in Kinematics of Linear Motion 8.4 Application of Kinematics of Linear Motion |
Format for Additional Mathematics Exam based on the Malaysia Certificate of Education is as follows:
* Paper 1 (Duration: 2 Hours): Questions are categorised into Sections A and B and are tested based on the student's knowledge to grasp the concepts and formulae learned during their 2 years of learning. Section A consists of 12 questions in which all must all be answered, whereas Section B consists of 3 questions and students are given the choice to answer 2 of the three questions only. Each question may contain from zero to three subsets of questions with marks ranging from 2 to 8 marks. The total weighting of the paper is 80 marks and constitutes 44% of the grade.
* Paper 2 (Duration: 2 hours 30 minutes): Questions are categorised into 3 sections: A, B and C. Section A contains 7 questions which must all be answered. Section B contains 4 questions where students are given the choice to answer 3 out of 4 of them. Section C contains 4 questions where students are only required to answer 2 out of 4 of the given questions. All Section C questions are based on the same chapters every year and are thus predictable. A question in Section C carries 10 marks with at 3 to 4 subquestions per question. This paper tests the student's ability to apply various concepts and formulae in real-life situations. The total weighting of the paper is 100 marks and constitutes 56% of the grade.
In 2020, the first batch of students learning the new syllabus, KSSM, will receive new Form 4 textbooks with new chapters which contain certain topics from A-levels.

## Additional Mathematics in Mauritius

In Mauritius, Additional Mathematics, more commonly referred to as Add Maths, is offered in secondary school as an optional subject in the Arts Streams, and a compulsory subject in the Science, Technical and Economics Stream. This subject is included in the University of Cambridge International Examinations, with covered topics including functions, quadratic equations, differentiation and integration (calculus).

## Additional Mathematics in Hong Kong

In Hong Kong, the syllabus of HKCEE additional mathematics covered three main topics, algebra, calculus and analytic geometry. In algebra, the topics covered include mathematical induction, binomial theorem, quadratic equations, trigonometry, inequalities, 2D-vectors and complex number, whereas in calculus, the topics covered include limit, differentiation and integration.
In the HKDSE, additional mathematics has been replaced by two Mathematics Extend Modules, which include a majority of topics in the original additional mathematics, and a few topics, such as matrix and determinant, from the syllabus of HKALE pure mathematics and applied mathematics, while notably missing analytic geometry, inequalities involving absolute values and rational functions, and complex numbers (of which only the basic arithmetic is covered in the Mathematics Compulsory part).