Further and Additional Maths

Many able home-ed students take GCSE maths early, and wonder what the next step might be. 
Additional Maths is a stepping-stone between GCSE and AS-level maths. You can move straight on to A-level maths and, as with GCSE, there is no coursework so all A-level maths exams are available to private candidates. 
However, this might complicate the situation if you intend to apply to university or go to sixth form, because universities sometimes make offers based only on A-levels yet to be taken, and for 3 taken simultaneously. If going to sixth form, they may insist you take three or more A-levels regardless of whether you already have some. In addition, if you want to study maths at university, some leading departments want to know your A-level UMS (percentage mark), so it may make sense to leave it until you're sure you'll do well. 
In the meantime, you could look at Additional Maths, and the UK Mathematics Trust activities, to broaden and deepen your mathematics education.
==Additional/Further Maths Specifications ==
Additional Maths is a stepping stone between GCSE / IGCSE and A-level, and is used to stretch able students who have completed their GCSE maths before they start the A-level course, and to keep them in practice at maths. Some schools enter their more able maths students for GCSE in Year 10 and then a further maths course in yr 11.
=== AQA Level 2 Certificate in Further Mathematics ===
AQA Level 2 Certificate in Further Mathematics page and specification
* Exams from: June 2020
* Available in June
* Specification code: 8365
==== Subject content ====
The AQA Level 2 Certificate in Further Mathematics comprises the following main topics:
# Number 
# Algebra 
# Coordinate Geometry (2 dimensions only) 
# Calculus 
# Matrix Transformations 
# Geometry
==== Assessment ====
100% exam based.
* Paper 1 - Non-Calculator - 1hr 45mins - 80 marks - 50%
** Written examination papers with a range of question types
** No calculator is allowed
** Covers all content
* Paper 2 - Calculator - 1hr 45mins - 80 marks - 50%
** Written examination papers with a range of question types
** Calculator allowed
** Covers all content
==== Resources ====
Hodder Level 2 Certificate in Further Mathematics Student Book 
CGP Revision resources - link is to a bundle but they are available separately
Integral Maths - paid subscription required 
Past papers can be accessed through the specification page or here.
Mr Barton Maths - check resources suit current spec.
Primrose Kitten Revision playlist
===OCR Free-Standing Maths Qualification (FSMQ)===
OCR Free-Standing Maths Qualification (FSMQ) - Additional Mathematics page and specification
This is a popular and well-respected qualification. It is accredited at Level 3, the same difficulty as AS-levels, although the content is smaller. It is graded A-E - as with AS levels, there is no A* grade. It carries up to 10 UCAS points for university entry. OCR says: "This qualification provides candidates with an introduction to the mathematics studied in AS and A Level GCE modules. It is designed as an enrichment programme for those students who have a thorough knowledge of the content of the Higher Tier of the National Curriculum for Mathematics. They should have achieved grade A*, A or B on the legacy GCSE or be expected to achieve a 9, 8 or 7 on the reformed GCSE (9-1) content. It is designed for those students who have sat, or are studying, the Higher Tier of the National Curriculum for Mathematics
*Exams from: June 2019
*Available in June
*Specification code: 6993
====Subject content====
The OCR FSMQ comprises the following main topics: 
*Algebra
*Enumeration
*Coordinate geometry
*Pythagoras and trigonometry
*Calculus
*Numerical methods
*Exponentials and logarithms
==== Assessment ====
100% exam based.
* Paper 1 - Calculator - 2hrs - 100 marks - 100%
** Written examination papers with a range of question types
** Calculator is allowed
** Covers all content
==== Resources ====
Hodder Student book 
Hodder Exam Practice book 
My Revision Notes revision guide
Free guides to support from OCR
Free scheme of work from Hodder
Integral Maths - paid subscription required
Past papers can be accessed through the specification page or here.
Mathsaurus - videos linked to chapters in the textbook
Primrose Kitten revision playlist 
===CAIE IGCSE Additional Maths===
Cambridge IGCSE Mathematics -Additional page and specification
CAIE says:It provides a strong foundation of mathematical knowledge both for candidates studying mathematics at a higher level and those who will require mathematics to support skills in other subjects. It is designed to stretch the most able candidates and provides a smooth transition to Cambridge AS & A Level Mathematics. 
*Exams from: June 2020
*Available in June and Autumn
*Specification code: 0606
Grading is A* - E (F and G are not awarded)
====Subject content====
The CAIE Additional Maths comprises the following main topics: 
*Functions
*Quadratic functions
*Equations, inequalities and graphs
*Indices and surds
*Factors of polynomials
*Simultaneous equations
*Logarithmic and exponential functions
*Straight line graphs
*Circular measure
*Trigonometry
*Permutations and combinations
*Series
*Vectors in two dimensions
*Differentiation and integration
====Assessment====
100% exam based.
* Paper 1 - Calculator - 2hrs - 80 marks - 50%
**Written examination papers with a range of question types
**Calculator is allowed
**Covers all content
*Paper 2 - Calculator - 2hrs - 80 marks - 50%
**Written examination papers with a range of question types
**Calculator is allowed
**Covers all content
====Resources ====
There are several different textbooks, workbooks, teacher books and revision books available for this specification - a list can be found here. There were changes in 2020 which involved publication of new textbooks, Books from 2020 will be valid to at least 2024 exams. Check whether answers to student books are included as for Cambridge they are often in the teacher's guide.
Past papers can be found on the specification pages or sites such as pastpapers.co and papacambridge. Each series (besides the March series - which is only sat in India) has 3 sets of papers available. These are because they were sat in different time zones. They are the same syllabus and useful for practice. There were changes to the content in 2020 (outlined at the back of the 2020-2022 spec) but papers from earlier will still be useful
===Edexcel International GCSE Further Pure Mathematics===
Edexcel International GCSE Further Pure Mathematics page and specification
*Exams from: June 2019
*Available in June and November 
*Specification code: 4PM1
====Subject content====
The Edexcel International GCSE Further Pure Mathematics comprises the following main topics: 
*Logarithmic functions and indices
*The quadratic function 
*Identities and inequalities 
*Graphs 
*Series 
*The binomial series 
*Scalar and vector quantities 
*Rectangular Cartesian coordinates 
*Calculus 
*Trigonometry
==== Assessment ====
100% exam based.
* Paper 1 - Calculator - 2hrs - 100 marks - 50%
** Written examination papers with a range of question types
** Calculator allowed
** Covers all content
* Paper 2 - Calculator - 1hr 45mins - 80 marks - 50%
** Written examination papers with a range of question types
** Calculator allowed
** Covers all content
==== Resources ====
Pearson publish a textbook and an online teachers guide for the specification. Details can be found here 
Past papers can be accessed through the specification page or here. Papers with an R suffix were available in a different time zone. They are the same specification and can be used as practice.
==Comments from home-educators==
One family's experience:
DS1 took his maths GCSE aged 12 because he wanted to. He got an A*, then just pottered with maths for a couple of years after GCSE, and instead focussed on other subjects. I felt that physics (which he did aged 13) and Chemistry (aged 14) would provide some decent practice. He started working through the A-level maths books aged 14 and completed 5 of the 6 units for single maths before we changed course - he didn't take any exams, but had done the work. The plan was that he would take single maths A-level in 2013 aged 15, then further maths in 2014, and go to sixth form already having those two A-levels plus possibly one other. Then, in autumn last year, we went round a couple of sixth forms, including a maths specialist grammar school, which DS1 liked.
I had lots of email correspondence with their head of maths, asking if it would be an issue if DS1 arrived there with single A-level maths, thinking that maybe if he didn't finish Further Maths at home, he could do it at sixth form. He replied that it would be better from their point of view if he arrived without any maths A-levels because it could be difficult to timetable further maths for someone who already had single maths - they're not separate subjects of course, and the students taking double maths will all be in the same classes together the rest of the time, so I think it would be difficult to separate out just the further maths lessons. Also, the school would still require students to do a minimum of 4 subjects even if they had already done some A-levels, so if he'd done his best subject at home, he would have to choose other subjects to make up his numbers. This was what made DS1 decide to leave A-level maths until sixth form; he said he definitely wanted to do his best subject at sixth form. I think there are several reasons. First,, just going to sixth form is going to be an upheaval - early mornings etc - so doing subjects he finds harder would add to that. Secondly, it would be a confidence boost to be studying something you know you can do well in. 
The Head of Maths said that they were discouraging their own students from doing early entry for A-level, no matter how able, because unis are now looking at UMS scores and not just grades, so it was vital to get a good grade first time round if the student might want to study maths at university. He suggested that it would be better for DS1 to keep his hand in doing the FSMQ (the OCR Additional Maths qualification) and perhaps the AQA additional maths and the UKMT challenges. He recommended the UKMT Mentoring Scheme, which we're on the waiting list for now - but you can access their materials online free (it's just the email mentor we're waiting for). We also had some great suggestions from members of the HE Exams yahoogroup, which are on the wiki page (below).
DS1 did the FSMQ (OCR Additional Maths); he didn't need long to prepare for it as it's intended as a filler for school children who've taken their GCSE maths early and want to go on to A-level maths. Rather than take another maths qualification in his last year of home-ed, he decided to do his own thing, look at maths from different approaches, and learn to play the guitar. This has all been very positive, and clearly didn't put off the school as he has a place for September.
I know someone (at school) whose school puts bright kids in for AS maths early, in yr11 (GCSE year). This has ended up really restricting the child's choice at sixth form because he too has found that most other places can't really accommodate him turning up with one AS-level and he'd have to repeat the material. This is not a problem if you're planning on doing A-levels from home, of course.
==UK Mathematics Trust / Maths Olympiad ==
The United Kingdom Mathematics Trust run individual maths competitions which are puzzle-solving exercises. It is a different approach to maths from the usual examination style and offers the opportunity for everyone to have fun, while stretching able students. Top scorers win certificates and the opportunity to go through to prestigious further rounds such as the Kangaroo and the Maths Olympiad. 
Entry for home-ed students:"Q. Can my home-educated child take part in UKMT competitions?
A. Yes. Young people educated at home may participate in our competitions through a school that is a registered UKMT centre, at the discretion of the school. Parents or carers of home-educated young people should contact a nearby school directly to make such arrangements."
Some other interesting material on the website was quite difficult to find so is listed here:
The UK Maths Trust Mentoring scheme has lots of past problems and solutions to stretch kids in maths.
==Other Maths Activities==
UKMT Individual Challenges and mentoring materials. The UKMT material is good because it's fresh and a different approach from standard 'textbook' maths.  See above for links.
The Problem Solver's Handbook, from UKMT
The Art of Problem Solving books, also from UKMT - these would probably be good for the proofs you're after, and it's an international approach so may work well with what your son has already used.
Project Euler - maths problems combined with programming problems - https://projecteuler.net/
"Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems."
Vedic Maths - DS1 enjoyed learning lots of shortcuts with this approach.  I wouldn't get too carried away with stories of its heritage because that part is probably made up - they are maths tricks, but useful ones and apparently very efficient.  See http://blogannath.blogspot.co.uk/2009/06/introduction-to-vedic-mathematics.html?m=1
The following books were recommended by a maths whizz on the HE-Exams list as being good material for mathsy teens, and the sort of thing which might be good to discuss at interviews: 
*James Gleick's book on chaos
* Euclid's Elements
* Principia
*Ian Stewart's Cabinet of Mathematical Curiosities
*C J Bradley's Introduction to Number Theory
*Surely You're Joking Mr Feynman
*The Man Who Loved Only Numbers
* The Man Who Knew Infinity (Ramanujan)
Category:Subjects
Category:Maths