Further Mathematics Resources

The Further Mathematics Teaching Resources page is one of a series of Further Maths pages and is updated with any new resources. Additionally any posts which include Further Mathematics resources are now categorised as Further Mathematics A Level.

To highlight some recent changes:

TLMathsIndex

Jack Brown has created thousands of videos covering the complete A Level specification; the easiest way to navigate the videos is through his website, TLMaths.com.

Jack also has Further Maths teaching videos and exam paper walkthroughs. I see many useful videos on Proof, including the use of Induction for proofs of divisibility. This collection is growing and currently there are also videos available on many Complex Numbers and Matrices topics.

amsp videos

From amsp this brilliant collection of short videos produced by the legacy Further Mathematics Support Program supports the Further Maths Specification. I have used many of these successfully in class and recommended them to students to support their studies. Look at any of the examination boards to see the coverage for the course.

OwenElton Further Pure
Owen (Owen134866 on TES) has a library of Further Mathematics teaching resources, these are really clearly structured with step by step examples.
Further Pure 1 Lesson PowerPoints
Further Pure 2 Lesson PowerPoints
Further Mechanics 1

Berwick - Statistics Notes

Joe Berwick has written a complete set of notes for Further Statistics 1, these are for the Edexcel and AQA specifications and include many very clear worked examples.

PowerPoint - Questions by topicSebastian Bicen (@BicenMaths) has created this really useful hyperlinked PowerPoint and pdf which provides Further Mathematics exam questions by topic. The exam papers included are as follows:
AS and A2 SAMs
AS and A2 Specimen
AS 2018
A2 Mock Paper

Jonny Griffiths Further RISPS

Jonny Griffiths investigative activities for the pure A Level Mathematics classroom are well known (details are included in the A Level resources pages, see RISPS.  He has now published Further Risps, forty rich tasks for the pure Further Mathematics classroom.

Two editions are available – a hard copy or pdf versionThe pdf version is now available free from www.further-risps.co.uk. On this page you can see two trial activities, Further Risps 1 and 2.

The pdf not only provides the forty problems but also full teachers notes for each. The notes for each task begin with the topic or topics covered, identify the type of task, for example, introductory and state any preliminary knowledge required. This is a valuable resource for teaching Further Mathematics.

Investigating the first Risp on Matrices at the session this certainly would help students with fluency in finding the determinant of a matrix. Rich tasks like this can provide students with a greater understanding than just a traditional exercise and will hopefully stick for longer!

 

Further Mathematics

Some updates to the series of pages on Further Mathematics.
Still a work in progress, but the pages to date are:

I have updated the Teaching Resources page with additional content and am adding additional posts I have written on Further Mathematics.

New pages have been added, see Notes and Examples, Legacy Papers as well as the posts on Calculus, Differential Equations and Linear Programming.

AQA Algebra Topic Test example

AQA Topic Test

OCR Calculus Check In Test

OCR A Section Check In Test


See also this series of pages and tutorials on calculators.


Watch this space…!

Further Maths – Further Calculus

Further Calculus

Included in the detailed content statements on the material which must be included in A level specifications in further mathematics which makes up approximately 50% of the total content of A level further mathematics, we see the section on Further calculus.

Given that this is a compulsory topic, teachers of Further Maths can usefully look at specimen and practice materials for all the examination boards. We can also use technology to demonstrate and check answers. Additionally there are many useful resources offering notes and examples which are freely available.

WolframAlpha Volumes of Revolution
WolframAlpha can very easily be used to check any volumes of revolution and also gives a clear visualisation of the solid formed; note the option to show the surface or solid formed.

Dr Frost - Further Calculus

Dr Frost – Further Calculus

On Dr Frost’s site, Further Calculus can be found here.

CIMT Reduction Formulae

CIMT – Further Calculus

The A Level notes from CIMT include some useful notes, activities and exercises for Further Mathematics, if we look at Chapter 8, Further Calculus in the Further Pure Mathematics section the we see that the chapter includes Volumes of revolution. We also have reduction formulae which is included in the optinal content for AQA, Edexcel and OCR Specification A.

For more notes, examples and exercises, not just for Calculus but many parts of the Further Maths specification, as well as CIMT, try AJ Hobson’s ‘Just the Maths’ or the Helm Project.

Just the Maths AJ Hobson

Many useful notes and examples can be found in AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths.

Helm Notes

HELM Project – Workbook 14, Applications of Integration 1

If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

(For easy reference these notes have been added to the Teaching Resources page for Further Mathematics.)

Improper Integrals University of Pennsylvania

Improper Integrals – University of Pennsylvania

Looking for resources on Improper Integrals, I came across this from the University of Pennsylvania, a clear presentation with worked examples.

Edexcel have a SolutionBank and GeoGebra interactives for their various texts for Maths and Further Maths freely available online. If we look at Core Pure Mathematics Book 2 for example, you can see the various chapter links. These take you to the interactives for each chapter and to full worked solutions for each exercise. Looking at Chapter 4 for example on Volumes of Revolution, we can see two GeoGebra Interactives allowing exploration of volumes of revolution around the x and y axes.

Volumes of Revolution

Edexcel Geogebra Interactive – Volumes of Revolution

 

 

 

Further Mathematics Mapping Documents

Unlike Mathematics where the subject content is the same for all the examination boards, for Further Mathematics only half of the subject content is compulsory. The core content can be found on pages 7-10. This prescribed subject content can also be seen in my file of Technology Resources for Mathematics and Further Mathematics (still an ongoing project which will be developed further):
maths & further maths technology by content  
Technology Resources

The content of the remaining 50% is not prescribed and has been defined in the specifications from the examination boards who could build from the applied content in A level Mathematics, introduce new applications, or extend further the core content further or any combination of these. With such choice, the specifictions are very varied; from AMSP, we can see a very useful summary of the structure and content of the various specifications.

For any change in specification, I have always found mapping documents helpful, for Further Mathematics where resources are perhaps less plentiful than for Mathematics, then a clear mapping to the legacy specifications can be useful for finding practice questions.

Mapping documents:
These are all very clear and include helpful notes.

  • OCR A scroll all the way down to Teacher Guides for a document mapping new content to the legacy specification
  • OCR B scroll down to Teacher Guides
  • Pearson – GCSE 2017 and GCSE 2008 content mapping. This spreadsheet maps the 2017 specification content for Maths (sheets 1 and 2) and Further Maths (sheets 3 and 4) to the old specification.

For AQA there is a useful summary of the subject content available and on Mohammed Ladak’s ‘MathedUp’ see his A Level Further Maths Takeaway, a wonderful source of exam questions by topic with mark schemes. AQA questions have been used here.

Looking for further questions on core topic such as Proof by Induction I know I can look at sample questions from all the Examination Boards; see Teaching Resources in the Further Maths series of pages.

To take a look at an optional example, earlier this year, teaching Dimensions (MEI) to the Further Mathematicians, having taught legacy M3 before, I did know where to find many questions, but for anyone new to the subject, quickly consulting the mapping document, we can see that the required legacy unit is indeed M3.
OCR B Dimensions
There are many questions available from the old M3 papers. Looking at the other examination boards, I can see that AQA has Dimensions on its first optional application.

AQA Dimensions

AQA Subject Content

So I know I can find Dimensions questions on AQA specimen and practice papers. I can also check MathedUp where we see questions and solutions on Dimensional Analysis.
MathedUp Dimensions Looking at Revise, a video on the subject is available which includes the AQA legacy reference. Challenge and Ans provide examination questions with very clear solutions.

MathedUp Dimensions example

MathedUp! Further Maths Takeaway

 

Into the Archives…

…For some Further Maths.

Looking for some revision examples for a revision session for my Further Mathematicians led me to some great questions in that treasure trove that is Edexcel’s Maths Emporium.

Edexcel Emporium Very Past Papers

Edexcel Emporium

Inside the Very Past Papers Cabinet, the A Level Papers, do indeed contain some very past papers! Exploring the resources I found a set of old AEB Pure Mathematics papers, very usefully with answers.
AEB papers with answers

 

 

These papers have several useful questions that could be used for Maths with some for Further Maths. Looking at the 1977 paper 1, I found a couple of useful questions or parts of questions, including the example above. Question 10  has a differential equation, Maclaurin series and a volume of revolution calculation – perfect for one of my revision session questions. Calculators were not allowed on this paper for the 1977 students.
tables
For 2019 students we can show them what the solid of revolution looks like on WolframAlpha!WolframAlpha volume of revolution
Returning to the top level of the Emporium, a trip down memory lane for me can be found inside the A Level cabinet. I began my teaching career with The University of London School Examinations Board for Maths and Further Maths. Past papers and Mark Schemes are available. There are some useful questions here for the current specifications.Further Paper 2 1988 qn paper
Further Paper 2 1988

Looking at Further Maths paper 2, I see some useful questions. Helpfully, inside the cabinets, the specifications are available. On Further Mathematics (372) I see Complex Numbers, Hyperbolic Functions and Differential Equations.
specifications

June 86
University of London June 1986 Mathematics 1

At the start of my teaching career, I really liked the first paper of the Mathematics A level from the University of London School Examinations Board – thirty multiple choice questions to complete in one hour, 15 minutes.

Multiple Choice Example

For questions 1 to 20, candidates had to select one answer from 5 and for questions 21-30 the instructions were as follows.
June 86 Mathematics 1

University of London June 1986 Mathematics 1

The pdf file here has the paper, followed by the exam board answers followed by notes from the 1986 version of me! These days I would illustrate with Desmos and/or WolframAlpha for example as well where appropriate.
Colleen Young answers

The first Further Maths paper was also a multiple choice paper in this style.
Note the comment from Graham Cummings below, there are further papers available in  Edexcel’s Emporium:

The Emporium has some 17 multiple-choice question papers from the period 1988-1992 – by no means a complete set, but they range across the Mathematics, Further Mathematics, Pure Mathematics and Applied Mathematics syllabuses. You can find them in the “Pre-C2000” cabinet within GCE AS/A Level.

Edexcel’s Emporium is a free website intended for the use of teachers of mathematics in secondary schools, wherever they might be and regardless of what awarding body they use; a valid centre email address and the school centre number are required.to register.