…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

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.

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.

For 2019 students we can show them what the solid of revolution looks like on WolframAlpha!
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.

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.

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.

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

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.

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.

Jonny Griffiths investigative activities for the pure A Level Mathematics classroom are well known (details are below). He has now published Further Risps, forty rich tasks for the pure Further Mathematics classroom.

I thoroughly enjoyed Jonny’s excellent session for London Maths, ATM and MA London Branch in May 2019, where teachers enjoyed working on these activities.

The pdf version of this resource is available free from www.furtherrisps.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! The tasks very much remind me of Colin Foster’s wonderful Mathematical Etudes.

For the A Level collection (all free, you can optionally purchase an e book version for £1) use RISPS (Rich Starting Points), these open-ended investigative activities are for the A Level Pure Mathematics classroom. Note the helpful index by topic, choose a personal favourite of mine, Risp 21 Advanced Arithmagons for example and you will see not only the task but also teachers notes.

Arithmagons provide a wonderful activity for making students of all ages think; I always tell my students that good mathematicians can go backwards!

(For a whole collection of Arithmagons activities see this post).
Also from Jonny Griffiths his companion sites on Statistics and Carom-Maths -activities to bridge the gap between A Level and University. You can find a resource collection from Jonny Griffiths on TES Resources.

Making Statistics Vital has some tasks which are excellent for the A Level specification, look at this task on World Wide Statistics for example which includes the task with answers and a spreadsheet with data for 191 countries.

…and on the subject of bargains – note the free/cheap Maths books available!

Subject content Mathematics: Differential Equations

Subject content Further Mathematics: Differential Equations

The study of Differential Equations is part of the prescribed core content of A Level further mathematics; this prescribed core comprising approximately 50% of each examination board’s content.

There are some very useful resources on this subject, particularly needed for further mathematics, where for many teachers they will be teaching content new on the specification.

OCR MEI Further Maths- Differential Equations Check in test

Teaching the MEI OCR specification, I do find the Delivery Guides and Check in tests useful, these provide both teaching guidance and example questions.

Dr Frost – Differential Equations

On Dr Frost’s excellent site you will find resources on Differential Equations, see Chapter 11, Integration for Mathematics and Chapters 7 and 8 on Further Mathematics.

We can use a little technology to check solutions to Differential Equations using WolframAlpha. The slides here illustrate the syntax for first and second order differential equations. Examples like this and more are available from WolframAlpha: Examples for Differential Equations.

Differential Equations on Slideshare.

On the subject of Technology, we can have a look at Edexcel’s helpful guide with links to numerous GeoGebra files: A Guide to using GeoGebra when teaching AS and A Level Mathematics.which includes Solving Differential Equations.

Pearson Secondary Maths – Solving Differential Equations

Notes and examples on Differential Equations.

• CIMT Section 18.5 on First Order Differential Equations
• Units 15.1 to 15.10 from AJ Hobson’s Just the Maths (individual pdf files hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’) are all on Differential Equations.
• mathcentre: Differential Equations Resources
  mathcentre: First Order Differential Equations
  mathcentre Second Order Differential Equations
•  A series of notes from the Helm Project. 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. 
  
  
  

  Helm Notes – First order ODEs

   

Very much a work in progress, but I have created an Excel version of the A Level Mathematics (age 16-18) subject content. This means it is easy to add notes and/or links to resources.

I have made a start by adding some links to ClassWhizz Calculator instructions and videos. Links have also been given to other resources.

I will be adding to this in the coming weeks and also over the next year, teaching the A Level content of the new specifications for the first time. More technology resources will be added by topic. Scroll down this page, Maths & Further Maths Technology by Content passed the first image.

AQA’s ‘A Level Maths The thinking behind great assessment (no login required for download) provides much food for thought whichever exam board you are using. See page 17 for example on what AQA expect students to be able to do with a calculator. Are your students aware of what they should be using a calculator for?

Mike Hadden – Excel Files