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.
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.

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 register.

Mathematical Miscellany #29

With Exam season already started, an exam themed miscellany this week…

Command Words GCSE EdexcelFrom Edexcel – I do like this recently added Teacher’s Guide to Command Words, not only do we have commentary on what is expected from students when a particular commonly used command word is used but examples of questions to exemplify the use of the word.

All the examination boards have very clear resources illustrating command words, something we can incorporate into our teaching.

On Naikermaths you will find several new A Level Mathematics Practice Papers. These papers are based on legacy Edexcel Exam Papers, but they have been edited to reflect the new A Level Specifications and include new questions to make them suitable for the new specification. Full solutions are provided.

What a treasure trove Edexcel’s Emporium is, log in and look for example at the themed practice papers for GCSE. This includes one mark questions and mark schemes (scroll down the alphabetical list), a good idea for any final revision sessions. Talking of one mark questions has given me an idea for Maths and Further Maths A Level revision sessions;  the very old Multiple choice papers I began my career with could be a very useful source of questions. Scroll down this post for an example.
Multiple Choice Example

Note the comment from Graham Cummings below, papers are 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.

A recent post from The Calculator Guide blog has very useful advice from AQA on the use of calculators in A Level Maths. I have recommended the Calculator Guide before and have shown many of the very helpful videos in both Maths and Further Maths classes. Check the resources here. Certainly, calculator use at A level needs to be taught explicitly. Note section 3 on Calculator Use from AQA’s excellent A-Level Maths The thinking behind great assessment.
Note my page here on Classwiz tutorials.

Crashmaths AS Countdown
With common content for all the examination boards at A Level, we have many resources for our students. On Crashmaths, see the AS and A Level Countdowns. The first AS exam was 15th May – all 10 worksheets are available.

CrashMaths Countdown

crashMATHS – A Level Countdown

The A Level countdown leads up to the first A Level Maths Exam on 5th June 2019 with the first worksheet published on 26th May. Each question sheet has 5  pure questions and 2 applied, one Mechanics and one Statistics; I like the way the pure section includes a mix of basic, problem and modeling questions.

From crashMATHS I have successfully used all the AS key skills checks this year with Year 13 and will do so again; they make great lesson starters. I see a section on End of Unit Tests under development, with a good set of Proof questions available now; certainly a space to watch.

Maths on Toast

Maths on Toast Bites

Maths on Toast is a family charity whose mission is to make maths a creative, enjoyable, human, social activity for families and communities. Maths on Toast want everyone to feel positive about maths – to feel that it is something they can do, and enjoy.

The charity holds regular library sessions and works with community groups. Now, to spread the word further, we have the launch of Maths on Toast Bites. I was delighted to attend the launch event, also attended by some of the children in the films. It is always so good to see children’s enjoyment in Maths.

The five films are now available on the website, use the dropdown menu and you can see from the children in the films how to make a hexaflexagon, construct a sturdy structure, make Platonic solids, step through a playing card and how to enjoy fun family maths.
Maths on Toast Bites Films

Maths on Toast ‘Bites’ …
Showing how maths is creative, fun and for everyone.

Maths on Toast – the family maths charity – today, May 16th, launches ‘Bites’, a series of short ‘How To’ creative maths videos presented by children aged between 8 and 11. Aimed at primary school children and their families, our group of enthusiastic ‘maths advocates’ demonstrates a range of fun, hands-on maths activities that families can try together.

The 5 short films, along with accompanying activity instructions, are available on Maths on Toast’s website.

Maths on Toast’s mission is to promote positive attitudes to maths. Bites will broaden children and their families’ view of what and where maths is, and show that maths can be creative family fun.

An instruction sheet for the activities demonstrated in each film is available to
download. Families can try them together at home – they could also be used in school or
community settings. Trying them, enjoying them, and being aware that this is maths,
will change the whole family’s perception of maths; learning together through play and
discovery will increase confidence and create positive memories of maths.

Lucy Davis, CEO, Maths on Toast
“Maths on Toast Bites illustrate perfectly that ‘maths is everywhere and for everyone’.
The films show fun, creative, hands-on activities that we hope will be shared far and
wide to inspire children and their families to look at maths in a different light and
explore the many possibilities that maths has to offer. The enjoyment of maths in its
many forms provide the first step to creating a positive attitude to maths that can help
counter maths anxiety and a lack of confidence in maths.”

Rashid Iqbal, CEO, The Winch
“Maths on Toast is showing that children love to learn when learning is child-centred,
and that we can nurture an intrinsic joy to mastering maths. I am always delighted to
see projects in action that assert the value of informal education and champion the
principles of community-driven pedagogical practices.”

Sebastian Thiel, Founder, Upshot Entertainment
“Working with Maths on Toast and the young people was beyond amazing! All of the
young people were fully engaged and found every maths game and activity extremely
fun. As someone who hated maths growing up, I was astonished to see how
fun maths can be, especially for young people.”

Mathematics & Further Mathematics – Risps

Jonny Griffiths Further RISPS

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.

Two editions are available – a hard copy or pdf version. The pdf version is now available free from 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.

Alan Hudson example

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.

Making Statistics Vital

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

Mathematical Miscellany #28

MEI Calculator Crunch

MEI – Calculator Crunch

This coming June MEI’ summer challenge, the Calculator Crunch will be helping Year 6s get ‘calculator-ready’ for secondary school, and additionally provide extra practice for Year 7s.

Note the dates – from 10 to 20 June, on each school day, MEIMaths will tweet a question a day for students to work on. PDFs of the questions will also be available from MEI’s page for display on whiteboards and to print as hand-outs. MEI has designed the problems not only to practise using a basic or scientific calculator but to develop mathematical thinking skills. Lesson plans will also be provided.

On the subject of MEI and Primary Maths, don’t forget their latest wonderful Sumaze! Primary app, one of a series of three apps in the series. All highly recommended, these problem-solving games cover a wide range of ages. Sumaze! which explores arithmetic, inequalities, the modulus function, logarithms, powers and primes is a problem-solving puzzle game suitable for A level mathematicians and above. Its sequel, Sumaze! 2, which explores fractions, decimals, percentages, primes and digits is suitable for those studying GCSE Mathematics and above.

For some more practise with calculators, perhaps try this calculator exercise from ADA Maths. Students can check that they can key in all the calculations correctly by answering the questions and checking their answers. ADA Maths is a free Mathematics Question Bank with no login required. The creator is Sam Powell, a mathematics teacher from the UK, working in Australia. All questions have answers, some have a fuller solution and a video explanation. Several categories of questions are available; look at Algebra for example and we see some nice use of colour in presenting a solution to a problem expanding double brackets (select Show Solutions) These examples could be helpful for students when revising.
ADA Project Algebra

Desmos AQA LDS Cars
In February of this year, I was very pleased to see that we can use Desmos for Statistics. Tom Bennison has written a large data set (AQA-Cars) resource using the Desmos statistics visualisation functions. It really is very easy to copy and paste into Desmos and display boxplots. Tom’s resource includes questions for students. It strikes me we should use the large data sets with younger year groups, using Desmos (or GeoGebra) makes it very easy to investigate such data when we are teaching boxplots and other Statistics topics.

Tom Bennison A Level Question a day

Tom Bennison – Year 13 Exam Question a Day

Tom has been busy, see also his Year 13 Exam Question a Day with questions up to June 13th. Tom has used Sample Assessment Materials from the awarding bodies’ websites so students can find mark schemes using the reference for any question. This will be a very useful resource for revision now and also in future. I do like the way we can easily see the topic for each question.

TeachItMaths Proof review

TeachitMaths Proof review sheet

Following my post on Proof resources, looking further through TeachIt Maths proof resources, I see many gems. Rather good for year 11 Higher GCSE students is this Higher Level Proof review sheet.

Maths Week London
In June we have for London teachers, Maths Week London. You can still register for this event, there are a limited number of welcome packs remaining.

Coming in November 2019, we have Maths Week England.