2406 Problems

2406 problems – that’s 1000 from 1000 Problems to Enjoy and 1406 problems from the Nrich site, the results of a search on the World Mathematics Championships.

The 1000 Problems site is an excellent collection of resources organised into collections on Number, Algebra, Shape and Space and Statistics. Within each category problems are organised by age and key words and a clear description are given for each problem. Extension questions are available for each category. For each problem a file can be downloaded which includes solutions.

1000 Problems Number Extension

1000 Problems – Number, Extension

The other 1406 Problems are the results of a search on The World Mathematics Championships on Nrich. A large collection of problems can be found on Nrich and many more sample problems from selected rounds provide a rich source of problems.

WMC Junior Questions
Looking at the Mathematician sample problems for example we see a collection of  mostly Junior (Junior level is aimed for those in Middle School/lower High School (in Grade 9 and below)) resources including mixed question sets and also some by topic.

 

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

 

Advanced Starters – Transum

Transum Advanced Starters

Chancing upon this tweet from Transum proved fortunate. Hidden gems indeed, I had not noticed the Advanced Starters before, some of which I think could be useful for students aiming at the highest GCSE grades as well as for Advanced Level students.
The problem, Find the Radius, illustrated in the tweet is very neat!

Looking at the Main Transum Starters page I see at the foot of the page we have various categories of starters including the Advanced Starters.
Transum Starters
I recommend exploring!


I wrote the above in February of last year and I see from Transum’s latest newsletter that The Advanced Starters collection has grown. I see many starters here I like, thinking about next year it’s a good time to check these. With linear courses, starters provide an ideal opportunity for review.

Looking at Coordinate Distance, I can never resist a Desmos page to illustrate a problem! This starter could be also be used to review some coordinate geometry – find the midpoint? Find the equation of the line?

Coordinate Distance - Transum Advanced Starter

Transum – Advanced Starters

Quad Midpoints - Advanced Starter

Transum – Advanced Starters

For more on Transum see this post which includes some great Venn Diagrams resources and a reminder that there is a very clear index for teachers and another for students. The extensive library of activities on Transum are all free to use.  Activities such as this Algebra resource on Completeing the Square can be checked as part of the free offering.
Completing the Square

A subscription offers teachers answers to for example the Advanced Starters and Exam Questions.

 

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.

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.


Naikermaths
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.