Mathematical Miscellany #18

crashMATHS has some very useful free resources for GCSE and A Level; the site is under development but you will find plenty of useful resources already. Checking for some additional resources on Completing the square for my Year 10 students I came across a GCSE worksheet from crashMaths; this has a good variety of questions including questions to stretch your students aiming at the highest grades. The worksheet solutions are noted as coming soon. For a topic such as completing the square, this is an ideal time to use technology to check our work. Any of Desmos, GeoGebra and WolframAlpha could be used. (Select for Desmos page.)
Completing the Square

crashMATHS AQA styleStaying with crashMATHS, note the A Level Practice Papers and mark schemes, this looks like a very valuable resource. There are several papers and mark schemes available with more coming. Versions are provided for AQA and Edexcel, these use the style of papers we see from the exam boards. Currently, there is nothing for OCR. For Mathematics the content is the same for all the boards and for Further Mathematics we have a prescribed core which must comprise approximately 50% of its content. This common content as we have with GCSE is very useful indeed as we can use resources from all the examination boards.

Continuing with an A Level theme we have a very interesting read, published 25th January 2018, “An evaluation of the item difficulty in AS and A level maths“. This compares the difficulty of items in sample AS and A level maths assessment materials from 2016 and 2017 with the A Level papers from 2015. The overall objective of the exercise was to compare the profile of item difficulty within the SAMs with that of the corresponding 2015 assessments, a question I believe is on the minds of Maths teachers!

SAMS & Legacy A Level comparison

Ofqual: Expected item difficulty Legacy & Reformed Specs



Clearly, we can look at the specimen materials ourselves and make our own judgement on the difficulty but this seems a robust study which used Comparative Judgement. This is a technique where each reviewer reviews many pairs of items and decides each time which item is more difficult to answer.

Items from the sample assessment materials submitted for 4 specifications,  AQA, MEI, OCR and Pearson were used.

The study shows slightly higher levels of expected difficulty for items from the sample assessments relative to the 2015 assessments but the increase in difficulty is small.  The paper states that ‘Such small differences can easily be accommodated by the setting of grade boundaries at awarding. The choice of specifications to teach should be based more on content and style as there is little appreciable difference in difficulty.’

For further reading on Comparative Judgement, look at the work of Daisy Christodoulou.


If you have not yet signed up for the new home of Edexcel’s Maths Emporium then do so! This is such a valuable resource. Latest additions include some great new GCSE maths practice papers. There is a wonderful set of practice papers by topic. Look first at GCSE Mathematics, then choose Cabinet 11 for the current specification. Under Practice Papers you will find the themed set – brilliant!
Emporium Resource

NCETM Secondary Assessment MaterialsTo finish this collection, from NCETM look at their Secondary Assessment materials which have been written to support teachers in making judgements on the degree to which pupils have mastered various components of the KS3 mathematics curriculum. This follows the primary Mastery Materials, which was published in 2015.

See further posts in the Mathematical Miscellany Series.

Searching for questions – part 3

This post is the third in a series of posts looking at sites where you can easily find resources by topic. For just the links in this series of posts, see this page which provides a useful list of such sites.
This is an ongoing project.

Don Steward

Don Steward has an extensive collection of activities to really make your students think and learn. Note the helpful topics menu on the right-hand side of the page. The example shown above can be found under Algebra Misconceptions.

With changes to examinations at both GCSE (UK examination taken at age 16) and A Level (UK examination taken at age 18), all the examination boards are offering help and support for teachers and students. Many excellent resources are available.


Edexcel Resources

Edexcel Teaching Resources

For example, we have Edexcel’s very helpful resources for teaching new content. For each topic, information, examples and exercises (with answers) are given.

Edexcel Resources New Content

Edexcel Resources

These resources are very helpful with plenty of examples and exercises, a very thorough treatment of each topic.

Edexcel Resources New Content example

Edexcel – Interpreting gradients of straight lines

Edexcel menuNote the menu on the left; you can easily navigate these resources choosing the Content type. Looking at Classroom tests, for example, you will find a complete collection of tests with answers by topic for both Foundation and Higher tiers.


Similarly, from AQA we have their very helpful Bridging the Gap resources. These resources by topic are also useful for ideas to teach younger students. Further outstanding resources from AQA including the wonderful Problem Solving resource are detailed on this AQA GCSE page.

Iterative techniques 1

OCR too provide us with many excellent teaching resources by topic.
OCR Resources

OCR GCSE 9-1 Check in tests

The Check in tests are excellent. Consider the language of functions for example, the first questions could be used with students as young as Year 7. Full details of the Check in tests can be found in the Teachers’ Guide. Each test is of a similar format in that Questions 1-5 cover procedural calculations (AO1), questions 6-8 require the ability to reason and communicate mathematically (AO2) and questions 9-10 relate to problem solving tasks (AO3). There is also an extension task. Very usefully (thank you OCR) the Check in tests are also available in Word Format.


Searching for questions – part 2

Searching for questions – part 1

This post is the second in a series of posts looking at sites where you can easily find resources by topic. For just the links in this series of posts, see this page which provides a useful list of such sites.


Nrich has extensive resources including problems, interactives and games. Full teachers’ notes are provided. The Resources for Teachers page has several useful links; note the curriculum mapping documents which suggest resources relating to the National Curriculum. There is a search by topic facility to find all the resources for a particular topic.

Nrich resources

Nrich Secondary

CIMT– The Centre for Innovation in Mathematics Teaching, from the University of Plymouth, an astonishing resource has free curriculum materials for all school-age students. For example, if we look at the Year 7 Course Material page as well as all the book chapters with their numerous worked examples and exercises there are also teachers’ resources including notes, slides, revision tests, activities and mental tests on each topic. The activities are varied and include investigative work.

CIMT Mental Test

CIMT Mental Test – Logic

M4THScomFrom Steve Blades, see for age 14-18. This site has an extensive library of resources. A great feature of these resources is very efficient use of space so cutting copying costs. Keep scrolling down the page – there are a lot of resources!

Exercises & solutions on David Smith's 'The Maths Teacher'

Exercises & solutions on David Smith’s ‘The Maths Teacher’

David Smith’s The Maths Teacher site has a great collection of videos for both GCSE (age 14-16) and A Level (16-18). Transcripts are available for each lesson, also exercises with worked solutions. Many of the GCSE videos would also be useful for younger students. There are also videos and written solutions to some Legacy A Level papers.

Teachit Maths KS3 Percentages code breaker

Teachit Maths KS3 Percentages code breaker

Teachit Maths though a subscription site offers its entire collection of activities as pdfs free. I have found many high quality resources here for all ages. Look at this activity on quadratic functions for example – this should really help understanding.

From Transum, well known for its Maths Starter of the Day, note the complete index of starters including the topic of the starter. The Transum site is easy to navigate, there is a clear Topic Index for Teachers also, for students a Maths Map with numerous activities to support their learning. The example shown here is one of the Algebra Activities found on this page. See this post for further examlples of some excellent resources including Venn Diagrams and Systematic Listing Strategies.


Transum – Algebra

 Transum Maths Map

The list of sites easily searchable for resources by topic can be found on this page.

Searching for questions – part 1

I thought it would be useful to put together a simple list of sites where you can easily find resources by topic.

For just the links in this series of posts, see this page which provides a useful list of great sites to search for resources by topic.

I want some questions on Sequences for Year 10 so I’ll use Sequences as an example.

Sequences Index Diagnostic QuestionsOn Diagnostic Questions, we can search all the Maths Questions. Under Algebra we will find Sequences. The questions are helpfully categorised as shown. For the links here to work, sign in to Diagnostic Questions

We can also use the excellent Quiz collections, do explore the variety available for all ages. The collections from the exam boards are so useful, for AQA, OCR and Pearson Edexcel, Higher and Foundation quizzes for Years 9, 10 and 11 are available. The UKMT collection includes themed topics at Junior, Intermediate and Senior level.

It is also possible to search all the Maths quizzes which have been created. As before, under Algebra we can find the Sequences quizzes.

Sequence Quizzes Diagnostic Questions

Sequence Quizzes – Diagnostic Questions

On Craig Barton’s site, you can use his Maths Topics Index page.
Sequences mrbarton topicsSelecting Sequences for example gives a further choice, if we choose Linear Sequences, we find a treasure trove of sequences resources all conveniently in one place!
mrbarton Topic Index

DrFrostOn we can browse all resources which include complete lessons as well as homework questions and assessments. Looking at Sequences we see resources for Year 7 through to GCSE/IGCSE Further Maths. Note also Dr Frost’s homework site.

gcse-inequality-example J HallFrom Jonathan Hall: mathsbot is superb. The home page provides a clear menu and the site is easy to navigate. Many of these resources allow you to generate questions by topic. See for example Differentiated questions and GCSE exam style questions.

The list of sites easily searchable for resources by topic can be found on this page and this will be added to in the next few days.

To be continued…