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.

Coordinate Geometry – Underground Maths

geometry-of-equationsExploring Algebra Review Questions from Underground Mathematics I came across some Coordinate Geometry questions I really like and yesterday spending a day with the very talented writing team and my fellow Underground Mathematics Champions we explored Straight Line Pairs, a question with much scope for exploration and possible methods of solution.

The image above straight-line-pairs-1has been created from the Printable/supporting materials.

My Year 11s will be looking at Coordinate Geometry this week and I have some other questions I would like them to try. It is possible to create pdf files for a collection of questions, see Saving Favourite Resources, one of Underground Mathematics’ How To Videos. (See the tutorials page I have in the Underground Maths series of pages – a work in progress).


You will find a whole collection of such questions if you look at Geometry of Equations. This includes many resources including Review questions. Note the Building Blocks resources. I think I’ll be using Underground Mathematics resources with ever younger students – Year 9 can try Lots of Lines! You will see from the the supporting materials that this has come from the brilliant Standards Unit (A10) collection. Students must sort the lines into six pairs, each pair matching one of the given descriptions.
Staying with the Building Blocks I do like Straight Lines where students must decide which of 17 equations are equations of a straight line.
Look at the list – a wonderful lesson in not jumping to conclusions here! Both my Year 9 and my Year 11 are going to be trying these this week!


Straight Lines reminded me of Line Pairs, I feel an extension for Year 11 coming on!


The Standards Unit – Mathematics

This is an outstanding resource – many excellent activities here for the secondary classroom.

This page has a permanent home under Resources  and has been recently updated.
There are flipcharts available for each lesson and note the excellent Professional Development materials.

Start by reading Improving Learning in Mathematics – Malcolm SwanThe following links give you access to the resources, also some IWB resources.

Home pageThe resources are hosted by Nottingham University, including all the pdf files very clearly indexed. Note that this sites includes the complete set of resources including the software; (though some software is too old to work!) Note that Tom Button has rewritten the Traffic Simulations in GeoGebra for A5: Interpreting Distance-Time Graphs with a Computer. The Professional Development resources are excellent. Look at PD1 ‘Getting Started’ for example, note PD1.4 on page 9 – 8 Principles for Effective Teaching.


All materials – hosted by National STEM Centre. Including:
Mostly Algebra   Mostly Number  Mostly Statistics   Mostly Shape and Space   Mostly Calculus   Others   

…or Craig Braton has the pdf files on mrbartonmaths

IWB Materials This website has come about as a result of the NCETM research project: Enabling enhanced mathematics teaching with some interactive whiteboards (September 2006- September 2008) and is supported by the IWB research team at Keele University and s supported by the IWB research team at Keele University and the Spire Maths team.

PowerPoint Files

MMT has written a set of PowerPoint files for almost all of these excellent resources, he has made them – in his own words ‘classroom ready’. MMT has provided the resources in an editable format so that they can be tweaked to your own requirements. Thank you MMT! These files are available directly from Dropbox; note they are also hosted on Craig Barton’s site.

Standards Unit MMT Files

Standards Unit A1:Expressions – MMT PowerPoint

Review Questions – Underground Mathematics

Underground Mathematics - QuadraticsI wrote earlier on the wonderful resources on Underground Mathematics from the University of Cambridge. Thinking about the new A level specifications  I believe this site will provide us with rich resources for these new specifications.

Each section includes Review Questions, look at Thinking about Algebra for example; scroll down the different resource types for the Review questions for this station.

Alternatively you can browse all the Review questions.

Review Questions - types
An excellent feature of Underground Mathematics is the excellent search facility; we could look at the Review questions by type. One can also search by line ( Number, Geometry, Algebra, Functions or Calcuus) and by Station.
Review Questions - search

See the example question below, for each review question you will find the question, a suggestion, the solution and sometimes suggestions for taking it further with for example GeoGebra resources.

Note the star by the title – if you choose to log on to the site (you don’t have to but it’s a very good idea!) you can save any favourite resources to your collection.

Review Question - Gemetry

I can never resist a quick illustration on Desmos! I think I’ll start an Underground Maths Desmos collection! Note the use of the hyperlink on Desmos to link to the question.
Select the image for the Desmos page.
Desmos illustration

I think we have a wonderful supply of excellent questions here to challenge our students and help them see connections between the various areas of the subject. These are ideal to use with A Level students; some are also useful for higher level GCSE students aiming at those top grades or Level 2 Further Maths students. Any student who wants to study Mathematics at university should certainly be using this site.

Happy Numbers and more…

Happy Numbers ResourcesContinuing my theme from last week I have added a new page to the Investigations series – one of my all time favourites – Happy Numbers.

I have also added further resources to the Investigations – Links and Resources page. From the National STEM Centre which is an incredible source of resources for STEM subjects we have

Investigations L Mottershead - National STEM CentreLorraine Mottershead’s book: Investigations a complete book by including solutions to the puzzles and investigations.

STEM Centre Investigations Kirby & PatillaAlso on the NMoational STEM Centre from David Kirby and Peter Patilla we have Maths Investigations – Pupils’ Workshop together with a comprehensive teachers’ guide.