One of my students told me recently about MadAsMathsby Dr Trifon Madas. She likes the Practice Papers, particularly the way the papers are rated according to their difficulty, see C1for example. All papers come with full solutions.

Not only do all the questions come with full solutions but most have very clear mark schemes too.

MadAsMaths mark scheme example

The papers cover the Pure Mathematics content of the UK A Level course. Note the Special Papers designed for extremely able students; ideal for students capable of the top grades. This is a really valuable collection of questions.

There are further questions and solutions available in the large collection of booklets, originally samples they are free to use. These are pdf files, if you zoom you will see a clear solution as illustrated in the image here. Some of these booklets are aimed at undergraduate students.

From the University of Cambridge comes Underground Mathematics which started in 2012 as the Cambridge Mathematics Education Project (CMEP). The site provides a library of rich resources for age 16+ students with the aim of “Enabling all students to explore the connections that underpin mathematics”. Underground Mathematics is being developed by the University of Cambridge, funded by a grant from the UK Department for Education. The resources are free for all users.

Note that you can also select individual elements on the map. Try Quadratics for example and check the station guide for information. Looking at the guide led me to Name that graph; as with all the resources on the site more than just the problem is provided, we also have printable resources, solutions and teachers notes.

UCLES A level Mathematics A 1, QP 840/1, 1975, Q2a

I thought I’d have a look at Review Questions on Algebra– so much to choose from! How about an A Level question from 1975?!As you can see, not only do we have the question, but a suggestion, then a solution. And if that’s not enough let’s take it further and generalise, exploring on GeoGebra as we do!

I’m tempted to try that one with my very able Year 10 set studying for the new Mathematics GCSE qualification. In fact I suspect we can find other useful questions here for our most able GCSE students.

Prepare to lose yourself for some hours / days in this treasure trove! This resource is frankly incredible and I applaud the team behind this. The site I know will be on my own go to list of places where I know I will always find high quality resources to make my students think and more than that has suggestions for exploring further on a given area and of course explore connected areas.

I predict rich and happy travels around these lines for students and teachers!
See the Underground Maths series of pages for further information.

World Book Day last Thursday inspired me to update the page on free books page; working further on that page has inspired me to create a further page.

Nelson Thornes GCSE Texts

STEM learning has an extensive library of free resources for Mathematics (and also for Computing, Design and Technology and Science) including textbooks; you can search the collection in various ways, a search for textbooks for ages 11-18 returns these results.You will find a real treasure trove in this collection hence this new page. which can be found on the Reading Menu. Do have a look at all the goodies available, although some of the books are old many have chapters on topics such as Venn Diagrams; GCSE books developed for Intermediate students could be rather useful as the new Foundation course looks more like the old Intermediate level!

Several books are available – answers included. Perhaps these will be handy when we get the new A Level specifications!

Checking some of the Nuffield National Curriculum books proved a distraction as the following activity in one of the texts (Number and Algebra) reminded me that I wanted to revisit iteration for a revision session with my Year 10 GCSE students.

. I thought these questions would provide a way to revise several topics – simultaneous equations, solving equations graphically and iteration. Graphical solution to equations is something that seems to puzzle students and it does not come naturally to them that you can solve an equation by rearrangement.

I decided we will look at question IV, first we can use algebra to form a cubic equation in x, then solve the cubic by trial and improvement – a familiar method that seems more intuitive to students than rearrangement.

Then we can rearrange the equation – plot the graphs and show that we still get the same result. The next step is to impress them with using the rearrangement in the form of an iterative techniques question – it’s a lot quicker than trial and improvement. This is a particularly able set of students so I have gone well beyond what we need here; I checked the rearrangement on Autograph hence some of the additional slides in the file I have included below in case it is useful to anyone.

I thought I would include this AQA practice question for the new GCSE as when we looked at iteration earlier this year the subscripts really bothered students; my colleague said exactly the same of his set.

Returning to a Nuffield text I mentioned earlier, I found this question in one of the exercises (an isolated question, I couldn’t see any more). Now what’s the probability of that?!