Transition Time

As we come to the end of an academic year and look to a new one, it will be a time of change for many students. Perhaps you have completed GCSEs or equivalent qualifications (UK age 15-16) and are about to start on your A Levels or perhaps you have completed those and are about to start studying Mathematics at university.

To be in a position to begin your new courses well you should be thoroughly familiar with the essentials of the work you have studied to date. At whichever level you are studying your Algebra should be at a standard where you can manipulate expressions with ease.

Some resources to help you prepare and will be useful reference material for you during your course…


OCR Transition Algebra Fractions

OCR – Bridging the Gap Student Guide

For students going on to A Level then the best thing you could do is use OCR’s brilliant guide for students Bridging the gap between GCSE and AS/A Level Mathematics – A Student Guide. With sections on Algebra, Trigonometry and graphs including examples, question practice on key topics and suggested reading before starting the A Level this will be so valuble for students.

You may also find these GCSE revision resources useful.The takeaways are really useful and Mohammed Ladak has picked out Transition Takeaways specifically chosen to help with A Level Maths preparation.

You could also look at Step Up to A Level Maths from The Centre of Innovation in Mathematics Teaching which helpfully lists skills you should be confident with and provides resources to support your study of these skills.

As you study your A level (16-18) course you may find some of the material in the section below useful.

For many challenging questions to really get you thinking, try the brilliant Underground Mathematics site.

Make sure you have some useful apps on your phone if you don’t have them already. Mathscard app from Loughborough university is free and a handy reference guide of mathematical facts and formulae. Every student should have the Desmos app (free) and you could also get the  WolframAlpha app (low cost).

Sign up to Brilliant and follow them on Facebook so Maths problems appear in your stream and hopefully distract you from trivia!

If you are preparing for university, then make sure your A Level knowledge is secure – perhaps check the Algebra Refresher from The Mathcentre which has many questions and the answers are at the end of the document. The The Mathcentre has an extensive collection of helpful resources for students of Mathematics.

Calculus workbook from Plymouth University

Carom MathsFor a collection of forty mathematics activities bridging between A Level and University, try Carom Maths from Jonny Griffiths.

Check the List of Activities, how much do you know about Inequalities for example? For a complete PowerPoint with information and questions on Inequalities, choose Carom 1-2: Inequalities.

For older students AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’is very useful as is the excellent Math Centre site which includes extensive resources. The quick reference leaflets which are available on numerous topics are very clearly written and succinct, see these for example on the Product Rule and the Quotient Rule. There are also teach yourself booklets, revision booklets, videos and diagnostic tests. See also these very clear notes with exercises from Plymouth University. There are many free courses available from The Open University and MIT .

The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles are ‘not ready yet’; for the sake of completeness the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

If you are studying or about to study at university then have a look at Kevin Houston’s ‘How not to get a good mathematics degree‘ and ‘How to get a good mathematics degree‘. He also has provided a pdf file you can download: 10 Ways to Think Like a Mathematician. Kevin Houston works at the University of Leeds in the UK.

From Professor Stephen Chew of Samford university, this series of 5 videos looks at how to get the most out of studying (any subject, not just maths). Part 1 includes ‘beliefs that make you stupid’!
How to get the most out of studying (Part 1),  Part 2,  Part 3 Part 4Part 5

The Open University has several helpful publications for students of Mathematics. Many of these resources would be helpful for students still at school.

From John Kerl – see these excellent tips for mathematical handwriting.

For older students Peter Alfeld wrote this guide on Understanding Mathematics for his students at Utah University.

And finally – check the 11 Commandments of Mathematics!

Wishing Mathematics students everywhere – whatever stage you are at a very successful year.


Another post to clarify the requirements for the UK GCSE Mathsematics specification. The constructions required are as follows:

Use the standard ruler and compass constructions

Math Open Ref Instructions
Note that all these Math Open Reference demonstrations are available as a printable atep by step instruction sheet, see for example the guide to bisecting an angle.

The AQA specification also includes a note that Constructing a 60° angle is required. This is shown here by constructing an equlateral triabgle.
​Students will be asked to use these constructions  to construct given figures and solve loci problems. Students should also know that the perpendicular distance from a point to a line is the shortest distance to the line.
The constructions shown here are all from the excellent Math Open Reference by John Page. (There are many more constructions given on the site which are not a GCSE requirement). Another very useful source for demonstrations comes from BBC Bitesize on Loci and Constructions which gives step by step diagrams and instructions.BBC Bitesize Constructions
The BBC site also has clear examples of solving problems using constructions.



Transformations of Graphs

Translations Graphs

In this post we will look at reflections and translations (the requirement for the UK GCSE syllabus). For GCSE students are required to sketch translations and reflections of a given function.

Experiment with the following Desmos pages to examine these transformations, f(x) + a, f(x + b), -f(x) and f(-x) where a and b are integers.


Reflections - Graphs


Equations of Transformed Graphs

Note that you can work out the equation of the transformed graphs.
Suppose we wish to work out the equation of  f(x) = x2−5x+4 after reflection in the x axis.
To reflect in the x axis the transformation is –f(x)
So f(x) becomes  –f(x) and we have  – (x2−5x+4) = – x2 +5x-4

To work out the equation of  f(x) = x2−5x+4 after reflection in the y axis.
To reflect in the y axis the transformation is f(–x)
So f(x) becomes  f(–x) so we replace x by –x
and we have (–x)2−5(–x)+4 =  x2 +5x+4

Suppose we wish to work out the equation of  f(x) = x2−3x+2 after a translation of 3 units to the left parallel to the x axis as shown in the first image in this post.
f(x) = x2−3x+2, we need f(x+3) for the equation of the transformed graph
f(x+3) = (x+3)2−3(x+3)+2 = x2+6x+9−3x−9+2 = x2+3x+2

Diagnostic Questions

GCSE Diagnostic Qns
To use the links here you will need to be logged in to the brilliant Diagnostic Questions site. Students, just in case your school does not use Diagnostic Questions you can create your own free account here. On Craig Barton’s Diagnostic Questions site you can find collections of GCSE 2017 examination questions from AQA, OCR and Pearson Edexcel (scroll down each of the pages linked to for numerous quizzes on different topics on the GCSE syllabus).

Craig Barton who is the creator of the excellent Diagnostic Questions site has compiled a collection of some of the worst answered new specification GCSE questions on Diagnostic Questions. See also his Question of the Week selection.

Each set consists of 10 questions chosen from all 3 awarding bodies. As well as the online version of each quiz, the design team have also created a booklet in case you wish to print it out.

Badly Answered GCSE Questions – Foundation 1
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 1
— Try the questions online or download the booklet

Foundation Badly Answered Questions – Set 2
Online       Paper

Higher Badly Answered Questions – Set 2
Online      Paper 

Badly Answered GCSE Questions – Foundation 3
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 3
— Try the questions online or download the booklet

You can also learn on the move with the free Diagnsotic Questions app.
GCSE 2017

Note that Advanced Level Questions are also available.

University Mathematics Preparation

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. For students studying for STEP papers try this excellent portal from which has (all free) access to STEP questions and solutions. Create an account, login and you have access to a complete library of resources.

The resources are very clearly presented. For each question you have access to a pdf with the question, Examiners’ Report and both an Official and thanks to Peter Mitchell a fully worked handwritten solution.

Note that you can also download a copy of  Stephen Siklos’ Advanced Problems in Mathematics and Core MathematicsAdvanced Problems in Mathematics is excellent preparation for ANY undergraduate Mathematics course.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 43 problems. Each problem has been given a title and a rough indication of the mathematical content which means you can pick out questions by topic.

Stephen Siklos Advanced Problems in Mathematics and Core Mathematics

See also, from Cambridge University, their STEP Support Programme.

And from Nrich, Prepare for University.


demos-pagesA project – note the new tab Demos – a place for some favourite demonstrations / simulations. Currently just two pages but note the numerous PhET simulations for a variety of subjects to explore.

The most recent addition is the PhET Projectiles Simulation.



Use this excellent PhET simulation to explore the path of a projectile. Try changing the angle to investigate the relationship between the angle of projection and the horizontal distance (range) travelled.

Have a look at these problems on Underground Maths to extend your thinking:
Where did it land? and Maximum Angle Throw. Note the questions given on these problems and things you might have noticed.
projectilesFor amusement try projectiles of different types!

You will find some very useful notes on Mechanics on Mr Barton’s website.

There are numerous PhET simulations covering Physics, Biology, Chemistry, Earth Science and Mathematics. Note the growing collection of HTML5 versions which will work across all platforms and devices. The Projectiles simulation here is currently a Flash resource.

You can download an app for iOS also for Android.
PhET Balancing Act working nicely on my phone!

Mathematics Challenges

UK Maths Challenges
You can practice for the UK Maths Challenges with these past papers. Questions and full solutions are provided. You can find British Mathematical Olympiad papers here.

You could generate a random quiz, using Mathster’s UKMT Mathematics Challenge Online Quiz. Choose Junior, Intermediate or Senior and one of three difficulty levels; you can also choose the number of questions, a time limit and the order the questions are presented in – random or in order of difficulty.

Note Geoff Smith’s advice for young mathematicans and as he says you will find thousands of questions are available at all levels on the Art of Problem Solving site.

Nrich have a series of short problems based on the UK Junior and Intermediate Challenges.Nrich Short Problems

Nrich publish new problems every month. Why not try and get a solution published on their website? There is a menu specifically for studentsYou can sign up for an Nrich student newsletter if you want to be notified of new developments on the site.

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish.

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish

Brilliant problem

Signing up to Brilliant (including an easy option for sign in for Facebook users) will allow you to join an international community and try numerous.questions at various levels.

Circles & Teddy Bears


Have a look at the teddy bear – can you identify all the equations from this list?
This problem comes from the excellent Underground Maths site.
You can view the Teddy Bear on Desmos by selecting the image above.

desmos-toggleNote that on Desmos you can choose to display or hide a graph.

As with all problems on the site you can see the question and a very full solution with all reasoning explained.

For further notes on the equation of a circle, check these Helm Project notes and exercises on the circle.