University Admissions Tests

An annual update on preparation for UK University Entrance Examinations for Mathematics; note that these resources provide challenging questions for any students anywhere studying Mathematics beyond age 16. (Teachers, see this page which includes a teacher webinar)

From Nrich, Prepare for University.

Nrich University prep
Nrich – Prepare for University

mat-step-aea-dr-frost

To further challenge yourself, MAT, STEP and AEA questions provide an excellent source of questions. Dr Jamie Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

For mark schemes see:


  • MAT (Maths Admissions Test), scroll down the page for past papers, solutions and additionally video solutions. For superb resources for these questions see these Underground Mathematics Review Questions where you will find not only the questions but suggestions and complete solutions.
  • STEP mark schemes can also be found on the Cambridge Assessment Preparing for STEP page. Note that STEP 1 will no longer exist from Summer 2021.
  • The AEA questions are legacy questions, extended solutions are currently available for the papers which have been used from the University of Warwick. Legacy papers and mark schemes are available on Colmanweb.

Underground Mathematics Review Menu

The Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions

TMUA is a newer admissions test only one question is available on the Underground Maths site, however, there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking this examination. Interestingly, Durham University states that “Those students already registered for MAT may substitute those results in place of our own test, if they do not wish to take both.”
Warwick University advise taking one of MAT, TMUA or STEP.

Oxford Admissions Test

TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification with its notes on logic and proof.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems.

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”


For further resources try UKMT Senior Maths Challenge Questions.

MadAsMaths

…and also see MadAsMaths with its many papers and solutions increasing in difficulty.


You can download a free pdf copy of  Stephen Siklos’ Advanced Problems in Mathematics. Whilst written to support students taking STEP examination papers, Advanced Problems in Mathematics is excellent preparation for any undergraduate Mathematics course.

Advanced Problems in Mathematics
Advanced Problems in Mathematics Problem Index

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems.

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.

STEP Database

………..

You will find free STEP soluions on MEI’s site. The STEP question papers are all available on the Cambridge Assessment website; note the STEP resources include a searchable database.  The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. The Sample Assessment Materials provides a valuable source of challenging questions (with mark schemes) for A Level mathematicians.

STEP papers here. Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.

Underground Maths STEP example
STEP CU

See also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules.

Reading

Plus Magazine  (University of Cambridge)

The Maths Careers site has sections for ages 11 through to adult.

For a list of recommended books for young people interested in Mathematics, try this list from Nrich which is grouped into three different categories: History of Mathematics, Recreational and Thinking Mathematically.

From Cambridge University this list of interesting mathematics books and internet sites is mainly intended for sixth-formers planning to take a degree in mathematics. The list includes some items which are suitable for less experienced readers so may well appeal to a wider audience. The list was last updated in September 2020. I see it includes Kevin Houston’s “How to Think Like a Mathematician, see also from Kevin Houston his page on the book which includes some solutions to problems in the book, also available are sample chapters on writing mathematics.

From Imperial College, their STEM book list has many recommendations in several categories including Mathematics. Imperial College says that the list is aimed at A Level students but is suitable for anyone looking to develop their STEM knowledge and have a great read.

See the Reference page also which includes a useful list of reference materials.

Currently free on Kindle:

White Rose Maths

If you have younger brothers and sisters, for Primary age children, these are all available as pdf files from White Rose Maths. The books are also available on Kindle.

If you like puzzles try Henry Ernest Dudeney’s – Amusements in Mathematics puzzle collection (with solutions). The first set of puzzles will offer a trip down memory lane for some of your parents who remember money – pre-decimal! There are several categories of puzzles available.

Integration

Firstly for some useful notes and examples on Integration:

Integration notes University of Plymouth

University of Plymouth Notes

(These sources of Notes can all be found on the Notes page.)

You can use Desmos and WolframAlpha to check your work and see some excellent visual representations.

As an example, we can find the total area bounded by f(x) = x4−3x3−4x2+12x, the x-axis, the line x=−1 and the line x=3.

We could use WolframAlpha for a quick check. The visual representation shows clearly that we are dealing with areas above and below the x-axis.
Definite Integral
Scrolling down the page we see that this query also returns the indefinite integral.
Indefinite Integral

For the total shaded area, we can change the limits of the query to evaluate each section.
See Calculus & Analysis for more examples of WolframAlpha queries.

Or we could turn to the excellent Desmos where we can very simply change the limits.
Definite Integration

Definite Integration 2

If you are unfamiliar with Integration with Desmos, turn to Learn Desmos: Integrals.
Note you can explore the graph shown in the video.

We could also look at this introduction to Integration on GeoGebra.
Integration

Proof Sorters

Nrich – IFF

From Nrich, try this Interactive Proof Sorter example which works on any device.

There are several more examples you can try:

SUMS – Steps to University for Mathematical Students

From AMSP have a look at this great new magazine which is aimed at high-attaining Year 12 (UK age 16-17) students. You can read more about the magazine and freely access the first issue here.

You will find out more about where Maths can take you in the future, learn more about the applications process and what it is like to study maths at university.

Of course check the Do some maths! section. There is some great advice on problem solving and puzzles to try.

Data Science

MEI’s Introduction to Data Science Short Course is now freely available from MEI; this is a self-study course aimed at A level Maths students and uses the large data sets for AQA, Edexcel, MEI and OCR.

MEI Data Science course contents

The course uses the large data sets for all the examination boards, note also this very useful page from MEI on the large data sets. Not only are there links to all the large data sets but resources using Desmos, Excel and GeoGebra.

Check this playlist on YouTube of the MEI large data set videos.

Reading

Plus Magazine  (University of Cambridge)

The Maths Careers site has sections for ages 11 through to adult.

For a list of recommended books for young people interested in Mathematics, try this list from Nrich which is grouped into three different categories: History of Mathematics, Recreational and Thinking Mathematically.

From Cambridge University this list of interesting mathematics books and internet sites is mainly intended for sixth-formers planning to take a degree in mathematics. The list includes some items which are suitable for less experienced readers so may well appeal to a wider audience. The list was last updated in September 2020. I see it includes Kevin Houston’s “How to Think Like a Mathematician, see also from Kevin Houston his page on the book which includes some solutions to problems in the book, also available are sample chapters on writing mathematics.

From Imperial College, their STEM book list has many recommendations in several categories including Mathematics. Imperial College says that the list is aimed at A Level students but is suitable for anyone looking to develop their STEM knowledge and have a great read.

See the Reference page also which includes a useful list of reference materials.

Currently free on Kindle:

White Rose Maths

If you have younger brothers and sisters, for Primary age children, these are all available as pdf files from White Rose Maths. The books are also available on Kindle.

If you like puzzles try Henry Ernest Dudeney’s – Amusements in Mathematics puzzle collection (with solutions). The first set of puzzles will offer a trip down memory lane for some of your parents who remember money – pre-decimal! There are several categories of puzzles available.

Polynomials

For notes and examples on polynomials, see the following resources:
From the mathcentre which has an extensive collection of very clear notes and other resources, the Algebra collection includes Polynomial Division.

Another reliable source is Tony Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’includes a document on Polynomials: Just the Maths Algebra(8) – Polynomials.

David Smith’s site ‘The Maths Teacher’ includes Algebraic Multiplication and Division under Algebra AS level.

In Paul’s Online Notes, Section 1-4 is on Polynomials; these start with a clear definition and include examples of polynomials and also expressions which are not polynomials .

Earlier I wrote about the excellent Helm Notes, this collection includes Solving Polynomial Equations which includes exercises with answers.

It is really useful to have a look at the graphical representation as well as the algebraic solution.
For example:
Show that x = −1 is a root of x3+11x2+31x+21 = 0 and locate the other roots algebraically. The graph on Desmos is here. You can check your algebraic solution, by looking at the graph.

Cubic example, solve f(x)=0

One of the pages in the series on Calculators is on Polynomials where you will find the following resources.

Division of  Polynomials from Mathisfun – a really clear explanation:

mathisfun-polynomial-division

Mathisfun
Mathisfun exercise

Note the exercise at the end so you can practice, fully worked answers are shown.

WolframAlpha can of course handle polynomials.

wolframalpha-polynomials

Transition Time 2021

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 – A Student Guide

For students going on to A Level then a really useful publication 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 is so valuable for students.
pdf format: bridging-the-gap-between-gcse-and-as-a-level-mathematics-a-student-guide


Dr Frost Maths is used by millions of students round the world. Amongst the extensive library of resources are videos for students age 8-18 explaining topics from scratch. There are longer Exam-topic videos, average length 10 minutes and shorter Key Skill videos which are about 2-4 minutes in length.


Hegarty Maths

Free lessons from Colin Hegarty recorded on YouTube are available to help GCSE students prepare for A Level Maths.


These Transition Takeaways from Mohammed Ladak have been 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.

AMSP – Transition To A Level Mathematics – Essential Skills

From AMSP – check Transition to A Level Mathematics – Essential 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).


Calculus workbook from Plymouth University

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 Math Centre – Product and quotient rules

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 Mathcentre has an extensive collection of helpful resources for students of Mathematics.

Carom Maths

For 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.

Calculus workbook from Plymouth University

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 .

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 very clear worked examples. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, 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!

Remember these ten eleven commandments for mathematicians!

Alternatively – a pdf file, the links can work faster from pdfs.

Also available as a poster: 11-commandments-mathematics

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