How about a little calculated colouring from Dr Tom Bennison.
This is designed for UK Maths A Level students, age 16-17.
You can access calculated colouring resources from previous years also.
For Christmas…
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.
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.
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.
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.
…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.
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.
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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.
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:
- GMAT Foundations of Math – 900+ Practice problems (see also GMAT Math Practice)
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:
University of Plymouth Notes
- Plymouth University – Indefinite Integration
- Plymouth University – Definite Integration
- Units 12-1 – 13.16 from AJ Hobson’s ‘Just the Maths’ offers very comprehensive coverage starting with the basics and going well beyond school Mathematics
- Choose Integration from the MathCentre topics
- Helm Workbooks – section 13
(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.
Scrolling down the page we see that this query also returns the 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.
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.
Puzzles by A-Level (16+) topic
For some great puzzles by Jonny Griffiths, check Digitisers. Clear instructions explaining the puzzles and notation used are given at the beginning of the document, we then have all the tasks by topic. Full solutions are provided.
Proof Sorters
From Nrich, try this Interactive Proof Sorter example which works on any device.
There are several more examples you can try:
- Nrich – The Quadratic Formula
- Nrich – Sum of an Arithmetic Sequence
- Nrich – Geometric Sequence
- Nrich – the Square Root of 2 Is Irrational (electronic version)
- Nrich – Curve Fitter a proof by contradiction, related problem Curve Fitter
- Underground Maths Proving the laws of logarithms
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.
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:
- GMAT Foundations of Math – 900+ Practice problems (see also GMAT Math Practice)
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.
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
Note the exercise at the end so you can practice, fully worked answers are shown.
WolframAlpha can of course handle polynomials.
Mathematics for Students 