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

WolframAlpha can of course handle polynomials.

For further notes & examples on operations with polynomials, see the following resources: From the mathcentrewhich has an extensive collection of very clear notes and other resources the Algebra collection includes Polynomial Division.

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.

Recommended Mathematics reading from the University of Oxford (scroll right down the page).

It seems appropriate to check some world records on books!Did you know that the first collection of crossword puzzles was published in the USA in 1924?

The following demonstrations show how to use a straight edge and a pair of compasses to do standard constructions.

On Corbett Maths you will find clear demonstrations and also practice questions on constructions. Look at this alphabetical list of contents, you will find several several items under Constructions. There are videos and also clearly worked answers to practice questions. Answers are all provided.

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° angleis required. This is shown here by constructing an equilateral triangle.

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.

The BBC site also has clear examples of solving problems using constructions.

The excellentMaths Careerssite is managed and maintained by the Institute of Mathematics and its Applications. If your students wonder where Mathematics is used they will find plenty of answers here. See for example Who employs mathematicians?

Also from Maths Careers, see this post with instructions on how to make this wonderful pair of linked Möbius hearts.

If you wish to get creative and try this I advise watching the Numberphile video carefully; following the instructions worked as you can see from my creation here! I can verify that unless you follow the instruction to make sure the twist in each strip is in a different direction you will end up with a mess! Quite an interesting mess but certainly not two hearts!….

Note the Desmos graphs on my strips. I created a file in Word valentine-mobius-hearts (or pdf: valentine-mobius-hearts) with Desmos images in a table. Adding dotted borders to the table gives guidelines for cutting. I began each cut by using the end of a paperclip to pierce the paper.

To create my strips I printed the document and then printed again on the reverse. I then cut out and trimmed the strips so there was no white space at the end – the picture here has been made using strips 10 cells long.

From the MEI Archives, the February 2015 edition of the MEI Monthly Maths Magazine includes some connections between maths and Valentine’s Day. On page 7 note the article “A Happy Ending” which includes references to some Numberphile videos, Professor Ron Graham discusses the Happy Ending Problem and from Dr Emily Riehl, The Stable Marriage Problem. We also have a great Parametric Heart spreadsheetfrom Think Maths.

This edition of the magazine includes some lovely activities which link paper folding and proof.…………………………………………………..

Are any of you thinking about applying for Maths at university? Or perhaps you have applied for this year. The Oxford Online Maths club should be of interest – all free and available indefinitely on YouTube – no sign up required.

Have you seen the details for Oxford University’s Oxford Online Maths Club? This is a new weekly maths livestream from the Mathematical Institute of Oxford University providing free super-curricular maths for ages 16-18.

The sessions start on 7th January at 16:30 UK time.

The content will include maths problems, puzzles, mini-lectures, and Q&A. The sessions are all free and require no sign-up. The livestreams will be available on YouTube indefinitely.

The sessions will be of particular interest to students who are about to start an undergraduate Mathematics (or joint honours) degree later this year, also for those who are thinking of applying later this year to start a Maths degree next year.

Full details are available here, and you can see what the session on 7th January will include on the YouTube channel.

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.

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.

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.

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”

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 asearchable 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.

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.