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 portalfrom stepmaths.co.uk 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.

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

A 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.
For amusement try projectiles of different types!

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

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.

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

Signing up toBrilliant (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.

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.

Note 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.

It struck me that it might be useful to think about my top recommendations for students. Using some categories again gives me the excuse to mention more than 10! All these resources are free to use.

Francesco Bondi’s art work on Desmos. Click the image to see the graph on Desmos.

For an online graph plotter try the excellent Desmos graphing calculator, it is very easy to use and allows you to save your graphs if you sign up. (Facebook is one option you can use to sign in to Desmos). You can see more examples of Desmos graphs hereand there is a helpful user manual you can download from Desmos. There are many creative users of Desmos, have a look at the selection of art work! Make sure you get Desmos on your phone and/or tablettoo.

Calculators

For checking your work WolframAlpha is so useful, it is free to use for checking answers for as many queries as you want (step by Step solutions require a subscription). The set of slideshows here show you the syntax for a variety of queries.

For more excellent calculators and tools for checking your work, try this collection.

Calculus workbook from Plymouth University

There are many sites with useful notes and examples online for all ages, you will find several on the Notes page, this Evernote shared notebook,Mathematics notes includes many links, several universities have very helpful resources which they have made available to all students. You do not have to be an Evernote user (though I’d recommend it highly), just select ‘View’ to access the notebook.

For reference materials see the various resources on the Reference page which includes links to online dictionaries.

If you like to watch videos to help you learn then you may find some useful resources on the Videos page. Though of course you need to actually do lots of questions!

The best way to learn Mathematics is of course to do Mathematics and there are some excellent sources of problems for students of all ages to try.

Underground Mathematics has an extensive collection of questions to get you really thinking about your Mathematics. Suggestions and full solutions are provided but as always make sure you really do everything you can first with the question.

There are several sites with questions and examples for students of all ages. See more posts with many more resources in the Questions Category.

For revision you can use questions and examples already mentioned, Underground Mathematics includes examination questions for students age 16+. Note the above question comes from an Oxford University Mathematics aptitude test; it is one of the many Review Questions.

Diagnostic Questions

The Revision pages include questions from UK 15-18 Mathematics Examinations. These all include very challenging questions as well as more routine practice.

For 17-18 Year olds, MadAsMaths includes some very challenging questions for those aiming at the top grades. My student who recommended the site went on to achieve an A* grade!

On the Challenges page you can see resources such as the UK Maths Challenges, Nrich, Underground Maths and Brilliant. Signing up toBrilliant (including an easy option for sign in for Facebook users) will allow you to join an international community and get free weekly, personalised problems. Questions at various levels are available. Follow Brilliant on Facebook.

We all like to play Games, many games are available to help you practise Mathematics, you can see a whole collection on Mathematics Games.

Are you wandering what mathematicians do or are thinking about a career in Mathematics? Aimed at anyone from age 11 to adult the Maths Careerssite will answer your questions.

The Maths Careers site offers you many articles to read, for further reading materials try Plus Magazine from The Millennium Mathematics Project – University of Cambridge or perhaps Math in the News from the Mathematical Association of America or Mathematical Momentsfrom the American Mathematical Society.

For valuable resources to support the techniques described here see the excellent downloadable materialson study strategies. Note how each strategy is backed up by research.

Obviously all these sites are those that I think are particularly good, I do know that many of my students use a lot of the sites I have mentioned here. You will find more recommendations on the Useful links pages. Students do let me know your own particular favourites.

Appropriate for New Year Resolution Time – a revised and checked 11 Commandments of Mathematics!
Also available as a poster: 11-commandments-mathematics

The Top Tools for Learning 2016, inlcuding top tools for Education. This is what the educators think. Students what do you think? What are your favourite tools for learning?

Note from Jane’s overview she has done a finer analysis for 2016 including the Top 100 Tools For Education (for use in primary and secondary (K12) schools, colleges, universities and adult education.)

Back in April, I wroye about my own choices for 2016and I am always interested to see where my own choices are in Jane’s list.

Students try an alternative to chocolate – mathematical Advent calendars up to Advanced Level, and note the Primary calendars from Nrich if you have little siblings!

December means Advent Calendars and Nrich have published two Advent Calendars, one for Primary and one for Secondary each containing twenty-four problem-solving activities, one for each day in the run-up to Christmas. The primary Calendar tasks focus on encouraging mathematical habits of mind and the Secondary tasks have been chosen to encourage mathematical creativity.

Alex Pett created his advent calendar complete with history and problems for each day. Alex has provided a pdf versionor use as aGoogle document. For an Activeinspire resource this versionalso has sound.

For some excellent GCSE (UK age 15-16) revision resources explore this collection. Thank you so much to the teachers who create these resources and make them freely available.

Jonathan Hall has a bank of GCSE questions.on his site; note that you can show solutions for these questions. Also on this site – you can create a practice paper with your chosen topics and the number of questions you want. If you would like a random collection you could also try his Revision Grid.

Mr Carter Maths– for lots of practice questions. Select Differentiated Topics to see the selection available. If you want to practise this new topic – Iteration; this is the direct link.

On Tanner Maths you will find Flashcardsfor the new specification – choose A4 or A5.

Many of these resources are also very useful. Check with your teacher the detailed content for your specification.

For many challenging questions to really get you thinking, try the brilliant Underground Mathematicssite. You can follow them on Facebook or Twitter too.

These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.

Note the various question types available; these include very challenging questions for students age 16+.

Select image for question & solution

The Oxford MAT collection includes an extensive selection of Multiple Choice Questions.

O/AO-level questionsare included. These questions provide excellent challenge for sudents aspiring to the top grades for examinations taken at age 15-16 and beyond..

Can we fully factorise x^{4}+4y^{4}? Starts with a Show that….
We could get very sophisticated and look at those quadratic factors too; useful for those studying the Level 2 Further Mathematics Qualification.

Can we simplify these simultaneous equations of degree 1 and 2? Solve simultaneous equations. We will need to factorise a quadratic in this problem with a coefficient which is not 1 for the square term. My students and I are fans of the Box Methodwhere a factorisation cannot easily be done by inspection.