# Constructions

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

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

# Transformations of 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.

Translations

Reflections

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

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.

Online       Paper

Online      Paper

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

Note that Advanced Level Questions are also available.

# Interactive Linear Algebra

From Immersive Math a Linear Algebra book with fully interactive figures. Have a look at the Table of Contents and you will see this is a very comprehensive coverage of Linear Algebra.

Immersive Math – from Chapter 2, Vectors

# 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 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.

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

And from Nrich, Prepare for University.

# Projectiles

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!

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.

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 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

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.

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.

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

# Top >10 Mathematics Websites for Students

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 here and 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 tablet too.

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 pagethis 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!

If your teacher is not using Diagnostic Questions shown here, you can sign yourself up as a student. Note the many Collections of Questions as well as questions by topic. (You need to be signed in for the link to work).

Brilliant – Level 5 example

On the Challenges page you can see resources such as the UK Maths Challenges, Nrich, Underground Maths and Brilliant. Signing up to Brilliant (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.

Try Bart Bonte’s logic game ‘Think Outside the Flock’. The game is on Math Playground which I know has many games students enjoy. For more logic games and puzzles try the Logic page.

Are you wandering what mathematicians do or are thinking about a career in Mathematics? Aimed at anyone from age 11 to adult the Maths Careers site 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 Moments from the American Mathematical Society.

The 11 Commandments of Mathematics is perhaps a good way to end this post – and note 10 on Learning Mathematics – think about how you learn not just what you learn. Have a look at these Study Strategies from The Learning Scientists.

For valuable resources to support the techniques described here see the excellent downloadable materials on 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.

# The 11 Commandments of Mathematics

Remember these ten eleven commandments for mathematicians!