Why not try **Factris**? This is a new App published in July 2017, developed by Richard Lissaman, of MEI.

# Category Mathematics

# Transition Time

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…

For students going on to A Level then the best thing you could do 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 will be so valuble for students.

You may also find these **GCSE revision resources** useful.The takeaways are really useful and Mohammed Ladak has picked out **Transition Takeaways** 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.

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).

Sign up to **Brilliant **and follow them on Facebook so Maths problems appear in your stream and hopefully distract you from trivia!

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

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.

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 HELM Project**. 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 clear worked examples. You can see on the list that a very small number of titles are ‘not ready yet’; for the sake of completeness 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 4, ****Part 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**!

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

# Constructions

Another post to clarify the requirements for the UK GCSE Mathsematics specification. The constructions required are as follows:

**perpendicular bisector of a line segment****constructing a perpendicular to a given line from a given point****constructing a perpendicular to a given line at a given point****bisecting a given angle**

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**.

**Constructing a 60° angle**is required. This is shown here by constructing an equlateral triabgle.

**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.

# 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

**Translations parallel to the x axis.**as illustrated above.- Or try translating this graph of
**f(x) = sin x parallel to the x axis**. **Translations parallel to the y axis**.**Graph of sin x translated parallel to y axis**

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) = x^{2}−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 – (x^{2}−5x+4) = – x^{2} +5x-4

To work out the equation of f(x) = x^{2}−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 = x^{2} +5x+4

Suppose we wish to work out the equation of f(x) = x^{2}−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) = x^{2}−3x+2, we need f(x+3) for the equation of the transformed graph

f(x+3) = (x+3)^{2}−3(x+3)+2 = x^{2}+6x+9−3x−9+2 = x^{2}+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.

Badly Answered GCSE Questions – Foundation 1

**— Try the questions online** or **download the booklet**

Badly Answered GCSE Questions – Higher 1

**— Try the questions online** or **download the booklet**

Foundation Badly Answered Questions – Set 2

**Online Paper**

Higher Badly Answered Questions – Set 2

**Online Paper **

**Badly Answered GCSE Questions – Foundation 3**

**— Try the questions online **or** download the booklet**

**Badly Answered GCSE Questions – Higher 3**

**— Try the questions online **or** download the booklet**

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.

# 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 Mathematics**. *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 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.

See also, from Cambridge University, their **STEP Support Programme**.

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.

You can download an app for **iOS** also for **Android**.

**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.

**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 students. **You 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

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**.