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 x^{3}+11x^{2}+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:

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

WolframAlpha can of course handle polynomials.

# Transition Time 2021

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 a really useful publication 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 is so valuable for students.**pdf format: bridging-the-gap-between-gcse-and-as-a-level-mathematics-a-student-guide**

**Dr Frost Maths** is used by millions of students round the world. Amongst the extensive **library of resources** are **videos for students age 8-18** explaining topics from scratch. There are longer Exam-topic videos, average length 10 minutes and shorter Key Skill videos which are about 2-4 minutes in length.

Free lessons from Colin Hegarty **recorded on YouTube** are available to help GCSE students prepare for A Level Maths.

These **Transition Takeaways** from Mohammed Ladak have been 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.

From AMSP – check **Transition to A Level Mathematics **– Essential 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).

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 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** .

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 very clear worked examples. For easy access to these resources, the **HELM Project Workbooks** are hosted by **Loughborough University’s Mathematics Learning Support Centre**. Alternatively, 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!

Remember these ~~ten~~ eleven commandments for mathematicians!

Alternatively – a pdf file, the links can work faster from pdfs.

Also available as a poster: **11-commandments-mathematics**

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

# Chi-Squared

**Chi Squared Calculator**From

**Social Science Statistics**this

**Chi Squared Calculator**is for a contingency table with up to 5 rows and columns. It is very simple to use, enter the group and category names, then the data, choose the significance level and calculate.

**A clear conclusion is then given:The chi-square statistic is 10.2061. The p-value is .037096.The result is significant at p < .05.**

To interpret the results remember to look at the observed data. the expected probabilities and the chi squared contributions.

### Note the **many other calculators** available on **Social Science Statistics.**

# Polynomials

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:

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 **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 with examples of Polynomial division: **Just the Maths Algebra(8) – Polynomials.**Another source is David Smith’s site ‘

**The Maths Teacher’**which includes Algebraic Multiplication and Division under Algebra AS level.