Mathematics – Transition Time

Notes from Mr Barton

As we start a new academic year, 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…

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!

For students going on to A Level then these GCSE revision resources will be 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 course you may find some of the material in the section below useful.

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.

Calculus workbook from Plymouth University

Calculus workbook from Plymouth University

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.

See this Evernote shared notebook: Mathematics notes for many more useful links. Several universities have created very helpful Mathematics support which they have made available to all students. (You do not need an Evernote account to view the notebook).

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 4Part 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.

New Academic Year Resolutions

New Academic Year


Happy New Academic Year  (using geoGreeting) (Click on the image).

Many of you will still be enjoying your holidays but holidays are perhaps a good time to make some resolutions for the next academic year.

Firstly, remember the ten eleven commandments!

To elaborate a little more on some of these:

5 – looking it up – there are several excellent resources online for you to look up definitions or find extra examples. Never rely on just one source if you are finding a topic tricky, it can be helpful to see explanations written by different authors.

6 – To learn Mathematics you need to do Mathematics. You can never do enough examples! There are plenty of questions here (with answers included). Have you looked at the problems on Brilliant? New problems are published regularly and are suitable for younger school students all the way through to university students. Or perhaps you might like to try some of the trickier problems from the challenges set by the University of Mississippi. (These problems from the Problem of the Week category are open to people of any age).

9 – if you want to practise your arithmetic you could play some games! If you enjoy Maths games there are many excellent free resources available.

10 – on writing the language of mathematics correctly – see  this clear guide to writing Mathematics from Dr Kevin P Lee and from John Kerl some excellent tips for mathematical handwriting, many of these tips these apply to students of all ages – do you distinguish carefully between a 1 and a 7 for example? Perhaps it is hard to tell whether you have written a 2 or a z or perhaps your 5s look a bit like a letter s?

Have a look at Peter Alfeld’s  guide to Understanding Mathematics which he wrote for his students at Utah University.

Are you familiar with all the excellent (free) resources online to help your studies? Have you tried the brilliant Desmos graphing calculator for example, or used WolframAlpha to check your work?

Some more thoughts for you.

If you are trying to get organised generally then some of the resources on this page might be useful. I recommend Evernote highly (it’s free).

Wishing you all a very happy and productive new academic year!


Desmos - function notation

Desmos – function notation

Desmos can be used very simply to illustrate function notation and note the use of Desmos as a calculator to evaluate the value of a function for a given input. (Select image to go to the Desmos graph page).

The University of Plymouth have a series of very clear notes with examples, exercises and solutionssee this booklet on Functions.

University of Plymouth Notes

University of Plymouth Notes

Helm Notes - Functions

Helm Notes – Functions

The Helm Project – Basic Functions and One-to-one and inverse functions 

Desmos and Functions

You can use the Desmos graphing calculator to support your study and understanding of functions. Using function notation you can enter composite functions as shown in the next two illustrations. Selecting each image will take you to the relevant Desmos graph page.

Composite Functions with Desmos

Composite Functions with Desmos – select image

Composite functions with Desmos

Composite functions with Desmos – select image

For further exploration of composite functions check the pages in this slideshow.

You can also easily explore transformations of graphs with Desmos.

Simon Singh – The Simpsons and Their Mathematical Secrets

Colleen Young:

Note the daily deal today for Simon’s book on Kindle.

Originally posted on Mathematics, Learning and Technology:

Simposon & Their Mathematical Secrets - Simon Singh

May 14th – get the book as a deal today
on Kindle:  
UK £1.09       USA $1.70

In Simon Singh’s ‘The Simpson’s and Their Mathematical Secrets”, now published in paperback Simon explains how writers have included mathematical jokes throughout the cartoon’s twenty-five year history. Here Simon Singh answers some questions I put to him.

I love the scene with Lisa surrounded by all her books including the one with Euler’s identity! I suppose if I had to pick a favourite mathematics reference in The Simpsons, then it would be that one – do you have a favourite?
It has to be Fermat’s last theorem, which appears in two episodes, namely “Treehouse of Horror VI” and “The Wizard of Evergreen Terrace”. My first book was all about Fermat’s notorious problem, so it is close to my heart and seeing it make cameo appearances in the world’s favorite TV…

View original 676 more words

Simultaneous Equations

MathisFun Easter Puzzle

MathisFun Easter Puzzle

An Easter puzzle from MathisFun – as an excuse for solving simultaneous equations. We could of course use algebra. Using the notation, h, q and t for the egg Horace wants, the egg with the small square pattern and the egg with the stripey pattern respectively.
We have:
h + 2q      =550 (1)
h + q   +t =600 (2)
—-2q  +t =500 (3)
Subtracting equation (3) from equation (2) gives
h−q=100 (4)

(1)−(4) gives 3q=450 so q=150 and h must be 250 ($2.50).

We could check our solution on WolframAlpha of course:

Select image for WolframAlpha query

Did you know that you can easily invert matrices and solve simultaneous equations using Excel?

Excel sim equations & matrices

Select image for Excel file

Select the image or this link for the Excel file. Excel simultaneous equations

To enter the MINVERSE function in the example above, select cells C7:E9, enter the MINVERSE function as shown then press CTRL + SHIFT + ENTER; similarly to enter the MMULT function in this example, select cells H7:H9, enter the MMULT function as shown and then press CTRL + SHIFT + ENTER. (Using formulae like these demonstrates a neat Excel technique, to learn more see MrExcel on Array Formulae).

For notes / examples / tutorials on Simultaneous Equations try the mathcentre resources or this workbook from Plymouth University. For more on solving three equations in three unknowns for older students see Simultaneous Linear Equations from AJ Hobson’s ‘Just the Maths’

Since it’s Easter an updated version of the best Easter Eggs of all – from WolframAlpha.

If you want some more puzzles, MathisFun has plenty more, or try some of the puzzles here.





Box Plots

Box&Whisker - WolframAlpha

Box&Whisker – WolframAlpha

Box Plots show summary information for data – the box shows the median and quartiles and the whiskers are extended to all points which are not outliers.

A very clear description for younger students can be found on BBC Bitesize. For A Level Students the Centre for Innovation in Mathematics Teaching includes Box Plots in Chapter 3 of their Statistics text (page 26 of the pdf file, which is page 76 of the text).

Box Plots are particularly useful when comparing one or more distributions;

CIMT notes

CIMT Statistics Text

To help older students think further about about outliers and skew, have a look at this excellent illustration of Nobel Prize data from Plotly. Who is that outlier for the Nobel Peace Prize?!

Plotly - Nobel Prize winners by field

Plotly – Nobel Prize winners by field

Note that it is possible to display points with Box Plots in Plotly, the following data and charts demonstrates skew well:


Box Plots & Skew

You can learn more about Plotly for graphing and sharing your data here: Learn and explore Plotly