HELM Notes

For a very comprehensive library of Notes and Examples – see the HELM Project.

Mathematics, Learning and Technology

I have referred to the HELM (Helping Engineers Learn Mathematics) notes many times over the years on this blog. They are referred to on the Notes and Examples page for Further Mathematics as well as in many individual blog posts, for example Mechanics – Dimensional Analysis, Differential Equations, and Further Calculus.

HELM Notes - Basic Algebra HELM (2008): Workbook 1: Basic Algebra

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. Whilst the workbooks cover the basic engineering mathematics and statistics teaching for first and second year students in a typical UK undergraduate engineering degree many of the workbooks include content appropriate for A Level Mathematics and particularly, Further Mathematics. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics…

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Learning Mathematics at Home

Are you studying Mathematics at home?

Several posts and pages here may be useful. See:

Back in 2014 I was using free samples from Bring on the Maths resources with my students. A whole series of activities where you must pick out the correct answers is provided. Activities are available for young children right through to Advanced Level.

I must recommend the free samples from Bring On the Maths, the Core 3 activity – Logarithmic equations worked really well with my Year 13 class and next week I’ll use the C4 Binomial Expansion resource when we are talking about the validity of a given expansion. I have used the Trigonometric Ratios resource before – and will again; there are several other great resources in that list of samples – explore!

Kangaroo Maths
Bring on the Maths Logarithmic Equations
Free access is available to the complete set of these resources on the old Kangaroo Maths site. Log in with the user and password as above to access all the resources. This will give access to all the activities for students from Key Stage 2 right through to A Level – note that this being the old site, the resources are aligned to earlier schemes of learning but still so useful.

Looking at the old level 5 and 6 Algebra resources for example, we could try Simplifying Expressions  or Constructing and solving equations.

Distance Learning will be demanding, it’s certainly a chance to read more or perhaps get away from screens and listen to a story. For as long as schools are closed, Audible are providing stories which are free to stream on a desktop, laptop, phone or tablet. An extensive collection of stories, including titles across different languages is available. No accounts or passwords are needed. See Audible’s ‘How to Use Stories‘ and start listening; note the menu where children can select by age.

White Rose
White Rose Maths has so many free high quality resources, see for example Secondary Schemes of Learning and Assessments for End of Block or End of Term. Check their Home Learning, the White Rose Maths Team has prepared a series of five maths lessons for each year group from Year 1-8. More will be added each week for the next few weeks. Every lesson comes with a short video showing  clearly and simply how to help children to complete the activity successfully.


The summer term includes work on consolidation for the year, see this collection from Nicola Whiston who has started a collection of Knowledge Organisers which followAlgebra KO extract Nicola Whiston

the White Rose Schemes of Learning, she is sharing the collection here, via Dropbox. These are really attractive.

CK-12 Foundation has created a resource page with hand-picked lessons in math and science popular during the month of March.

From Bobby Seagull, “The Magic of Numbers: Why Everyone Should Love Maths”.

And of course you will need a break, maybe a puzzle? Try Futoshiki and other puzzles.
It’s certainly a chance to read more or perhaps get away from screens and listen to a story. For as long as schools are closed, Audible are providing stories which are free to stream on a desktop, laptop, phone or tablet. An extensive collection of stories, including titles across different languages is available. No accounts or passwords are needed. See Audible’s ‘How to Use Stories‘ and start listening; note the menu where you can select by age.

Stay safe everyone.

Futoshiki and other puzzles

Mathematics, Learning and Technology

FutoshikiI first came across and enjoyed a Futoshiki puzzle in the Guardian puzzles section.

Work out the digit that goes in each cell. In this 4×4 example, the digits from 1 to 4 must appear exactly once in each row and column.

Initially, some digits might be revealed and additionally, the board might also contain some inequalities between the board cells; these inequalities must be respected and following them will help you find the missing digits.

Futoshiki2In the above puzzle, difficulty level, easy, we see that the fourth column already has a 1 and a 3, the remaining digits 2 and 4 can only be placed one way as we have to follow the inequality sign.

On Futoshiki.org from Vlad Daskalu, you can generate puzzles of sizes from 4×4 to 9×9 and choose one of 4 difficulty levels.

Other sites offering these puzzles include Brain Bashers where you can play…

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Linear Programming

For students studying Decision Mathematics try Linear Programming from Transum. This provides you with a reminder of inequalities and gives you 5 problems to try.

If you need some notes and further activities on Linear Programming then try these from CIMT which provide an excellent introduction as well as exercises with answers.

Transum Linear Programming

This earier post post looks at the use of Technology to interpret solutions to such problems and includes online utilities and instructions on the use of the Excel Solver. Examination questions are used to illustrate the use of these tools.

University Admissions Tests

An annual update on preparation for UK University Entrance Examinations for Mathematics; note that these resources provide challenging questions for any students anywhere studying Mathematics beyond age 16.

From Nrich, Prepare for University.
Nrich University prep


To further challenge yourself, MAT, STEP and AEA questions provide an excellent source of questions. Dr Jamie Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

For mark schemes see:

Underground Mathematics Review MenuNote the Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutionsTMUA is a newer admissions test only one question is available on the Underground Maths site, however, there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking this examination. Interestingly, Durham University states that “Those students already registered for MAT may substitute those results in place of our own test, if they do not wish to take both.”
Warwick University advise taking one of MAT, TMUA or STEP.

Oxford Admissions Test

TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems (probably just STEP I initially).

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”

For further sources try UKMT Senior Maths Challenge Questions.
…and also see MadAsMaths with its many papers and solutions increasing in difficulty.
c1qn Madas

You can download a free copy of  Stephen Siklos’ Advanced Problems in Mathematics and Core Mathematics. Whilst written to support students taking STEP examination papers, Advanced Problems in Mathematics is excellent preparation for any undergraduate Mathematics course.

Advanced Problems in Mathematics

Advanced Problems in Mathematics Problem Index

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems.

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.

STEP DatabaseYou will find free STEP and AEA (Advanced Extension Award) paper solutions on MEI’s site. The STEP question papers are all available on the Cambridge Assessment website; note the STEP resources include a searchable database.  The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. Note that AEA papers can be found here. and STEP papers here note the

Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.
Underground Maths STEP

Underground Maths STEP example

STEP CUSee also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules.

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…

OCR Transition Algebra Fractions

OCR – Bridging the Gap – A Student Guide

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

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.

Calculus workbook from Plymouth University

Carom MathsFor 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 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.

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).

Plymouth University notes

Plymouth University workbook

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.

Factorisation of Quadratic Expressions

When factorising quadratic expressions do you check coefficients first? If the coefficient of x2 and the constant are prime for example you should be able to just write down the factorisation without needing an elaborate method.

Some students have difficulty with the splitting the middle term method; you might like an alternative – try the box method.

For instructions on the method:

Quadratic Factorisation Box method (pdf file)

Working on Quadratic Grids from Underground Mathematics will help you develop and understand the method.


Perhaps even simpler is Lyszkowski’s method which avoids the manipulation required by conventional methods.
Lyszkowski's method

Comparing the two methods with an example:
Lyszkowski & box comparison

We could have a look at the general case for the box method :
Box method general

and for Lyszkowski’s method:
Lyszkowski general
Have a look at this series of videos on Factorising Quadratic Expressions from Exam Solutions. You could try the examples given with the various methods presented.

I would be interested to hear student views, which methods do you like?