Simultaneous Equations

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?

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 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.

WolframAlpha – a little fun!  

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.

Comparing the two methods with an example:

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

and for Lyszkowski’s method:

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?

 

Happy New Year

At the beginning of the year, time to contemplate some resolutions perhaps – an annual update – all links checked and some new ones added: 

Firstly, remember the ten eleven commandments!

Also available as a poster: 11-commandments-mathematics

To elaborate a little more on some of these and provide some further links:

1 – reading and understanding the problem
See this document from The University of California, Berkeley for a succinct guide to Polya’s problem-solving strategy and for a more detailed guide with examples try this excellent publication from Arizona State University which is exceptionally clear. 

2 – on Algebra, you will find many examples to help – see Notes. See also this post on Transition Time which includes some very helpful documents with lots of examples.

See for example this brilliant guide for students from OCR –  Bridging the gap between GCSE and AS/A Level Mathematics – A Student Guide.

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?

9 – on Arithmetic, many tests for University Admissions require competence with estimation and mental arithmetic. Put your calculator away sometimes – perhaps have a look at this Open Learn (free) course on Rounding and Estimation.
If you want to practise your arithmetic you could play some games!

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?

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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.

• Are you guilty of making any of the classic mistakes?
• How are your problem-solving skills? There is plenty of good advice available – see this publication from Arizona State University for example
• If you are studying 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.

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 2019!

 

University Preparation

An annual update on UK University Entrance Examinations for Mathematics:

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, 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 free copy of  Stephen Siklos’ Advanced Problems in Mathematics and Core Mathematics. Advanced Problems in Mathematics is excellent preparation for any undergraduate Mathematics course. (Alternative source – free pdf)

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.

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An alternative (also free) source for worked solutions to STEP papers comes from MEI.

Also available are Advanced Extension Award worked solutions. 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 STEP resources including the searchable database.

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

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See 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.

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From Nrich, Prepare for University.

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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:

• MAT (Maths Admissions Test), see the right-hand menu, question papers, followed by solutions as separate documents. For superb resources for these questions see these Underground Mathematics Review Questions where you will find not only the questions but suggestions and complete solutions.
• AEA questions and mark schemes.
• For STEP, see the beginning of this post for mark schemes and full worked solutions. STEP mark schemes can also be found on the Cambridge Assessment Preparing for STEP page.

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Note the Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions. TMUA 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 MAT, 2007, Q1D

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”

 

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For further sources try UKMT Senior Maths Challenge Questions.

…and also see MadAsMaths with its many papers and solutions increasing in difficulty.

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Maclaurin Series

AJ Hobson – Just the Maths

For some clear notes and examples on Maclaurin Series see Differentiation Applications 5  from 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’).

For further notes, designed for students, try the HELM Project. This project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. 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.

Use Desmos to see how Maclaurin series approximate functions – use the sliders to use an increasing number of terms of the series. Have a look at f(x) = cos x.

Try these further Desmos pages:
f(x) = sin x       f(x) = sinh x     f(x) = cosh x       

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 – 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.

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!

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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.

Famous Equations

Do you have a favourite equation? Peter Alfeld of The University of Utah has a collection he thinks are important or intriguing. My personal favourite, Euler’s identity, is his first on the list.

                    e^iπ = -1

For more on beautiful equations try the 17 equations that changed the world on the World Economic Forum where for each equation we have the following information:

• What does it mean?
• History
• Importance
• Modern use

Or from BBC Earth, have a look at “What is the most beautiful equation?”

For UK school age students note the famous equation poster contest from the excellent Maths Careers site. This competition closes on Friday 8th June; winners in each age group will win an Android tablet. There will also be five ‘highly commended’ certificates awarded in each group.

 

 

 

Differential Equations

You can check solutions to Differential Equations using WolframAlpha. The slides here illustrate the syntax for first and second order differential equations. Examples like this and more are available from WolframAlpha: Examples for Differential Equations.

Differential Equations on Slideshare.

Notes and examples on Differential Equations.

• CIMT Section 18.5 on First Order Differential Equations
• Units 15.1 to 15.10 from AJ Hobson’s Just the Maths (individual pdf files hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’) are all on Differential Equations.
• mathcentre: Differential Equations Resources
  mathcentre: First Order Differential Equations
  mathcentre Second Order Differential Equations
•  A series of notes from the Helm Project. 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. 
  
  
  

  Helm Notes – First order ODEs

 

 

 

Underground Mathematics

A reminder that the excellent Underground Mathematics Resources are all still available, now hosted by Nrich.

Have you looked at the many Review Questions? Choose your question type and level by topic, all with very full solutions.

Linear Programming

For Linear Programming there are some excellent ways of using technology to help your learning. This post will look at online utilities from zweigmedia and the use of the Excel Solver.

This example from MEI (MEI Decision 1 Legacy Specimen Paper) will be used to illustrate the use of these tools.

A vet is treating a farm animal. He must provide minimum daily requirements of an antibiotic, a vitamin and a nutrient. He has two types of medicine available, tablets and liquid. The table summarises what the medicines contain and the requirements.

The tablets cost £0.38 each and liquid medicine costs £1 per dose.
The vet wants to find the cheapest way to treat the animal.
We have x as the number of tablets which the vet prescribes per day, and y is the number of doses of liquid medicine, giving us the linear programming problem:
Minimise    0.38x + y
Subject to
3x +   2y   ≥ 18
2x +   4y   ≥ 16
10x + 50y ≥ 100

Firstly we will look at zweigmedia’s Linear Programming Grapher. This is intuitive to use. Note the option to see examples. Fraction mode is selected here.
Selecting Solve gives a clear solution illustrated graphically.

Linear Programming Grapher – zweigmedia

We see that the optimal solution, if x and y can take any values, is a cost of £3.20,
with x = 20/3 and y = 2/3.

We could also solve the problem using the Simplex Tool. As well as a solution it is possible to see the simplex tableaus. The Simplex Tool is needed for any problems with more than two variables.

If we wish to restrict x and y to integers, we can turn to Excel and use Solver. This allows a choice of making x and y integer or not.

Using the same example, we can set up the problem in Excel.
Vet Problem with Excel Solver is the Excel spreadsheet with annotation to explain the answer report and for full instructions on setting up the spreadsheet, use this file:
Solver Excel to solve LP problems – Colleen Young

Set up the problem in Excel

and use Solver..

Excel Solver – Answer Report

We could choose to restrict x and y to integers with Solver.

giving us the solution:

For a further example, this time a maximisation problem from Edexcel have a look at this Excel file, Badges LP Edexcel. Each sheet in the workbook is fully annotated to explain the method.