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.
Vet Problem - table
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.

LP vet problem

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.

LP Vet Graphical Solution

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.
Simplex Tool zweigmedia

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

Excel Solver

Set up the problem in Excel

and use Solver..

Excel Solver Answer Report

Excel Solver – Answer Report

We could choose to restrict x and y to integers with Solver.
Excel Solver specify integer solution
giving us the solution:
Excel Solver Answer Report Integer

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.
Edexcel LP Problem Excel Solver
Edexcel LP Problem Excel Solver Graphical Solution






Casio FX991EX-ClassWiz

Are you getting used to a new ClassWiz calculator?
There are guides, manuals and videos available to help.

On the Casio website, you will find quick guides for Casio Calculators including the ClassWiz.

Classwiz guide

Casio ClassWiz quick guide

You can find an excellent calculator guide for the Casio FX991EX-ClassWiz on Dr Frost’s site.
Calculator Guide Dr Frost

For the complete manual from Casio see this page, note the calculator is listed under fx-570EX / fx-911EX on the Casio manuals page.

From The Calculator Guide on YouTube see this comprehensive playlist on the Casio Classwiz fx-99EX. For example see this introductory video on using Statistics mode to find the mean, variance and other statistical summary data.

Other examples include further statistical videos such as Normal Distribution calculations, integration and differential calculations and using Complex mode.

Matrix Calculations.

Correlation and Regression

Some useful resources to help you study Correlation and Regression.

This GeoGebra applet allows you to move points and watch the effect on the line of best fit.


Why not try to plot the points, draw a lines of best fit and then compare your line with the computer. This works very well on a phone, use this link.

Correlation coefficients and lines of best fit can also be studies with this PhET simulation on Least Squares Regression.
correlation-line-of-best-fit PhET

Choose from a range of examples or choose Custom to add your own points and guess then check the correlation coefficient. You can also draw your own line of best fit and compare it to the theoretical line of best fit. Note the option to include residuals for both your own attempt and the line of best fit.

We can check Regression Calculations using this Linear Regression calculator from Social Science Statistics.

On the subject of correlation coefficients, we can play a game to see how well we can guess the correlation coefficient! Guess the Correlation Coefficient.

From Cambridge PhD student, Omar Wagih ‘Guess the Correlation‘, a rather addictive game with a purpose – Omar Wagih is collecting the data on the guesses collected and using it to analyse how we perceive correlations in scatter plots. Select About to read the rules and further details.

For ranked data you must be able to calculate Spearman’s rank correlation coefficient from raw data or summary statistics. Again, Social Science Statistics, offers us a calculator which is useful for checking work.

Calculation details provide a useful check on work.
Note Social Science Statistics also has a calculator for calculating the Pearson correlation coefficient.



Firstly for some useful notes and examples on Integration:

Integration notes University of Plymouth

University of Plymouth Notes

(These sources of Notes can all be found on the Notes page.)

You can use Desmos and WolframAlpha to check your work and see some excellent visual representations.

As an example, we can find the total area bounded by f(x) = x4−3x3−4x2+12x, the x-axis, the line x=−1 and the line x=3.

We could use WolframAlpha for a quick check. The visual representation shows clearly that we are dealing with areas above and below the x-axis.
Definite Integral
Scrolling down the page we see that this query also returns the indefinite integral.
Indefinite Integral

For the total shaded area, we can change the limits of the query to evaluate each section.
See Calculus & Analysis for more examples of WolframAlpha queries.

Or we could turn to the excellent Desmos where we can very simply change the limits.
Definite Integration

Definite Integration 2

If you are unfamiliar with Integration with Desmos, turn to Learn Desmos: Integrals.
Note you can explore the graph shown in the video.

We could also look at this introduction to Integration on GeoGebra.


University Preparation

An updated version of an earlier post ….

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 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 MathematicsAdvanced 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.

Stephen Siklos Advanced Problems in Mathematics and Core Mathematics

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

Underground Maths STEP example

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

And from Nrich, Prepare for University.
Nrich University prep

See also, a related post 16+ Challenge Questions. Also, Multiple Choice Questions which includes Oxford Admissions test questions.
Oxford Admissions Test

Order of Operations

There are several resources to help demonstrate the correct order of operations in a calculation. cimt-order-of-operations

From CIMT, one of their interactive sections gives examples and exercises.

Of course, you can always check answers on WolframAlpha!
Check answers on WolframAlpha

On Transum there are self marking exercises on Order of Operations (note the three levels available).

Transum Order of Operations

Transum – Order of Operations

game from mathFROG allows you to practice against the clock.

Maths Frog

Graspable Math

Try Graspable Math which offers a highly innovative interface for mathematical notation. You can read the Graspable Math story here.

You can learn a great deal about Graspable Math simply by experimenting, selecting Explore Algebra takes you to the interface which is intuitive; you can also find plenty of help and tutorials on the Learn section of the site, note the Gesture Library as well as the video tutorial collection.  There is a YouTube channel here.

Graspable Math is very easy to use, I decided I would solve an equation and wanted to show all the steps. I have used the method of selecting and holding the = sign to start as you can see illustrated in the video above; I was then able to enter an operation to apply to both sides of the equation.

equation to solve

equation to solve 2

We can also illustrate the solution graphically by inserting a graph to open a GeoGebra window.
equation & graph

Each expression has a circle at the end – simply drag that to the GeoGebra window. You will sometimes see more than one circle at the end of an expression, select to separate expressions hence showing all steps clearly.


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 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 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.

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.