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.

geogebra-correlation

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.
linear-regression

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.
guess-the-correlation

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.
spearmans-rank-calculator

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

 

Integration

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.
Integration

 

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 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 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.
STEP CU


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.
underground-mathematics

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.

Constructions

bisect-angle-math-open-ref
Another post to clarify the requirements for the UK GCSE Mathsematics specification. The constructions required are as follows:

Use the standard ruler and compass constructions

Math Open Ref Instructions
Note that all these Math Open Reference demonstrations are available as a printable atep by step instruction sheet, see for example the guide to bisecting an angle.

The AQA specification also includes a note that Constructing a 60° angle is required. This is shown here by constructing an equlateral triabgle.
​Students will be asked to use these constructions  to construct given figures and solve loci problems. Students should also know that the perpendicular distance from a point to a line is the shortest distance to the line.
The constructions shown here are all from the excellent Math Open Reference by John Page. (There are many more constructions given on the site which are not a GCSE requirement). Another very useful source for demonstrations comes from BBC Bitesize on Loci and Constructions which gives step by step diagrams and instructions.BBC Bitesize Constructions
The BBC site also has clear examples of solving problems using constructions.

 

 

Transformations of Graphs

Translations Graphs

In this post we will look at reflections and translations (the requirement for the UK GCSE syllabus). For GCSE students are required to sketch translations and reflections of a given function.

Experiment with the following Desmos pages to examine these transformations, f(x) + a, f(x + b), -f(x) and f(-x) where a and b are integers.

Translations

Reflections - Graphs

Reflections

Equations of Transformed Graphs

Note that you can work out the equation of the transformed graphs.
Suppose we wish to work out the equation of  f(x) = x2−5x+4 after reflection in the x axis.
To reflect in the x axis the transformation is –f(x)
So f(x) becomes  –f(x) and we have  – (x2−5x+4) = – x2 +5x-4

To work out the equation of  f(x) = x2−5x+4 after reflection in the y axis.
To reflect in the y axis the transformation is f(–x)
So f(x) becomes  f(–x) so we replace x by –x
and we have (–x)2−5(–x)+4 =  x2 +5x+4

Suppose we wish to work out the equation of  f(x) = x2−3x+2 after a translation of 3 units to the left parallel to the x axis as shown in the first image in this post.
f(x) = x2−3x+2, we need f(x+3) for the equation of the transformed graph
f(x+3) = (x+3)2−3(x+3)+2 = x2+6x+9−3x−9+2 = x2+3x+2

Diagnostic Questions

GCSE Diagnostic Qns
To use the links here you will need to be logged in to the brilliant Diagnostic Questions site. Students, just in case your school does not use Diagnostic Questions you can create your own free account here. On Craig Barton’s Diagnostic Questions site you can find collections of GCSE 2017 examination questions from AQA, OCR and Pearson Edexcel (scroll down each of the pages linked to for numerous quizzes on different topics on the GCSE syllabus).

Craig Barton who is the creator of the excellent Diagnostic Questions site has compiled a collection of some of the worst answered new specification GCSE questions on Diagnostic Questions. See also his Question of the Week selection.

Each set consists of 10 questions chosen from all 3 awarding bodies. As well as the online version of each quiz, the design team have also created a booklet in case you wish to print it out.

Badly Answered GCSE Questions – Foundation 1
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 1
— Try the questions online or download the booklet

Foundation Badly Answered Questions – Set 2
Online       Paper

Higher Badly Answered Questions – Set 2
Online      Paper 

Badly Answered GCSE Questions – Foundation 3
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 3
— Try the questions online or download the booklet

You can also learn on the move with the free Diagnsotic Questions app.
GCSE 2017

Note that Advanced Level Questions are also available.