Further Mathematics A Level

A Level Further Mathematics
Resources are not always so easy to find for Advanced Level Further Mathematics – so a project started and which I will work on in the next few weeks – a new series of Further Math pages to have the Further Mathematics information and resources in one place.

You will find content on each of the pages here. With compulsory Pure Content for Further Mathematics for all the awarding organisations, it is useful to look at material from all the examination boards for that content – specimen materials and topic tests for example.

The exam boards are all working hard to provide materials for teachers and students. Some of the resources linked to here such as practice and mock papers are recently published. Edexcel’s Teacher Support materials for Further Mathematicsfor example, were published on 20th February.

Watch this space….!

Statistics with Desmos

Normal Distribution

Desmos now provides several functions for computing statistical properties from lists of data.

It is also possible to perform basic statistical tests, count combinations and permutations, and work with distributions. The supported functions can be found on page 11 of the Desmos User Guide. Part of the Learn Desmos series, you can see many functions demonstrated in this series of graphs.

Also very helpful is this Desmos article, from which we see that the Normal, Poisson and Binomial distributions are all supported.

With a requirement for Advanced Level being the use of technology to explore data, the Desmos statistical functions will provide a very valuable addition to our lesson toolbox.

dotplot and boxplotIn this article on Data Visualisations, we see that boxplot, dotplot and histogram functions are available.

Boxplots Activity Builder

Boxplots – Desmos Activity Builder

This activity builder includes excellent tasks for students to explore boxplots.

Bob Lochel has uploaded this helpful video illustrating the use of these functions.

For Valentine’s Day …

An annual update, ideas and resources for Valentine’s Day …

From Perton Maths Department, try these Valentine puzzles.

From @OCR_Maths, we could try this puzzle.
OCR Valentine Puzzle
OCR often share some great maths puzzles, look out for them.

The excellent Maths Careers site is managed and maintained by the Institute of Mathematics and its Applications. If your students wonder where Mathematics is used they will find plenty of answers here. See for example Who employs mathematicians?  


For Valentine’s day we have an appropriate article from Maths Careers, did you know that 6 is a kissing number?! And for your Further Mathematicians, What is the equation for a heart?


Also from Maths Careers, see this post with instructions on how to make this wonderful pair of linked Möbius hearts.


If you wish to get creative and try this I advise watching the Numberphile video carefully; following the instructions worked as you can see from my creation here! I can verify that unless you follow the instruction to make sure the twist in each strip is in a different direction you will end up with a mess! Quite an interesting mess but certainly not two hearts!….
Note the Desmos graphs on my strips. I created a file in Word valentine-mobius-hearts (or pdf: valentine-mobius-hearts) with Desmos images in a table. Adding dotted borders to the table gives guidelines for cutting. I began each cut by using the end of a paperclip to pierce the paper.

To create my strips I printed the document and then printed again on the reverse. I then cut out and trimmed the strips so there was no white space at the end – the picture here has been made using strips 10 cells long.

NASA Valentines
For an alternative source of Valentine’s cards, we can turn to NASA! Take your pick from this post or these fabulous NASA images. (And don’t forget that NASA provides us with many Mathematics resources.)

Transum Valentine Puzzle
Here’s a Valentine logic starter from Transum.

This Valentine Relay from Chris Smith is excellent as are all the other relays in this excellent set of resources. You can find more excellent resources from Chris on TES and follow him on Twitter here.valentine-relay

Valentine's Treasure Hunt

From Maths4Everyone, try this Valentine’s Day Maths Treasure Hunt. Students have to work out the costs of gifts, meals and other outings for Valentine’s Day.

You can find a whole collection of Treasure Hunts from Maths4Everyone here.

Also on TES resources, from a favourite TES author of mine, Andy Lutwyche, you will find this Valentine’s card Area and Perimeter problem.


Think Maths Parametric Heart

From the MEI Archives, the February 2015 edition of the MEI Monthly Maths Magazine includes some connections between maths and Valentine’s Day. On page 7 note the article “A Happy Ending” which includes references to some Numberphile videos, Professor Ron Graham discusses the Happy Ending Problem and from Dr Emily Riehl, The Stable Marriage Problem. We also have a great Parametric Heart spreadsheet from Think Maths.

This edition of the magazine includes some lovely activities which link paper folding and proof.

Remaining with the loving theme you can express your feelings for WolframAlpha!

I Love YOU

and from the WolframAlpha archives, Computing Valentine’s Day.

Factorisation of Quadratic Expressions

When factorising quadratic expressions I always encourage students to check coefficients first, if the coefficient of x2 and the constant are prime for example they clearly do not need elaborate methods.

Some students have difficulty with the splitting the middle term method; if one must have a recipe to follow – try the box method.

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


PhET Sims Area Model Algebra

We could illustrate on the brilliant Area Model Algebra from PhET Sims.

For instructions on the method:

Quadratic Factorisation Box method (pdf file)

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

Lyszkowski's method

Lyszkowski’s method

I came across the method through Colin Foster’s wonderful Mathematical Etudes site. To quote Colin Foster:

Mathematical Etudes are creative, imaginative and thought-provoking ways to help learners of mathematics develop their fluency in important mathematical procedures. They are an alternative to traditional, tedious exercises.

Note the Etudes by topic at the foot of the page; Number, Algebra, Geometry, Probability & Statistics are available. Looking at Algebra for example, under Solving Equations we see Connected Quadratics which is where I found Lyszkowski’s method of factorising quadratics.

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
Try this excellent lesson plan of  Colin Foster’s on Quadratic Equations. I really like the starting activity in this lesson which should promote a deeper understanding of factorising quadratic expressions.