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.

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.