# Further Mathematics Conference 2021 resources

During the spring term 2021, the AMSP held an online professional development conference dedicated to supporting teachers with:

• the teaching of AS and A level Further Mathematics
• preparing students for university admissions tests.

The presentation slides used in the welcome plenary can be viewed here.

When viewing recordings, please note that some sessions used Desmos activities to collect responses and many of these links will no longer be active. Where sessions used Desmos activities that can be translated to use in the classroom, we've included links to editable versions within the presentations.

## Using GeoGebra Classroom in Further Maths (Pure Maths)

GeoGebra is a very powerful dynamic graphing tool that can be used to visualise many of the concepts in Further Maths such as complex numbers, matrices, 3D Geometry and polar curves. The new GeoGebra Classroom feature allows you to easily share activities with classes of students and review their interactions and responses. This is an ideal tool to suit learning in classrooms or remotely. In this session we will demonstrate how to create and assign GeoGebra Classroom activities as well as considering how you can integrate this into your teaching.

This session is suitable for teachers who have some basic familiarity with plotting curves in GeoGebra.

## Why every mathematics teacher needs to know about admissions tests… and what they need to know

More and more universities are making use of mathematics admissions tests either as a compulsory requirement or to reduce offers. In some cases this is to discriminate between A* candidates; in others, it is to encourage all A/A* students to engage in more challenging problem solving, mathematical thinking and communication. This session aims to give an overview of the current situation with pointers towards how you can help your students engage in higher level problem solving, and how this can benefit their A level studies, their university applications and their transition to university.

This session is suitable for all teachers of A level Mathematics/Further Mathematics.

## Energy efficiency: a graphing approach to modelling work and energy calculations (Mechanics)

In this session we will consider the energy calculations in a series of increasingly accurate models of similar systems, looking at conversion from one energy type to another, work done and energy lost. We’ll use a graphing approach which encourages students to consider how the energy in a closed system can vary from one form to another, to ensure that energy added or removed from a system is accounted for correctly and to smooth the transition from a discrete to a continuous approach to these topics.

This session is suitable for teachers who already have an understanding of the work-energy principle.

## Problem solving in Further Maths

Problem solving is not only one of the overarching themes for A level Maths, it is also a part of A level Further Maths. In this session we will look at how problem-solving skills are tested on Further Maths papers. There will then be an opportunity to try out some simple activities and questions that can be used with students to help them develop their problem-solving skills.

This session is suitable for A level teachers with some knowledge of the topics taught in Further Maths.

## Where do students go wrong with proof by induction? (Pure Maths)

When it comes to proof by induction, examiners’ reports often start with statements such as “the majority of students clearly knew how to structure a proof”. However, things generally go downhill from there, and many students struggle to provide an adequate solution that gains full credit. In this session we’ll look at where students go wrong and share some thoughts on why they go wrong, focusing particularly on divisibility.

This session is suitable for all teachers of A level Mathematics/Further Mathematics.

## Linking complex numbers and matrices (Pure Maths)

In this session we will explore and discuss the interconnectedness of two fundamental topics that students meet when studying AS level Further Mathematics. We will look at activities that both new and experienced teachers can use in the classroom for these topics.

This session is suitable for teachers who have some basic familiarity with complex numbers and matrices: i.e. know how to add and multiply them.

## Using probability generating functions to prove key results in statistics (Statistics)

Probability generating functions or PGFs are an alternative way of writing a discrete probability distribution as a polynomial function in a dummy variable rather than as a table. They are a powerful tool in establishing results in statistics, which can be otherwise quite difficult or long-winded to prove. As well as deriving the mean and variance of common discrete distributions, they can be used to work out the distribution of sums of two or more independent random variables. In this session we will explore their properties, see why the sum of two Poisson variables is also Poisson and unpick the relationship between the geometric and negative binomial distributions.

This session is suitable for teachers who are familiar with at least some of the common discrete distributions. No prior knowledge of PGFs is assumed.

## The problem-solving skills required for success in admissions tests

Admissions tests are being taken into consideration for an increasing number of undergraduate maths courses. They not only allow universities to differentiate between able candidates, they also allow students to encounter the level of thinking that is part of a maths rich degree. This session looks at the difference between the problem-solving thought processes needed at A level and those needed for the TMUA, MAT and STEP examinations. The session will look at examples of questions from admissions tests that most students will be able to access before answering the question “how hard can it get?” by looking at a challenging question from STEP III. The session will show how parts of questions and ideas from questions can be incorporated into A level Maths and Further Maths lessons to help students develop those skills that will help them settle into a maths rich undergraduate course.

This session will include materials that require a good knowledge of A level Further Maths as well as some at a more general A level standard.

## Modelling with differential equations (Pure Maths)

Differential equations are one of the most powerful mathematical tools for making sense of the world but some students complete Further Maths without fully gaining a sense of this. In this session we will start by posing the question ‘What changes and why?’ to help us develop mathematical models that lead to differential equations, and then inspect their solutions. Graphing software will be used to aid understanding by showing the results predicted by these models.

This session is suitable for teachers who have some basic familiarity with the solution of first and second order differential equations.

## The travelling salesperson problem: A tour of the main points (Discrete/Decision/Modelling with Algorithms)

It is possible for students to become proficient in the various procedures required for the travelling salesperson problem without ever grasping the bigger picture. What is the point of a lower bound that is clearly not a tour? How does the classical problem differ from the practical problem? This session aims to tie several threads together into a holistic understanding.

This session is suitable for teachers who are already familiar with the basics of the travelling salesperson problem.