Introduction to Generating Functions for A-Level Maths Students
Learn the fundamentals of generating functions and experience studying maths at a university level.
Duration
3 weeksWeekly study
3 hours
Introduction to Generating Functions for A-Level Maths Students
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Discover what it’s like to study mathematics at University
This three-week course from Manchester Grammar School will give you an opportunity to study a university-level topic in-depth.
As an A-level mathematics student, you’ll be able to further your learning through the study of generating functions and obtain a sense of what it’s like to study maths at university.
Explore the capabilities of generating functions as a problem-solving tool
Generating functions is a powerful mathematical tool that can be applied to solve problems in many different contexts.
On this course, you’ll be introduced to generating functions and its applications in mathematics. In the first week of the course, you’ll define generating functions and work with various mathematical problems to understand the types of problems generating functions can solve.
Learn how to use generating functions for probability distributions and more
This course will cover roughly one topic per week and is structured to provide you with an in-depth understanding of how to build generating functions for a wide range of mathematical problems.
You’ll learn how to derive the nth term for a selection of recurrence relations, solve combinatorial problems, work with standard probability distributions, and more, all using generating functions.
Decide your academic trajectory with help from an expert educator
The lead educator on this course is an experienced A-Level teacher who has studied generating functions at a high level during his own undergraduate and postgraduate studies.
With the insight and expertise of your lead educator, you will have an understanding of what it means to study maths at a higher level, and be better equipped to make a decision in pursuing maths at university.
Syllabus
Week 1
Introduction
Introduction to the course
I will introduce the course and give you some mathematical problems to try. These problems will be referred to later in the course and will give you an idea of the types of problems generating functions can help us to solve.
Introduction to generating functions
In this activity, generating functions will be formally defined. This is the most important lesson of the entire course because all future activities will build upon the ideas introduced here.
Recurrence relations
In this activity we will consider how generating functions can be used to solve recurrence relations.
Week 2
Combinatorics and generating functions
Combinatorics overview
A brief recap of the basics of combinatorics
Building generating functions
In this activity we will learn how to build generating functions that help us to solve combinatorics problems.
Solving combinatorics problems
In this activity we will learn how to solve a variety of combinatorics problems using generating functions.
Week 3
Probability and generating functions
Probability generating functions
In this activity we will learn how to build generating functions for probability distributions.
Sums of random variables
In this activity we will consider an important result which enables us to quickly find the generating function for the sum of random variables.
Random walks (advanced)
In this activity we will introduce a process known as random walks and a related generating function.
Closing remarks
A final chance for you to share your thoughts with other learners and closing remarks from the course creator.
Learning on this course
On every step of the course you can meet other learners, share your ideas and join in with active discussions in the comments.
What will you achieve?
By the end of the course, you‘ll be able to...
- Explain what generating functions are and why they are useful
- Apply generating functions to find the nth term for sequences defined using recurrence relations
- Solve combinatorics problems
- Apply generating functions in the context of combinatorics
- Apply generating functions in the context of probability distributions
Who is the course for?
This course is designed for A-Level mathematics students who are considering applying to study mathematics at university, but anyone with an interest in mathematics is welcome to join the course.
What software or tools do you need?
We will be using Wolfram Alpha in the course to do simple mathematical calculations (the free version is absolutely fine).
Who will you learn with?
Mathematics Teacher at The Manchester Grammar School with a particular interest in Mechanics and Probability. I am always on the lookout for interesting ways to introduce topics in the classroom.
Who developed the course?
Manchester Grammar School
Our history dates back to the time of Henry VIII, when The Manchester Grammar School was founded in 1515 by Hugh Oldham, Bishop of Exeter, to provide ‘godliness and good learning’ to the poor boys of Manchester.
The School proceeded to build a reputation as one of the country’s leading educational establishments, a position it still holds today as an independent day school.
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