This unit introduces the concepts of probability and chance. The basic concept of theoretical probability is also introduced (events, frequency etc.). Practical examples are used to illustrate different concepts and the basic laws and rules governing probability are outlined (e.g. Laplace's rule and the law of large numbers). The unit ends with an introduction to the probability of compound events and focuses on examples of the union and intersection of events.

The Dutch artist Mauritis Cornelis Escher (1898-1972) is, perhaps, one of the most highly respected artists in the world of mathematics. He is, without a doubt, one of the most successful contemporary "mathematical artists". His work reflects a deep understanding of geometry. However, the mathematical content of his work is much closer to you than you think. Just look around you, perhaps with a slightly different perspective to normal, and you will start to recognise different areas of maths ...

These pages aim to present some of the most important mathematical problems in chronological order. It is not intended to be an exhaustive explanation of each problem but to simply show the evolution of mathematical thought throughout history. In each section there is an outline of the most relevant historical events of the era and the maps indicate the area of interest, both from a mathematical point of view.

An envelope is an object which is used a lot in everyday life. We have all used envelopes to put letters or important documents into and as students we have often had to use them. However, as well as being very useful objects, envelopes are also interesting to analyse from a mathematical point of view. Their geometrical shape is of interest firstly to people who have to make them and secondly to maths students who can use them to review a few interesting concepts such as line symmetry and sim...

This unit focuses specifically on "Simulation as a formal model". Most of the ideas and content of this unit have been taken from the last three units of Martin Gardner's book "Wheels. Life and other Mathematical Amusements".(W.H. Freeman and Company 1983), which specifically focus on the Game of Life. It uses a simple game to simulate social or biological behaviour based on a simple mathematical model Despite its simplicity, we shall look in more detail at its far reaching implications.

In this unit we will learn how to construct and 'define' star-shaped polygons. The first part involves a description of how to construct them, using a ruler and compass, although we will be using the computer to draw them. The second part is somewhat more abstract as we discover which conditions we need to produce a star-shaped polygon and which properties it has. In both parts we use algorithms to give constructive explanations, i.e. we define star-shaped polygons as shapes obtained by follo...

In this unit we will focus on transformations on the plane in a very intuitive and practical way. The idea is to try and "visualise" the movements before conceptualising them. In the last section, Symmetry, we shall look at how the three types of transformation relate to each other. The first section about vectors is an introduction to them and is necessary in order to fully understand the notion of translation.

The Greeks were very interested in measuring things by comparing them to a set unit. They found that there was a relation between pairs of measurements such as the ratio or the quotient. They were also interested in proportion e.g. when two ratios are equal. They were also interested in "beauty". They referred to the most beautiful proportion as golden ratio (from gold) and the number represented by the symbol F is the golden number. It appears in nature and the Greeks were the first to apply...

Before beginning this unit we assume that students have already worked with some functions, in particular functions which deal with proportion. Through the activities in this unit we expect the students to build on their existing knowledge to find out more about functions and learn how to interpret graphs of functions. The activities focus on plotting points from coordinates and reading straight line graphs, i.e. being able to identify different types of graphs depending on their gradient and...

This unit deals with some formulae connected to proportionality between segments and Thales' theorem. Rather than giving rigorous, step-by-step demonstrations, this unit offers the advantage of verifying conclusions and properties in a straightforward manner. To conclude, different examples of similarity in triangles and polygons in general is also focused on, using an interactive approach.

This unit deals with quadratic equations. It looks at how to interpret them geometrically and focuses on a formula that can be used to solve them. At the end of the unit there are some different exercises and problems that can be solved by using these equations.

This unit introduces students to the concepts of arithmetic and geometric progression. It also demonstrates how to find the terms in a progression, the general or nth term and the sum of terms.

Throughout history geometric shapes have been used as decorative motifs. Although many objects such as containers, fabrics, doors, windows and other bits of architecture have all been decorated using regular geometric designs we are going to focus specifically on walls and floors. Walls and floors are decorated using mosaics. A mosaic is the complete covering of a plane area with pieces usually called floor tiles that fit together exactly without covering each other or leaving any gaps betwee...

This unit is intended for students in their third year of secondary education. It aims to reinforce their ability to connect algebraic terms to data, which is usually given in the form of a table. It also helps them to match tables of values to given graphs and introduces them to the basic characteristics of graphs. Furthermore, there are activities which encourage students to compare two graphs and give a detailed interpretation of them. In addition, the data used in the examples is based o...

This unit is designed to show students, in their third year of secondary education, how to recognise arithmetic and geometric series, how to use certain strategies to solve problems and to try and work out general rules to express these series using symbols. The activities in this unit are designed to be varied, interesting and motivating but they also lend themselves to small classroom discussions where pupils can compare their ideas and strategies used during the activities. This will hopef...

Counting was one of the first mathematical activities that primitive man had to learn how to do. Expressing a quantity with symbols meant a great step forward as they were the first number systems. However, they were awkward to work with. The decimal system and place-value notation, which we use nowadays, was thought up in India (VI - IX Century) and the Arabs introduced it to the Western world (X century) where it started to be used in the XIII century. We use the decimal number system as we...

Pythagoras' Theorem is one of the most well-known statements in the world of mathematics. It has been discovered and used by different civilizations during different periods. The Egyptians and the Hindus used it to help construct buildings. The Chinese (who called it "kou ku") also used it to help with construction and later on applied it to astronomy and algebra. There are over two hundred ways of proving Pythagoras' Theorem and there are even books which are specially written to explain the...

In this unit we are going to focus on polygons and their properties. We are going to find out about their angles, area and how they relate to the circumference of a circle in an active way.

Through language we are able to communicate with each other. The most common forms of language are: oral, written and graphical. Nowadays, graphics represent one way of expressing and transmitting information. The media often express information by means of tables and graphs as they offer a fast, visual interpretation as well as showing how certain variables depend on others. Newton (1642-1727) was first to deal with the concept of a function. he used the term "fluents" or "flowing quantities...

Geometry is an essential parts of the game of billiards. Every shot in a pool game is based on forming perfect angles. This unit is purely theoretical. It probably won't turn you into a billiards champion but you will see the game in a different light.