Coincidences
Carom Maths provides this resource for teachers and students of A Level mathematics. This thought provoking presentation introduces the mathematics of coincidences, challenges students to produce a spreadsheet which shows the probabilities of sharing a birthday, as well as highlighting the coincidence of sharing friends on social…
Descent
Carom Maths provides this resource for teachers and students of A Level mathematics. This activity explores an example of the Descent method. This method was popularised by the famous mathematician Pierre de Fermat and has since been used by many other number theorists. The aim of this demonstration is to show that the equation…
Repunits
Carom Maths provides this resource for teachers and students of A Level mathematics. A repunit, which stands for repeated unit, is a number like 11, 111, or 1111 that contains only the digit 1 and extends to number bases other than base ten. This presentation challenges students to prove that any repunit having a composite (non-prime)…
Transformation of Shapes
This interactive resource, produced by the University of Leicester, is designed to enable students to explore transformations of shapes including translation by a vector, stretches, rotations around a point, reflection in the axes and reflections in the lines y=x and y=-x. The first activity draws a parallelogram which is translated…
Vectors Introduction
This interactive resource, produced by the University of Leicester, is designed to enable students to explore vectors, beginning with the definition of a vector followed by the algebra of vectors and the scalar product. The opening slides of the presentation explain the difference between scalar and vector quantities followed…
Quadratic Equations
This interactive resource, produced by the University of Leicester, is designed to enable students to explore what is meant by a quadratic equation, the meaning of the coefficients of a quadratic equation and to be able to solve quadratic equations. An introduction page gives examples of where quadratic equations can be found…
Transformation of Functions
This interactive resource, produced by the University of Leicester, is designed to enable students to explore transformations of functions including translations, stretches, reflection in the axes and rotations. The first activity uses function notation to explore translations parallel to the x axis and translations parallel…
The Normal Distribution
These Making Statistics Vital resources are provided for those studying the S2 Module of A Level Statistics and present students with the opportunity to investigate problems, which require a different approach to those questions usually posed, on the topic of the Normal Distribution. The coffee problem – poses the problem…
Binomial Distribution
This Making Statistics Vital resource consists of independent activities, covering aspects of this area of statistics for those studying the S1 Module of AS Level Statistics. The tasks are designed to encourage students to apply their knowledge of binomial distributions to solve challenging problems, giving them the opportunity to…
Sampling and Hypothesis Testing Using the Normal Distribution
These Making Statistics Vital resources are produced for those studying the S2 Module of A Level Statistics provide practical experience of sampling techniques and highlight the importance of using suitable sampling procedures. Sampling – Students are given 100 natural numbers and are asked to estimate the mean by taking…
The Distribution of Prime Numbers
Carom Maths provides this resource for teachers and students of A Level mathematics. In this activity the distribution of prime numbers, proved by Hadamard and de la Vallée Poussin in 1896, is investigated using an Excel spreadsheet program, Autograph and the Prime Number Theorem. The resource is designed to explore…
The ABC Conjecture
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation begins by introducing students to the technique used to find the radical of an integer before progressing to the ABC conjecture, which is recognised by mathematicians as being an important unsolved problem in number theory. The…
The Four Colour Problem
Carom Maths provides this resource for teachers and students of A Level mathematics. Colouring maps, so that no two countries sharing a border are shaded with the same colour, is the focus of this presentation, which includes both the history and mathematical proofs to this problem. This activity is designed to explore aspects…
The Exponential Function
This interactive resource, produced by the University of Leicester, is designed to enable students to explore the nature of the exponential function and to explore the derivative of the exponential function. An introduction page sets out some basic information about exponential functions leading to a definition of the exponential…
Triominoes
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation shows how, when placing triominoes onto a chessboard, there is always one empty square. An algebraic proof is developed to show that the empty square will always appear in the same location, or in one of its rotations. The…
Quadratic Reciprocity
Carom Maths provides this resource for teachers and students of A Level mathematics to explore aspects of the subject which may not normally be encountered, to encourage new ways to approach a problem mathematically and to broaden the range of tools that an A Level mathematician can call upon. This presentation on Quadratic reciprocity…
Geodesics
Carom Maths provides this resource for teachers and students of A Level mathematics. A geodesic is the shortest distance between two points and here two problems involving cuboids and cylinders are investigated. The resource is designed to explore aspects of the subject which may not normally be encountered, to encourage new…
Adding Again and Again...
Carom Maths provides this resource for teachers and students of A Level mathematics. Ergodic maths is a fairly new branch of mathematics which concerns itself with repeated processes and some of the basic theorems are introduced here. The activity is designed to explore aspects of the subject which may not normally be encountered,…
The Logistic Map and Chaos
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation provides a spreadsheet, which models a population of mice, and allows students to vary the input and thus change the behaviour of the model. There are some surprising discoveries to be made when the results are analysed. The…
The Cross-ratio
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation looks at the theorem of cross-ratio, of four complex numbers, which is of great interest in a field of mathematics known as projective geometry and has an ancient history. The activity is designed to explore aspects of the…
Infinity
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation highlights the concept of infinity and its implications, as well as introducing the mathematician Georg Cantor and the arguments relating to infinity that he came up with. The resource is designed to explore aspects of the…
The Propositional Calculus
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation challenges students to use the rules they have been given to develop a logical system and deduce theorems which prove that the system is complete and consistent and introduces the idea of truth tables to achieve this aim. The…
Triangle Centres
Carom Maths provides this resource for teachers and students of A Level mathematics. A triangle can have more than one centre and this presentation demonstrates the application of vectors, in three different situations, to show that the circumcentre, the centroid and the orthocentre of a triangle are indeed positioned at the centre…
Elliptic Curves
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation uses the fact that 1729 is the smallest number that can be expressed as two cubes in two different ways to introduce the topic of Elliptic curves, which are used more and more in the field of number theory. The activity is…
Zeroes of a Recurrence Relation
Carom Maths provides this resource for teachers and students of A Level mathematics. The Fibonacci sequence is an example of a linear recurrence relation (LRS). A matrix is used to calculate future terms, as well as running the sequence backwards to see how many zeroes appear. Algebra is used to prove the maximum number of zeroes…
Differentiation Exercise
This interactive resource, produced by the University of Leicester, is designed to enable students to explore the differential function of a polynomial and conclude by forming a generalisation. An instruction page explains to students that they should draw a polynomial. Then by sliding the slider the gradient of the curve is plotted,…
Could Pi Be 3?
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation investigates the value of π in different types of geometry and provides a link for students to experiment with Hyperbolic geometry. The classic definition of a distance function, or metric, is given and a version of the Manhattan…
Wallpaper Patterns
Carom Maths provides this resource for teachers and students of A Level mathematics. Wallpaper patterns all have one thing in common; they are all produced by repeating a single fundamental tile by translation, that one basic rule gives rise to a range of possible patterns, each with their distinctive symmetries. Here the number…
Buffon's Needle
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation introduces an explanation and a simulation of Buffon's Needles experiment, that yields an approximation for π, before developing the experiment further by asking what would happen if the shape of the needles were altered. The…
Hyperbolic Geometry
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation introduces Euclid's axioms and theorems and provides historic information on the development of geometric principles, as well as discussing hyperbolic geometry. A link to Non-Euclid software, from Joel Castellanos's…
Curves of Constant Width
Carom Maths provides this resource for teachers and students of A Level mathematics. This presentation introduces the Reuleaux Triangle and shows how to construct a curve which has a constant width, as well as Barbier's Theorem and the Isoperimetric Inequality. Three dimensional shapes are also investigated to consider if…
Cyclotomic Polynomials
Carom Maths provides this resource for teachers and students of A Level mathematics. Cyclotomic polynomials are explained in this presentation, which uses a regular polygon to illustrate the complex roots of unity and establish which of those are primitive, before demonstrating some intriguing algebra. The activity is designed…