# Leonhard Euler

This list of resources is to accompany the biograph of Leonhard Euler posted in the Mathematics Resources Group in April 2013.

The resources are all connected withh the work of Euler and show that thew work he did is still used in mathematics lessons today.

### Mensuration

Designed to improve confidence in mathematics, these resources from the Centre for Innovation in Mathematics Teaching, focus on mensuration and were developed particularly for primary teachers and those non-specialists who teach mathematics in the lower secondary years.

Section B activity 7.8 provides inspiration for exploring Euler's Formula investigating the connection between the number of edgaes, the number of faces and the number of vertices of a solid.

I like the extensions in this tasks which asks the question whether Euler's formula still holds when the solid contains a hole.

### Crystal Shapes 1

This cre8ate resource considers the nets of crystals. The investigation explores Euler’s formula, which is the relationship between the number of vertices, edges and faces for each crystal, before students test their conjecture on other solids.

This is useful when students are creating, using and testing formulae and demands students can substitute numbers into a formula.

### Chinese Postman Problems

In this activity students use the Route Inspection Problem, to solve practical problems.

The College Open Day problem provides an introduction to the concept, and asks students to investigate the minimum distance someone would have to travel to deliver leaflets along all the streets near to a college, starting at and returning to College.

The second problem, called Easter Parade, requires students to find an Eulerian trail for a network with four odd nodes

### Königsberg Bridges

The bridges problem, devised by Leonard Euler living in the town of Königsberg, Russia which was divided into four different areas by the river Pregel. There were seven bridges. Euler asked whether it was possible to walk around Königsberg crossing each bridge once but no bridge more than once. This classic network problem is a great illustration of how mathematics can be used to prove what is and is not possible and explain why.

The activity concludes with nice extension activities looking at connections between areas, vertices and edges.

### Working for Efficiency

This Cre8ate maths topic looks at simplified versions of three different network problems that are encountered in practical logistical planning.

Paper rounds offers the opportunity for reasoning and proof as the arguments needed to establish Euler's theorem are within their grasp. It also, along with Cable connections and Deliveries, offers opportunities for the mathematical skills of planning, being systematic, recording and logical experiment.

The algorithmic thinking developed is picked up in key stage 5 in the decision maths curriculum.

### Focus Year 7/8 Shape and Space Extension

Chapter two of the text book investigates networks. Tasks involve finding traversable networks, explores the Königsberg Bridges problem and the Travelling Salesman problem.

Whilst this features in an extension book aimed at year 7/8 the ideas may be adapted for students studying decision mathematics in later years.

### The Exponential Function

This investigation, aimes at advanced level students, explores the exponential function leading to students developing a vaule for e, Euler's number, 'one of the most important numbers in mathematics and physics'

### 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 resource, definitely aimed at extension work for A level students looks at Euler's theorem when dealing with the number of numbers that are coprime with a number n.