Learners explore paths, cycles and trails before meeting Euler trails: special routes that use every edge exactly once. Carefully chosen examples help distinguish ideas that can initially look very similar.

This lesson includes:

• clear definitions of paths, cycles and trails;
• examples showing the difference between repeating vertices and repeating edges;
• practice tracing and recording routes;
• an introduction to Euler trails and Euler circuits;
• puzzles requiring both experimentation and explanation;
• answers and discussion;
• supplementary examples with worked solutions.

Suitable for learners aged 10 and over. It works best after Lessons 1 and 2 but can also be used independently with some adult guidance.

This is Lesson 3 of the six-part Beyond Curriculum: Graphs course.

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This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi .

The challenge sounds simple: can you walk through the city and cross each of its seven bridges exactly once? Learners replace areas of land with vertices and bridges with edges, turning a complicated map into a much simpler mathematical graph.

This lesson develops:

• translating maps and routes into graphs;
• counting the degree of each vertex;
• recognising the importance of odd and even degrees;
• testing possible routes systematically;
• explaining why some route puzzles are impossible;
• mathematical reasoning through a famous historical problem.

The download also includes further practice and worked solutions in the supplementary examples booklet.

Suitable for learners aged 10 and over. Lesson 1 is helpful but not essential.

This is Lesson 2 of the six-part Beyond Curriculum: Graphs course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi .

In this introductory lesson, learners discover how graphs can represent connections between places, people and objects. They learn the meanings of vertex, edge and degree, practise translating everyday situations into graphs, and investigate how the structure changes when edges are added or removed.

This lesson includes:

• a clear introduction to vertices and edges;
• practice counting vertices, edges and degrees;
• real-world examples of networks;
• opportunities to draw and interpret graphs;
• answers and discussion;
• a supplementary booklet containing further examples and worked solutions.

Suitable for learners aged 10 and over. No previous knowledge of graph theory is required.

This is Lesson 1 of the six-part Beyond Curriculum: Graphs course.

☕ Free to download

This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi .