Graphs can represent roads, railways, computer connections, delivery routes and many other systems. When the edges carry numbers, the aim is often to find a route with the smallest total weight.

In this lesson, learners explore:

• weighted graphs and what their weights can represent;
• calculating and comparing the weights of complete paths;
• finding shortest routes by inspection;
• using Dijkstra’s algorithm step by step;
• recording tentative distances and settled vertices;
• understanding why the cheapest next edge need not lead to the cheapest complete route;
• investigating how closures and changed weights affect a network.

The download includes the main printable lesson and a supplementary booklet containing additional examples, answer checks and complete model solutions.

Suitable for learners aged 10 and over. Familiarity with paths and basic graph terminology is helpful.

This is Lesson 6 and the conclusion of the 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 .

Learners replace each region with a vertex and join vertices whose regions share a border. This transforms a visual map-colouring problem into a graph-colouring problem that can be explored systematically.

This lesson introduces:

• turning a map into an adjacency graph;
• proper colourings;
• the chromatic number of a graph;
• finding colourings that use as few colours as possible;
• explaining why fewer colours cannot work;
• the statement and history of the Four Colour Theorem;
• applications such as timetable and scheduling problems.

The download includes the main lesson and a supplementary examples booklet with further problems, answer checks and model solutions.

Suitable for learners aged 10 and over. No previous experience of graph colouring is required.

This is Lesson 5 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 .

This complete printable course guides learners from the basic language of vertices, edges and degrees to Euler trails, map colouring and shortest-path algorithms. Each lesson develops the ideas through diagrams, questions and investigations, while the accompanying Examples booklets provide additional problems, answer checks and complete worked solutions.

This complete course includes:

• Lesson 1: What is a graph?
• Lesson 2: The Bridges of Königsberg
• Lesson 3: Paths, cycles and Euler trails
• Lesson 4: Drawing without lifting your pencil
• Lesson 5: Colouring maps
• Lesson 6: Networks and shortest paths
• a combined Complete Course teaching booklet;
• concise answers and discussion;
• six supplementary Examples booklets;
• 48 additional examples and six challenges;
• complete worked solutions.

Suitable for learners aged 10 and over. No previous knowledge of graph theory is required. Younger learners may benefit from working alongside an adult, particularly in the later lessons.

☕ 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.

This is the complete six-part Beyond Curriculum: Graphs course from Sumthing Else.

Instead of relying on trial and error, learners count the degree of each vertex and use the number of odd vertices to predict whether a drawing is possible. They then decide where a successful route must begin and end.

This lesson explores:

• representing drawings as graphs;
• counting odd- and even-degree vertices;
• deciding whether a one-stroke drawing is possible;
• choosing the correct starting and finishing vertices;
• distinguishing crossings from marked vertices;
• explaining failed attempts mathematically;
• repairing and designing one-stroke drawings.

The download includes the main printable lesson together with additional examples, answer checks and complete model solutions.

Suitable for learners aged 10 and over. Familiarity with vertices, edges and degrees is recommended.

This is Lesson 4 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 .

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.

☕ 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 .

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 .