Mathematics 7–10 · Year 10
Networks from the school map: degrees, Euler trails and Euler's formula
Measurement and space
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The idea
A map of buildings and paths becomes a network of vertices and edges, and counting the vertices of odd degree decides whether every path can be walked exactly once, as Euler showed for the bridges of Königsberg.
Safety card
Hazards
- walking through busy areas at recess
Controls
- stay on marked paths
- walk in pairs
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.
What you need
- A site map of the school (or of a local park) showing buildings and the paths between them
- Tracing paper and pencil
How to do it
- Mark each building or junction as a vertex and each path as an edge on tracing paper over the map; label the vertices.
- Count the degree of each vertex and the totals V, E and F (regions, counting the outside) for the drawing; check V − E + F = 2 for the planar network.
- Count the vertices of odd degree; if there are none, plan a walk that uses every path once and returns; if there are two, plan one that starts and ends at those two; if more, explain why none exists.
- Walk the planned route at recess with the map and tick off edges; record any edge walked twice.
- Compare with the Königsberg network (four vertices of degrees 5, 3, 3, 3) and state why no trail exists there.
What you should see
Every network satisfies the handshake rule (the sum of degrees is twice the number of edges) and every connected planar drawing satisfies V − E + F = 2; whether the school map has an Euler trail is decided by counting its odd vertices: with 0 or 2 the planned walk covers every path exactly once, and with 4 or more the walk must repeat at least one path. Königsberg has four odd vertices and so has no trail. The learner knows it worked when the counted degrees sum to 2E, Euler's formula balances, and the planned walk succeeds exactly when the odd-vertex count is 0 or 2.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A network needs to be drawn to scale.
- A trail exists whenever the network is connected.
- Crossing edges in a drawing count as vertices.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Introduction to networks (Path); page read 2026-09-22MA5-NET-P-01
- Australian Curriculum v9AC9M10SP02
Sources
The pages the author read to write this activity.