Mathematics 11–12 · Year 11

The school grounds as a network: minimum spanning tree and shortest path from measured paths

Networks, paths and trees: Network concepts (Mathematics Standard, Year 11); Networks, paths and trees: Shortest paths and spanning trees (Mathematics Standard, Year 11)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

Buildings joined by measured paths form a weighted network, in which a minimum spanning tree connects every building with the least total length and a shortest path may use edges the tree does not include.

What you need

A trundle wheel or a 50 m tape, or the distance tool in NSW Spatial Services SIX Maps
A site plan or SIX Maps aerial view of the school with 6 to 8 buildings marked
Coloured pencils and a spreadsheet

How to do it

Choose 6 to 8 buildings as vertices and measure the length of every direct footpath between them in metres.
Draw the weighted network and list the degree of each vertex.
Find the minimum spanning tree with Kruskal's or Prim's algorithm; it is the least total length of cable that would join every building along existing paths.
Find the shortest path from the front gate to the library by Dijkstra's method.
Check whether the shortest path lies inside your spanning tree, and describe a real reason the shortest path might not be the best one (stairs, a locked gate, a road crossing).

What you should see

Worked check on the model's test network (vertices A to F, edges AB 40, AC 65, BC 30, BD 55, CD 45, CE 70, DE 35, DF 60 and EF 25 m, total 425 m): the minimum spanning tree uses EF, BC, DE, AB and CD for a total of 175 m. The shortest distance from A to F is 155 m, reached by two routes that tie, A-B-D-F and A-B-D-E-F; both use BD, which is not in the spanning tree, and the route from A to F inside the tree is 175 m. The learner's own network gives its own totals, which a classmate can check by running the algorithm again. The learner knows it worked when the tree has one fewer edge than there are vertices, contains no cycle, and no single swap of edges shortens it.

What changes

What you change: the weighted network: the buildings chosen as vertices and the measured footpath lengths as weights
What you measure: the total length of the minimum spanning tree and the shortest gate-to-library distance
What you keep the same: the same 6 to 8 buildings for every algorithm
only existing footpaths counted as edges
every path measured with the same trundle wheel, tape or map tool

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The minimum spanning tree contains the shortest route between any two vertices; in the test network the shortest A to F route is 20 m shorter than the route in the tree.
A minimum spanning tree can include a cycle if the edges are short; a tree has no cycles by definition.

Safety card

Low riskLearners carry it out

Hazards

measuring near car parks or driveways
walking backwards with a trundle wheel

Controls

measure only on footpaths inside the school
work in pairs, one watching ahead

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).

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 Standard 11–12 Syllabus (2024), Year 11 focus area Networks, paths and trees; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-11-07
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities