Vocabulary
Vertex/node, edge/link, path, degree, connected component.
Learning goals: what you will learn
- explain the central idea: Interpret networks as vertices and edges, analyse connectedness and use network diagrams in practical situations.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
You need basic graph-reading skills, logical reasoning and comfort following multi-step routes. No advanced graph theory is assumed.
Concept teaching
A vertex represents an entity/location and an edge represents a relationship/connection. A network is connected if every vertex can be reached from every other through some path. Always state what the vertices and edges mean in the real context.
Connected
There is a path between every pair of vertices.
Degree
Number of edges incident to a vertex.
Euler polyhedra
For convex polyhedra: V−E+F=2, equivalently F+V=E+2.
Worked examples
1. Translate a transport network
Stations are vertices; direct rail links are edges. Track length may be attached as an edge weight if distance matters.
2. Connectedness
Vertices A-B-C form one component; D-E form another with no link between components. The whole network is disconnected.
3. Degree
If A connects to B,C,D, then degree(A)=3.
4. Seven Bridges idea
Represent each land region as a vertex and each bridge as an edge. The problem becomes one about traversing edges, not geographic shape.
5. Euler formula cube
Cube: V=8,E=12,F=6. V−E+F=8−12+6=2.
6. Home network
Devices/routers can be vertices and communication links edges. A router failure can disconnect a section; the network diagram makes that vulnerability visible.
7. Weighted route
A→B→D costs 4+5=9; A→C→D costs 2+9=11. Shorter weighted route is A-B-D.
8. Food web
Species can be vertices and feeding relationships directed edges. Direction matters because “eats” is not symmetric.
9. Airline comparison
A hub airport has high degree because it connects directly to many locations; high degree may indicate connectivity but not necessarily shortest travel time.
10. Kinship network context
Network diagrams can represent relational structures, but First Nations kinship systems are culturally specific and should be represented only from an appropriate source/context, not invented as a generic graph exercise.
Common misconceptions and corrections
- Not defining what vertices/edges represent.
- Assuming physical drawing distance equals an edge weight.
- Confusing connected with complete: connected does not mean every vertex links directly to every other.
- Applying Euler's polyhedron formula to an inappropriate network.
- Flattening culturally specific relationships into invented generic categories.
Guided practice
Create a five-vertex network with at least six edges. Label what each vertex and edge represent, find each vertex degree, decide whether the network is connected and identify one bridge-like link whose loss would matter.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- Define vertex and edge for a flight network.
- What does connected mean?
- Find degree of a node with 5 incident edges.
- If a network has two components, is it connected?
- Verify Euler formula for tetrahedron V=4,E=6,F=4.
- Choose shorter route: weights 3+8 or 6+4.
- Why can a transport map distort physical distance without changing topology?
- Give one example of a directed edge.
- What might high-degree vertex represent in social/transport network?
- Why should a First Nations kinship network use an authentic source?
Check answers and explanations
- Airports=vertices; direct flights=edges.
- Every vertex can be reached from every other by some path.
- 5.
- No.
- 4−6+4=2.
- 6+4=10 is shorter than 11.
- Topology represents connection structure, not necessarily map-scale geometry.
- “A follows B” or “species A eats species B”.
- A hub/highly connected entity.
- Kinship systems are culturally specific; accuracy and authority matter.
Reasoning and problem-solving task
Model a small school-bus network with at least six stops. Define vertices and edges, identify the degree of each stop, decide whether the network is connected, and explain one vulnerability if a road link becomes unavailable.
Important questions and answers
- Define vertex and edge for a flight network.
- Airports=vertices; direct flights=edges.
- What does connected mean?
- Every vertex can be reached from every other by some path.
- Find degree of a node with 5 incident edges.
- 5.
Assessment-style questions
A delivery network has six depots and weighted road links. Explain how connectedness, vertex degree and edge weights answer three different operational questions, and why a visually short edge need not have the smallest weight. [5 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- define vertex and edge meanings
- look for isolated components
- count incident edges for degree
- do not infer physical distance from drawing position
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Use transport and dependency examples where visual distance is deliberately misleading. Ask students to state the network definition before answering route questions.
For parents and carers
Use a train map or family of connected devices. Ask what the dots and lines mean and whether every point can reach every other point.
Curriculum alignment
Australian Curriculum: AC9M10SP02 — Year 10 Space. Interpret networks as vertices and edges, analyse connectedness and use network diagrams in practical situations.
Victoria: VC2M10SP02 — Level 10 Space
NSW: Stage 5 Path — Introduction to networks
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10SP02 | VC2M10SP02 — Level 10 Space | Stage 5 Path — Introduction to networks |
| Guided and independent practice | Builds fluency and application for AC9M10SP02 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10SP02: interpret networks and network diagrams used to represent relationships in practical situations and describe connectedness.
- E1: model authentic situations and identify meanings of vertices/edges. Examples 1,6,8.
- E2: analyse connectedness, including Seven Bridges. Examples 2,4.
- E3: represent polyhedra as networks and demonstrate Euler F+V=E+2. Example 5.
- E4: model social, LAN, electrical or wireless networks and practical connection/router problems. Example 6.
- E5: represent travel, food webs, metabolic or biological structures. Examples 8–9.
- E6: represent First Nations Australians’ kinship systems and significance of Country/Place relationships using culturally authoritative material. Example 10.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
Networks are less uniformly placed internationally; closest alignments occur in discrete mathematics, graph theory, computational thinking and modelling strands in Australian states, UK/Canadian/NZ secondary mathematics and enrichment curricula.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: How the Königsberg bridge problem changed mathematics - Dan Van der Vieren
TED-Ed — Replacing a real route problem with vertices and edges in a network.
As you watch: Which details of the city matter to the network, and which can be ignored?
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Try it: Draw a network for four rooms linked by doorways; explain whether every room can be reached.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Interpret networks and network diagrams used to represent relationships in...
Mapped skill: interpret networks and network diagrams used to represent relationships in practical situations and describe connectedness
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10SP02 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10SP02 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-NET-P-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M10SP02 — Networks and Connectedness — AC9M10SP02
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