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Interpret practical networks

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Five trail locations A to E. Two-way links AB, BC, CA, CD and DE. Only marked vertices are junctions. Not a scale map.ABCDE

This trail network uses a vertex for each location and an edge for each direct two-way trail. What does the edge C–D mean?

Year 10, Chapter 7, 4 difficulty levels, No calculatorBuild a worksheet

Networks and connectedness

Read the key first. A network records specified relationships; the drawing is not automatically a scale map

  1. Vertices and edges Vertices represent objects or places. Edges represent the stated relationship between their endpoints
  2. Adjacency and reachability Adjacent vertices share an edge. A reachable vertex may require a route with several edges
  3. Connected groups In a two-way network, a component includes every vertex reachable from one of its vertices. Include isolated vertices
  4. Closures and alternatives Remove only the closed edge, then trace again. An alternative route may preserve connectedness
  5. Directions and weights Follow arrows for one-way relationships. Add the stated weights along the selected route; fewer edges need not mean less time
  6. Limits of the model Changing the layout preserves endpoint relationships. Crossings are not junctions unless marked. Unrecorded distances, times and conditions cannot be inferred

Worked example

Compare the two displayed routes from A to C. Weights are fixed travel times in minutes, with no extra transfer delay

Two-way travel times: AB 4 minutes, BC 7 minutes and AC 15 minutes.4715ABC
\[4+7=11\]
  1. \(\displaystyle A\to B\to C\)This route uses AB and BC
  2. \(\displaystyle 4+7=11\text{ min}\)Add only the two used weights
  3. \(\displaystyle 11<15\)The two-edge route is quicker than the direct service under the stated assumptions. A transfer delay could change this comparison

Common mistakes

Confusing reachability and direct links
✕\(\displaystyle \text{reachable}=\text{adjacent}\)✓\(\displaystyle A\to B\to C\)
Trace how the destination is reached before calling it a direct neighbour
Ignoring isolated vertices
✕\(\displaystyle \text{no edges}\Rightarrow\text{omit}\)✓\(\displaystyle \text{isolated}=1\text{ group}\)
The vertex remains a location in the model even when no link reaches it
Assuming fewer links means quicker
✕\(\displaystyle \text{fewer links}\Rightarrow\text{quicker}\)✓\(\displaystyle 4+7<15\)
Compare total times, not just the number of links

Difficulty levels

introInterpret a key and direct neighbours
standardCount groups, trace access and add route weights
challengeAnalyse closures, arrows and redrawing
examReconnect groups and evaluate missing model information

Questions about this topic

Does an edge crossing create a connection?+

Only when the model marks or defines a junction there. A crossing in the drawing alone does not change the endpoint relationships

Is connected the same as directly linked?+

No. A connected two-way network allows routes between every pair of vertices, possibly via intermediate vertices

Can I move the vertices on the page?+

Yes, if every label, endpoint pair, direction and weight is preserved. Explain any marked junctions clearly

Do I need a shortest-path algorithm here?+

This lesson interprets practical networks and connectedness and compares stated small routes. Formal optimisation algorithms are separate later content

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 7: Geometry and similarity

In this chapter

Angle facts and parallel linesTriangle angle sum, special triangles and the exterior angle theoremQuadrilateral angle sums and special-quadrilateral propertiesPolygon angle sums and regular polygonsCongruent triangle testsTests for similar triangles7F Similar triangle tests: Similar figures and scale factors7G Applied triangle similarity: Applied similarity and scaleSimilar solids and scaling volume7H Similar solids and map scale: Similar figures and scale factors7H Similar solids and map scale: Applied similarity and scaleInterpret practical networks