Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 2 - Networks and connectedness

Networks and connectedness

Represent practical networks and use vertices, edges, paths and connectivity to compare routes and structures.

Updated 2026-07-26 - 12 min read

Networks and connectedness is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Represent networks with vertices and edges

Vertices represent objects or locations and edges represent defined connections; geometry and drawing scale are irrelevant unless weights encode them. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by represent networks with vertices and edges and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A crossing is a vertex only when the network definition marks a connection there.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

A transport network has 6 stations and 7 direct links. How many vertices and edges?

Step 1 - identify the governing idea: Vertices represent objects or locations and edges represent defined connections; geometry and drawing scale are irrelevant unless weights encode them.

Step 2 - apply it to this evidence: Stations are vertices and direct links are edges.

Result: 6 vertices and 7 edges

The relationship is visible in the working: Stations are vertices and direct links are edges. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 7 vertices and 6 edges — It does not agree with the required relationship: Stations are vertices and direct links are edges.
  • 13 vertices — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 42 edges — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Two drawn edges cross with no marked junction. What does the crossing mean?

Step 1 - identify the governing idea: Vertices represent objects or locations and edges represent defined connections; geometry and drawing scale are irrelevant unless weights encode them.

Step 2 - apply it to this evidence: Only defined or marked vertices create network connections.

Result: No connection is implied

The relationship is visible in the working: Only defined or marked vertices create network connections. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • A mandatory vertex — It does not agree with the required relationship: Only defined or marked vertices create network connections.
  • Both edges end — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The network is disconnected — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

What is the degree of a vertex joined by 4 incident edges?

Step 1 - identify the governing idea: Vertices represent objects or locations and edges represent defined connections; geometry and drawing scale are irrelevant unless weights encode them.

Step 2 - apply it to this evidence: Vertex degree counts incident edges.

Result: 4

The relationship is visible in the working: Vertex degree counts incident edges. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 8 — It does not agree with the required relationship: Vertex degree counts incident edges.
  • 3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Trace paths and connectivity

A path follows adjacent edges; a network is connected when every vertex can be reached from every other vertex. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by trace paths and connectivity and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Connectedness may use multi-edge paths; direct all-to-all links describe a complete network.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

A-B, B-C and C-D are edges. Is there a path from A to D?

Step 1 - identify the governing idea: A path follows adjacent edges; a network is connected when every vertex can be reached from every other vertex.

Step 2 - apply it to this evidence: Each consecutive pair is joined by an edge.

Result: Yes: A-B-C-D

The relationship is visible in the working: Each consecutive pair is joined by an edge. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • No — It does not agree with the required relationship: Each consecutive pair is joined by an edge.
  • Only A-D is a path — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only if A and D are adjacent — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

What makes a network disconnected?

Step 1 - identify the governing idea: A path follows adjacent edges; a network is connected when every vertex can be reached from every other vertex.

Step 2 - apply it to this evidence: Disconnected components have no path between them.

Result: At least one vertex cannot be reached from another

The relationship is visible in the working: Disconnected components have no path between them. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • It has more vertices than edges — It does not agree with the required relationship: Disconnected components have no path between them.
  • An edge crosses another — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It has a cycle — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Which added edge connects components {A,B} and {C,D}?

Step 1 - identify the governing idea: A path follows adjacent edges; a network is connected when every vertex can be reached from every other vertex.

Step 2 - apply it to this evidence: It joins one vertex from each component.

Result: B-C

The relationship is visible in the working: It joins one vertex from each component. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • A-B — It does not agree with the required relationship: It joins one vertex from each component.
  • C-D — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A-A — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare weighted routes

For weighted networks, add edge weights along each valid route and compare totals under the stated objective. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare weighted routes and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A route with more low-weight edges can beat one short high-weight edge.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Route A has weights 4,6,3; Route B has 8,4. Which is shorter?

Step 1 - identify the governing idea: For weighted networks, add edge weights along each valid route and compare totals under the stated objective.

Step 2 - apply it to this evidence: A totals 13 and B totals 12.

Result: Route B

The relationship is visible in the working: A totals 13 and B totals 12. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Route A — It does not agree with the required relationship: A totals 13 and B totals 12.
  • They are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It cannot be determined — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A direct edge weighs 20; a two-edge route weighs 7 and 8. Which route is lighter?

Step 1 - identify the governing idea: For weighted networks, add edge weights along each valid route and compare totals under the stated objective.

Step 2 - apply it to this evidence: 7+8=15, which is 5 less than 20.

Result: The two-edge route by 5

The relationship is visible in the working: 7+8=15, which is 5 less than 20. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The direct edge by 5 — It does not agree with the required relationship: 7+8=15, which is 5 less than 20.
  • They are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The direct edge because it has fewer edges — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Why might the shortest-distance route not be the fastest?

Step 1 - identify the governing idea: For weighted networks, add edge weights along each valid route and compare totals under the stated objective.

Step 2 - apply it to this evidence: Distance and travel time can rank routes differently.

Result: Edge weights may represent different quantities

The relationship is visible in the working: Distance and travel time can rank routes differently. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Networks cannot use time — It does not agree with the required relationship: Distance and travel time can rank routes differently.
  • Shortest always means fastest — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Vertices have no location — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Two drawn edges cross with no marked junction. What does the crossing mean?
  2. What makes a network disconnected?
  3. A direct edge weighs 20; a two-edge route weighs 7 and 8. Which route is lighter?

Answers

  1. No connection is implied — Only defined or marked vertices create network connections.
  2. At least one vertex cannot be reached from another — Disconnected components have no path between them.
  3. The two-edge route by 5 — 7+8=15, which is 5 less than 20.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources