Mathematics Standard • Year 12 • Module 6 • Lesson 3
Drawing Activity Networks — Skill Drill
Build fluency in drawing AOA networks from precedence tables: events, arrows, dummies, event numbering and the unique-event rule.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 In an AOA network, an activity is drawn as ____________ and an event as ____________ (one word each).
Q1.2 Complete the unique-event rule: "No two activities can share both the same ____________ event and the same ____________ event."
Q1.3 When numbering events: every arrow must go from a ____________ -numbered event to a ____________ -numbered event.
Q1.4 A dummy activity has duration ____________ and is drawn as a ____________ arrow.
2. Worked example — drawing an AOA network step by step
Follow each line of working. Every step has a reason on the right.
Problem. Draw the AOA network for: A(3, —), B(4, A), C(2, A), D(5, B), E(1, C), F(3, D and E).
Step 1 — Place the start event and draw start activities.
Event 1 → A → Event 2
Reason: A has no predecessor, so it starts from event 1.
Step 2 — Draw activities that start after A.
Event 2 → B → Event 3 Event 2 → C → Event 4
Reason: B and C both need only A; they leave event 2. Each needs its own end event to satisfy the unique-event rule.
Step 3 — Draw activities that depend on B and on C separately.
Event 3 → D → Event 5 Event 4 → E → Event 5
Reason: D needs B (so starts at event 3); E needs C (so starts at event 4). Both end at event 5 because F needs both.
Step 4 — Draw the final activity from the merge event.
Event 5 → F → Event 6 (finish)
Reason: F needs both D and E, so it starts where they converge (event 5).
Step 5 — Check the rules.
• Every arrow goes from a lower-numbered event to a higher-numbered event. OK.
• No two activities share both a start and an end event. OK.
• No dummies are needed in this network.
Conclusion. The network has 6 events numbered 1-6 and 6 activities. No dummies required.
3. Faded example — fill in the missing steps
Draw the AOA network for A(2, —), B(3, A), C(2, A), D(4, B and C). State whether a dummy is needed and why. Fill the blanks. 4 marks
Step 1 — Start event: Event ____ → A → Event ____.
Step 2 — Activities after A: Event 2 → B → Event ____ Event 2 → C → Event ____.
Step 3 — Check the unique-event rule: B and C must have ________ end events because if they shared the same end event, D (which needs both) would still need them to converge somewhere — and they would break the unique-event rule.
Step 4 — Converge for D: Add a ________ activity from event ____ to event ____ so that both B and C "arrive" at the same event for D. Then D goes from event ____ to event ____.
Conclusion. Dummy needed? ____________ Why? __________________________________________.
4. Graduated practice — draw and check AOA networks
Sketch each network in the space provided using arrows and numbered circles. State the number of events and the number of dummies for each.
Foundation — single-idea questions (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | True or false: in AOA, an event is drawn as a circle (node). | |
| 4.2 1 | True or false: arrows in an AOA network can go from higher-numbered to lower-numbered events. | |
| 4.3 1 | A network has 5 activities and no parallel pairs. What is the minimum number of events? | |
| 4.4 1 | What style of arrow is a dummy activity drawn with? |
Standard — typical HSC difficulty (6 questions)
Sketch each network. Label events 1, 2, 3, ... so all arrows go forward.
4.5 Draw the AOA network for A(2, —), B(3, A), C(4, B). State the number of events. 2 marks
4.6 Draw the AOA network for A(3, —), B(2, A), C(4, A), D(5, B and C). State whether any dummies are needed. 2 marks
4.7 Draw the AOA network for A(2, —), B(3, A), C(4, A), D(2, B), E(3, C), F(1, D and E). State the number of events. 2 marks
4.8 Draw the AOA network for A(2, —), B(3, A), C(2, A), D(4, B), E(3, B and C). One dummy is needed — explain where and why in one short sentence. 2 marks
4.9 A student draws an AOA network with two activities sharing the same start event and the same end event. Identify which rule is broken and state the one-line fix. 2 marks
4.10 A network has events numbered 1, 2, 3, 4, 5 and an arrow drawn from event 4 to event 2. Identify which rule is broken and state the one-line fix. 2 marks
Extension — multi-dummy networks (2 questions)
4.11 Draw the AOA network for A(2, —), B(3, A), C(2, A), D(4, B), E(3, B and C), F(1, D and E). State the number of events and the number of dummies. 3 marks
4.12 Draw the AOA network for A(1, —), B(2, —), C(3, A), D(2, A and B), E(4, C and D). Number every event. How many dummies are needed and where? 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — AOA notation
Activity = arrow. Event = circle (numbered node).
Q1.2 — Unique-event rule
"No two activities can share both the same start event and the same end event."
Q1.3 — Event numbering
Arrows always go from a lower-numbered event to a higher-numbered event. This rules out cycles and keeps the diagram time-ordered.
Q1.4 — Dummy properties
Duration = 0. Drawn as a dashed arrow. It carries precedence only, not work.
Q3 — Faded example (A, B, C, D)
Step 1: Event 1 → A → Event 2.
Step 2: Event 2 → B → Event 3; Event 2 → C → Event 4.
Step 3: B and C must have different end events (unique-event rule).
Step 4: Add a dummy from event 3 to event 4 so that the merge for D happens at event 4. Then D goes from event 4 to event 5.
Conclusion: Dummy needed = Yes. Why? Because B and C cannot share both the same start event (2) and the same end event — the dummy gives them distinct end events while still letting D wait for both.
Q4.1 — Event shape
True. Events are drawn as circles (nodes) and contain an integer label.
Q4.2 — Arrow direction
False. Arrows must always go from a lower-numbered to a higher-numbered event.
Q4.3 — Minimum events for a chain of 5
6 events. A chain of n sequential activities needs n + 1 events.
Q4.4 — Dummy arrow style
Dashed arrow (with duration 0 and no label other than "dummy" or 0).
Q4.5 — Three-activity chain
1 → A → 2 → B → 3 → C → 4. Number of events = 4. No dummies.
Q4.6 — Two parallel activities then a merge
1 → A → 2; 2 → B → 3; 2 → C → 4. Add a dummy 3 → 4 (or, equivalently, place B and C at the same end event by drawing a dummy on one side). Then 4 → D → 5. 1 dummy needed to make B and C converge while preserving the unique-event rule.
Alternative valid drawing: 1 → A → 2; 2 → B → 3; 2 → C → 3 only if you accept they share both events — but this breaks the unique-event rule, so the dummy version is the canonical one.
Q4.7 — Six-activity network
1 → A → 2; 2 → B → 3; 2 → C → 4; 3 → D → 5; 4 → E → 5; 5 → F → 6. Number of events = 6. No dummies.
Q4.8 — Partial-dependency dummy
1 → A → 2; 2 → B → 3; 2 → C → 4; 3 → D → 5; (dummy 3 → 4); 4 → E → 6. One dummy from event 3 to event 4 forces E to wait for B AND C while leaving D depending on B only.
Q4.9 — Same start AND same end
The unique-event rule is broken. Fix: insert a dummy on one of the two activities to give it a different end event (or different start event) so the diagram unambiguously labels which arrow is which.
Q4.10 — Backward-numbered arrow
The numbering rule is broken — arrows must go from a lower-numbered event to a higher-numbered event. Fix: renumber the events so that the activity goes from a smaller number to a larger number (or, if the table allows, swap the order).
Q4.11 — Two-dummy practice
1 → A → 2; 2 → B → 3; 2 → C → 4; 3 → D → 6; dummy 3 → 4; 4 → E → 5; dummy 5 → 6; 6 → F → 7. Events = 7. Dummies = 2. (The first dummy makes B and C converge for E while leaving D needing only B; the second dummy lets F wait for both D and E by giving them a common end event.)
Q4.12 — Two start activities and a partial dependency
1 → A → 3; 2 → B → 4; dummy 3 → 4 (so D can wait for A and B at event 4 while C, which only needs A, leaves event 3 separately); 3 → C → 5; 4 → D → 5; 5 → E → 6. Dummies = 1. (You can equivalently use a single start event 0 and add dummy arrows 0 → 1 and 0 → 2; this version keeps two separate start events.)