Mathematics Standard • Year 12 • Module 6 • Lesson 1
Introduction to Project Networks — Skill Drill
Build fluency in the language of project networks: activities, events, precedence, dummies, AOA vs AON — one term, one diagram and one path at a time.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete each definition with one short phrase.
An activity is ____________________________________________.
An event is ____________________________________________.
An immediate predecessor of activity B is ____________________________________________.
Q1.2 In AOA notation, activities are drawn as ____________ and events are drawn as ____________.
A dummy activity is drawn as a ____________ arrow and has duration ____________.
Q1.3 State the one-line rule that forces us to use a dummy activity in some networks. Rule: __________________________________________________________________.
2. Worked example — precedence table and minimum project time
Follow each line of working. Every step has a reason on the right.
Problem. A small project has activities A(4), B(3), C(2), D(5). A has no predecessor. B and C both need A. D needs both B and C. Write the precedence table and find the minimum project time.
Step 1 — Write the precedence table.
A(4, —) B(3, A) C(2, A) D(5, B and C)
Reason: list activity, duration, then only the immediate predecessors.
Step 2 — List every path from start to finish.
Path 1: A → B → D Path 2: A → C → D
Reason: D needs BOTH B and C, so it starts only after the slower of the two paths finishes.
Step 3 — Add the durations on each path.
Path 1 = 4 + 3 + 5 = 12 Path 2 = 4 + 2 + 5 = 11
Reason: project time on any path is the sum of its activity durations.
Step 4 — Take the maximum path length.
Minimum project time = max(12, 11) = 12 days
Reason: D cannot start until B AND C finish, so the longer path wins.
Conclusion. The project takes a minimum of 12 days, controlled by the path A → B → D.
3. Faded example — fill in the missing steps
A project has activities P(2), Q(4), R(3), S(2), T(5). P has no predecessor. Q and R need P. S needs Q. T needs R and S. Find the minimum project time. 4 marks
Step 1 — Precedence table:
P(__, __) Q(__, __) R(__, __) S(__, __) T(__, __ and __)
Step 2 — List every path from start to finish.
Path 1: P → Q → S → T Path 2: P → R → T
Step 3 — Add the durations on each path.
Path 1 = __ + __ + __ + __ = ____ Path 2 = __ + __ + __ = ____
Step 4 — Take the maximum:
Min project time = max( __ , __ ) = ____ days
Conclusion. Minimum project time = ____ days, controlled by path ____________.
4. Graduated practice — vocabulary, tables and minimum times
Show your working in the space below each part. For every minimum-time question, write at least one path-sum line.
Foundation — single-idea recall (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | True or false: an event has zero duration. | |
| 4.2 1 | True or false: a dummy activity is drawn as a solid arrow. | |
| 4.3 1 | In a precedence table, do you list all predecessors or only the immediate predecessors? | |
| 4.4 1 | What letter normally labels the start activity (the one with no predecessor) in our worked examples? |
Standard — typical HSC difficulty (6 questions)
For network questions, write the path-sums clearly before stating the minimum time.
4.5 A project has A(3), B(2,A), C(4,A), D(1,B), E(2,C,D). Write the precedence table in (activity, duration, immediate predecessors) form. 2 marks
4.6 For the project in Q4.5, list every path from start to finish. 2 marks
4.7 For the project in Q4.5, find the minimum project time. 2 marks
4.8 Activities: M(5,—), N(3,M), O(4,M), P(2,N,O). Find the minimum project time. 2 marks
4.9 In the AOA network of Q4.8, can N and O run in parallel? In one short sentence, explain why or why not. 2 marks
4.10 A project has A(2,—), B(3,—), C(4,A), D(5,B), E(1,C,D). Find the minimum project time. 2 marks
Extension — combine ideas (2 questions)
4.11 A house-build project has: Site prep S(2,—), Foundation F(4,S), Walls W(6,F), Roof R(3,W), Electrical E(4,W), Plumbing P(3,W), Drywall D(5,E,P), Finish N(2,D,R). Write the precedence table and find the minimum project time, showing every path. 3 marks
4.12 A small software project has: Design U(3,—), Backend B(6,U), Frontend F(5,U), Test T(2,B,F), Deploy D(1,T). The team wants the project finished in 11 working days — is this possible without changing any activity? Show the path-sum that decides this. 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 — Definitions
Activity: a task that takes time (and resources) to complete — drawn as an arrow in AOA.
Event: a point in time when one or more activities start or finish — drawn as a numbered circle (node), zero duration.
Immediate predecessor of B: an activity that must finish before B can start (no intermediate activities in between).
Q1.2 — AOA notation
Activities are drawn as arrows (arcs); events are drawn as circles (nodes). A dummy is drawn as a dashed arrow with duration 0.
Q1.3 — The unique-event rule
No two activities can share both the same start event and the same end event. When the precedence table forces this, we insert a dummy to give one of the activities a different start or end event.
Q3 — Faded example (P, Q, R, S, T)
Step 1: P(2, —), Q(4, P), R(3, P), S(2, Q), T(5, R and S).
Step 2: Path 1 P → Q → S → T; Path 2 P → R → T.
Step 3: Path 1 = 2 + 4 + 2 + 5 = 13; Path 2 = 2 + 3 + 5 = 10.
Step 4: Min project time = max(13, 10) = 13.
Conclusion: minimum project time = 13 days, controlled by path P → Q → S → T.
Q4.1 — Event duration
True. An event is a point in time, so its duration is 0. Time passes during activities, not at events.
Q4.2 — Dummy arrow style
False. A dummy is drawn as a dashed arrow (and labelled with duration 0). Solid arrows are reserved for real activities.
Q4.3 — Predecessor scope
Only the immediate predecessors. Listing all ancestors is redundant and can produce incorrect network diagrams.
Q4.4 — Start-activity label
In all our worked examples the start activity is the one with the empty predecessor column (typically A). A project may also have more than one start activity.
Q4.5 — Precedence table for the small network
A(3, —), B(2, A), C(4, A), D(1, B), E(2, C and D).
Q4.6 — Paths from start to finish
Path 1: A → B → D → E. Path 2: A → C → E.
Q4.7 — Minimum project time
Path 1 = 3 + 2 + 1 + 2 = 8. Path 2 = 3 + 4 + 2 = 9. Min time = max(8, 9) = 9 days. Controlled by A → C → E.
Q4.8 — M, N, O, P network
Path M → N → P = 5 + 3 + 2 = 10. Path M → O → P = 5 + 4 + 2 = 11. Min time = max(10, 11) = 11 days.
Q4.9 — Parallel N and O
Yes. N and O share the same single predecessor (M) and neither one depends on the other, so they can run in parallel after M finishes.
Q4.10 — Two start activities A and B
Path A → C → E = 2 + 4 + 1 = 7. Path B → D → E = 3 + 5 + 1 = 9. Min time = max(7, 9) = 9 days. (Two start activities is allowed: the project starts when either can begin.)
Q4.11 — House-build network
Table: S(2, —), F(4, S), W(6, F), R(3, W), E(4, W), P(3, W), D(5, E and P), N(2, D and R).
Paths from S to N:
S → F → W → R → N = 2 + 4 + 6 + 3 + 2 = 17.
S → F → W → E → D → N = 2 + 4 + 6 + 4 + 5 + 2 = 23.
S → F → W → P → D → N = 2 + 4 + 6 + 3 + 5 + 2 = 22.
Min project time = max(17, 23, 22) = 23 days, controlled by S → F → W → E → D → N.
Q4.12 — 11-day finish for software project
Path U → B → T → D = 3 + 6 + 2 + 1 = 12. Path U → F → T → D = 3 + 5 + 2 + 1 = 11.
Min project time = max(12, 11) = 12 days. No — 11 days is not possible without changing any activity, because the longer (critical) path is 12 days.