Mathematics Standard • Year 12 • Module 6 • Lesson 1
Introduction to Project Networks — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on project networks: vocabulary, precedence tables, parallel work and one structured multi-part response.
1. Short-answer questions
1.1 A project has activities R(3, —), S(4, R), T(2, R), U(5, S and T). Write the precedence table and calculate the minimum project time. 3 marks Band 3
1.2 Explain in one or two sentences why a dummy activity is sometimes needed in an AOA project network. Then give one short concrete example (in words) where omitting a dummy would make the network ambiguous. 3 marks Band 3-4
1.3 A project network has the precedence table A(2, —), B(3, —), C(4, A), D(5, B), E(2, C and D).
(a) State the start activities and the finish activity.
(b) List every path from a start activity to the finish activity, with its total duration.
(c) State the minimum project time and identify the controlling path. 4 marks Band 4
2. Extended response
2.1 A community-hall fit-out project in regional NSW is described by the precedence table below (durations in days).
Site clear SC(2, —)
Floor lay FL(4, SC)
Wall paint WP(3, SC)
Lighting install LI(5, WP)
Furniture in FI(2, FL and LI)
Opening event OP(1, FI)
(a) List every path from SC to OP, with its total duration.
(b) State the minimum number of days from SC to OP, and name the controlling path.
(c) The committee asks whether reducing Floor lay (FL) from 4 days to 2 days will bring the opening forward. Use your path-sums to justify your answer in one clear sentence. 6 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — both paths SC → FL → FI → OP and SC → WP → LI → FI → OP correctly identified.
• 1 mark — correct duration sum on each path (with units).
Part (b) — 2 marks
• 1 mark — correct minimum project time (numeric).
• 1 mark — the controlling path is named explicitly.
Part (c) — 2 marks
• 1 mark — applies the change to the FL path-sum (4 → 2) and recomputes that path.
• 1 mark — explicit conclusion sentence that compares the new FL-path total to the other (controlling) path and states whether the opening moves forward.
Your response:
Stuck on (c)? FL is only on one of the two paths. Compare the recomputed FL-path total to the WP-LI path total — the larger one still controls the finish.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Small precedence table and minimum time (3 marks)
Sample response.
Table: R(3, —), S(4, R), T(2, R), U(5, S and T).
Path R → S → U = 3 + 4 + 5 = 12 days.
Path R → T → U = 3 + 2 + 5 = 10 days.
Minimum project time = max(12, 10) = 12 days, controlled by R → S → U.
Marking notes. 1 mark — correct precedence table. 1 mark — both path-sums correct. 1 mark — correct minimum time stated with units AND the path named. A bare "12" with no path or no working scores 1/3.
1.2 — Why dummies are needed (3 marks)
Sample response. A dummy activity (dashed arrow, duration 0) is needed because the AOA rule says no two activities can share both the same start event and the same end event; if two activities would otherwise share both, a dummy is added to separate them.
Example: activities B and C both start after A and both finish before D. If we draw them with the same start and end events, we cannot tell them apart on the diagram. Adding a dummy after C gives it a different end event, restoring uniqueness.
Marking notes. 1 mark — clear statement of the unique-event rule (or partial-dependency rationale). 1 mark — names the dummy's properties (dashed, duration 0). 1 mark — concrete worked example in words. Answers that only define a dummy without saying why it is needed score 1/3.
1.3 — Two start activities and a merge (4 marks)
(a) Sample response. Start activities = A and B (no predecessors). Finish activity = E (no successors).
(b) Sample response. Path A → C → E = 2 + 4 + 2 = 8 days. Path B → D → E = 3 + 5 + 2 = 10 days.
(c) Sample response. Minimum project time = max(8, 10) = 10 days, controlled by B → D → E.
Marking notes. (a) 1 mark — both starts and the finish correctly named. (b) 1 mark — both paths and sums correct with units. (c) 1 mark — correct minimum time. 1 mark — controlling path named explicitly. A response giving only "10 days" without naming the controlling path scores 1/2 on part (c).
2.1 — Community-hall fit-out (6 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Paths from SC to OP.
Path 1: SC → FL → FI → OP = 2 + 4 + 2 + 1 = 9 days. [part of 1 mark — path identified.]
Path 2: SC → WP → LI → FI → OP = 2 + 3 + 5 + 2 + 1 = 13 days. [1 mark for both paths identified; 1 mark for both sums correct with units.]
(b) Minimum project time and controlling path.
Minimum project time = max(9, 13) = 13 days. [1 mark — correct numeric value.]
Controlling path: SC → WP → LI → FI → OP. [1 mark — controlling path named.]
(c) Effect of shortening FL from 4 to 2 days.
If FL drops from 4 to 2, Path 1 becomes SC → FL → FI → OP = 2 + 2 + 2 + 1 = 7 days. [1 mark — applied the change to the correct path and recomputed.]
Path 2 is unchanged at 13 days and is still the longest.
Conclusion: No — reducing FL from 4 to 2 days does NOT bring the opening forward. The controlling path is still SC → WP → LI → FI → OP at 13 days, because FL is not on the controlling path. [1 mark — explicit conclusion comparing recomputed Path 1 to Path 2 with a clear yes/no answer.]
Total: 6/6.
Band descriptors for marker.
Band 3: Identifies one path correctly and a partial sum; does not list both paths or does not state a clear minimum. ≈ 2-3 marks.
Band 4: Both paths and sums correct, correct minimum time, but no controlling path named OR no analysis of FL change. ≈ 4 marks.
Band 5: Full numerical analysis including recomputed FL path-sum, but the conclusion is just "13 days, no" without explaining why FL doesn't matter. ≈ 5 marks.
Band 6: Complete — both paths listed, controlling path named, FL change applied to the correct path, AND a clear conclusion sentence that explicitly links the answer to the fact that FL is not on the controlling path. 6/6.