Mathematics Standard • Year 12 • Module 6 • Lesson 2

Activity Networks and Precedence Tables — Skill Drill

Build fluency in building, reading, and verifying precedence tables — turn worded project descriptions into clean activity / duration / immediate-predecessor rows.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 A precedence table has three columns. Name them in order:

Column 1: ____________ Column 2: ____________ Column 3: ____________.

Q1.2 Define each term in one short phrase.

A start activity is __________________________________________.

A finish activity is __________________________________________.

Two activities are concurrent (parallel) when ______________________________________.

Q1.3 Tick the rule that makes the predecessor column of a precedence table valid:
□ list every activity that comes before
□ list only the immediate predecessors
□ list any one of the predecessors.

Stuck? Revisit lesson § Key Ideas — Precedence table, Immediate predecessor, Start activity.

2. Worked example — building a precedence table from a description

Follow each line of working. Every step has a reason on the right.

Problem. A small bakery sets up the morning. Tasks (minutes):
Switch ovens on (5) — no predecessor.
Mix dough (15) — needs Switch ovens on.
Prove dough (20) — needs Mix dough.
Set tables (10) — no predecessor.
Bake (25) — needs Prove dough and Switch ovens on.
Open shop (5) — needs Bake and Set tables.
Write the precedence table.

Step 1 — List every activity with its duration.

SO(5), MD(15), PD(20), ST(10), BK(25), OP(5)

Reason: rename to short codes (SO, MD, PD, ST, BK, OP) to keep the table compact.

Step 2 — For each activity, ask "what must finish immediately before this can start?"

SO: — MD: SO PD: MD ST: — BK: PD, SO OP: BK, ST

Reason: record only direct precedents — not the whole ancestor chain.

Step 3 — Remove redundant ancestors.

BK lists PD and SO, but PD already needs MD, which needs SO. So BK's SO is redundant. BK: PD only.

Reason: SO → MD → PD → BK already chains SO to BK. Listing SO again is redundant and could cause errors when drawing the network.

Step 4 — Write the cleaned-up table.

SO(5, —), MD(15, SO), PD(20, MD), ST(10, —), BK(25, PD), OP(5, BK and ST)

Conclusion. The cleaned table uses only immediate predecessors. Start activities: SO and ST. Finish activity: OP.

3. Faded example — fill in the missing predecessors

A research project has: Literature review LR(5) — no predecessor. Ethics approval EA(2) — no predecessor. Design experiment DE(3) — needs Literature review and Ethics approval. Collect data CD(7) — needs Design experiment. Analyse AN(4) — needs Collect data. Write report WR(6) — needs Analyse. Fill in the precedence column of the table below. 4 marks

LR(5, ____________ )

EA(2, ____________ )

DE(3, ____________ )

CD(7, ____________ )

AN(4, ____________ )

WR(6, ____________ )

Then state: Start activities = ____________ Finish activity = ____________ Parallel pair = ____________.

Stuck? Revisit lesson § Reading Tables — start = no predecessor; finish = not a predecessor of anything; parallel = same predecessors.

4. Graduated practice — build, read and verify tables

Show your working in the space below each part. For "errors" questions, name the rule that has been broken.

Foundation — single-idea questions (4 questions)

Q	Problem	Answer
4.1 1	From this table, name the start activity: A(3, —), B(4, A), C(2, A), D(1, B and C).
4.2 1	From the same table, name the finish activity.
4.3 1	From the same table, name the parallel pair.
4.4 1	True or false: a project can have more than one start activity.

Standard — typical HSC difficulty (6 questions)

Show your reasoning, especially when fixing or verifying tables.

4.5 Build the precedence table from: "Tasks W(4), X(3), Y(2), Z(5). W is first. X and Y both need W. Z needs X and Y." 2 marks

4.6 Build the precedence table from: "Tasks A(2), B(3), C(4), D(2), E(1). A first. B needs A. C needs A. D needs B. E needs C and D." 2 marks

4.7 Given table A(3, —), B(2, A), C(4, A), D(5, B), E(2, C), F(3, D and E). Name all start activities, all finish activities, and any pair of parallel activities. 2 marks

4.8 A student writes the table A(3, —), B(4, A), C(2, A and B). Is the table correct? If not, fix the C row and explain in one short sentence. 2 marks

4.9 A student writes the table P(3, —), Q(2, P), R(4, Q), S(1, R), and they say "P, Q, R, S are all parallel". Decide if the statement is true or false, with one short reason. 2 marks

4.10 A student writes X(2, Y), Y(3, Z), Z(1, X). Identify the rule this table breaks and explain in one sentence why no project can have this structure. 2 marks

Extension — multi-step verification (2 questions)

4.11 A house-build foreman lists: SP(2, —), FN(4, SP), FR(8, FN), RF(5, FR), PL(4, FR), EL(3, FR), DW(6, PL and EL), PT(4, DW and RF). Build the precedence table, then state every start activity, every finish activity, and the activities that can run strictly in parallel after FR. 3 marks

4.12 Verify the table A(3, —), B(4, A), C(2, A), D(5, B), E(3, C), F(2, D and E), G(2, B and E). Identify (a) any start activities, (b) any finish activities, (c) any redundant predecessor entries, (d) any pairs that can run in parallel. 3 marks

Stuck on 4.12? "Redundant" means a predecessor that is already reached through another predecessor in the chain.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For Q4.1 my start activity is the one with the dash (—) in the predecessor column, not the alphabetically first letter.

For Q4.3 my parallel pair are the two activities that share the same single predecessor and don't depend on each other.

For every "build a table" question, I wrote the activity, the duration, and only the immediate predecessor(s).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Three columns of a precedence table

Column 1: Activity (name or code). Column 2: Duration. Column 3: Immediate predecessor(s).

Q1.2 — Definitions

Start activity: an activity with no predecessors — it can begin at the start of the project.
Finish activity: an activity that is not an immediate predecessor of any other activity.
Concurrent (parallel): two activities with the same set of immediate predecessors and no dependency on each other, so they can run at the same time.

Q1.3 — Predecessor-column rule

The correct rule is "list only the immediate predecessors". Listing all ancestors is redundant; listing only one is incomplete.

Q3 — Faded example (research project)

LR(5, —), EA(2, —), DE(3, LR and EA), CD(7, DE), AN(4, CD), WR(6, AN).

Start activities = LR and EA. Finish activity = WR. Parallel pair = LR and EA (both have no predecessor and don't depend on each other, so they can start together).

Q4.1 — Start activity

A — it has — (no predecessor) in column 3.

Q4.2 — Finish activity

D — no other activity lists D as a predecessor.

Q4.3 — Parallel pair

B and C — both have A as their only predecessor and neither depends on the other.

Q4.4 — Multiple start activities

True. A project can have any number of start activities (each one is an activity with no predecessor).

Q4.5 — WXYZ table

W(4, —), X(3, W), Y(2, W), Z(5, X and Y).

Q4.6 — ABCDE table

A(2, —), B(3, A), C(4, A), D(2, B), E(1, C and D).

Q4.7 — Read the table

Start activities: A. Finish activities: F. Parallel pair: B and C (same single predecessor A, independent of each other).

Q4.8 — Redundant predecessor

The table is incorrect. C's row lists A and B, but B already depends on A, so A is reached through B. Cleaned: C(2, B). The redundancy could cause an incorrect arrow in the AOA diagram.

Q4.9 — "All parallel" claim

False. P, Q, R, S form a strict chain (P → Q → R → S). Each activity must wait for the previous one to finish, so they cannot run in parallel — they are sequential.

Q4.10 — Cyclic dependency

The table breaks the rule that no activity can be its own predecessor (directly or indirectly). Here X needs Y, Y needs Z, Z needs X — a cycle. No activity can ever start, so the project is impossible to schedule.

Q4.11 — House-build table

Table as given: SP(2, —), FN(4, SP), FR(8, FN), RF(5, FR), PL(4, FR), EL(3, FR), DW(6, PL and EL), PT(4, DW and RF).
Start activities: SP. Finish activities: PT.
Strictly parallel after FR: RF, PL and EL all share predecessor {FR} and none depend on each other. (RF can also run in parallel with DW once DW starts, but the strict triple is RF/PL/EL just after FR.)

Q4.12 — Verification table

(a) Start: A. (b) Finish: F and G (neither is a predecessor of anything).
(c) No redundant predecessors — every listed predecessor is direct.
(d) Parallel pairs: B and C (same predecessor A). D and E are not parallel (D needs B, E needs C, different predecessors). F and G are not strictly parallel (F needs D and E, G needs B and E — different sets), but G could start as soon as B and E are done.