P6 Question Bank: Diversity & Difficulty Gap Analysis
What's missing, what to add, and sample questions at every difficulty level
24 March 2026
Current Bank
274
questions
Question Types
15 / 30
covered of total
Difficulty
0%
advanced questions
Gap to Close
~200
new questions needed
I. Current P6 Bank Profile
The current 274-question bank covers 15 subtopics with 55 templates. All questions are 4-option MC, single-part, forward-reasoning only.
Difficulty Distribution
Basic
204 (74.5%)
Intermediate
70 (25.5%)
Advanced
0 (0%)
The bank covers the floor (TSA-level) but misses the entire upper half
HKAT, PSLE, and UK SATs all test backward reasoning, cross-topic integration, and comparison problems. We have none.
II. Critical Gaps (what real HK assessments test that we don't have)
Gap
Severity
Why It Matters
Questions Needed
Backward reasoning
CRITICAL
HKAT, PSLE, SATs all test "find original from result." Standard P6.
20–30
Cross-topic integration
CRITICAL
HKAT signature 跨範疇 section. Separates strong from average.
20–30
Comparison / optimization
CRITICAL
PSLE tests ticket combos, deal comparison. Real-life math IS comparison.
15–20
Sequential percentages
CRITICAL
HK school 高階 level. "+20% then −20% ≠ 0" is a classic.
15–20
Estimation / approximation
HIGH
TSA tests this. UK SATs test this. In the HK curriculum.
10–15
Pattern recognition
HIGH
Competition staple + appears in school textbooks and HKAT.
10–15
"Explain why" (open-ended)
HIGH
HKAT Section B requires explanations. Can't replicate with MC only.
10–15
Multi-part structured
HIGH
HKAT has (a)/(b)/(c) parts that build on each other.
15–20
III. Proposed 4-Tier Difficulty Rubric
Based on TSA, HKAT, PSLE, UK SATs, and HK publisher tiered objectives.
Level
Steps
Topics
Reasoning
HK Equivalent
BASIC
1–2
1
Forward only
TSA, 基礎
INTERMEDIATE
2–3
1–2
Mostly forward, simple backward
HKAT Section A, 進階
ADVANCED
3–4
2–3
Backward, comparison, explain
HKAT Section B, 高階
COMPETITION
4+
2+
Non-routine strategy
HKMO, 奧數
Current code limitationdifficulty.js has no path to "advanced" for any p6- prefixed question. The generic rule caps at intermediate (steps ≤ 2 → basic, else → intermediate). Adding the advanced band requires new overrides.
IV. How HK P6 Compares Internationally
Feature
HK TSA
HK HKAT
SG PSLE
UK SATs
Hardest difficulty
Low
Medium–High
High (top 15%)
Medium–High
Working backwards
Rare
Occasional
Systematic
Frequent
Cross-topic
Some
Significant (跨範疇)
Common
Moderate
Explain reasoning
No
Yes (Section B)
No
Partial
Optimization problems
Rare
Occasional
Common
Occasional
Our bank currently sits at TSA level.
To serve HKAT prep — what HK parents actually buy tutoring for — we need to add backward reasoning, cross-topic, and multi-part questions.
V. Provenance + Syllabus Sieve
Every imported family should be traceable. The table below marks where each candidate family came from and whether it overlaps with the local atomic graph and the HK curriculum. HKAT and HK school tiered objectives are treated as local evidence in this amended pass.
Family
Sample IDs
Main source
Atomic graph
HK syllabus
Decision
Backward percentage / recover original
BR-1, BR-3
HKAT + HK school tiered objectives
Yes (`percent-reverse`)
Yes (`6N4`, `6A1`)
ADOPT
Backward average / required score
BR-2
HKAT + UK SATs reasoning style
Yes (`find-missing-value`)
Yes (`6D1`)
ADOPT
Sequential percentages / changing base
SP-1, SP-2
HK school tiered objectives
Yes (`percent-sequential`)
Yes (`6N4`)
ADOPT
Data reading + percentage
CT-1
HKAT cross-domain
Yes
Yes (`6D2`, `6D3`, `6N4`)
ADOPT
Discount + equation
CT-2
HKAT
Yes
Yes (`6N4`, `6A1`)
ADOPT
Speed + ratio comparison
CT-3
Internal synthesis / PSLE-style comparison
Yes
Partial (`6M4` only)
ADAPT
Deal comparison / least-cost package
CO-1, CO-2
PSLE
No
No
OUT
Estimation / reasonableness
EST-1, EST-2
TSA + UK SATs
No
Partial
ADAPT
Pattern / sequence families
Not in core samples
UK SATs + competition
No
No
OUT
Pre-processing rule for SG / UK imports
If a family has no overlap with both the atomic graph and the HK curriculum, treat it as out-of-syllabus for the core P6 bank. Keep it only as inspiration, enrichment, or a teacher-validated stretch lane.
Why this matters
This lets us isolate later issues by source. If a family causes complaints, we can immediately tell whether it came from HKAT, PSLE, UK SATs, or our own synthesis, instead of debugging blindly.
VI. Sample Questions — Filling the Gaps
12 sample questions across 6 gap categories, at difficulty levels currently missing from the bank. Each includes bilingual stems, worked solutions, and error-modelled traps. Use the provenance table above to see which external system each family came from.
Gap 1: Backward Reasoning CRITICAL GAP
BR-1ADVANCEDp6-percent-backward
A jacket is on sale at 25% off. The sale price is $450. What was the original price?
一件外套以七五折出售，售價為 $450。求原價。
A. $600 ✓
B. $562.50
C. $337.50
D. $575
Solution:
25% off means sale price = 75% of original.
Let x = original price. 0.75x = 450
x = 450 ÷ 0.75 = $600
B
Added 25% to sale price (450 × 1.25) — treats discount as markup
C
Subtracted 25% from sale price (450 × 0.75) — applied discount twice
D
Arithmetic error with wrong formula
Trap B catches the most common error: students who add 25% instead of dividing. This tests whether they understand $450 is 75% of the unknown, not the base.
BR-2ADVANCEDp6-average-backward
David scored 78, 85, 92, and 71 on his first four maths tests. What is the minimum score he needs on the fifth test to achieve an average of at least 82?
A warehouse had 3000 boxes. 27% were moved out yesterday. Today, 40% of the remaining boxes were moved out. How many boxes are still in the warehouse?
倉庫原有 3000 箱貨物。昨天搬走了 27%，今天又搬走了餘下的 40%。倉庫還剩多少箱？
A. 1314 ✓
B. 990
C. 1800
D. 1890
Solution:
After yesterday: 3000 × 0.73 = 2190
After today: 2190 × 0.60 = 1314
B
Added 27% + 40% = 67% and subtracted from 3000 — can't add sequential %
C
Applied 40% to original 3000 instead of remainder
D
Applied combined percentage incorrectly
Trap B is the #1 student error: adding sequential percentages. The second percentage applies to the remainder, not the original. Directly from HK school 高階 level objectives.
SP-2ADVANCEDp6-percent-sequential
A shop increases the price of a bag by 20%, then offers a 20% discount on the new price. The original price was $500. What is the final price?
一間商店將一個手袋的售價提高 20%，然後以新價格的八折出售。手袋原價 $500，求最終售價。
A. $500 (equal to original)
B. $480 (less than original) ✓
C. $520
D. $400
Solution:
After 20% increase: 500 × 1.20 = $600
After 20% discount: 600 × 0.80 = $480
$480 < $500 — the final price is LESS than the original!
A
Most common error: "+20% then −20% = 0 change" feels intuitive but is wrong
C
Applied 20% increase but only 10% discount
D
Applied 20% discount to original, not increased price
The central insight: "+20% then −20% ≠ no change" is a famously counterintuitive result. The 20% decrease applies to a larger base. This is a thinking question, not just calculation.
Gap 3: Cross-Topic Integration CRITICAL GAP
CT-1ADVANCEDp6-cross-data-percentage
A class of 40 students took a Science test. The results: 12 scored A, 16 scored B, 8 scored C, and the rest scored D. What percentage of students did NOT score A?
Calculated % who DID score A (12/40) — misread the question
C
Gave the count (28) as if it were a percentage
D
Only counted B+C=24, forgetting D-grade students
CT-2ADVANCEDp6-cross-discount-equation
A stationery shop has a 25% off sale. A teacher buys 15 identical notebooks and pays $157.50 in total. Using an equation, find the original price of one notebook.
Divided total by 15 ($10.50) — got sale price, forgot to recover original
C
Divided total by 12 instead of 11.25
D
Used 80% instead of 75% for the discount
Directly from HKAT practice papers (Pan Lloyds demo). Requires setting up an equation with discount × quantity. Trap B is the "half-answer" — student stops at the sale price.
CT-3ADVANCEDp6-cross-speed-ratio
Mei Ling cycles to school at 12 km/h and walks home at 4 km/h. The school is 6 km from her home. What is the ratio of her cycling time to her walking time?
Brilliant trap: used speed ratio (12:4 = 3:1) instead of time ratio — direction is inverted
C
Confused distance ratio with time ratio
D
Arithmetic error simplifying
Speed ratio is 3:1 but time ratio is the inverse (1:3). Students who grab the speed ratio without computing get it backwards. Tests the inverse relationship between speed and time.
Gap 4: Comparison / Optimization CRITICAL GAP
CO-1ADVANCEDp6-comparison-deals
A swimming pool charges:
• Single ticket: $50 per person
• Group of 3: $120
• Group of 5: $180
8 friends want to go swimming. What is the least amount they need to pay?
The student made a decimal-place error: 8.5 instead of 85
85 km/h is reasonable for a car; 8.5 km/h is walking speed
A
Accepted the claim without verifying
C
Multiplied instead of dividing
D
Accepted unreasonable answer with flawed reasoning
Tests meta-cognitive skill: checking whether an answer "makes sense." Also requires real-world knowledge (8.5 km/h ≈ walking speed, not driving). TSA and PSLE both value this.
Verdict: What to Build
Priority 1 — build now: Backward reasoning, sequential percentages, data-plus-percentage cross-domain, and discount-plus-equation. These are the cleanest families because they survive the provenance sieve and map to both the atomic graph and HK syllabus.
Priority 2 — build as stretch with flags: Speed-plus-ratio and estimation / reasonableness. Keep them visible as stretch or teacher-validate until school-book references confirm them.
Do not put into the core bank yet: PSLE-style deal optimization, pattern sequences, and competition families. These are useful inspiration, but they currently fail the overlap test and should stay out of the main P6 path.
Implementation path: (1) Add provenance metadata fields, (2) Add the SG/UK preprocessing sieve before generation, (3) Add "advanced" band to difficulty.js, (4) Create generator files for the approved families, (5) Add trap builders and competency-map nodes where missing. The sample questions above can serve as seed templates.
Track (3) — awaiting: Leslie/Renee to share school exercise books and teacher-created question variations for additional reference material and ground-truth calibration.
References
[1] HKEAA TSA Papers — Territory-wide System Assessment, P6 Math.
TSA format and difficulty baseline
[2] Pan Lloyds HKAT Demo Paper — Pre-S1 HKAT practice materials.
Cross-domain and equation-required questions
[3] EPH HKAT Practice — Educational Publishing House sample.
Multi-part structured questions