Quadratics OS (A-Math) — Full Operating System Page

Unified Almost-Code (Z0–Z3) + Failure Atlas + Sensors + Repairs + Traps v0.2

PAGE-ID: EDK-LAN-AMTH-QUAD-OS-v0.2
Role: LAN.TopicOS + Directory + FailureAtlas + SensorPack + RepairLoops + TrapGovernance
Purpose: make Quadratics reliable (not just “can do some questions”) under O-Level load.

0) Definition Lock

Quadratics OS := the capability system for recognising quadratic structure, selecting the right tool, executing transformations cleanly, and validating solutions under traps and time pressure.

P3 Quadratics condition:

Can solve routine + unfamiliar quadratic variants with correct method choice, clean algebra, correct roots/constraints, and stable pace.

1) Quadratics OS Scope (what is included)

Core families:

  • quadratic expressions (expand/factorise/simplify)
  • quadratic equations (solve using the right tool)
  • completing the square (vertex form)
  • discriminant reasoning (nature of roots)
  • quadratic graphs (shape/vertex/intercepts)
  • quadratic inequalities (sign logic)
  • application problems involving quadratics (basic modelling)

2) Z0–Z3 Directory (skills → failure modes → sensors → repairs)

Z0 — Kernel Atoms (Non-negotiables)

Node Q0.1 — Factorisation Fluency (Forms)

NodeID: QUAD-Z0-FACT-FORMS
Skills: common factor; trinomials; special products; regrouping patterns
Failure modes: wrong factor pair; sign error; cannot spot structure (B2/B3)
Sensors: Foundation Fluency; Method Selection (structure recognition)
Repairs:

  • Fluency Lab (micro factor sets)
  • Variation Ladder V0→V2 on factor forms
  • “no expansion if factorable” rule

Trap: TR1 sign trap (± confusion).

Node Q0.2 — Expansion & Simplification Discipline

NodeID: QUAD-Z0-EXP-SIMP
Skills: expand cleanly; collect like terms; simplify before solving
Failure modes: algebra explosion; wrong collection; careless slips (B2/B5)
Sensors: Messiness/Hesitation; Error Recurrence
Repairs:

  • Algebra Kernel Daily 15
  • Chain discipline (1 operation/line)
  • Repair & Retest ≤ 7 days

Node Q0.3 — Algebraic Fractions / Restrictions (if present)

NodeID: QUAD-Z0-FRAC-REST
Skills: simplify rational expressions with quadratics; denom restrictions
Failure modes: illegal cancellation; forget denom ≠ 0 (TR2)
Sensors: Trap activation; Error Recurrence
Repairs:

  • Restriction-first rule
  • Trap Lab: domain constraints

Z1 — Method Signals (Choose Tool Correctly)

Node Q1.1 — Solve Quadratic Equations (Tool Selection)

NodeID: QUAD-Z1-SOLVE-TOOLS
Tools: factorise; complete square; quadratic formula
Failure modes: chooses wrong tool; wastes time; partial factorisation (B3)
Sensors: Method Selection; Bandwidth (structure recognition)
Repairs:

  • Method Signal Lab (no solving): “signal → tool”
  • Decision policy (below)

Signal map (minimal):

  • easily factorable integers → factorise
  • non-factorable / messy coefficients → quadratic formula
  • need vertex / minimum/maximum / turning point → complete square

Node Q1.2 — Discriminant / Nature of Roots

NodeID: QUAD-Z1-DISC-ROOTS
Skills: interpret b²−4ac; link to root count and graph intersection
Failure modes: wrong substitution; misinterpretation (B1/B2)
Sensors: Concept Sensor (B1) + Error Recurrence
Repairs:

  • micro drills: compute discriminant quickly
  • explain: “>0 two roots, =0 one root, <0 no real roots” (and what it means in graphs)

Node Q1.3 — Graph Structure (Vertex / Axis / Intercepts)

NodeID: QUAD-Z1-GRAPH-STRUCT
Skills: shape; axis of symmetry; vertex; intercepts
Failure modes: wrong vertex; wrong axis; scale reading errors (B4/B5)
Sensors: Representation Sensor
Repairs:

  • graph reading lab: vertex/intercepts extraction
  • connect forms: factor form ↔ intercepts, vertex form ↔ turning point

Z2 — Transfer & Multi-Step Chains (Survive Variation)

Node Q2.1 — Completing the Square Pipeline

NodeID: QUAD-Z2-COMP-SQ
Skills: convert ax²+bx+c; interpret vertex form
Failure modes: wrong half-b; sign errors; messy steps (B2/TR1)
Sensors: Foundation Fluency; Error Recurrence
Repairs:

  • chain discipline lab
  • micro “complete square” sets
  • trap mini: sign distribution

Node Q2.2 — Quadratic Inequalities

NodeID: QUAD-Z2-INEQ
Skills: sign chart reasoning; intervals from roots; include/exclude endpoints
Failure modes: wrong interval; flips inequality; misses endpoint logic (B1/B3/TR4)
Sensors: Method Selection; Transfer; Error Recurrence
Repairs:

  • inequality sign-chart lab
  • variation ladder across >, ≥, <, ≤
  • final answer form checklist (interval notation / set statement)

Node Q2.3 — Mixed-Form Quadratics (connect representations)

NodeID: QUAD-Z2-MIX-FORMS
Skills: move between standard/factor/vertex; choose best form
Failure modes: expands when should factor; cannot see purpose of form (B3)
Sensors: Bandwidth; Method Selection
Repairs:

  • form-choice drills: “goal → best form”
  • mixed set weekly

Z3 — Timed Stability + Trap Audit (Exam Reliability)

Node Q3.1 — Timed Quadratics Mini-Paper

NodeID: QUAD-Z3-TIMED
Skills: pace + clean working under time
Failure modes: slow algebra; panic switching; careless spikes (B6→B5)
Sensors: Pace Budget; DriftRisk
Repairs:

  • Timed Stability Lab (smaller scope)
  • “skip/return” policy
  • reserve audit time

Node Q3.2 — Quadratics Trap Audit Checklist

NodeID: QUAD-Z3-TRAP-AUDIT
Traps:

  • TR1 sign trap in expansion/factor
  • TR3 extraneous roots (after squaring / manipulations)
  • TR4 final form trap (wrong set/interval, wrong requested quantity)
  • TR2 domain restrictions (if denominators/logs introduced)

Audit stack:

  1. substitute root back into equation (when feasible)
  2. check sign consistency
  3. check required answer form (roots vs x-intercepts vs vertex vs range)
  4. check domain restrictions if any were introduced

3) Quadratics Sensors (Topic-specific pack)

Minimum sensors to track:

  • S1 Structure Recognition Speed (Bandwidth)
  • S2 Factorisation Fluency (Foundation)
  • S3 Tool Selection Accuracy (Method)
  • S4 Error Recurrence (repair effectiveness)
  • S5 Pace Budget Stability (timed)
  • S6 Trap Activation Rate (drift)

4) Quadratics Labs (Topic-specific drill set)

  • Lab-Q1: Factorisation Fluency micro sets (daily 10 min)
  • Lab-Q2: Signal→Tool classification (no solving)
  • Lab-Q3: Completing-square chain lab
  • Lab-Q4: Inequality sign-chart lab
  • Lab-Q5: Timed mini-paper + audit routine
  • Lab-Q6: Trap lab (curated mistakes)

5) Quadratics Gauge (Topic Scorecard)

Weekly:

  • Precision __ /5
  • Bandwidth (structure recognition) __/5
  • Speed __/5
  • DriftRisk: Low/Med/High
  • Top failure bucket: B1–B6

Readiness targets (default):

  • Precision ≥ 3
  • Bandwidth ≥ 3
  • Speed ≥ 3
  • DriftRisk not High

6) Operator Checklist (Copy-Paste Block)

[Quadratics OS Operator Checklist v0.2]
1) Identify dominant leak: factor fluency vs tool selection vs chain errors vs pace vs traps
2) Run the right lab:
   - fluency leak → Lab-Q1 daily
   - tool selection leak → Lab-Q2
   - chain leak → Lab-Q3
   - inequality leak → Lab-Q4
   - pace leak → Lab-Q5
   - trap leak → Lab-Q6
3) Retest within 7 days
4) Only expand to harder variants after DriftRisk drops

Start here if you want the full sequence:

Vocabulary OS Series Index:
https://edukatesg.com/vocabulary-os-series-index/

Fence English Learning System: 

eduKateSG Learning Systems: 

Recommended Internal Links (Spine)

Start Here for Lattice Infrastructure Connectors