Problem Set Synthesis

Create interleaved problem sets that weave together multiple mathematical topics. This skill produces synthesis-level study materials that force the brain to recognize which techniques to apply, building stronger neural connections than siloed practice.

When to Use

  • Student preparing for comprehensive exams
  • Creating mastery-level practice materials
  • Building study plans that span multiple days
  • Combining topics that are usually taught separately

Workflow

Phase 1: Gather Context

  1. Read existing tutorials — understand question formats and difficulty levels
  2. Review concept pages — identify key formulas and techniques
  3. Check lecture sources — align with course content
  4. Identify target topics — list all areas to interleave

Phase 2: Design Interleaved Problems

Each problem should require 2-4 different techniques from different topics.

Good interleaving:

  • Geometry + Series + Integration (Problem uses circle equation, requires series expansion of sqrt, integrates using trig sub)
  • Taylor series + Trigonometric identities + Definite integrals

Bad interleaving:

  • 10 geometry problems, then 10 series problems (not interleaved—just sequential)

Phase 3: Create Problem Structure

Each problem should have:

  1. Context/Story — real-world application (physics, engineering, architecture)
  2. Progressive parts (a), (b), (c)... — each part uses different technique
  3. Verification steps — compare approximate (series) to exact (integration) answers
  4. Difficulty ramp — start accessible, build to challenging

Phase 4: Write Problem Set

Create file in wiki/synthesis/:

---
title: "[Course] Mastery Set — Interleaved [Topics]"
date: YYYY-MM-DD
course: [Course Code] [Subject]
tags: [mathematics, interleaved-practice, topic1, topic2, mastery]
---

# [Course]: [Subject] — Interleaved Mastery Problem Set

**[X]-Day Intensive Study Plan**
**Topics:** [List all topics]

## How to Use This Set

Each problem intentionally **mixes multiple concepts**...

## The Mastery Problems

### Problem 1: [Descriptive Name] [Topic1 + Topic2 + Topic3]

[Context/setup]

**(a)** [Geometry part]

**(b)** [Series part]

**(c)** [Integration part]

**(d)** [Connection/verification part]

Phase 5: Add Supporting Materials

  1. Techniques summary table — show which problems use which techniques
  2. Formula reference — key equations for all topics
  3. Study schedule — day-by-day breakdown
  4. Related resources — wikilinks to concept pages

Problem Templates

Template 1: The Physical Object

A [parabolic mirror/cooling tower/elliptical orbit] has equation [conic section equation].

(a) Find [geometric property]
(b) A [physical phenomenon] occurs at [condition]. Calculate [value].
(c) Expand [function] as [series type] to approximate [quantity].
(d) The exact answer requires [integral]. Evaluate using [technique] and compare.

Template 2: The Parametric Motion

A particle moves along [parametric equations involving trig functions].

(a) Show the path lies on [conic section].
(b) Find the Maclaurin/Taylor series for [component] up to [term].
(c) Find slope/area/arc length using parametric derivatives.
(d) Evaluate the resulting integral using [trig technique].

Template 3: The Approximation Verification

Consider [function with integral].

(a) Find the [series expansion] up to [term].
(b) Approximate [definite integral] by integrating series term-by-term.
(c) Evaluate the integral exactly using [substitution/technique].
(d) Calculate percentage error. How many terms needed for [accuracy]?

Quality Checklist

  • [ ] Each problem uses ≥2 distinct topics
  • [ ] Problems increase in difficulty
  • [ ] Real-world contexts where possible
  • [ ] Verification steps (approximate vs exact)
  • [ ] Formula reference included
  • [ ] Study schedule provided
  • [ ] All wikilinks resolve to concept pages
  • [ ] Matches tutorial question style

Example Execution

User: "I want to study Mathematics II for 5 days, covering Geometry, Series, Taylor-Maclaurin, and Trigonometric Integration. Create interleaved problems."

Agent:

  1. Read Tutorial 12 (Power Series + Circle Geometry) — see how questions combine topics
  2. Read Tutorial 3 (Trigonometric Integrals) — understand integration techniques
  3. Read concept pages for all four topics
  4. Design 15 problems, each mixing 2-4 topics:
    • Problem 1: Circle + Maclaurin + Trig integration (area formula)
    • Problem 2: Parabola + Binomial series + Trig substitution (arc length)
    • ...etc
  5. Create file: wiki/synthesis/FAD1014 Mastery Set — Interleaved Mathematics II.md
  6. Update index.md synthesis section
  7. Append to log.md

Integration with Academic Wiki

Problem sets are synthesis documents — they live in wiki/synthesis/ and link together concepts from multiple sources.

Update academic-wiki skill:

  • Add synthesis folder to directory structure
  • Add problem-set-synthesis as related skill
  • Reference in tips section

Tips

  • Interleave, don't batch — never group all problems of one type together
  • Verification builds confidence — always include "check your answer" steps
  • Physical contexts help — mirrors, orbits, pendulums, arches
  • Progressive difficulty — parts (a)→(d) should build complexity
  • Formula overload — provide reference sheet; problems should test application, not memorization
  • Time estimates — consider adding suggested time per problem for exam practice