Imagine math as a kitchen: we're not rushing a stew, but we are arranging an elegant service — concentrated, sensual, and utterly deliberate. For a keen 15-year-old, here is an accelerated, savoury plan using the AoPS Introduction series (Prealgebra → Introduction to Algebra) with Introduction to Geometry running concurrently. Alcumus and the free video library are your mise en place.
Overview & tempo
Goal: finish Prealgebra in an accelerated phase, then move into Introduction to Algebra, while studying Introduction to Geometry throughout both stages. Total tempo: roughly 6–9 months, assuming 8–12 focused hours per week. Faster or slower depending on comfort and contest appetite.
Stage 1 — Finish Prealgebra (≈8–12 weeks)
- Read each section’s problems first — attempt them before any solution. This mirrors the book’s pedagogy and sharpens instincts.
- Work through solutions in the text. Pause to practice the algebraic techniques introduced.
- Use Alcumus to target weak areas (fractions, exponents, ratios) — a daily spoonful of practice keeps skills bright.
- Weekly milestone: complete the set of chapter problems and 20–30 Alcumus problems; every 3–4 weeks, take a cumulative checkpoint test (timed, mixed topics).
Stage 2 — Begin Introduction to Algebra (≈12–20 weeks) while continuing Introduction to Geometry (ongoing)
Start Intro Algebra once basic linear equations, ratios, exponents, and integers feel fluent. Geometry runs alongside to build spatial reasoning and proof habit — you’ll alternate focus days.
- Algebra focus days (3–4 per week): new sections, attempt problems-first, then study solutions and subtleties (factorizations, quadratics, functions).
- Geometry focus days (2–3 per week): tackle problems from Introduction to Geometry — similar triangles, congruence, circles, power of a point — aim for visual proofs and diagrams.
- Use Alcumus topic sets for both algebra and geometry to reinforce procedural fluency; watch aligned videos after attempting problems to consolidate.
Discrete math, number theory & probability
AoPS weaves counting, probability and number theory through Prealgebra and beyond. Reserve one weekly session to tackle these problems and Alcumus sets — they boost combinatorial thinking and contest readiness.
Practical weekly schedule (sample)
- Mon: Algebra new section + 15 Alcumus problems
- Tue: Geometry problems + video walkthrough
- Wed: Problem set review (mixed) + solutions study
- Thu: Algebra practice (problem set) + Alcumus
- Fri: Geometry constructions/proofs + Alcumus
- Sat: Discrete math / timed mixed quiz
- Sun: Rest or light review (errors, concept notebook)
Assessment & tips
Every 4 weeks: a timed mixed exam. Keep an error log. Use the solutions manual when stuck, but first attempt reconciling your work with a peer or video explanation. Read problems first, savor struggle, then taste the solution — that's where real learning blooms.
With gentle persistence and the AoPS ecosystem (books, Alcumus, videos), this 15-year-old will move from arithmetic confidence to elegant algebraic and geometric thinking — all served, as Nigella might whisper, with style and pleasure.