Year Math Prospectus for a 15-year-old — ACARA v9-aligned Algebra & Geometry with Alcumus, Financial Projects, and Contest Prep

A 1-year, ACARA v9-aligned course prospectus for a 15-year-old: Introduction to Algebra and Geometry using AoPS Alcumus and Rusczyk texts, with business/finance mini-projects, contest prep, and clear weekly rhythms.

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IN THE COURT OF LEARNING
Case: ACARA v. Curiosity — Yearly Prospectus
Counsel for the Student-Plaintiff: Instructor, Alcumus, and Rusczyk

Brief — Overview (Ally McBeal cadence: brisk, slightly theatrical)

This one-year course (for a 15‑year‑old) adjudicates two chief counts: (1) Introduction to Algebra + Alcumus and (2) Introduction to Geometry + Synthesis — each paired with applied finance/business projects. (Cue small musical aside: "Oh, the proofs!" — delivered with wink.)

Statement of Objectives (Goals)

  • Algebra: linear equations & inequalities, systems, functions, basic polynomials and factoring, algebraic problem solving.
  • Geometry: Euclidean reasoning, congruence & similarity, area & volume, circle theorems, coordinate geometry, proof-writing.
  • Financial & Business: cost/revenue/profit modelling, break-even analysis, compound interest, simulated small business or portfolio.
  • Mathematical proficiencies per ACARA v9: understanding, fluency, problem-solving and reasoning across Number & Algebra; Measurement & Geometry; and Financial contexts.

Required Reading & Tools

Primary texts: Rusczyk — Introduction to Algebra (aim: half to two-thirds completed this year) and Introduction to Geometry (key proof chapters). Daily practice: AoPS Alcumus (algebra → geometry tracks). Tools: Desmos, GeoGebra, Google Sheets/Excel.

Weekly Rhythm & Assessment

  1. Daily: 30–60 min Alcumus adaptive practice (build fluency and error logs).
  2. 3 short lessons per week: guided reading, worked examples, and one written proof per week.
  3. Weekly: one increasing-difficulty problem set, one timed problem session, and one applied mini-report (finance/business sketch or geometry synthesis).
  4. Monthly: project checkpoint. Mid-year and end-of-year capstones.

Applied Mini-Projects (Concrete Ideas)

Algebra Business Tie-in: choose a small product, model cost = fixed + variable * units, revenue = price * units; compute break-even point, per‑unit profit, and sensitivity to price changes (use spreadsheets and linear graphs).

Geometry & Finance Capstone: run a simulated small business or savings portfolio. Present profit graphs, compound growth curves, break-even visuals, and geometric proofs or diagrams that justify design choices (e.g., packaging area minimisation).

Alignment to ACARA v9 (explicit)

  • Number & Algebra — linear relationships, algebraic techniques and polynomial manipulation.
  • Measurement & Geometry — Euclidean geometry, area/volume, coordinate geometry and reasoning.
  • Financial contexts — modelling money, interest and basic business mathematics.
  • Mathematical proficiencies — reasoning (proofs), fluency (Alcumus practice), problem solving (contest-style & project work), understanding (explanations & error logs).

Success in Two Years (Relief Sought)

Master Prealgebra + Introduction to Algebra conceptually and via problem solving; complete Introduction to Geometry with solid proof skills; be contest-ready (AMC 10/12 targets); and have practical financial literacy: budgets, spreadsheets, and a polished capstone business/portfolio model.

Closing Motion

Adopt a steady daily Alcumus habit, pair it with Rusczyk reading, alternate algebra and geometry modules, and embed weekly finance mini-projects. Verdict: curiosity sustained, skills sharpened, and real-world math made deliciously useful.

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