It is a truth, which I dare hope will be readily acknowledged by the most modest pupil, that a young mind of thirteen is equally capable of mathematical refinement and of sensible application to affairs of finance. Presented here, in a style politely reminiscent of Mrs. Austen's most agreeable compositions, is a prospectus for a three-term course, prepared with due regard to the principles of ACARA v9 and to the books by Mr. Richard Rusczyk which shall be our chief companions.
Course Overview
This course invites the student to pursue algebra and geometry with both rigorous thought and worldly application. Across three terms we traverse:
- Term 1: Linear and quadratic reasoning, functions, graphing, and an introduction to complex numbers.
- Term 2: Deeper study of polynomials, sequences and series, inequalities, and optimization — with deliberate connections to financial modelling and stock-market thinking.
- Term 3: Geometry and proof, analytic geometry, and trigonometry, closing with projects that unite spatial reasoning and quantitative analysis.
Alignment with ACARA v9
This curriculum is aligned to the ACARA v9 mathematical strands and mathematical proficiencies by developing:
- Fluency in Number and Algebra: manipulating expressions, solving equations and systems, understanding functions and polynomials.
- Understanding in Measurement and Geometry: 2D and 3D measurement, analytic geometry, and formal proofs.
- Reasoning with Patterns, Chance and Data when modelling financial situations and interpreting stock and economic news.
- Working mathematically: problem solving, modelling, communicating, and using technology.
Textbooks & Resources
- Primary texts: Introduction to Algebra, 2nd edition, Richard Rusczyk; Introduction to Geometry, Richard Rusczyk.
- Technology: Graphing calculator or Desmos; spreadsheet software (Google Sheets or Excel) for financial modelling; an online stock market simulator (eg. Investopedia Simulator).
- Supplementary: Selected news articles on markets and economics, curated for age-appropriate discussion.
Term-by-Term Detail
Term 1 — Algebraic and Geometric Perspectives
Objectives: Students shall master linear equations and systems, solve quadratics by factoring and formula, sketch and interpret graphs in the Cartesian plane, work with functions, and receive an introduction to complex numbers.
- Weeks 1–2: Algebraic expressions and linear equations; modelling simple real-life relationships (cost = fixed + variable).
- Weeks 3–4: Systems of equations; graphical and algebraic solutions; application to problems such as break-even analysis.
- Weeks 5–6: Quadratics: factoring, completing the square, quadratic formula; parabolas and their real-world interpretations.
- Weeks 7–8: Functions: definitions, domain and range, function notation, and sketching transformations.
- Weeks 9–10: Brief introduction to complex numbers and their geometric interpretation on the complex plane.
Assessment examples: a short project modelling a simple budget using linear equations; problem sets from AoPS; a written test on quadratics and graphing.
Term 2 — Polynomials, Sequences, and Financial Applications
Objectives: To explore polynomial behaviour, radical expressions, conic sections, inequalities and optimization. To begin forming foundations of financial mathematics including sequences and compound growth.
- Weeks 1–2: Polynomial arithmetic and factor theorem; solving higher-degree equations and sketching polynomial graphs.
- Weeks 3–4: Radical and rational expressions; simplifying and solving; application to domain and asymptote ideas.
- Weeks 5–6: Sequences and series: arithmetic and geometric sequences; compound interest as a geometric sequence; introduction to present and future value concepts.
- Weeks 7–8: Inequalities, optimization techniques (completion of the square, derivative intuition via discrete optimization problems), and quadratic/linear programming ideas at an accessible level.
- Weeks 9–10: Conic sections and their equations; discussion of orbits, lenses, and optimisation problems with geometric constraints.
Finance integration: Students will build a simple spreadsheet model of an investment with periodic contributions, simulate stock-portfolio returns using historical simplified data, and read a short finance news item each week to extract quantitative claims for scrutiny.
Assessment examples: a spreadsheet-based financial modelling task; an extended written problem set on polynomials and inequalities; an oral presentation linking mathematical results to a market story.
Term 3 — Geometry, Proof and Analytical Methods
Objectives: To develop formal proof skills, measurement of 2D and 3D shapes, analytic geometry methods, and trigonometry useful for modelling and later calculus study.
- Weeks 1–3: Logic and proof techniques — direct proof, contradiction, and induction at an introductory level; Euclidean geometry problems from AoPS.
- Weeks 4–5: Study and measurement of polygons, circles, polyhedra; surface area and volume computations and reasoning.
- Weeks 6–7: Analytic geometry: lines, circles, conic intersections, distance and midpoint formulae, and coordinate proofs.
- Weeks 8–10: Trigonometry — unit circle, sine and cosine rules, solving triangles and applications to periodic phenomena and simple harmonic ideas.
Capstone: A project that requires geometric reasoning and algebraic modelling — for instance, designing a fair division of resources using analytic methods or modelling cyclical market behaviour with trigonometric ideas.
Assessments & Evidence of Learning
- Formative: weekly problem sets, in-class quizzes, and reflective learning journals.
- Summative: end-of-term exams, a spreadsheet financial modelling project, a geometry proof portfolio, and an oral defence of a chosen modelling project.
- Mathematical communication: students will submit written solutions in clear proof-style and produce short presentations connecting mathematics to financial news.
Sample Weekly Lesson Structure (step-by-step)
- Warm-up (10 minutes): a brief problem to recall prior knowledge and provoke curiosity.
- Direct instruction (15–25 minutes): concise explanation of a new concept, with an example solved together.
- Guided practice (20 minutes): paired or small-group problems, with teacher circulating to scaffold.
- Application (15–20 minutes): a real-world or contest-style problem — e.g., a mini financial modelling task or an AoPS challenge problem.
- Reflection & homework (5 minutes): a sentence or two describing what was learned and one problem to attempt at home.
Differentiation for Years 10–12
Although our student is thirteen, the course is scaffolded so that:
- Year 10 level: emphasis on fluency and problem solving with support and concrete examples.
- Year 11 extension: deeper proof work, more challenging polynomial and sequence problems, and enriched modelling complexity.
- Year 12 extension: introduction to limits and an informal bridge to calculus, rigorous optimization problems, and more advanced statistics relevant to finance.
How This Prepares for Careers in Finance and Economics
Algebra and geometry are the language of modelling: linear systems describe budgets, quadratics model certain profit curves, polynomials and sequences underpin forecasting, and trigonometry and analytic geometry support data visualisation and spatial reasoning. By learning to translate news into quantitative questions, and by building spreadsheet models, the student acquires habits and tools prized in careers of economics, financial analysis, and data science.
Example: A Brief Financial Illustration
Suppose one invests a modest sum each month. The formula for the accumulated value of regular contributions uses geometric sequences: we compute each contribution's growth and sum them. Learning sequences and series thus becomes immediately practical — and will later extend to continuous compounding when calculus is introduced.
Recommended Assessments for Reporting
- Knowledge and Understanding: written exams covering procedures and definitions.
- Problem Solving and Modelling: graded projects using spreadsheets and open-ended problems.
- Reasoning and Communication: proof portfolio and oral presentation on a finance-related modelling task.
Final Remarks in a Gentle Tone
It is hoped that, under these studies, a young scholar will grow both in mathematical strength and in worldly discernment. The lessons are fashioned to cultivate precision of thought, delight in problem solving, and an ability to engage with the economic discourse of our day. Should the pupil apply themselves with steady good humour, they shall find both profit and pleasure in the numbers.
Sincerely,
Your Mathematician-in-Residence, faithfully committed to instruction and to the charming manner of explanation.