📚 Pre-U CAIE Mathematics: Project Writing Framework and Exemplar Essay | Pre-U CAIE 数学：项目论文写作框架与范文
For Cambridge Pre-U Mathematics candidates, the final component — Paper 3: Project — is a unique opportunity to conduct independent mathematical research and present findings in a formal report. Unlike timed examinations, this coursework element assesses your ability to explore a mathematical topic in depth, model real-world phenomena, and communicate reasoning with clarity and rigour. This article provides a comprehensive framework for structuring your project report, along with an annotated exemplar excerpt to guide your writing.
对于剑桥 Pre-U 数学的考生来说，最终部分——试卷三：项目——是进行独立数学研究并以正式报告呈现的独特机会。与限时考试不同，这项课程作业评估的是你深入探究数学主题、对现实世界现象建模，以及清晰严谨地传达推理过程的能力。本文提供了构建项目报告的全面框架，并附有带注释的范文摘录，以指导你的写作。
1. Understanding the Pre-U Mathematics Project | 理解 Pre-U 数学项目
The Pre-U Mathematics (9794) project requires you to produce a written report of 2000–3000 words on a mathematical topic of your choice. The project should demonstrate independent thought, appropriate use of mathematical techniques beyond the standard syllabus, and the ability to structure a coherent argument. You are expected to apply mathematical modelling, analysis, or historical exploration, and to reflect critically on the process.
Pre-U 数学（9794）项目要求你就自选的数学主题撰写一篇 2000-3000 字的书面报告。项目应体现独立思考、超越常规教学大纲的数学技巧的恰当运用，以及组织连贯论证的能力。你需要应用数学建模、分析或历史探究，并对过程进行批判性反思。
Examiners look for a clear research question, a logical development of ideas, accurate notation, and a well-founded conclusion. This component carries significant weight, so mastering the report structure is essential.
考官看重清晰的研究问题、合乎逻辑的思路发展、准确的符号以及有充分依据的结论。这部分占比较大，因此掌握报告结构至关重要。
2. Choosing a Suitable Topic | 选择合适的主题
A strong project starts with a well-defined, manageable question. Avoid topics that are either too trivial or excessively broad. Ideal projects often come from applying calculus, differential equations, probability, or numerical methods to real-world scenarios, such as modelling the spread of a disease, analysing the optimal angle of a projectile, or investigating the mathematics of encryption.
一个优秀的项目始于明确且可控的问题。避免过于简单或范围过大的主题。理想的项目往往来自将微积分、微分方程、概率或数值方法应用于现实场景，例如模拟疾病传播、分析抛射物的最佳角度，或研究加密数学。
You might also consider a historical or philosophical investigation, such as tracing the development of complex numbers or comparing different proofs of the infinite nature of primes. Whatever you choose, ensure there is enough mathematical depth and scope for personal input.
你也可以考虑历史或哲学性质的探究，比如追溯复数的发展历程，或比较素数无穷性的不同证明。无论选择何种主题，都要确保其具备足够的数学深度和个人发挥空间。
3. Overall Structure of the Report | 报告整体结构
A typical Pre-U project report should contain the following sections, though slight variations are acceptable depending on the nature of the investigation:
典型的 Pre-U 项目报告应包含以下部分，不过根据研究性质允许稍有变化：
Title Page → Abstract → Introduction → Background Theory / Literature Review → Methodology → Analysis and Results → Discussion → Conclusion → References → Appendices. Each section serves a distinct purpose, guiding the reader from the initial question to the final insight.
标题页 → 摘要 → 引言 → 背景理论 / 文献综述 → 方法 → 分析与结果 → 讨论 → 结论 → 参考文献 → 附录。 每个部分都有独特的目的，引导读者从最初的问题走向最终的见解。
Adhering to this structure not only enhances clarity but also aligns with the marking criteria, which reward organisation, analysis, and communication.
遵循此结构不仅能提高清晰度，还契合评分标准，后者看重组织、分析和沟通能力。
4. Title Page and Abstract | 标题页与摘要
The title page should include the project title, your candidate number, centre number, and the date. The title itself must be informative and concise, e.g., ‘Modelling Cooling Rates Using Newton’s Law of Cooling: An Experimental Investigation’.
标题页应包含项目标题、考生号、中心号和日期。标题本身应信息明确且简洁，如“使用牛顿冷却定律模拟冷却速率：一项实验探究”。
The abstract is a 150–200 word summary of your entire project. It should briefly state the research question, methodology, key findings, and main conclusion. Write it last, so it accurately reflects the completed report.
摘要是对整个项目的 150-200 字总结。它应简要陈述研究问题、方法、关键发现和主要结论。最后撰写摘要，以便准确反映已完成报告的内容。
5. Introduction and Rationale | 引言与研究依据
In the introduction, you set the scene. Begin by outlining the real-world context or the mathematical puzzle that motivated your investigation. Then, clearly formulate your central research question and any subsidiary questions. Explain why this topic is interesting mathematically and what you hope to achieve.
在引言部分，你需要铺陈背景。首先概述激发你研究的现实背景或数学谜题。然后，清晰地阐述中心研究问题和任何辅助问题。解释为什么该主题在数学上令人感兴趣，以及你期望达成什么目标。
For example: ‘This project aims to model the temperature decay of a hot liquid under ambient conditions, using Newton’s Law of Cooling. The objective is to determine how well this first-order linear differential equation predicts empirical data, and to discuss potential sources of discrepancy.’
例如：“本项目旨在利用牛顿冷却定律模拟热液体在环境条件下的温度衰减。目标是评估这个一阶线性微分方程对经验数据的预测效果，并讨论潜在的差异来源。”
6. Background Theory and Mathematical Preliminaries | 背景理论与数学预备知识
Here you present the mathematical tools and theorems relevant to your investigation. This section should not simply copy textbook material; instead, explain the concepts in your own words and show why they are applicable. Include key definitions, derivations, and notation.
在此部分，你要展示与研究相关的数学工具和定理。不应照搬教科书内容；而要用自己的语言解释概念，并说明其适用性。包括关键定义、推导和符号。
If your project involves differential equations, you might derive the general solution and discuss the role of constants. For a statistical project, outline probability distributions and test statistics. Use clear, well-structured equations:
如果你的项目涉及微分方程，可以推导通解并讨论常数的作用。对于统计项目，则概述概率分布和检验统计量。使用清晰、结构良好的方程：
dT/dt = −k(T − Tₑ) ⇒ T(t) = Tₑ + (T₀ − Tₑ) e⁻ᵏᵗ
此处方程显示为居中加粗： dT/dt = −k(T − Tₑ) ⇒ T(t) = Tₑ + (T₀ − Tₑ) e⁻ᵏᵗ
Always define each symbol: T(t) represents temperature at time t, Tₑ is ambient temperature, k is a positive constant, and T₀ is initial temperature.
始终定义每个符号：T(t) 表示 t 时刻的温度，Tₑ 为环境温度，k 是正常数，T₀ 是初始温度。
7. Methodology: Data Collection and Modelling Approach | 方法：数据收集与建模方法
Describe how you obtained your data or developed your model. If you conducted an experiment, detail the apparatus, procedure, and how you ensured reliability (e.g., repeated trials, controlled variables). If the project is purely theoretical, explain the mathematical approach, such as setting up a system of equations or running a simulation.
描述你如何获得数据或建立模型。如果进行了实验，详细说明仪器、步骤以及如何确保信度（例如，重复试验、控制变量）。如果项目纯属理论，解释数学方法，例如建立方程组或运行模拟。
For a modelling investigation, specify any simplifying assumptions you made (e.g., ignoring heat loss due to evaporation, assuming a well-stirred liquid). Justify these assumptions mathematically and acknowledge their limitations.
对于建模探究，明确指出你做出的任何简化假设（例如，忽略蒸发造成的热损失、假设液体搅拌均匀）。从数学上论证这些假设的合理性，并承认其局限性。
8. Analysis and Results | 分析与结果
Present your processed data, calculations, and any graphs or tables. This section is the core of your project. Use appropriate mathematical techniques to analyse the data, such as regression to estimate the cooling constant k, or solving an integral. Show key steps clearly.
展示处理后的数据、计算以及任何图表。这部分是项目的核心。使用适当的数学技巧分析数据，例如回归以估算冷却常数 k，或求解积分。清晰展示关键步骤。
When comparing model predictions to actual data, calculate percentage errors or use the coefficient of determination R². Display results in a well-labelled table:
将模型预测与实际数据进行比较时，计算百分比误差或使用决定系数 R²。在标注清晰的表格中展示结果：
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