📚 Mastering the Mathematics Investigation Paper: Structure and Sample | 论文写作框架与范文
Writing a mathematics investigation paper is an essential skill for Year 10 CAIE Further Mathematics students. It goes beyond solving equations to developing a structured, evidence‑based argument. This guide provides a clear framework, practical writing tips, and a detailed exemplar to help you produce a high‑quality paper that meets the expectations of the curriculum.
撰写数学探究论文是 Year 10 CAIE 进阶数学学生的一项核心技能。它不仅仅是解方程，更要求建立起一个结构严谨、基于证据的论证。本指南提供了清晰的写作框架、实用技巧以及一篇详细的范文，帮助你完成符合课程要求的高质量论文。
1. Understanding the Purpose of a Mathematics Investigation | 理解数学探究的目的
A mathematics investigation asks you to explore a mathematical situation, identify patterns, test conjectures, and communicate your findings logically. It is not about simply getting the right answer; it is about demonstrating the process of mathematical thinking, from posing a question to justifying conclusions.
数学探究要求你探索一个数学情境，识别规律，检验猜想，并有条理地呈现你的发现。它并不只是得出正确答案，而是展现数学思维的全过程——从提出问题到证明结论。
Your paper should read like a story of discovery: you state a problem, develop a method, collect and analyse data, and finally reflect on what you have learned. The CAIE marking criteria often reward clarity of communication, appropriate use of notation, and the ability to generalise results.
你的论文应该像一部探索故事：陈述问题、设计方法、收集与分析数据，最后反思所得。CAIE 的评分标准通常看重表达清晰、符号使用恰当以及归纳推广的能力。
2. Selecting a Suitable Topic | 选择合适的题目
Choose a topic that is accessible yet rich enough for mathematical exploration. It could be drawn from algebra, geometry, number theory, or modelling. For instance, ‘Investigating the maximum area of a rectangle with a fixed perimeter’ or ‘Exploring patterns in Pascal’s triangle and their links to combinations’ are excellent Year 10 level investigations.
选择一个既易于入手又具有丰富数学内涵的题目。它可以从代数、几何、数论或建模中产生。例如，“探究固定周长的矩形的最大面积”或“探索帕斯卡三角形中的规律及其与组合数的联系”，都是非常适合 Year 10 水平的探究选题。
Ensure your topic allows you to vary parameters, collect numerical data, and identify a general algebraic rule. Avoid topics that are too broad or that lead only to a single numerical answer with no scope for analysis.
确保你的题目能让你调整参数、收集数值数据并归纳出通用的代数法则。避免范围过大或只能得出唯一数值答案、没有分析空间的题目。
3. Formulating a Clear Research Question | 形成清晰的研究问题
A well‑defined research question guides your entire paper. It should be specific, focused, and phrased as a question you will attempt to answer. For example: ‘For a fixed perimeter, what dimensions of a rectangle maximise the area, and can this be generalised to other polygons?’
一个明确的研究问题将引领整篇论文。它应当具体、集中，并以你要尝试回答的疑问句形式提出。例如：“在周长固定的情况下，怎样的矩形尺寸能使面积最大？这一结论能否推广到其他多边形？”
State the question in your introduction and keep returning to it throughout your analysis. A good question often begins with ‘How’, ‘What is the relationship’, or ‘Can we prove that…’. Avoid vague starters like ‘I want to look at…’ without a clear objective.
在引言中明确陈述该问题，并在整个分析过程中反复回顾它。一个好的问题常以“如何”“是什么关系”或“能否证明……”开头。避免使用目标不清晰的模糊开头，如“我想看看……”。
4. Structuring Your Paper: The Essential Sections | 论文结构：基本组成部分
A strong investigation paper typically follows a logical structure: Introduction, Methodology, Results and Analysis, Conclusion, and References. Each section has a distinct role, and the flow from one to the next should feel natural.
一篇优秀的探究论文通常遵循一个逻辑结构：引言、方法、结果与分析、结论和参考文献。每个部分都有其独特作用，且各部分之间的衔接应当流畅自然。
You may also include sub‑sections such as ‘Preliminary explorations’, ‘Generalisation’, and ‘Evaluation’. Use clear headings to signpost your reader through the investigation. A typical word count for a Year 10 paper is 1500–2500 words, but always check your specific task requirements.
你还可以加入诸如“初步探索”“推广”和“评估”等子章节。使用清晰的标题引导读者跟随你的探究。Year 10 论文的典型篇幅为 1500–2500 字，但要始终以具体任务要求为准。
5. Writing the Introduction | 撰写引言
The introduction sets the scene. Begin with a brief context for your investigation, state your research question clearly, and explain why it is of mathematical interest. You might also outline what you hope to discover and mention any constraints you are aware of.
引言设定背景。开篇简要交代探究的情境，清晰陈述你的研究问题，并解释其数学趣味所在。你也可以概述期望发现什么，并提及你意识到的任何限制条件。
For example: ‘A farmer wishes to enclose a rectangular field using 100 metres of fencing. Intuitively, a square might give the largest area, but is this always true? This investigation aims to prove the condition for maximum area and explore how the result changes for other shapes.’
例如：“一位农民想用 100 米篱笆围成一个矩形田地。直觉上正方形可能给出最大面积，但这总是成立吗？本探究旨在证明最大面积的条件，并探索当换成其他形状时结论如何变化。”
6. Presenting Methodology and Mathematical Processes | 陈述方法与数学过程
Describe clearly how you carried out your investigation. This includes defining variables, setting up equations, and explaining any algebraic manipulations. Use precise mathematical language and standard notation.
清楚描述你是如何开展探究的，包括定义变量、建立方程以及解释所有代数变形。使用精确的数学语言和标准符号。
For instance, if investigating rectangle area with fixed perimeter P, you could write: ‘Let the length be l and the width be w, with 2l + 2w = P. Express w in terms of l: w = ½P − l. Then the area A is given by A = l(½P − l) = ½Pl − l².’ Show every step of reasoning.
例如，如果探究固定周长 P 的矩形面积，你可以这样写：“设长为 l，宽为 w，则 2l + 2w = P。将 w 用 l 表示：w = ½P − l。从而面积 A 为 A = l(½P − l) = ½Pl − l²。”要展示推理的每一步。
If you use technology such as spreadsheets or graphing software to generate data, mention what you did and attach screenshots in an appendix if allowed. Never simply state results without explaining the underlying mathematics.
如果你使用电子表格或绘图软件等工具生成数据，要说明你的操作，并在允许的情况下将截图附在附录中。切勿只陈述结果而不解释背后的数学原理。
7. Results and Data Analysis: Using Tables and Graphs | 结果与数据分析：使用表格和图表
Present your findings in organised tables and graphs. A table of values for length, width, and area helps to illustrate a pattern before formal algebraic proof. Label all axes, provide units, and give each table or figure a clear title.
以有序的表格和图表呈现你的发现。一张列出长、宽和面积的数值表，有助于在进行正式代数证明前直观展示规律。标注所有坐标轴，提供单位，并为每个表格或图形给出清晰的标题。
For a perimeter of 20, you might produce a table like:
| Length l | Width w (10 − l) | Area A |
|---|---|---|
| 1 | 9 | 9 |
| 2 | 8 | 16 |
| 3 | 7 | 21 |
| 4 | 6 | 24 |
| 5 | 5 | 25 |
| 6 | 4 | 24 |
Comment on the symmetry: ‘The area increases as l approaches half the perimeter divided by 2, and then decreases symmetrically, suggesting a maximum when l = w = 5.’
对对称性进行评论：“面积随着 l 接近周长一半的一半而增大，然后对称地减小，这表明当 l = w = 5 时面积最大。”
When converting your data into a graph, use a smooth curve if the relationship is continuous. A parabola for A = ½Pl − l² will illustrate the maximum turning point visually. Ensure you describe what the graph reveals about the relationship.
将数据转换为图形时，若关系是连续的，应使用平滑曲线。A = ½Pl − l² 的抛物线将直观展示最大值转折点。确保你描述图形所揭示的关系。
8. Drawing Conclusions and Evaluating | 得出结论与评估
Your conclusion must directly answer the research question. State the general result clearly: ‘For any fixed perimeter P, a rectangle encloses maximum area when it is a square with side length P/4.’ Show the algebraic proof using differentiation or completing the square as appropriate for your level.
你的结论必须直接回答研究问题。清晰陈述一般性结果：“对任意固定周长 P，当矩形为边长为 P/4 的正方形时，包围的面积最大。”根据你的水平，使用求导或配方法给出代数证明。
At Year 10 CAIE Further Mathematics, completing the square is an excellent method: rewrite A = ½Pl − l² as A = −(l − P/4)² + (P/4)². Since the square term is non‑positive, the maximum occurs when l = P/4. This justifies the observation from the table and graph elegantly.
在 Year 10 CAIE 进阶数学中，配方法是一个极佳的方法：将 A = ½Pl − l² 改写为 A = −(l − P/4)² + (P/4)²。由于平方项非正，当 l = P/4 时取得最大值。这巧妙地证明了表格和图形的观察结果。
Beyond the conclusion, include an evaluation section where you reflect on limitations, such as assuming the rectangle’s sides are aligned with axes, and suggest further exploration, like investigating regular polygons or three‑dimensional shapes.
在结论之外，加入一个评估部分，反思局限性（例如假设矩形的边与坐标轴平行），并提出进一步探究方向，如研究正多边形或三维图形。
9. Referencing and Academic Honesty | 参考文献与学术诚信
Acknowledge any sources you used, including textbooks, websites, or software. Use a consistent referencing style (such as APA or MLA) and provide a list at the end of your paper. Even if you only consulted a single textbook, cite it properly.
标明你使用的所有资料来源，包括教材、网站或软件。采用一致的引用格式（如 APA 或 MLA），并在文末列出参考文献。即使你只参考了一本教材，也要规范引用。
Plagiarism is a serious academic offence. Always write in your own words and clearly distinguish between your own ideas and those you have read. If you include a direct quotation, enclose it in quotation marks and provide a page number. Your teacher will value original thought and honest effort above all else.
抄袭是严重的学术不端行为。始终用自己的语言撰写，并明确区分自己的观点与他人的观点。如果你引用原句，用引号标出并提供页码。老师最看重的永远是原创思考和诚实努力。
10. Common Pitfalls to Avoid | 常见误区避免
One frequent mistake is jumping straight into complex algebra without first exploring simpler cases numerically. Always begin with small, concrete examples to develop intuition before generalising. Another pitfall is presenting calculations without explaining what they mean or why you did them.
一个常见误区是直接跳到复杂的代数推导，而不先用数值探索简单情况。始终从简单具体的例子入手培养直觉，再推广。另一个误区是只展示计算过程，却不解释它们的意义或这样做的原因。
Also avoid leaving your research question unanswered. Every section should contribute to answering it. If you find an interesting side path, either tie it back to the main question or save it for the ‘further exploration’ section. Finally, poor formatting, such as missing labels on axes or equations not centred, can make your paper look unprofessional.
同样要避免让你的研究问题悬而未答。每一部分都应为回答该问题服务。如果你发现了有趣的旁支，要么将其与主问题联系起来，要么留到“进一步探究”部分。最后，格式不当（例如坐标轴缺少标签或方程未居中）会使你的论文看起来不专业。
11. Exemplar Extract: Investigating Maximum Area of a Rectangle | 范文节选：探究矩形的最大面积
The following extract illustrates how to integrate the above elements into a coherent section of an investigation. It shows the transition from numerical experimentation to algebraic proof.
以下节选展示了如何将上述要素融入一篇探究的连贯段落中。它示范了从数值实验到代数证明的过渡。
Investigation extract
Let us consider a fixed perimeter of 24 cm. Denote the length by x cm and the width by y cm, so 2x + 2y = 24 ⇒ y = 12 − x. The area A(x) = x(12 − x) = 12x − x².
探究节选
考虑一个固定周长 24 cm。设长度为 x cm，宽度为 y cm，则 2x + 2y = 24 ⇒ y = 12 − x。面积 A(x) = x(12 − x) = 12x − x²。
We first generate a table of integer values for x from 1 to 11.
我们首先生成 x 从 1 到 11 的整数取值表。
| x | y = 12 − x | A = x(12 − x) |
|---|---|---|
| 1 | 11 | 11 |
| 2 | 10 | 20 |
| 3 | 9 | 27 |
| 4 | 8 | 32 |
| 5 | 7 | 35 |
| 6 | 6 | 36 |
| 7 | 5 | 35 |
| 8 | 4 | 32 |
The table shows a clear maximum at x = 6, y = 6, giving an area of 36 cm². This suggests that a square yields the maximum area. To prove this algebraically, we complete the square:
表格显示在 x = 6, y = 6 时取得最大值，面积为 36 cm²。这表明正方形给出最大面积。为了从代数上证明，我们使用配方法：
A(x) = 12x − x² = −(x² − 12x) = −[(x − 6)² − 36] = 36 − (x − 6)²
Since (x − 6)² ≥ 0 for all real x, the expression is always less than or equal to 36. Equality holds when x − 6 = 0, i.e., x = 6. Thus the area is maximised when the rectangle is a square of side 6 cm. This logic extends to any perimeter P: the area is A(l) = ½Pl − l² = −(l − P/4)² + (P/4)², maximum when l = P/4.
由于对所有实数 x 均有 (x − 6)² ≥ 0，该表达式始终小于等于 36。等号在 x − 6 = 0，即 x = 6 时成立。因此当矩形为边长 6 cm 的正方形时面积最大。这一推理可推广至任意周长 P：A(l) = ½Pl − l² = −(l − P/4)² + (P/4)²，最大值在 l = P/4 处取得。
Notice how the extract moves logically from a specific numerical example to a general algebraic proof, which is the hallmark of a strong investigation.
注意该节选如何从具体的数值例子合乎逻辑地过渡到一般代数证明，这正是优秀探究的标志。
12. Final Tips for Success | 成功要诀
Start early and draft your paper section by section. Write mathematics in complete sentences, even when equations are present; for instance, ‘The area is given by A = lw’ is better than simply ‘A = lw’ standing alone. Use a thesaurus to vary your language, but keep terminology precise.
尽早开始，逐节撰写草稿。即便有方程，也要用完整的句子的形式书写数学内容；例如，“面积由 A = lw 给出”比孤立地写“A = lw”更好。使用同义词来丰富语言，但术语务必精确。
Read your paper aloud to check for clarity and flow. Ask a peer to review it and point out any steps that are confusing. Finally, be proud of your own mathematical voice—your investigation should reflect your reasoning and curiosity. With this framework and consistent practice, you can craft a paper that truly stands out.
大声朗读你的论文，检查清晰度和流畅度。请同伴评审，指出任何令人困惑的步骤。最后，要为你自己的数学声音感到自豪——你的探究应反映出你的推理和好奇心。借助这个框架，加上持续练习，你一定能写出真正脱颖而出的论文。
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