📚 Writing a Mathematical Investigation: Framework and Exemplar | 数学探究论文写作：框架与范文
In the WJEC Level 2 Further Mathematics course, students are often required to produce a mathematical investigation. This type of assignment assesses your ability to explore a topic independently, apply mathematical reasoning, and communicate your findings in a structured way. A high-quality investigation demonstrates not only correct calculations but also clear planning, logical flow, and critical reflection. The following guide provides a complete framework for writing a successful mathematical investigation, together with a worked exemplar on Pythagorean triples.
在 WJEC Level 2 进阶数学课程中，学生通常需要完成一篇数学探究论文。这种任务考查你独立探索主题、运用数学推理并用清晰结构表达结论的能力。一篇高质量的探究不仅要展示正确的计算，还要体现清晰的规划、严谨的逻辑和批判性反思。以下指南提供了撰写数学探究论文的完整框架，并附上一份关于勾股数的范文。
1. Understanding the Mathematical Investigation | 理解数学探究
A mathematical investigation is a structured piece of writing in which you choose a mathematical idea, pose a question, gather evidence, and draw conclusions. Unlike routine exercises, an investigation asks you to think like a mathematician: to make conjectures, test them, and explain why patterns occur.
数学探究是一篇结构化的文章，在文中你选择一个数学概念，提出问题，收集证据并得出结论。与常规练习不同，探究要求你像数学家一样思考：提出猜想，进行检验，并解释模式产生的原因。
Your work is assessed against criteria such as planning, use of mathematical language, accuracy of reasoning, and presentation. It is essential to keep the reader in mind and make every step clear.
你的作品将依据计划、数学语言的运用、推理的准确性和展示等方面进行评分。始终考虑读者的感受并清晰说明每一步骤至关重要。
2. Choosing a Suitable Topic | 选择合适的课题
The first and most important stage is selecting a topic that is rich enough to investigate but not so broad that it becomes unmanageable. Good investigations often arise from playful exploration of numbers, shapes, or sequences.
第一步也是最重要的一步是选择一个足够丰富但又不过于宽泛以致失控的课题。好的探究常常源于对数字、图形或序列的有趣探索。
For further mathematics, you might consider topics such as properties of number patterns, iterative processes, matrix transformations, geometric sequences linked to fractals, or the history behind a famous theorem. Always check that the topic allows you to collect numerical evidence and generalise.
在进阶数学中，可以考虑数字模式的性质、迭代过程、矩阵变换、与分形相关的几何数列或某个著名定理背后的历史。务必确保所选课题能够让你收集数值证据并进行推广。
For our exemplar, we choose 'Generating Pythagorean Triples' because it connects number theory, algebra, and proof, all of which are central to the WJEC further mathematics syllabus.
在我们的范文中，我们选择“勾股数的生成”，因为它将数论、代数与证明联系起来，这些都是 WJEC 进阶数学大纲的核心内容。
3. Planning Your Research | 制定研究计划
Before writing, draft a plan that outlines your aim, the data you need to gather, and the methods you intend to use. A strong plan includes stages such as background reading, experimentation, pattern spotting, and proving any conjectures.
在动笔之前，先草拟一份计划，概述你的目标、需要收集的数据以及打算使用的方法。一份扎实的计划应包括背景阅读、实验、发现规律以及证明猜想等阶段。
ew the Pythagorean theorem; (2) use a spreadsheet to generate triples; (3) discover Euclid' a systematically; (5) prove algebraically that the generated triples satisfy a² + b² = c².
对于勾股数探究，计划可以如下：(1) 回顾勾股定理；(2) 使用电子表格生成勾股数；(3) 发现欧几里得公式；(4) 系统检验该公式；(5) 用代数方法证明生成的勾股数满足 a² + b² = c²。
4. Conducting Mathematical Research | 进行数学研究
Research in a school investigation typically involves a blend of empirical exploration and formal reasoning. You may use technology such as dynamic geometry software or Python scripts, but always explain the process so that someone else can replicate it.
学校探究中的研究通常包括实证探索和形式推理的结合。你可以使用动态几何软件或 Python 脚本等技术，但必须解释过程，以便他人能够重现。
Keep a log of your findings, including false starts and surprising intermediate results. These reflections strengthen your final write-up because they show genuine mathematical thinking.
记录你的发现，包括错误尝试和令人惊讶的中间结果。这些反思能让最后的论文更有说服力，因为它们展示了真实的数学思维。
5. Structure of the Investigation Paper | 论文结构概览
A typical mathematical investigation follows the IMRaD model: Introduction, Methodology, Results, and Discussion, together with a Conclusion and References. Adapting this structure to pure mathematics helps you present your reasoning logically.
典型的数学探究论文遵循 IMRaD 模式：引言、方法、结果与讨论，外加结论和参考文献。将此结构应用于纯数学有助于逻辑地呈现你的推理过程。
- Introduction – states the problem and your aim.
- Methodology – explains how you generated data and the algebraic tools used.
- Results – presents patterns, tables, and algebraic expressions.
- Discussion – interprets the significance, limitations, and possible extensions.
- Conclusion – summarises the main findings.
引言 – 陈述问题与目标；方法 – 说明数据生成方式和所用代数工具；结果 – 展示模式、表格和代数表达式；讨论 – 阐释意义、局限性及可能的拓展；结论 – 总结主要发现。
6. Writing the Introduction and Aim | 撰写引言与目的
Your introduction should hook the reader and provide the necessary background. Begin with a real-world connection or a historical note. After the context, clearly state the aim as a specific research question, such as: 'How can we generate all primitive Pythagorean triples using elementary number theory?'
引言应吸引读者并提供必要背景。可以从与现实世界的联系或历史趣闻开头。在背景之后，明确陈述具体的研究问题作为目标，例如：“如何用初等数论生成所有原始勾股数？”
Avoid making the introduction too long; four or five sentences are usually sufficient. End the introduction with a roadmap that previews the sections to come.
引言不宜过长，四五句话通常足够。在引言末尾用路线图预览接下来的章节。
7. Presenting Methodology and Results | 撰写方法与结果
In the methodology section, describe your approach step by step rather than simply listing equations. For the Pythagorean triple exemplar, you might explain how you used the Euclid formula: a = m² − n², b = 2mn, c = m² + n², with integers m > n > 0.
在方法论部分，一步步描述你的方法，而不是简单罗列方程。在勾股数范文中，可以解释如何使用欧几里得公式：a = m² − n², b = 2mn, c = m² + n²，其中整数 m > n > 0。
Present your results clearly using tables. For instance, a table of triples for small values of m and n helps the reader see the pattern instantly.
用表格清晰呈现结果。例如，针对较小的 m 和 n 列出勾股数的表格，可以帮助读者立即看出模式。
| m | n | a | b | c | a² + b² = c²? |
|---|---|---|---|---|---|
| 2 | 1 | 3 | 4 | 5 | 9+16=25 ✓ |
| 3 | 2 | 5 | 12 | 13 | 25+144=169 ✓ |
| 4 | 1 | 15 | 8 | 17 | 225+64=289 ✓ |
| 4 | 3 | 7 | 24 | 25 | 49+576=625 ✓ |
After the table, guide the reader to notice that swapping m and n simply swaps a and b, and that primitive triples occur when m and n are coprime and not both odd.
在表格之后，引导读者注意到交换 m 和 n 仅仅交换了 a 和 b，并且当 m 和 n 互质且不全为奇数时出现原始勾股数。
8. Discussing and Analysing Findings | 讨论与分析
The discussion is where you demonstrate deeper understanding. Explain why Euclid's formula works: substitute the expressions into a² + b² and simplify to show it equals (m² + n²)². This algebraic proof is the centrepiece of a high-scoring investigation.
讨论部分是展示深层理解的地方。解释欧几里得公式为何有效：将表达式代入 a² + b²，化简后显示它等于 (m² + n²)²。这一代数证明是高评分探究的核心。
(m² − n²)² + (2mn)² = m⁴ − 2m²n² + n⁴ + 4m²n² = m⁴ + 2m²n² + n⁴ = (m² + n²)²
You should also discuss limitations: Euclid's formula generates all primitive triples, but non-primitive triples require a scaling factor. Reflect on how the investigation could be extended, for example, by exploring triples in different number systems or linking to the unit circle.
同时还应讨论局限性：欧几里得公式生成了所有原始勾股数，但非原始勾股数需要一个缩放因子。反思探究如何拓展，例如探索不同数系中的勾股数或与单位圆建立联系。
9. Conclusion and Referencing | 结论与参考文献
Summarise the key outcomes in a short paragraph, avoiding new information. Restate the research question and confirm how it was answered. For the exemplar: 'Euclid's formula successfully generates all primitive Pythagorean triples; the algebraic proof confirms its validity.'
用简短段落总结主要成果，避免引入新信息。重申研究问题并确认如何得到解答。以范文为例：“欧几里得公式成功生成了所有原始勾股数；代数证明验证了其正确性。”
Always include a reference list citing any sources used, such as textbooks or online resources. Proper referencing demonstrates academic honesty.
务必附上参考文献列表，引用所用资料，如教材或在线资源。规范引用体现了学术诚信。
10. Exemplar Section: Introduction and Aim | 范文展示：引言与目标
The following is a condensed version of the introduction from the exemplar investigation 'Generating Pythagorean Triples'. Notice how it moves from a historical hook to a precise mathematical aim.
以下是范文探究“勾股数的生成”引言的浓缩版。注意它如何从历史引子过渡到精确的数学目标。
Introduction: ' it was the Greek mathematician Euclid who provided a general formula. The aim of this investigation is to understand how Euclid' ples and to prove that the formula always produces a valid triple satisfying a² + b² = c².'
引言：“勾股数，例如 (3,4,5)，已被研究超过三千年。古巴比伦人将其记录在泥板上，但希腊数学家欧几里得给出了通用公式。本探究旨在理解欧几里得公式如何生成所有原始勾股数，并证明该公式总能产生满足 a² + b² = c² 的有效三元组。”
11. Exemplar Section: Methodology and Key Results | 范文展示：方法与关键结果
c, and then checked the condition a² + b² = c². The key results are summarised as follows:
在方法部分，学生解释了系统性的操作：他们为 m 和 n 设定两列，用电子表格计算 a、b、c，然后检查 a² + b² = c² 的条件。关键结果总结如下：
Patterns observed:
utive, a and b are consecutive odd and even numbers (e.g. m=3, n=2 gives a=5, b=12, c=13).
2. All primitive triples have exactly one even leg and an odd hypotenuse.
3. The product of the two legs is always a multiple of 12.
观察到的模式：
1. 当 m 与 n 连续时，a 和 b 为连续的奇数与偶数（例如 m=3, n=2 得到 a=5, b=12, c=13）。
2. 所有原始勾股数恰好有一个偶数的直角边和奇数斜边。
3. 两直角边的乘积总是 12 的倍数。
The student then shows the algebraic verification displayed earlier, confirming that the pattern is not accidental but a consequence of the formula.
随后，学生展示了前文中的代数验证，确认这些模式并非巧合而是公式的结果。
12. Exemplar Section: Discussion and Conclusion | 范文展示：讨论与结论
itive triple (6,8,10) directly; it must be obtained by scaling (3,4,5) by a factor of 2. This observation leads to the idea of a generic triple generator: a = k(m² − n²), b = k(2mn), c = k(m² + n²), where k is a positive integer.
范文的讨论解释了该方法为何有效及其局限性。例如，公式不会直接生成非原始勾股数 (6,8,10)；必须通过将 (3,4,5) 按系数 2 缩放得到。这一观察引出了通用勾股数生成器的概念：a = k(m² − n²)，b = k(2mn)，c = k(m² + n²)，其中 k 为正整数。
The conclusion succinctly summarises the main finding and suggests further work: investigating Pythagorean triples in three dimensions or exploring their link to trigonometric identities.
结论简洁地总结了主要发现，并建议进一步研究：探究三维空间中的勾股数或探索其与三角恒等式的联系。
The exemplar demonstrates that a well-structured investigation, supported by precise mathematical language and a clear chain of reasoning, meets the highest criteria of the WJEC mark scheme.
该范文表明，一篇结构良好的探究，辅以准确的数学语言和清晰的推理链条，能够满足 WJEC 评分标准的最高要求。
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