📚 Writing a Mathematical Investigation Paper: Framework and Sample | Year 7 OCR 进阶数学论文写作框架与范文
Writing a mathematical investigation paper might feel daunting at first, but it is one of the best ways to develop logical thinking and clear communication in mathematics. In Year 7 OCR Further Maths, you are encouraged to go beyond just getting the right answer – you need to explain how you think, record your discoveries, and present your reasoning in a structured way. This article provides a complete writing framework, followed by a full sample paper and analysis, so you can learn exactly what makes a high-quality maths paper.
写一篇数学探究论文起初可能令人生畏，但这是培养逻辑思维和清晰表达数学思想的最佳方式之一。在 Year 7 OCR 进阶数学中，你不仅要得出正确答案，更需要解释你的思考过程、记录你的发现，并以有条理的方式呈现推理。本文提供一套完整的写作框架，并附上一篇完整的范文与评析，让你精准把握一篇优质数学论文的要领。
1. What is a Mathematical Investigation Paper? | 什么是数学探究论文？
A mathematical investigation paper is a short report where you explore a mathematical question, look for patterns, form conjectures, and try to prove or justify your ideas. It is not a textbook exercise – there is often no single correct path, and part of the challenge is deciding how to organise your exploration and what to record. In OCR Further Maths, such papers help you demonstrate the 'reason mathematically' and 'problem solving' skills that are assessed throughout the course.
数学探究论文是一份简短的报告，你需要在其中探索一个数学问题、寻找规律、提出猜想，并尝试证明或论证自己的想法。它不同于课本练习——通常没有唯一的正确路径，挑战之一就在于如何组织你的探索过程并决定记录哪些内容。在 OCR 进阶数学中，这类论文能够帮助你展示贯穿课程始终的‘数学推理’与‘解决问题’能力。
A good investigation paper reads like a story: you set the scene, describe what you did, show what you found, and reflect on whether your findings make sense. Even at Year 7 level, your paper can follow the same basic structure used by professional mathematicians when they publish research, just in a simpler form.
一篇好的探究论文读起来像一个故事：你先做好铺垫，描述你做了什么，展示你的发现，然后反思这些发现是否合理。即使在 Year 7 阶段，你的论文也可以沿用专业数学家发表研究时所使用的基本结构，只是形式更简单而已。
2. Choosing an Engaging Topic | 选择一个有趣的主题
Start with a question that genuinely sparks your curiosity. It could come from a pattern you noticed in a number sequence, a shape you drew, or a puzzle you encountered. For OCR Further Maths, strong topics often involve number theory (e.g. triangular numbers, square numbers, figurate numbers), geometry (e.g. angles in polygons, tessellations), or probability (e.g. fair games with dice). Avoid topics that are too broad – 'Investigating triangles' is too vague, whereas 'Why is the sum of exterior angles of any polygon 360°?' gives a clear focus.
从一个真正能激发你好奇心的提问开始。它可以源于你在数列中注意到的模式、你画出的一个图形，或者你遇到的一个谜题。对于 OCR 进阶数学来说，有力的选题通常涉及数论（如三角形数、平方数、形数）、几何（如多边形内角、镶嵌问题）或概率（如公平的骰子游戏）。避免过于宽泛的主题——‘探究三角形’就太空泛，而‘为什么任意多边形的外角和总是 360°？’则给出了清晰的焦点。
A well-chosen topic allows you to collect some data by hand, spot a pattern, make a prediction, and then test that prediction further. If you can also create a visual proof or connect the idea to another part of maths (e.g. linking triangular numbers to combinations or algebra), your paper will be even stronger.
一个精心选择的主题让你能够手工收集一些数据、发现规律、做出预测，然后进一步检验预测。如果你还能构造一个可视化证明，或者将这一想法与数学的其他部分联系起来（例如将三角形数与组合或代数联系），你的论文就会更出色。
3. The Structure of a Maths Paper: Overview | 数学论文结构总览
Most mathematical investigations use a simplified version of the IMRaD structure (Introduction, Methods, Results, and Discussion). For Year 7, we can break this down into six key sections: Title, Abstract, Introduction, Investigation & Methods, Results & Patterns, and Conclusion (including References if needed). Thinking about these sections before you start writing will save you from producing a disorganised collection of notes.
大多数数学探究都使用简化的 IMRaD 结构（引言、方法、结果和讨论）。对于 Year 7 而言，我们可以将其分解为六个关键部分：标题、摘要、引言、探究与方法、结果与规律、结论（必要时包含参考文献）。在动笔之前先想清楚这些部分，可以避免你写出一堆杂乱无章的笔记。
Each section has a distinct job. The introduction tells the reader what you are investigating and why. The methods section shows what you actually did – sample calculations, diagrams, tables you created. The results section presents the patterns and formulas you found, often with evidence. The conclusion answers your original question and suggests what you might investigate next.
每个部分都有明确的任务。引言告诉读者你要探究什么以及为什么。方法部分展示你实际做了什么——样例计算、图示、你绘制的表格。结果部分呈现你发现的模式和公式，通常附有证据。结论部分回答你最初的问题，并提示下一步还可以探究什么。
4. Title and Abstract | 标题与摘要
The title should be specific and descriptive. Instead of 'Maths Project', use something like 'Investigating the Sum of Two Consecutive Triangular Numbers' or 'A Pattern in the Fibonacci Sequence'. The abstract is a short paragraph (3–4 sentences) that summarises your whole paper: what you investigated, the key method, the main finding, and one interesting connection or implication. Even though the abstract appears first, many writers find it easiest to write it last, when they know exactly what their paper says.
标题应该具体且具有描述性。不要用‘数学课题’这样的标题，而要用‘探究两个连续三角形数之和的规律’或‘斐波那契数列中的一个模式’。摘要是用一小段话（3–4 句）概括整篇论文：你探究了什么、关键方法、主要发现以及一个有趣的联系或引申。尽管摘要出现在最前面，但许多作者觉得最后写摘要最容易，因为他们那时已经准确知道论文的内容了。
In the sample paper later in this article, the abstract tells the reader immediately that the investigation is about triangular numbers, uses a visual and algebraic approach, and reveals a connection to square numbers. This helps a reader decide whether the paper is worth their time.
在本文后面的范文中，摘要立刻告诉读者：该探究是关于三角形数的，使用了可视化与代数方法，并揭示了与平方数的联系。这有助于读者判断论文是否值得阅读。
5. Introduction: Setting the Scene | 引言：铺垫背景
The introduction should grab the reader's attention and explain why the topic is interesting. Begin with a simple definition or example of the mathematical objects you are studying. Pose your central question clearly. For instance: 'What is the quickest way to find the 100th triangular number without adding 1 + 2 + 3 + ... + 100?' This gives your investigation purpose and direction. You can also mention any prior knowledge you have that might help, but avoid long historical digressions unless they are directly relevant.
引言应当抓住读者的注意力，并解释为什么该主题有趣。用你正在研究的数学对象的简单定义或例子开头。清晰地提出你的核心问题。例如：‘不把 1+2+3+…+100 全部相加，最快求出第 100 个三角形数的方法是什么？’这为你的探究赋予了目的和方向。你还可以提及任何可能有用的已有知识，但除非直接相关，否则不要做冗长的历史溯源。
A strong introduction also hints at the structure of the paper, e.g. 'In this investigation, I will first record the first few triangular numbers, then search for a rule, test it, and finally try to prove it using a diagram.' This road map helps the reader follow your thinking.
有力的引言还会暗示论文的结构，例如：‘在本探究中，我将首先记录前几个三角形数，然后寻找规律，加以检验，最后尝试用图示证明。’这种路线图能帮助读者跟上你的思路。
6. Methods and Investigation Process | 方法与探究过程
This section is the 'story' of your investigation. Describe, step by step, what you did and why. If you drew dot diagrams, say so and include clear sketches or photographs. If you made a table of values, show how you filled in the numbers and what you were looking for. The key is to be systematic: list your initial observations, then explain how you refined them into a conjecture.
这一部分是你探究的‘故事’。逐步描述你做了什么以及为什么这么做。如果你画了点阵图，要说明并附上清晰的草图或照片。如果你制作了数值表格，要展示你是如何填写数字的，以及在寻找什么。关键是保持系统性：列出你的初步观察，然后解释你如何将它们提炼成猜想。
g data in a table, calculating differences or ratios, and spotting a pattern. For example: ' ting number. This suggests a recursive rule: Tₙ = Tₙ₋₁ + n. But I wanted a direct formula.'
在 Year 7 阶段，你的方法通常涉及生成例子、用表格整理数据、计算差或比，并发现模式。例如：‘我写下了前 10 个三角形数：1、3、6、10、15、21、28、36、45、55。我发现每个新数都是前一个数加上下一个自然数。这暗示了一个递推规律：Tₙ = Tₙ₋₁ + n。但我更想要一个直接公式。’
7. Results and Data Presentation | 结果与数据展示
Present your main findings clearly, using tables, lists, or diagrams where appropriate. If you discovered a formula, state it prominently and test it against several values to show it holds. Avoid dumping raw data without commentary; instead, pick two or three illuminating examples and explain what they show. A well-designed table can reveal patterns far more effectively than a block of text.
清晰呈现你的主要发现，酌情使用表格、列表或图示。如果你发现了一个公式，要醒目地陈述它，并用若干数值加以检验以表明其成立。避免不加解说地堆砌原始数据；相反，选取两三个有启发性的例子，并解释它们说明了什么。一张设计良好的表格比一大段文字更能有效揭示规律。
sual result, such as two copies of a triangular arrangement forming an n × (n+1) rectangle. Always label your tables and figures so the reader knows exactly what they are looking at.
例如，一张对比 n、第 n 个三角形数以及乘积 n(n+1)/2 的表格，能够立刻展示等价关系。你还可以加入一个可视化结果，比如把两份三角形点阵拼成一个 n×(n+1) 的矩形。永远给你的表格和图注上标签，让读者确切知道他们在看什么。
8. Discussion and Analysis | 讨论与分析
Discussion is where you explain what your results mean and why they make sense. If you found a formula, try to justify it with reasoning or a proof – at Year 7 level, a visual proof or a simple algebraic manipulation is perfect. Discuss any surprises or things that didn't work; for example, you might have first made an incorrect guess and then corrected it. This shows mathematical resilience and deeper understanding.
讨论部分用来解释你的结果意味着什么，以及它们为何合理。如果你发现了一个公式，试着用推理或证明来论证——在 Year 7 层面，一个可视化证明或简单的代数变换就非常理想。讨论任何出乎意料之处或不成功的地方；例如，你可能最初做出了错误猜测然后修正了它。这体现了数学韧性和更深的理解。
You can also make connections here: 'The triangular number formula reminds me of the formula for combinations, because choosing 2 items from n+1 gives the same sequence.' Such links demonstrate that you are thinking like a mathematician, not just following a recipe.
你还可以在此处建立联系：‘三角形数公式让我联想到组合公式，因为从 n+1 个物品中选 2 个会产生相同的序列。’这样的联想表明你正像数学家一样思考，而不仅仅是在按部就班。
9. Conclusion and References | 结论与参考文献
The conclusion should briefly restate your main finding and answer the question you posed in the introduction. It is also the place to reflect on what you have learned and suggest a next step: 'I could extend this by investigating tetrahedral numbers or looking at pentagonal numbers.' If you used any books, websites, or tools that were essential to your work, list them in a short references section – this shows academic honesty and gives credit to sources.
结论应简要重申你的主要发现，并回答你在引言中提出的问题。它也是反思所学内容并提出下一步设想的地方：‘我可以将其扩展到研究四面体数或五边形数。’如果你使用了任何对你的工作至关重要的书籍、网站或工具，请在简短的参考文献部分列出——这体现了学术诚信，也向来源致谢。
Even for a short Year 7 paper, a reference might look like: 'Enzensberger, H. M. (1997) The Number Devil. London: Granta Books.' This small habit prepares you for more formal academic writing later on.
哪怕是一篇简短的 Year 7 论文，参考文献也可以是这样的格式：‘Enzensberger, H. M. (1997) The Number Devil. London: Granta Books.’这个小习惯能为今后更正式的学术写作做好准备。
10. Sample Paper: Investigating Triangular Numbers | 范文：探究三角形数的模式
Title: Investigating Patterns in Triangular Numbers
标题：探究三角形数的模式
Abstract her number sequences. Using a pairing method, the formula Tₙ = ½ n (n + 1) is derived and proved. The investigation also reveals that the sum of two consecutive triangular numbers is a square number (Tₙ₋₁ + Tₙ = n²) and links the findings to combinations.
摘要：本文通过探索点阵的可视化排列、代数模式以及与其他数列的联系，探究了三角形数序列。运用配对法推导并证明了公式 Tₙ = ½ n (n + 1)。研究还揭示了两个连续三角形数之和是一个平方数（Tₙ₋₁ + Tₙ = n²），并将发现与组合问题建立了联系。
Introduction: Triangular numbers are numbers that can be represented as equilateral triangles made of dots. The first four triangular numbers are 1, 3, 6 and 10. My aim was to find a rule that gives the nth triangular number directly, without adding all the numbers from 1 to n. I started by drawing dot diagrams for the first few terms and recording the totals.
引言：三角形数是指能用点摆成等边三角形的数。前四个三角形数是 1、3、6 和 10。我的目标是找到一个规则，能直接给出第 n 个三角形数，而不必从 1 累加到 n。我首先绘制了前几项的点阵图，并记录下总数。
Method formula, I imagined copying each triangle and rotating it to form a rectangle. For n = 4, the 10-dot triangle plus its copy forms a 4 × 5 rectangle, giving 20 dots in total, so half is 10. Generalising: two copies of the nth triangular number make an n × (n+1) rectangle. Thus Tₙ = n(n+1) ÷ 2 = ½ n (n+1). I tested this formula for n = 1 to 10 and it matched perfectly.
方法：我画出点阵三角形并点数：1、3、6、10、15、21、28、36、45、55。我发现每一项都是加上下一个自然数。为了寻找直接公式，我设想复制每个三角形并旋转，使其拼成一个矩形。对于 n=4，10 个点的三角形与其复制品拼成一个 4×5 的矩形，总共 20 个点，所以一半是 10。推广开来：两个第 n 三角形数可拼成一个 n×(n+1) 的矩形。因此 Tₙ = n(n+1) ÷ 2 = ½ n (n+1)。我用 n=1 到 10 检验该公式，结果完全吻合。
Results: The first ten triangular numbers are shown in Table 1.
结果：表 1 展示了前十个三角形数。
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Tₙ | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 |
| ½n(n+1) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 |
Table 1: Comparison of n, Tₙ, and ½ n (n+1) | 表1：n、Tₙ 与 ½ n (n+1) 的对比
Tₙ = n². This is proved algebraically: ½(n-1)n + ½n(n+1) = ½n[(n-1)+(n+1)] = ½n(2n) = n².
我还发现 T₃ + T₄ = 6 + 10 = 16 = 4²，T₄ + T₅ = 10 + 15 = 25 = 5²。一般地，Tₙ₋₁ + Tₙ = n²。代数证明为：½(n-1)n + ½n(n+1) = ½n[(n-1)+(n+1)] = ½n(2n) = n²。
Discussion: The rectangle method provides a convincing visual proof of the formula. The connection to square numbers was a surprising bonus. I also realised that triangular numbers are the same as 'n+1 choose 2' in combinatorics, which makes sense because choosing 2 items from n+1 items can be visualised geometrically. My initial struggle was to see the rectangle idea – I only got it after physically rearranging dots on squared paper.
讨论：矩形法为公式提供了一个令人信服的可视化证明。与平方数的联系则是一个意料之外的惊喜。我还意识到三角形数在组合数学中相当于‘n+1 选 2’，这很合理，因为从 n+1 个物品中选 2 个可以几何直观地表示。我最初遇到的困难是想到矩形的思路——在方格纸上实际摆弄点阵之后，我才最终明白。
Conclusion rn was proved visually and algebraically, and a link to square numbers was established. My investigation could be extended by exploring tetrahedral numbers (the 3D version) or by studying pentagonal numbers and their formulas.
结论：本次探究成功找到了第 n 个三角形数的直接公式：Tₙ = ½ n (n + 1)。该模式得到了可视化与代数证明，并与平方数建立了联系。我还可以通过探索四面体数（三维版本）或研究五边形数及其公式，对此加以扩展。
References: Enzensberger, H. M. (1997) The Number Devil. London: Granta Books. (I used the chapter on triangular numbers as inspiration.)
参考文献：Enzensberger, H. M. (1997) The Number Devil. London: Granta Books.（我借鉴了其中关于三角形数的章节作为灵感。）
11. Sample Paper Analysis: Why It Works | 范文分析：为什么这篇论文好
The sample paper demonstrates every element of a successful investigation. The title is precise, the abstract summarises the whole journey, and the introduction sets a clear goal. Notice how the method section tells a story of
Published by TutorHao | Year 7 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图