📚 Year 9 Edexcel Maths: Paper Writing Framework and Exemplar | 数学：论文写作框架与范文
In the Year 9 Edexcel Mathematics curriculum, students often tackle extended investigations or mini-research projects that require them to present mathematical thinking in a structured written form. Being able to communicate reasoning clearly, support arguments with evidence and reflect on findings is just as important as getting the correct answer. This guide provides a complete framework for writing a mathematical paper, together with a detailed exemplar exploring a pattern in triangular numbers.
在 Year 9 Edexcel 数学课程中，学生经常需要完成扩展探究或小型研究项目，并以结构化的书面形式呈现数学思考。清晰地交流推理过程、用证据支持论点并对发现进行反思，与得到正确答案同样重要。本指南提供数学论文写作的完整框架，并配有一篇探究三角形数规律的详细范文。
1. What is a Mathematical Investigation Paper? | 什么是数学探究论文？
A mathematical investigation paper is a short report that presents a question, a method of exploration, results and a reasoned conclusion. It is not just a collection of calculations; it tells a story of discovery, supported by data, diagrams and algebraic proof where appropriate.
数学探究论文是一份简短的报告，它提出一个问题、展示探索方法、结果和有理有据的结论。它不只是一堆计算，而是用数据、图示和适当的代数证明来讲述一个发现的故事。
In Year 9, a paper typically asks you to spot a pattern, test it with examples and then explain why it works. The emphasis is on clarity, logical structure and personal reflection.
在 Year 9，论文通常要求你发现一个模式，用例子检验它，然后解释它为什么成立。重点在于清晰性、逻辑结构和个人反思。
2. Overall Structure of the Paper | 论文的整体结构
A well-organised paper follows a predictable structure so that the reader can easily follow your thinking. The standard sections are: Introduction, Method, Results, Conclusion and References.
一篇条理清晰的论文遵循可预测的结构，以便读者轻松跟上你的思路。标准部分是：引言、方法、结果、结论和参考文献。
Some papers also include a separate Discussion section, but at Year 9 level it is often merged with the Conclusion. Always check your teacher’s specific requirements.
有些论文还包括单独的讨论部分，但在 Year 9 阶段通常与结论合并。务必检查老师的特定要求。
3. Writing the Introduction | 撰写引言
The introduction sets the scene. It should explain what question you are investigating, why it is interesting, and what you hope to achieve. Start with a broad context and then narrow down to your specific line of enquiry.
引言设定情境。它应该解释你正在探究什么问题、为什么有趣以及你希望实现什么。从较广的背景入手，然后收窄到具体的探究问题。
For example: ‘I have always been fascinated by number patterns and wanted to see if there is a hidden link between triangular numbers and square numbers.’ Mention any prior knowledge or initial observations.
例如：“我一直对数字模式着迷，想看看三角形数和平方数之间是否藏有某种联系。” 提及任何先验知识或初步观察。
4. Describing Your Method | 描述方法
The method explains exactly how you carried out your investigation. Be specific enough that someone else could repeat your work. Describe the software used, the tables constructed, or the algebraic tools employed.
方法部分准确地解释你是如何进行探究的。要足够具体，让他人能够重复你的工作。描述使用的软件、构建的表格或采用的代数工具。
If you used a spreadsheet to generate values, say so. If you drew diagrams, mention their purpose. Avoid vague statements like ‘I looked at some numbers’.
如果你使用电子表格生成数值，请说明。如果你画了图，说明其目的。避免诸如“我看了一些数字”这样的模糊表述。
5. Presenting Results | 展示结果
This section lets the data speak. Use tables, lists and diagrams to display your findings clearly. Every table or chart should have a title and numbered labels so it can be referred to in the text.
这部分让数据说话。使用表格、列表和图示清晰展示你的发现。每个表格或图表都应有标题和编号，以便在正文中引用。
After displaying results, briefly describe what the reader should notice. For instance: ‘Table 1 shows that the sum of two consecutive triangular numbers appears to be a perfect square for all tested cases.’
在展示结果后，简要描述读者应注意什么。例如：“表 1 显示，两个连续三角形数之和在所有测试案例中似乎都是一个完全平方数。”
6. Writing the Conclusion | 撰写结论
The conclusion summarises what you have discovered, answers the original question and evaluates the investigation. State whether you have found a general rule, and provide a proof or explanation if possible.
结论总结你的发现，回答原始问题并评价整个探究。说明你是否找到了一个普遍规律，如果可能的话提供证明或解释。
Always reflect on limitations and suggest extensions. For example: ‘This proof only works for natural numbers. In future I would like to test the pattern with negative integers or explore square pyramidal numbers.’
始终反思局限性并提出扩展方向。例如：“这一证明仅适用于自然数。未来我想用负整数检验这个模式，或探究四棱锥数。”
7. Referencing and Presentation | 参考文献与排版
Even a short mathematical paper should credit any sources used, such as textbooks, websites or class notes. Include a simple reference list at the end with author, title and date if available.
即使是一篇简短的数学论文，也应注明所使用的任何来源，如教科书、网站或课堂笔记。文末附上一个简单的参考文献列表，包含作者、标题和日期（如有）。
Presentation matters: use a clear font, consistent spacing, page numbers and labelled diagrams. A neat paper shows respect for the reader and reflects your effort.
排版很重要：使用清晰的字体、一致的间距、页码和带标注的图示。整洁的论文体现了对读者的尊重，也反映了你的投入。
8. Worked Example: Investigating Patterns in Triangular Numbers | 范文实例：探究三角形数的规律
The following exemplar applies the framework to a specific investigation: ‘Does the sum of two consecutive triangular numbers always produce a perfect square?’ This question is accessible, rich in pattern and easily tested with Year 9 algebra.
以下范文将上述框架应用于一个具体的探究：“两个连续的三角形数之和是否总是得到一个完全平方数？”这个问题易于理解、模式丰富，并可用 Year 9 代数轻松检验。
Triangular numbers are numbers that can form an equilateral triangle: 1, 3, 6, 10, 15, … The n-th triangular number, denoted Tₙ, is given by Tₙ = n(n+1)/2. We will explore the sum Tₙ + Tₙ₊₁.
三角形数是能排列成等边三角形的数：1, 3, 6, 10, 15, … 第 n 个三角形数记为 Tₙ，由 Tₙ = n(n+1)/2 给出。我们将探究 Tₙ + Tₙ₊₁ 的和。
9. Example Introduction | 范文引言
While practising number patterns, I noticed that 3 + 6 = 9 and 6 + 10 = 16, both of which are square numbers. This made me curious: is the sum of any two consecutive triangular numbers always a perfect square? In this investigation, I aim to test this pattern for several cases and, if true, prove it algebraically.
在练习数字模式时，我注意到 3 + 6 = 9 和 6 + 10 = 16，两者都是平方数。这让我感到好奇：任意两个连续三角形数之和是否总是一个完全平方数？在本次探究中，我旨在用多个实例检验这一模式，并在成立的情况下用代数方法证明它。
I already know the formula for the n-th triangular number and understand basic expansion and factorisation. I expect to find a clear relationship linking triangular and square numbers.
我已经知道第 n 个三角形数的公式，并理解基本的展开和因式分解。我期望找到一个将三角形数和平方数联系起来的清晰关系。
10. Example Method and Results | 范文方法与结果
I first generated the first ten triangular numbers using the formula Tₙ = n(n+1)/2 and recorded them in a table. Then I calculated the sum of consecutive pairs and compared each sum with the square of (n+1).
我首先用公式 Tₙ = n(n+1)/2 生成了前十个三角形数，并将它们记录在表格中。然后我计算了连续数对的和，并将每个和与 (n+1)² 进行比较。
| n | Tₙ | Tₙ₊₁ | Tₙ + Tₙ₊₁ | (n+1)² |
|---|---|---|---|---|
| 1 | 1 | 3 | 4 | 4 |
| 2 | 3 | 6 | 9 | 9 |
| 3 | 6 | 10 | 16 | 16 |
| 4 | 10 | 15 | 25 | 25 |
| 5 | 15 | 21 | 36 | 36 |
Table 1: Comparing the sum of consecutive triangular numbers with (n+1)² for n = 1 to 5.
表 1：比较连续三角形数之和与 (n+1)²（n 从 1 到 5）。
From the table it is evident that Tₙ + Tₙ₊₁ = (n+1)² for all listed values. This strongly suggests a general rule. I then tested it further with n = 10: T₁₀ = 55, T₁₁ = 66, sum = 121, and (10+1)² = 121, confirming the pattern holds for a larger number.
从表格中可以明显看出，对于所有列出的数值，Tₙ + Tₙ₊₁ = (n+1)²。这强烈暗示了一个普遍规律。我随后用 n = 10 进一步检验：T₁₀ = 55, T₁₁ = 66, 和为 121，而 (10+1)² = 121，证实该模式在较大的数上也成立。
To prove the pattern, I used algebra. Substituting the formula for triangular numbers:
为了证明这一模式，我使用了代数。代入三角形数公式：
Tₙ + Tₙ₊₁ = n(n+1)/2 + (n+1)(n+2)/2
Combining the fractions gives a common denominator of 2:
合并分数，得到公分母 2：
= [n(n+1) + (n+1)(n+2)] / 2
Factor out (n+1) from the numerator:
从分子中提取公因式 (n+1)：
= (n+1)(n + n + 2) / 2
Simplify the bracket:
化简括号：
= (n+1)(2n+2) / 2
Factor 2 out of the second bracket and cancel with the denominator:
从第二个括号中提出 2，并与分母中的 2 约分：
= (n+1)(2)(n+1) / 2 = (n+1)(n+1) = (n+1)²
Thus, the sum of two consecutive triangular numbers is always the square of (n+1).
因此，两个连续三角形数之和总是 (n+1) 的平方。
11. Example Conclusion and Reflection | 范文结论与反思
This investigation has confirmed that the sum Tₙ + Tₙ₊₁ equals (n+1)² for all positive integers n. The algebraic proof shows exactly why the pattern works, not just that it happens. The discovery links two fundamental families of numbers – triangular and square numbers – in a simple yet elegant way.
本次探究证实，对于所有正整数 n，Tₙ + Tₙ₊₁ 的和等于 (n+1)²。代数证明清楚地说明了模式为何成立，而不仅仅是它发生了。这一发现以一种简单而优美的方式将两类基本数族——三角形数和平方数——联系了起来。
I feel my investigation was successful because I combined numerical testing with a formal proof. However, I only considered natural numbers. A limitation is that I did not explore whether a similar relationship exists for negative values of n, which would require a different interpretation of triangular numbers. In the future, I would like to investigate sums of three consecutive triangular numbers or extend the idea to pentagonal numbers.
我觉得本次探究是成功的，因为我将数值检验与形式化证明结合了起来。然而，我只考虑了自然数。一个局限性是，我没有探究当 n 取负值时是否存在类似的关系，而这需要对三角形数作不同的解释。未来，我想探究三个连续三角形数的和，或者把这一思想扩展到五边形数上。
12. Final Tips for Success | 成功的最后建议
Start early and keep a journal of your rough work. This makes writing the method section much easier. Always read your paper aloud to check for clarity – if you stumble, the sentence probably needs rewriting.
尽早开始，并记录你的草稿工作。这会让撰写方法部分变得容易得多。始终大声朗读你的论文以检查清晰度——如果读起来磕绊，句子可能就需要重写。
Ask a friend to read your introduction without seeing the rest: can they predict what you will investigate? If yes, your aim is clear. Finally, celebrate your curiosity – a mathematical paper is a chance to think like a real mathematician.
让一位朋友只看你的引言：他们能预测出你将要探究什么吗？如果可以，你的目标就是清晰的。最后，庆祝你的好奇心——一篇数学论文是一个像真正的数学家那样思考的机会。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导