📚 Year 9 Cambridge Maths: Writing Framework and Model Investigation | Year 9 剑桥数学：论文写作框架与范文
In Year 9 Cambridge Mathematics, you will often be asked to carry out a mathematical investigation and present your findings in a written report. This is not just about getting the right answer — it is about showing how you think mathematically, how you explore a problem, and how you communicate your reasoning clearly. This article gives you a step-by-step framework for writing an investigation report and provides a model investigation to follow.
在 Year 9 剑桥数学课程中，你常常需要完成一项数学探究，并以书面报告的形式呈现你的发现。这不仅仅是为了得到正确答案——更重要的是展示你的数学思维过程、如何探索一个问题，以及如何清晰地表达你的推理。本文为你提供了撰写探究报告的分步框架，并配上一篇范文供你参考。
1. Understanding the Purpose of a Mathematical Investigation | 理解数学探究的目的
A mathematical investigation is an open-ended exploration of a pattern, relationship or problem. Instead of simply applying a formula, you are expected to ask questions, make conjectures, test them with examples and draw conclusions. The written report allows you to demonstrate your ability to reason logically and to present your work in a structured way.
数学探究是对某个规律、关系或问题进行开放式探索的过程。它不是简单地套用公式，而是要求你提出问题、做出猜想、用例子进行验证并得出结论。书面报告则让你有机会展示逻辑推理能力，并以结构化的方式呈现你的工作。
2. The Structure of a Maths Investigation Report | 数学探究报告的结构
Although each investigation is unique, most reports follow a clear structure. This makes it easier for your teacher to see how you moved from a question to a conclusion. A strong report typically includes: title, introduction, problem definition, method, results, analysis, conclusion and references or appendices.
尽管每个探究都各不相同，但大多数报告都遵循一个清晰的结构。这有助于老师看清楚你是如何从一个问题逐步走向结论的。一份好的报告通常包含：标题、引言、问题定义、方法、结果、分析、结论以及参考文献或附录。
3. Title and Introduction | 标题与引言
Your title should be clear and describe the focus of the investigation. The introduction sets the scene: explain why the topic is interesting, what you want to find out, and briefly what mathematical ideas you will use. For example, 'In this investigation, I will explore number patterns in Pascal's Triangle and look for hidden sequences.'
标题应当清晰明了，能概括探究的焦点。引言部分则为报告定下基调：说明为什么这个主题有趣、你想探究什么，并简要提及你将用到哪些数学思想。例如：“本次探究中，我将探索帕斯卡三角中的数字规律，并寻找其中隐藏的数列。”
4. Defining the Problem and Hypothesis | 定义问题与假设
State the exact question you are investigating. This might be something like: 'Do the shallow diagonals of Pascal’s Triangle sum to the Fibonacci numbers?' Then make a conjecture — an educated guess — about what you think you will discover, based on initial observations or prior knowledge.
清晰说明你正在探究的具体问题，例如：“帕斯卡三角中的浅对角线之和是否等于斐波那契数？” 然后根据初步观察或已有知识提出一个猜想——即有根据的预测——说明你预计会发现什么。
5. Method: How You Carried Out the Work | 方法：如何执行工作
Describe the steps you took in your investigation. Did you draw diagrams, use a spreadsheet, write a short program, or work entirely by hand? Be specific about the tools and techniques you used so that someone else could repeat your work.
描述你在探究中采取的步骤。你是否画了图、使用了电子表格、编写了一个小程序，还是完全用手工计算？请具体说明你使用的工具和方法，以确保他人也能重复你的工作。
6. Results: Presenting Data and Calculations | 结果：呈现数据与计算
Present what you found in a clear, organised way. Use tables to show number patterns, diagrams to illustrate geometric relationships, and lists to summarise sequences. Every piece of data should be labelled so the reader can follow your thinking.
用清晰、有条理的方式展示你的发现。你可以用表格展示数字规律，用图形说明几何关系，用列表归纳数列。每个数据都应加上标注，让读者能够跟上你的思路。
7. Analysis and Discussion | 分析与讨论
Do not just describe what happened — explain why you think it happened. Comment on any patterns, anomalies or surprises. Try to justify your findings with mathematical reasoning. If something did not work as expected, discuss what you learned from that.
不要仅仅描述发生了什么，还要解释你认为它为什么发生。对任何规律、异常或意外发现加以评述。尝试用数学推理来证实你的发现。如果有事情没有如预期那样进行，也请讨论你从中学到了什么。
8. Conclusion and Evaluation | 结论与评估
Sum up your main findings and state whether your hypothesis was supported. Reflect on the strengths and limitations of your investigation. Suggest questions you could explore next or ways the investigation could be extended.
总结你的主要发现，并说明你的假设是否得到支持。反思探究的优点与局限性。提出接下来可以探索的问题或可以扩展的方向。
9. Referencing and Appendices | 参考文献与附录
If you used any books, websites or other sources, list them in a reference section. Any large tables, full data sets or extra calculations that would clutter the main report can be placed in an appendix.
如果使用了任何书籍、网站或其他资源，请将它们列在参考文献部分。任何可能使正文显得杂乱的大型表格、完整数据集或额外计算都可以放进附录中。
10. Model Investigation Report | 范文报告
Below is a sample investigation report on number patterns in Pascal’s Triangle, written following the framework described above. Use it as a guide for your own writing.
下面是一份关于帕斯卡三角数字模式的探究报告范文，按照上述框架撰写。你可以用它来指导自己的写作。
Title: Investigating Fibonacci Numbers in Pascal’s Triangle
标题：探究帕斯卡三角中的斐波那契数
Introduction
The aim of this investigation is to explore a famous connection between Pascal’s Triangle and the Fibonacci sequence. Pascal’s Triangle is formed by adding the two numbers above to get each new number. The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, … where each term is the sum of the two before it. I had heard that the sums of shallow diagonals in the triangle give the Fibonacci numbers, and I wanted to test this for myself.
引言
本探究的目的是探索帕斯卡三角与斐波那契数列之间的一个著名联系。帕斯卡三角是通过将上方两个数字相加得到每个新数字而形成的。斐波那契数列以 1, 1, 2, 3, 5, 8, … 开始，每一项是前两项之和。我曾听说法三角形的浅对角线之和能得到斐波那契数，我想亲自验证这一点。
Problem and Hypothesis
The question is: Does the sum of numbers along the shallow diagonals of Pascal’s Triangle give the Fibonacci sequence? My hypothesis is that if I sum the shallow diagonals, I will obtain 1, 1, 2, 3, 5, 8, … exactly.
问题与假设
问题是：帕斯卡三角中浅对角线上的数字之和是否等于斐波那契数列？我的假设是，如果我把浅对角线上的数字加起来，将恰好得到 1, 1, 2, 3, 5, 8, …。
Method
I constructed the first eight rows of Pascal’s Triangle by hand on squared paper. I then drew shallow diagonals (diagonals that go from right to left across the triangle, with a slope of 1). For each diagonal I added all the numbers it passed through and recorded the sum.
方法
我在方格纸上手工构造了帕斯卡三角的前八行。然后我画出了浅对角线（从右向左横穿三角形、斜率为 1 的对角线）。对于每条对角线，我把它经过的所有数字相加，并记录下总和。
Results
Here are the first few rows of Pascal’s Triangle I built:
结果
以下是我构建的帕斯卡三角的前几行：
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1
Row 7: 1 7 21 35 35 21 7 1
1 of row 1: sum = 1. The third diagonal passes through 1 (row 2) and 1 (row 1): sum = 2. The fourth: 1 (row 3) + 2 (row 2) = 3. The fifth: 1 (row 4) + 3 (row 3) + 1 (row 2) = 5. The sixth: 1 (row 5) + 4 (row 4) + 3 (row 3) = 8. I continued and obtained the sequence: 1, 1, 2, 3, 5, 8, 13, 21.
接着我计算了浅对角线的和。第一条对角线只经过顶部的 1：和为 1。第二条经过第 1 行最左边的 1：和为 1。第三条经过第 2 行的 1 和第 1 行的 1：和为 2。第四条：第 3 行的 1 + 第 2 行的 2 = 3。第五条：第 4 行的 1 + 第 3 行的 3 + 第 2 行的 1 = 5。第六条：第 5 行的 1 + 第 4 行的 4 + 第 3 行的 3 = 8。我继续计算，得到的数列为：1, 1, 2, 3, 5, 8, 13, 21。
Analysis
The sequence of sums exactly matches the Fibonacci numbers up to 21. This confirms my hypothesis for the first eight diagonals. The pattern works because the process of forming shallow diagonals effectively mimics the Fibonacci recurrence — each sum can be expressed as the sum of the two previous shallow diagonal sums. I also noticed that every number in the triangle is used exactly once in this set of diagonals, which is a neat property.
分析
这些和组成的数列与斐波那契数列一直到 21 都完全吻合。这证实了我对前八条对角线的假设。这个模式之所以成立，是因为形成浅对角线的过程实际上模拟了斐波那契递推关系——每个和都可以表示为前两个浅对角线之和。我还注意到，三角形中的每个数字在这组对角线中都被恰好使用了一次，这是一个很奇妙的性质。
Conclusion and Evaluation
I have shown that the shallow diagonals of Pascal’s Triangle produce the Fibonacci numbers for the rows I investigated. The investigation was straightforward but effective because I could verify the pattern by hand. However, my method only covered eight rows; a more rigorous proof would use algebra or combinatorics. Next, I would like to explore other hidden sequences in the triangle, such as triangular numbers or powers of two.
结论与评估
我已经证明，在我所考察的几行中，帕斯卡三角的浅对角线给出了斐波那契数。这次探究简单但有效，因为我能够通过手工计算验证该规律。不过，我的方法只涵盖了八行；更为严格的证明需要用到代数或组合数学。接下来，我希望探索三角形中的其他隐藏序列，比如三角形数或 2 的幂。
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