📚 Year 7 CAIE Mathematics: Essay Writing Framework and Sample Essay | Year 7 CAIE 数学：论文写作框架与范文
In Year 7 CAIE Mathematics, you are often asked to complete an investigation or a problem-solving task and then write up your findings in a clear, structured report. This is sometimes called a mathematical essay or an investigative write-up. It is not a story – it is a chance to show your thinking, your method, and your conclusions using mathematical language and evidence. In this article, you will learn a reliable framework for writing such essays and explore a full worked example on polygon interior angles.
在 Year 7 CAIE 数学中，你经常会被要求完成一项探究或解决问题的任务，然后以清晰、结构化的报告形式写下你的发现。这有时被称为数学小论文或探究报告。它不是一个故事——它是你使用数学语言和证据展示你的思维、方法和结论的机会。在本文中，你将学习一个可靠的写作框架，并通过一个完整的范例来探索多边形内角和的规律。
1. What Is a Mathematical Essay? | 什么是数学小论文？
A mathematical essay in Year 7 is a short structured piece of writing that explains how you approached a mathematical problem or investigation. It can cover topics from number, algebra, geometry or statistics. The main purpose is to communicate your reasoning clearly, so that anyone can follow your steps and understand your discoveries.
Year 7 的数学小论文是一篇简短、结构化的文章，用以解释你是如何处理一个数学问题或探究的。它可以涵盖数、代数、几何或统计等主题。其主要目的是清晰地传达你的推理过程，让任何人都能跟上你的步骤并理解你的发现。
This type of writing is not a long essay in the humanities sense; it focuses on logical flow, evidence (like calculations, diagrams and tables) and precise language. CAIE expects you to organise your work so that you can ‘explain your reasoning’ and ‘present your arguments clearly’. Learning this skill early will help you in later years when you tackle more complex investigations.
这类写作并非人文意义上的长篇论文；它侧重于逻辑流程、证据（如计算、图表和表格）以及准确的语言。CAIE 希望你组织自己的作品，以便能够“解释你的推理”并“清晰地陈述你的论点”。尽早学习这项技能将有助于你在以后处理更复杂的探究。
2. The Overall Essay Structure | 论文的整体结构
Every well-organised mathematical essay follows a standard structure, which makes your thinking easy to follow. The framework below works for most Year 7 investigations. It has five main parts: an introduction, a method or plan, a results or working section, a discussion or analysis, and a conclusion. You may also include a title and a short list of resources if needed.
每一篇组织良好的数学小论文都遵循一个标准结构，这使你的思维易于理解。下面的框架适用于大多数 Year 7 探究。它包含五个主要部分：引言、方法或计划、结果或计算过程、讨论或分析以及结论。如果需要，你还可以加上标题和简短的参考资料列表。
You should think of this framework like a map. It guides the reader from what you wanted to find out, through what you did, to what you discovered. Each part has a specific job to do, and you will practise writing them in the following sections.
你应该把这个框架想象成一张地图。它引导读者从你想查明的内容，到你做了什么，再到你发现了什么。每个部分都有特定的作用，你将在以下小节中练习撰写它们。
3. Choosing a Topic and Forming a Question | 选题与形成问题
In many CAIE classroom tasks, your teacher will give you a starting point – maybe a pattern to explore or a statement to test. When you choose your own angle, try to turn it into a clear, answerable question. For example, ‘Do all quadrilaterals have an interior angle sum of 360°?’ or ‘How does the number of sides affect the interior angle sum in regular polygons?’ are strong questions because they can be tested.
在许多 CAIE 课堂任务中，你的老师会给你一个起点——也许是一个需要探索的模式或一个需要检验的说法。当你自己选择角度时，试着把它变成一个清晰、可回答的问题。例如，“所有四边形的内角和是否都是 360°？”或“边数如何影响正多边形的内角和？”都是很好的问题，因为它们可以被检验。
A good topic should link to what you have learned in class but still let you discover something new. Avoid questions that are too vague, like ‘What is special about geometry?’. Instead, zoom in on a specific property or rule. Write down your question at the top of your plan, because it will define your whole essay.
一个好的主题应该与你课堂上学过的内容相联系，但同时能让你发现一些新东西。避免过于模糊的问题，例如“几何有什么特别之处？”相反，应聚焦于一个特定的性质或规则。在你的计划开头写下你的问题，因为它将定义你的整篇论文。
4. Writing the Introduction | 引言写作
The introduction sets the scene. Start by stating what your investigation is about and why it is interesting. Include the question you are trying to answer. You can also mention any background knowledge you already have, such as ‘I already know that the interior angles of a triangle add up to 180°.’ Keep it short – three or four sentences are enough in Year 7.
引言是搭建背景的部分。首先说明你的探究是关于什么以及它为何有趣。包含你试图回答的问题。你也可以提到你已经掌握的任何背景知识，例如“我已经知道三角形的内角和是 180°。”保持简短——在 Year 7 中，三到四句话就够了。
Use precise mathematical vocabulary right from the start. Instead of ‘the space inside a shape’, say ‘interior angle’. This shows that you are thinking like a mathematician. Below is an example opening: ‘This investigation explores the relationship between the number of sides of a polygon and the sum of its interior angles. I will test whether a single formula can predict the sum for any polygon, starting with triangles and quadrilaterals.’
从一开始就使用精确的数学词汇。比如，不要说“形状内的空间”，而要说“内角”。这表明你像数学家一样思考。下面是一个开场范例：“本探究探索多边形的边数与其内角和之间的关系。我将检验是否有一个公式可以预测任何多边形的内角和，从三角形和四边形开始。”
5. Describing the Method Clearly | 清楚地描述方法
Your method section is like a recipe. It must tell the reader exactly what you did so they could repeat your investigation. Use a numbered list or step-by-step sentences. Include details such as which polygons you drew, how you measured angles (with a protractor) and how you recorded your data. Mention any checks you made, like measuring twice to reduce error.
你的方法部分就像一份食谱。它必须准确地告诉读者你做了什么，以便他们可以重复你的探究。使用编号列表或逐步描述的方式。包括细节，例如你画了哪些多边形、你是如何测量角度（用量角器）的，以及你如何记录数据。提及你所做的任何检查，比如测量两次以减少误差。
Always write in the past tense because you are describing work you have already completed. For example: ‘I drew five regular polygons on plain paper: an equilateral triangle, a square, a regular pentagon, a regular hexagon and a regular heptagon. I measured all interior angles with a protractor to the nearest degree and recorded the sum for each shape.’
始终使用过去时，因为你描述的是你已经完成的工作。例如：“我在白纸上画了五个正多边形：等边三角形、正方形、正五边形、正六边形和正七边形。我用量角器测量了所有内角，精确到度，并记录了每个形状的内角和。”
6. Collecting and Presenting Data | 收集与呈现数据
Once you have gathered your measurements or calculations, you need to show them clearly. A table is often the best tool because it organises numbers in a way that patterns become visible. Every table must have a title and labelled columns. In CAIE mark schemes, clear presentation earns marks for communication.
一旦你收集了测量值或计算结果，你需要清晰地展示它们。表格通常是最好的工具，因为它以能够呈现模式的方式组织数字。每个表格都必须有一个标题和带标签的列。在 CAIE 评分方案中，清晰的呈现能为交流部分赢得分数。
For an investigation on interior angles, you might create a table like the one shown in the sample essay later. Remember to include units (degrees) and use consistent decimal places if you are rounding. If you have diagrams, you can mention them in the text, but in Year 7 you can often summarise your findings in a table alone.
对于内角和的探究，你可以创建一个类似后面范文所示的表格。记得包含单位（度），并在四舍五入时使用一致的小数位。如果你有图示，可以在文中提及，但在 Year 7 中，你通常可以仅用一张表格来总结你的发现。
7. Calculating and Using Formulas | 计算与公式运用
When you spot a pattern, try to express it as a formula. In geometry investigations, you might find the formula for the sum of interior angles: Sum = (n − 2) × 180°, where n is the number of sides. Write this formula in your essay, explain what each letter represents, and show how you derived it from your results.
当你发现一个模式时，尝试将其表达为一个公式。在几何探究中，你可能会找到内角和的公式：内角和 = (n − 2) × 180°，其中 n 是边数。在你的论文中写下这个公式，解释每个字母代表什么，并展示你是如何从你的结果中推导出来的。
Always test your formula on two or three examples that were not in your original data. This is called verification. For instance, if you derived (n − 2) × 180°, test it for an octagon (n = 8) to see whether it gives 1080°. Showing this step demonstrates that you are thinking scientifically and checking your own work.
始终在你的原始数据之外的两个或三个例子中检验你的公式。这称为验证。例如，如果你推导出 (n − 2) × 180°，用八边形 (n = 8) 检验它是否得到 1080°。展示这一步表明你在科学地思考并检查自己的成果。
8. Discussion and Analysis | 讨论与分析
In the discussion, you connect your results back to your original question. Explain what the patterns mean and why they might happen. For example, you can talk about how any polygon can be divided into (n − 2) triangles, each contributing 180°, and that this geometric fact explains the formula.
在讨论部分，你将结果与你最初的问题联结起来。解释这些模式意味着什么以及它们为什么会发生。例如，你可以谈论任何多边形如何被分成 (n − 2) 个三角形，每个三角形贡献 180°，而这个几何事实解释了该公式。
You should also reflect on any surprises or sources of error. If your measured sums differed slightly from the theoretical formula, mention why – perhaps because of protractor accuracy or hand-drawn shapes. This shows that you understand that measurements can have limitations, which is an important mathematical skill.
你还应该反思任何意外之处或误差来源。如果你的测量总和与理论公式略有不同，说明原因——或许是因为量角器的精度或手绘形状的误差。这表明你理解测量可能存在局限性，这是一项重要的数学技能。
9. Writing a Strong Conclusion | 写出有力的结论
Your conclusion should be brief and directly answer the investigation question. Restate your main finding clearly. For example: ‘My investigation confirms that the sum of interior angles for any convex polygon can be found using the formula (n − 2) × 180°.’ Then mention any extension ideas, such as ‘It would be interesting to explore interior angles of concave polygons next.’
你的结论应该简短，并直接回答探究问题。清晰地重述你的主要发现。例如：“我的探究证实，对于任何凸多边形，内角和都可以用公式 (n − 2) × 180° 求出。”然后提及任何扩展想法，例如“下一步探索凹多边形的内角和将很有意义。”
Do not introduce brand new information in the conclusion. It is better to keep it to two or three sentences. Remember that a strong ending leaves the reader sure that you have answered your question and understood the mathematics behind it.
不要在结论中引入全新的信息。最好将内容控制在两到三句话。记住，一个有力的结尾会让读者确信你已经回答了你的问题并理解了其背后的数学原理。
10. Sample Essay: Investigating Interior Angles of Polygons | 范文：探究多边形内角和
Title: Investigating the Sum of Interior Angles in Polygons | 标题：多边形内角和的探究
Introduction
In this investigation, I wanted to find out whether there is a rule that connects the number of sides of a polygon to the total sum of its interior angles. I already knew that a triangle’s interior angles sum to 180° and that a quadrilateral’s sum is 360°. My question was: Can we predict the interior angle sum for any polygon, from a pentagon to a decagon?
在这项探究中，我想查明是否存在一个规则，能将多边形的边数与其内角总和联系起来。我已经知道三角形的内角和为 180°，四边形的内角和为 360°。我的问题是：我们能否预测任何多边形（从五边形到十边形）的内角和？
Method
I drew five convex polygons on plain paper: an equilateral triangle, a square, a regular pentagon, a regular hexagon and a regular heptagon. For each shape, I used a protractor to measure every interior angle to the nearest degree. I repeated each measurement twice to improve accuracy. Then I added up the angles to get the total sum. I recorded the number of sides and the measured sums in a table.
我在白纸上画了五个凸多边形：等边三角形、正方形、正五边形、正六边形和正七边形。对于每个形状，我用量角器测量每个内角，精确到度。我重复了两次测量以提高准确性。然后我将角度相加得到总和。我将边数和测量的总和记录在表格中。
Results
Table 1 below shows the data I collected. I also included an extra column for the theoretical sum, which I researched later to compare.
下文的表 1 显示了我收集的数据。我还额外增加了一个理论总和的列，我是在之后查阅并用于对比的。
| Number of sides (n) | Polygon Name | Measured Sum (°) | Theoretical Sum (°) (n − 2) × 180° |
|---|---|---|---|
| 3 | Triangle / 三角形 | 181 | 180 |
| 4 | Quadrilateral / 四边形 | 359 | 360 |
| 5 | Pentagon / 五边形 | 538 | 540 |
| 6 | Hexagon / 六边形 | 718 | 720 |
| 7 | Heptagon / 七边形 | 898 | 900 |
Table 1: Comparison of measured and theoretical interior angle sums | 表 1：测量内角和与理论内角和的对比
I noticed that my measured sums were very close to the theoretical values, only off by 1–2 degrees. This small error is likely due to hand-drawing and reading the protractor.
我注意到我的测量总和非常接近理论值，只相差了 1–2 度。这个微小的误差很可能是由于手绘和读取量角器造成的。
Discussion
From the table, I could see that every time the number of sides increased by 1, the sum of interior angles went up by about 180°. I recalled that a polygon can be split into triangles by drawing diagonals from one vertex. A pentagon splits into 3 triangles, a hexagon into 4, and so on. The number of triangles is always (n − 2). Since each triangle contributes 180°, the formula must be (n − 2) × 180°. I tested this rule on an octagon (n = 8): (8 − 2) × 180° = 6 × 180° = 1080°, which matches known facts.
从表格中我可以看出，边数每增加 1，内角和就增加大约 180°。我想起多边形可以通过从一个顶点画对角线来分割成三角形。五边形分成 3 个三角形，六边形分成 4 个，以此类推。三角形的数量总是 (n − 2)。由于每个三角形贡献 180°，公式一定是 (n − 2) × 180°。我在八边形 (n = 8) 上检验了这个规则：(8 − 2) × 180° = 6 × 180° = 1080°，这与已知事实相符。
This geometric explanation is stronger than just looking at numbers because it shows why the pattern works. The small differences in my measurements did not affect the overall conclusion.
这种几何解释比仅仅观察数字更强，因为它展示了模式为何有效。我的测量中的微小差异并没有影响总体结论。
Conclusion
My investigation proved that the sum of interior angles of a convex polygon with n sides can be calculated using the formula (n − 2) × 180°. The measured data supported this rule, and the triangle-splitting idea explained it logically. In the future, I would like to test the formula for concave polygons and explore the sum of exterior angles, which might always be 360°.
我的探究证明，具有 n 条边的凸多边形的内角和可以用公式 (n − 2) × 180° 计算。测量数据支持了这一规则，三角形分割的想法在逻辑上解释了它。将来，我想测试该公式在凹多边形上的情况，并探索外角和，它可能总是 360°。
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