📚 Year 7 WJEC Further Maths: Writing Framework and Model Paper | Year 7 WJEC进阶数学：论文写作框架与范文
Writing about mathematics helps you organise your thoughts, communicate reasoning clearly, and deepen your understanding of concepts. In Year 7 WJEC Further Mathematics, you will often produce a short investigative paper. This guide provides a complete framework for structuring your work, alongside a detailed model paper to show you what an excellent piece of mathematical writing looks like.
书面呈现数学思考能帮助你整理思路、清晰地表达推理过程，并深化对概念的理解。在Year 7 WJEC进阶数学中，你常常需要完成一篇简短的探究性论文。本指南提供了完整的写作框架，并附上一篇详细的范文，向你展示一篇优秀数学论文的样子。
1. Understanding the Task | 理解任务要求
Before you start writing, read the assignment brief carefully. Identify the mathematical concept you need to investigate, whether it involves number patterns, shape properties, data handling, or algebraic rules. Clarify what the final output should include: a clear problem statement, your method of investigation, collected evidence, and a reasoned conclusion.
在开始写作之前，请仔细阅读任务要求。明确你需要探究的数学概念，可能涉及数字规律、图形性质、数据处理或代数规则。弄清楚最终成果应包含哪些部分：清晰的问题陈述、探究方法、收集的证据以及有理有据的结论。
Look for key instruction words such as 'explore', 'justify', 'generalise', or 'compare'. These words tell you the depth expected. For instance, 'generalise' means you must move beyond specific examples to state a rule using words or algebraic symbols. Keep the marking criteria handy so you know where to focus your effort.
留意题目中的关键指令词，如“探究”、“论证”、“概括”或“比较”。这些词暗示了任务的深度要求。例如，“概括”意味着你必须超越具体例子，用文字或代数符号陈述一条规律。把评分标准放在手边，这样你就知道该在哪些地方重点发力。
2. Choosing a Focused Title | 选择一个聚焦的标题
Your paper needs a title that tells the reader exactly what you explored. Avoid vague headings like 'Maths Project' or 'Number Work'. Instead, use precise phrases such as 'Investigating the Sum of the First n Odd Numbers' or 'The Relationship between the Area and Perimeter of Rectangles'. A good title frames the entire investigation.
你的论文需要一个能精确告诉读者你探究了什么的标题。避免诸如“数学项目”或“数字练习”这样模糊的标题。应使用精确的短语，例如“探究前n个奇数之和”或“矩形面积与周长的关系”。好的标题能为整个探究定下基调。
A focused title also helps you stay on track. If you find yourself collecting data that does not relate to the title, you may be going off topic. Write the title at the top of your planning page and refer back to it whenever you draft a new section.
一个聚焦的标题还能帮助你保持正轨。如果你发现自己在收集与标题无关的数据，你可能偏离了主题。把标题写在计划页的顶部，每当你起草新部分时都回看它一下。
3. Structuring Your Paper | 构建论文结构
A well-organised paper follows a logical sequence. The standard structure includes: Introduction, Method, Results, Analysis, Conclusion, and References. You might also include an appendix for raw data tables. Below is a table showing the purpose of each section.
一篇组织良好的论文遵循逻辑顺序。标准结构包括：引言、方法、结果、分析、结论和参考文献。你还可以包含一个附录来放置原始数据表。下表展示了每个部分的功能。
| Section 部分 | Purpose 目的 |
|---|---|
| Introduction 引言 | State the problem, give background, and outline your aim. |
| Method 方法 | Explain how you collected data or tested ideas. |
| Results 结果 | Present your findings using tables, graphs, and lists. |
| Analysis 分析 | Identify patterns, explain relationships, and suggest rules. |
| Conclusion 结论 | Summarise your discovery and reflect on limitations. |
Each section should flow naturally into the next. Use short paragraphs and clear headings so the reader can follow your thinking without confusion. Even within a section, present ideas in the order you carried them out.
每个部分都应当自然地衔接到下一个。使用短小的段落和清晰的标题，让读者能够毫无障碍地跟随你的思路。即使在同一个部分里，也请按照你实际操作的顺序来呈现想法。
4. Writing a Strong Introduction | 撰写有力的引言
The introduction sets the scene. Start by explaining what mathematical idea you are investigating and why it is interesting. For example, you might say, 'I have noticed that adding two consecutive triangular numbers always gives a square number. This paper explores whether this is always true and tries to explain why.' This shows personal curiosity.
引言要交代背景。首先解释你正在探究什么数学概念，以及它为什么有趣。例如，你可以说：“我注意到两个连续的三角形数相加总是得到一个平方数。这篇论文探讨这一规律是否总是成立，并尝试解释其原因。”这展示了你的个人好奇心。
Then state your investigation question clearly, using words like 'What happens when…?' or 'Is it always true that…?'. Finally, briefly mention your method: 'I will test cases using small numbers, look for a pattern, and then try to form a general rule.' Keep the introduction concise, around three to five sentences.
然后清晰地陈述你的探究问题，使用“当……时会发生什么？”或“……是否总是成立？”等句式。最后，简要提及你的方法：“我将使用小数字进行测试，寻找规律，然后尝试形成一个通用规则。”引言要简洁，大约三到五个句子即可。
5. Describing Your Method | 描述你的方法
In the method section, you explain step by step what you did. Use words like 'first', 'next', and 'then' to show order. If you drew diagrams, measured lengths, or used a spreadsheet, mention it. You should give enough detail that someone else could repeat your investigation.
在方法部分，你需要一步步解释你做了什么。使用“首先”、“接着”和“然后”等词语来表明顺序。如果你画了示意图、测量了长度或使用了电子表格，请提及。你应该提供足够的细节，以便他人能够重复你的探究。
For example: 'I started by listing the first ten square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. I then calculated the differences between consecutive terms: 3, 5, 7, 9, 11, 13, 15, 17, 19. I recorded these in a table and looked for a growing pattern.' Avoid evaluating the results here; save that for analysis.
例如：“我首先列出了前十个平方数：1, 4, 9, 16, 25, 36, 49, 64, 81, 100。然后我计算了相邻两项的差：3, 5, 7, 9, 11, 13, 15, 17, 19。我把这些记录在表格中，并寻找递增的规律。”此处不要评估结果，那是分析部分的事情。
6. Presenting Results Clearly | 清晰地呈现结果
Results should be organised so patterns are easy to spot. Use numbered tables and labelled diagrams. A table comparing the term number n with its value An is very effective. For instance:
结果应当有组织地呈现，以便规律易于识别。使用带编号的表格和带有标注的图表。一张将项数n与对应值An进行比较的表格非常有效。例如：
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Triangular number Tₙ | 1 | 3 | 6 | 10 | 15 |
Alongside tables, you can describe any visual patterns. For example: 'The differences between consecutive square numbers form the sequence of odd numbers: 3, 5, 7, 9, …'. Always link your words to the data shown in the table.
除了表格，你还可以描述任何视觉规律。例如：“连续平方数之间的差构成了奇数列：3, 5, 7, 9, …”。始终将你的描述与表格中显示的数据联系起来。
7. Analysing the Data and Finding Rules | 分析数据并寻找规则
This is where you show your mathematical thinking. Move from 'I see a pattern' to 'Here is the rule that explains it'. Use phrases like 'I noticed that…', 'This suggests that…', and 'Therefore the nth term can be expressed as…'. You might present a rule in words first, then in symbols.
这里是你展示数学思维的地方。从“我看到一个规律”转向“这是解释该规律的规则”。使用诸如“我注意到……”、“这表明……”和“因此第n项可以表示为……”等短语。你可以先用文字表达规则，然后用符号表示。
r observing that the sum of consecutive odd numbers gives square numbers, you might write: 'The sum of the first n odd numbers is n². In symbols: 1 + 3 + 5 + … + (2n - 1) = n².' rule with a new n value to check if it works. This verification strengthens your analysis.
例如，在观察到连续奇数之和给出平方数之后，你可以写道：“前n个奇数之和为n²。用符号表示：1 + 3 + 5 + … + (2n - 1) = n²。”用一个新的n值检验你的规则，看它是否成立。这种验证会增强你的分析力度。
t each symbol means and how the rule connects to the diagram or table. Substituting values alongside a diagram makes abstract formulae much clearer.
如果你的探究涉及某种关系，如y = 3x + 1，请解释每个符号的含义，以及该规则如何与示意图或表格相关联。在图表旁代入数值，能让抽象的公式变得清晰得多。
8. Writing a Reflective Conclusion | 撰写反思性结论
Your conclusion should summarise the main finding without introducing new information. Restate your investigation question and answer it directly. If you found that the product of two consecutive even numbers is always divisible by 8, say so clearly.
你的结论应当总结主要发现，而不要引入新信息。重新陈述你的探究问题并直接回答它。如果你发现两个连续偶数的乘积总是能被8整除，那就清楚地说明这一点。
Also reflect on what you might do differently. For example: 'If I had more time, I would test larger numbers to see if the pattern holds.' Acknowledge any difficulties, such as a problem where the pattern broke down, and suggest a possible reason. This shows a high level of mathematical maturity.
同时反思你可能会做哪些不同的事情。例如：“如果我有更多时间，我会测试更大的数字以检验该规律是否仍然成立。”承认遇到的任何困难，比如规律在某处失效了，并提出可能的原因。这显示出高水平的数学成熟度。
9. A Model Paper: 'Investigating the Sum of Consecutive Odd Numbers' | 范文：“探究连续奇数之和”
Title: Investigating the Sum of Consecutive Odd Numbers Starting from 1
标题：探究从1开始的连续奇数之和
Introduction
his project tests if the sum of the first n odd numbers always equals n² and explains why.
引言
我想知道是否存在一个简单的规则来求连续奇数的和。当我计算1 + 3时，得到4，即2²。接着1 + 3 + 5 = 9 = 3²。本项目检验前n个奇数之和是否总是等于n²，并解释其原因。
Method
I recorded the sums in a table alongside n, the count of terms. I also drew dot diagrams arranging odd numbers as L-shapes around a square.
方法
我列出了奇数：1, 3, 5, 7, 9, 11, 13, 15, 17, 19。我累加它们：1, 1+3=4, 1+3+5=9，以此类推。我将和记录在表格中，同时记录项数n。我还用点图画出了奇数的L形排列，围绕成一个正方形。
| n (number of terms) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Sum Sₙ | 1 | 4 | 9 | 16 | 25 |
| n² | 1 | 4 | 9 | 16 | 25 |
Results and Analysis
odd number can be written as 2n - 1. So the sum of the first n odd numbers is 1 + 3 + 5 + … d border to a square, keeping the shape a square. This visual proof confirms that Sₙ = n².
结果与分析
表格显示，前五个值中Sₙ与n²完全吻合。我注意到第n个奇数可以写成2n - 1。因此前n个奇数之和为1 + 3 + 5 + … + (2n - 1)。点图规律表明，每一个新增的奇数都为一个正方形添上L形边框，使得图形始终保持正方形。这一视觉证明证实了Sₙ = n²。
Conclusion
iagram also gives a convincing geometric reason. In the future, I would test larger n values using a spreadsheet to confirm the rule holds for numbers beyond those drawn by hand.
结论
本探究表明，对于n = 1到5，前n个连续奇数之和总是等于n²。图示还提供了一个令人信服的几何解释。将来，我会使用电子表格测试更大的n值，以确认该规则对手工绘制范围之外的数字同样成立。
10. Mastering Mathematical Vocabulary and Tone | 掌握数学词汇与语气
Using precise language makes your paper professional. Replace everyday words with mathematical ones: use 'consecutive' instead of 'next to each other', 'product' instead of 'answer when multiplied', and 'term' instead of 'number in the list'. A list of useful terms includes: sum, difference, factor, multiple, sequence, pattern, generalise, verify, and conjecture.
使用精确的语言让你的论文显得专业。用数学词汇替换日常用语：用“consecutive”代替“挨着的”，用“product”代替“相乘的结果”，用“term”代替“列表中的数”。一些有用的术语包括：和、差、因数、倍数、数列、规律、概括、验证和猜想。
Maintain a neutral but engaged tone. Avoid phrases like 'It was really cool!' Instead, write 'The pattern was unexpected and mathematically interesting.' Write in the first person if the task allows, but keep the focus on the mathematics, not your feelings. Always explain why a step was taken.
保持中立但投入的语气。避免使用“太酷了！”这类说法， 而要写“这个规律出乎意料，在数学上很有趣。”如果任务允许，可以使用第一人称，但要把焦点放在数学上，而不是你的感受上。始终解释为什么要采取某个步骤。
11. Checking, Editing, and Referencing | 检查、编辑与参考文献
Before submitting, read your paper aloud to catch awkward sentences. Check that all headings match the structure and that every table has a title. Verify all calculations, and ensure any algebraic rule you propose is tested with at least two new examples not already in your results table.
提交之前，大声朗读论文以发现不通顺的句子。检查所有标题是否与结构一致，每张表格是否有标题。核实所有计算，并确保你提出的任何代数规则都用至少两个不在结果表格中的新例子检验过。
If you used a textbook, website, or class notes to help you, list them in a short references section. A simple format is: Author, Title, Year. For websites, include the URL and date accessed. Even in Year 7, acknowledging sources shows academic honesty.
如果你参考了教科书、网站或课堂笔记，请在一个简短的参考文献部分列出它们。简单的格式为：作者，书名，年份。对于网站，请注明网址和访问日期。即使在Year 7，注明出处也体现了学术诚实。
12. Final Thoughts on Developing Your Paper | 关于完善论文的最后思考
Writing a mathematical paper is a skill that grows with practice. Start with a clear question, gather evidence systematically, and let the data lead you to a rule. The framework in this guide gives you a reliable template to follow, while the model paper demonstrates how each section comes alive with real mathematical exploration.
撰写数学论文是一项随着练习而增长的技能。从一个清晰的问题开始，系统地收集证据，让数据引领你找到规律。本指南中的框架为你提供了一个可靠的模板，而范文则展示了真实数学探究如何让每个部分生动起来。
Remember that even professional mathematicians revise their work several times. Your first draft is just the beginning. Leave time to refine your explanation, add diagrams, and tighten your reasoning. With this framework and model, you are well equipped to produce a paper that shows both rigorous thinking and clear communication.
请记住，即使专业数学家也会多次修改他们的作品。你的初稿仅仅是个开始。留出时间来完善解释、添加图示并收紧推理。有了这个框架和范文，你已充分准备好撰写一篇既能展现严密思维又能清晰表达的论文。
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