📚 Framework and Sample Paper for Year 7 Edexcel Further Mathematics | Year 7 Edexcel 进阶数学：论文写作框架与范文
In Year 7 Edexcel Further Mathematics, students are often encouraged to write short investigative papers. This style of writing helps you go beyond textbook exercises and develop skills in reasoning, communication, and structured argument. This article provides a clear framework for building such a paper and includes a full sample paper on perfect numbers, showing how each section should be written.
在 Year 7 Edexcel 进阶数学课程中，学生常常被鼓励撰写简短的探究性论文。这种写作形式能帮助你超越课本练习，培养推理、表达和有结构的论证能力。本文提供了一套清晰的论文构建框架，并附上一篇关于完全数的完整范文，展示每个部分应当如何撰写。
1. Understanding the Purpose | 理解论文目的
A mathematical paper is not simply a record of answers. It tells a story: you pose a question, gather evidence through exploration, spot patterns, and then explain what you have discovered. For Year 7 Further Mathematics, the paper shows that you can think like a mathematician, not just calculate.
数学论文不仅仅是答案的记录。它讲述一个故事：你提出一个问题，通过探索收集证据，发现模式，然后解释你发现了什么。对于七年级进阶数学而言，论文证明你能够像数学家一样思考，而不仅仅是计算。
2. Choosing an Engaging Topic | 选择有趣的课题
A good topic is narrow enough to explore thoroughly in a short paper, yet rich enough to reveal patterns. Examples include: ‘Are there infinitely many prime numbers?’, ‘How do triangular numbers relate to handshakes?’, and ‘Investigating perfect numbers below 100’. The topic should arise from curiosity, not just from a textbook heading.
好的课题范围要窄，足以在短论文中深入探索，但又足够丰富以揭示规律。例如：“质数是否无限多？”、“三角形数与握手问题有何关联？”、“探究100以下的完全数”。课题应来源于好奇心，而不仅仅是课本的标题。
3. The IMRaD Structure | IMRaD 结构
Most short mathematical papers follow the IMRaD structure: Introduction, Method, Results, and Discussion, sometimes followed by a Conclusion. This framework makes your writing clear and logical. The Introduction says what you will investigate and why. The Method explains how you collected data. The Results show what you found, often with tables. The Discussion analyses patterns and answers your question. The Conclusion summarises the main insight.
大多数短篇数学论文遵循 IMRaD 结构：引言（Introduction）、方法（Method）、结果（Results）、讨论（Discussion），有时后面还有结论（Conclusion）。这个框架使写作清晰且富有逻辑。引言说明你将研究什么及其原因。方法解释你如何收集数据。结果展示你的发现，常借助表格。讨论分析规律并回答你的问题。结论总结主要见解。
4. Writing the Introduction | 撰写引言
The introduction should capture interest and state the research question clearly. Begin with a surprising fact or a simple definition, then narrow down to exactly what you will investigate. Avoid vague statements like ‘I am going to write about numbers.’ Instead, write: ‘This paper explores whether regular patterns exist in the sums of proper divisors of integers from 1 to 100.’
引言应吸引兴趣并清晰陈述研究问题。从一个令人惊讶的事实或简单定义开始，然后收窄到你要具体探究的内容。避免含糊的表述，例如“我要写关于数字的文章”。而应写成：“本文探究从1到100的整数的真因数之和是否存在规律。”
5. Describing the Method | 描述方法
The method section should allow someone else to repeat your work. List the steps you took: how you chose the range of numbers, how you found divisors, how you recorded results. Use precise language. For example: ‘For each integer n from 2 to 30, I listed all proper divisors. I then calculated the sum of these divisors and compared the sum to n. I recorded the outcome in a table.’
方法部分应让他人能够重复你的工作。列出你采取的步骤：如何选择数字范围，如何找出因数，如何记录结果。使用精确的语言。例如：“对于从2到30的每个整数 n，我列出其所有真因数。然后计算这些因数之和，并与 n 进行比较。我将结果记录在一张表格中。”
6. Presenting Results | 展示结果
Results are best shown with a well-labelled table. It should include columns for the number, its proper divisors, the sum of those divisors, and a verdict (perfect or not). Always give the table a title and refer to it in the text. Avoid simply writing a list of numbers without explanation. A clear table makes patterns jump out.
结果最好用标注清晰的表格展示。表格应包含数字、其真因数、因数之和以及判定（是否完全数）等列。务必为表格加上标题并在正文中提及。避免只是罗列数字而没有解释。清晰的表格能让规律跃然纸上。
7. Discussion and Analysis | 讨论与分析
Here you explore what the results mean. Look for patterns: are all perfect numbers even? Could there be odd perfect numbers? Can you express the perfect numbers you found in terms of powers of 2 and primes? This is the section where you connect your findings to bigger mathematical ideas. Do not worry if you cannot fully explain a pattern; observing and describing it is already strong mathematical practice.
在这里你探究结果的含义。寻找规律：是否所有完全数都是偶数？可能存在奇数完全数吗？你能否用2的幂和质数表示你找到的完全数？在这个部分，你将发现与更大的数学思想联系起来。如果无法完全解释某个规律，不必担心；观察和描述它已经是很好的数学实践。
8. Crafting a Conclusion | 打造结论
The conclusion answers the original question without introducing new information. Restate the main finding in one sentence, mention any limitations, and suggest what could be explored next. For example: ‘This investigation confirmed that 6 and 28 are the only perfect numbers under 30, and both are even. A natural next step would be to explore whether all perfect numbers must be even.’
结论回答最初的问题，不引入新信息。用一句话重申主要发现，提及任何局限性，并建议下一步可探索的方向。例如：“本次探究确认6和28是30以内仅有的完全数，且均为偶数。很自然的下一步是探究是否所有完全数都必须是偶数。”
9. Sample Paper Part 1: Introduction and Method | 范文第一部分：引言与方法
Title: An Exploration of Perfect Numbers
标题：完全数探究
Introduction
引言
Perfect numbers have fascinated mathematicians for over 2000 years. A perfect number is a positive integer that equals the sum of its proper divisors. Proper divisors are all positive divisors of the number except the number itself. For example, the number 6 has proper divisors 1, 2, and 3. Since 1 + 2 + 3 = 6, the number 6 is perfect. The next perfect number known to ancient Greeks is 28. In this paper, I investigate how many perfect numbers exist between 1 and 30, and I search for patterns in their properties.
完全数吸引数学家已有两千多年。一个完全数是一个正整数，等于其真因数之和。真因数是该数除了自身以外的所有正因数。例如，数字6的真因数为1、2和3。因为1 + 2 + 3 = 6，所以6是完全数。古希腊人已知的下一个完全数是28。在本文中，我探究1至30之间存在多少个完全数，并寻找其性质的规律。
Method
方法
I chose to examine all integers from 2 to 30. I started at 2 because 1 has no proper divisors. For each number n, I wrote down all its factors smaller than n. I then added these factors to get the sum S. If S equalled n, I recorded n as a perfect number. I organised my results in a table with columns for the number, its proper divisors, the sum, and a label of perfect or not. I double-checked each calculation to avoid errors.
我选择考察2到30的所有整数。我从2开始，因为1没有真因数。对于每个数字 n，我写出所有比 n 小的因数。然后将这些因数相加得到总和 S。如果 S 等于 n，我便将 n 记为完全数。我将结果整理在一张表格中，列分别有数字、其真因数、总和以及是否为完全数的标注。我对每个计算都进行了复核，以避免错误。
10. Sample Paper Part 2: Results, Discussion and Conclusion | 范文第二部分：结果、讨论与结论
Results
结果
Table 1 below shows a selection of the numbers tested and their divisor sums. The full table included all numbers from 2 to 30, but for readability I display representative entries here.
下表1显示了部分测试数字及其因数之和。完整表格包含了2到30的所有数字，但为便于阅读，我在此展示有代表性的条目。
| Number (n) | Proper Divisors | Sum (S) | Perfect? |
|---|---|---|---|
| 2 | 1 | 1 | No |
| 6 | 1, 2, 3 | 6 | Yes |
| 10 | 1, 2, 5 | 8 | No |
| 12 | 1, 2, 3, 4, 6 | 16 | No |
| 28 | 1, 2, 4, 7, 14 | 28 | Yes |
| 30 | 1, 2, 3, 5, 6, 10, 15 | 42 | No |
The only numbers in the range 2 to 30 whose proper divisor sum equalled the number itself were 6 and 28. Both are even. No odd perfect number was found.
在2到30的范围内，仅有的真因数之和等于自身的数是6和28。两者均为偶数。未发现奇数完全数。
Discussion
讨论
Finding only 6 and 28 prompted the question: can perfect numbers be expressed in a special form? I noticed that:
仅发现6和28不禁使人发问：完全数能否用一种特殊形式表示？我注意到：
6 = 2 × 3 = 2¹ × (2² − 1)
6 = 2 × 3 = 2¹ × (2² − 1)
28 = 4 × 7 = 2² × (2³ − 1)
28 = 4 × 7 = 2² × (2³ − 1)
In each case, one factor is a power of 2 and the other is one less than a power of 2, and that second factor is prime. This matches a formula known from ancient Greek mathematics: if 2ⁿ⁻¹(2ⁿ − 1) is evaluated with (2ⁿ − 1) being a prime number, the result is a perfect number. Testing this idea:
在每一种情况下，一个因数是2的幂，另一个是2的幂减1，并且第二个因数是质数。这与古希腊数学中已知的一个公式相符：若2ⁿ⁻¹(2ⁿ − 1)中(2ⁿ − 1)为质数，结果则为完全数。验证这一想法：
When n = 2: 2¹ × (2² − 1) = 2 × 3 = 6
当 n = 2: 2¹ × (2² − 1) = 2 × 3 = 6
When n = 3: 2² × (2³ − 1) = 4 × 7 = 28
当 n = 3: 2² × (2³ − 1) = 4 × 7 = 28
When n = 5: 2⁴ × (2⁵ − 1) = 16 × 31 = 496
当 n = 5: 2⁴ × (2⁵ − 1) = 16 × 31 = 496
The next perfect number, 496, is indeed greater than 30, which explains why it did not appear in my table. The formula also predicts 8128 when n = 7. This pattern suggests there are infinitely many even perfect numbers linked to primes of the form 2ⁿ − 1, although this is still an unproven conjecture. I also noted that no odd perfect number has ever been found, and many mathematicians believe none exists.
下一个完全数496确实大于30，这就解释了它为何未出现在我的表格中。该公式还预测当 n = 7 时得到8128。这一规律表明，有无限多个偶完全数与形如2ⁿ − 1的质数相关联，尽管这仍是一个未被证明的猜想。我还注意到，至今从未发现奇数完全数，许多数学家认为它们根本不存在。
Conclusion
结论
My investigation confirmed that 6 and
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