📚 Mastering Mathematical Writing: Framework and Sample for Year 7 WJEC Advanced Maths | 掌握数学写作：Year 7 WJEC进阶数学论文框架与范文
Writing in mathematics is not just about finding the right answer – it is about explaining the journey to that answer in a clear, logical way. For Year 7 students studying WJEC Advanced Maths, learning how to structure and present a mathematical investigation or proof is a key skill. This article provides a complete writing framework, practical tips, and annotated samples to help you excel in mathematical writing tasks.
数学写作不仅仅是找到正确答案，更在于以清晰、有逻辑的方式解释得到答案的过程。对于学习WJEC进阶数学的Year 7学生来说，学会如何构建和呈现数学探究或证明是一项关键技能。本文提供一套完整的写作框架、实用技巧以及带注释的范文，帮助你在数学写作任务中脱颖而出。
1. Understanding the Importance of Mathematical Writing | 理解数学写作的重要性
Mathematical writing trains you to think precisely. When you write down a solution step by step, you are not only recording your reasoning but also checking it for gaps or mistakes. This skill helps you communicate ideas to others and to your future self, just like a scientist writing a lab report.
数学写作训练你精确思考。当你一步步写下解题过程时，你不仅在记录自己的推理，还在检查其中是否有漏洞或错误。这项技能帮助你向他人以及未来的自己传达想法，就像科学家撰写实验报告一样。
In WJEC Advanced Maths, marks are often awarded for clear explanation, not just for the final number. A well-written paper shows the examiner that you truly understand the concept, not that you just guessed or memorised a method.
在WJEC进阶数学中，分数通常会给到清晰的解释，而不仅仅是最终的数字。一篇写得很好的论文向考官展示了你真正理解了概念，而不是仅仅猜测或死记硬背方法。
Good mathematical writing also develops your ability to construct a coherent argument, a skill that is valuable across all subjects and in everyday life when you need to persuade or inform.
优秀的数学写作也能培养你构建连贯论证的能力，这项技能在所有学科以及日常生活中需要说服或告知他人时都很有价值。
2. Requirements of WJEC Advanced Maths Papers | WJEC进阶数学论文的要求
WJEC assessment objectives for mathematical writing include: stating the problem clearly, defining any variables or assumptions, showing a logical sequence of steps, and presenting a conclusion that refers back to the original question. You are expected to use accurate mathematical notation and vocabulary.
WJEC对数学写作的评估目标包括：清晰地陈述问题，定义任何变量或假设，展示合乎逻辑的步骤序列，并给出一个能回扣原题的结论。你还需要使用准确的数学符号和词汇。
At Year 7 Advanced level, tasks may involve investigating patterns, solving real-life problems, or proving a simple statement. The examiners look for systematic working, not just the answer. For instance, if you are asked 'Prove that the sum of three consecutive numbers is a multiple of 3', they want to see algebraic reasoning, not just an example.
在Year 7高阶水平，任务可能涉及探究规律、解决现实问题或证明一个简单的命题。考官看重的是系统化的推导过程，而不仅仅是答案。例如，如果题目是“证明三个连续整数的和是3的倍数”，他们希望看到代数推理，而不仅仅是一个例子。
Your writing should be concise but complete. Avoid filling space with irrelevant information; every sentence should move the argument forward.
你的写作应该简洁但完整。避免用无关信息填充篇幅，每一句话都应该推动论证的进展。
3. Basic Structure: Introduction, Body, Conclusion | 论文基本结构：引言、正文、结论
Every piece of mathematical writing needs a clear structure. Think of it as a three-part framework:
每篇数学写作都需要清晰的结构。可以把它看作一个三部分的框架：
- Introduction: Restate the problem in your own words, and mention what you are going to do.
- 引言：用自己的话重述问题，并说明你要做什么。
- Body: Present your logical reasoning. This can include tables of values, algebraic manipulations, diagrams, and step-by-step calculations. Every step should be explained.
- 正文：呈现你的逻辑推理。可以包括数值表格、代数变换、图表和逐步计算。每一步都应加以解释。
- Conclusion: Summarise your findings, give the final answer, and relate it back to the original problem. If relevant, comment on the validity or any limitations.
- 结论：总结你的发现，给出最终答案，并将其与原始问题联系起来。如果相关，可以评论有效性或任何局限。
For a short investigation, this structure might be just a few paragraphs. For a longer project, each part might be a full section. The key is that the reader can follow your thinking without getting lost.
对于一项简短探究，这个结构可能只是几个段落。对于更长的项目，每个部分可能是一个完整的节。关键是让读者能够跟上你的思路而不会迷失。
4. How to Write Clear Assumptions and Definitions | 如何写清楚假设和定义
Many mathematical problems require you to state assumptions. For example, if you are calculating the cost of fencing a garden, you might assume the garden is a perfect rectangle. Clearly stating this shows you are aware of the real-world limits of your model.
许多数学问题要求你陈述假设。例如，如果你在计算给花园围篱笆的费用，你可能会假设花园是一个完美的矩形。清楚地说明这一点表明你意识到模型的现实局限。
Define all variables early. Instead of writing 'let the number be something', define it precisely: 'Let n represent the first integer.' This removes ambiguity. If you use units, include them: 'Let the length be L metres.'
尽早定义所有变量。与其写“设这个数为某个东西”，不如精确定义：“设 n 表示第一个整数。”这消除了歧义。如果使用单位，也要写进去：“设长度为 L 米。”
Assumptions also apply to the scope of your investigation. For instance, 'I will only consider positive whole numbers for this pattern.' This prevents your work from being marked down for not covering negative cases if the question only implied positives.
假设也适用于你的探究范围。例如，“本人仅考虑正整数来研究该规律。”这避免了因为问题只是暗示了正数而你却没有覆盖负数而被扣分。
5. Using Logical Connectives | 使用逻辑连接词
Logical flow makes your writing readable. Use connectives like 'therefore', 'hence', 'because', 'since', 'this implies', 'it follows that'. These words show the relationship between your statements.
逻辑流畅使你的写作易读。使用“因此”、“所以”、“因为”、“由于”、“这意味着”、“由此可得”这样的连接词。这些词展示了你的陈述之间的关系。
For example, instead of '3 odd numbers. Sum is odd', write: 'Let the three odd numbers be a, b, c (2p+1)+(2q+1)+(2r+1)=2(p+q+r)+3, which is odd because 2(p+q+r) is even and even + 3 = odd.'
例如，与其写“3个奇数。和为奇数”，不如写：“设三个奇数为 a, b, c。由于每个奇数可以写成2k+1的形式，其和为 (2p+1)+(2q+1)+(2r+1)=2(p+q+r)+3，该和为奇数，因为2(p+q+r) 是偶数，而偶数 + 3 = 奇数。”
Use a mix of logical words, not just 'then'. Repeating 'then, then, then' makes the explanation sound robotic. Try 'next', 'consequently', 'as a result', 'we can see that'.
要使用多种逻辑词汇，而不仅仅是“然后”。重复“然后，然后，然后”会让解释听起来像机器人。尝试使用“接下来”、“从而”、“结果”、“我们可以看到”。
6. Presenting Mathematical Proofs or Solution Steps | 呈现数学证明或解法步骤
When writing a proof or a multi-step solution, display each step on a new line and justify it. Use the format:
在写证明或多步骤解法时，每一步都另起一行并给出理由。使用以下格式：
Statement + Reason
陈述 + 理由
Example (prove sum of two even numbers is even):
示例（证明两个偶数之和为偶数）：
Let the two even numbers be 2m and 2n, where m, n are integers. (Definition of even)
设这两个偶数为 2m 和 2n，其中 m, n 为整数。（偶数的定义）
Their sum = 2m + 2n = 2(m + n). (Factorising)
它们的和 = 2m + 2n = 2(m + n).（因式分解）
Since m + n 2 × (an integer), which is even. (Closure of integers under addition, definition of even)
由于 m + n 是整数，其和为 2 ×（整数），即为偶数。（整数加法封闭性，偶数的定义）
Always end a proof with a concluding sentence or the symbol QED (quod erat demonstrandum) to show you have reached the end. For exam purposes, a clear sentence is enough.
证明结束时总是写一句总结性的话或 QED 符号，表明你已经完成证明。对于考试来说，一句清晰的话就足够了。
7. Proper Use of Diagrams and Mathematical Notation | 图表和数学符号的正确使用
y words. However, a diagram alone is not a proof; you must still describe it. For example, 'As shown in Figure 1, triangle ABC is right-angled at B. By Pythagoras' theorem, AC² = AB² + BC².'
图表应用尺子清晰地绘制并标注。一幅好图可以代替大量文字。但是单单一张图并不能构成证明，你仍需对它进行描述。例如，“如图1所示，三角形 ABC 在 B 处为直角。根据毕达哥拉斯定理，AC² = AB² + BC²。”
Mathematical notation must be correct and consistent. Use standard symbols: ∠ for angle, △ for triangle, ⇒ for square root. If you are typing, use the available symbols or clear alternatives (e.g., 'sqrt(x)' if you cannot type √). In handwritten work, take care with Greek letters and indices.
数学符号必须正确且一致。使用标准符号：∠ 表示角，△ 表示三角形，⇒ 表示推出，∴ 表示因此，± 表示正负，√ 表示平方根。如果是打字，使用可用的符号或清晰的替代（例如，无法输入 √ 时可用 sqrt(x)）。手写时要注意希腊字母和上下标。
Never use a symbol for the first time without defining it. For instance, 'Let the cost be £C' or 'Let the area be A cm²'.
第一次使用某个符号时，一定要先定义。例如，“设费用为 £C”或“设面积为 A cm²”。
8. Writing Conclusions and Reflections | 撰写结论和反思
A strong conclusion does more than restate the answer. It interprets the result. Ask yourself: What does this answer mean in the context of the problem? Is it reasonable? Could there be another way to solve it?
强有力的结论不仅仅是复述答案，它还解读了结果。问问自己：这个答案在问题语境中意味着什么？它合理吗？是否有其他解法？
In an investigation, reflect on patterns you found and whether they always hold. You might write: 'The pattern holds for all positive integers, but further testing could confirm whether it also works for negative integers.' Or 'The total cost is £128, which seems high because we assumed premium materials; a different assumption would change the result.'
在一项探究中，要反思你发现的规律是否始终成立。你可以写：“该规律对所有正整数成立，但进一步检验可以确认它是否也适用于负整数。”或者“总费用为 128 英镑，这个数值看似较高，因为我们假设使用的是高档材料；不同的假设会改变结果。”
End with a clear final statement that directly answers the problem. Example: 'Therefore, the sum of three consecutive integers is always a multiple of 3.'
最后写一个清晰的最终陈述，直接回答问题。例如：“因此，三个连续整数之和总是 3 的倍数。”
9. Sample 1: Solving a Geometry Problem | 范文一：解决一个几何问题
Problem: A rectangle has length (2x + 3) cm and width (x − 1) cm. Find an expression for its perimeter and prove that the perimeter is always even for integer x > 1.
题目：一个矩形的长为 (2x + 3) cm，宽为 (x − 1) cm。求其周长的表达式，并证明当整数 x > 1 时周长恒为偶数。
Introduction: In this problem, I will find a formula for the perimeter of the given rectangle and then prove that the perimeter is always an even number when x is an integer greater than 1. I will define my variable and use properties of rectangles.
引言：本题中，我将求出给定矩形的周长公式，并证明当 x 为大于 1 的整数时周长恒为偶数。我将定义变量并利用矩形的性质。
Body: and the width be W = (x − 1) cm. The perimeter P of a rectangle is given by P = 2(L + W).
正文：设长为 L = (2x + 3) cm，宽为 W = (x − 1) cm。矩形周长 P 的公式为 P = 2(L + W)。
Substitute the expressions: P = 2[(2x + 3) + (x − 1)] = 2(3x + 2) = 6x + 4 cm.
代入表达式：P = 2[(2x + 3) + (x − 1)] = 2(3x + 2) = 6x + 4 cm。
teger), which is the definition of an even number. Hence, the perimeter is always even for the given condition.
现在，我们需要证明对于任意整数 x > 1，P 是偶数。注意到 P = 6x + 4 = 2(3x + 2)。因为 x 是整数，3x + 2 也是整数。因此，P = 2 ×（整数），这正是偶数的定义。所以，在给定条件下周长恒为偶数。
Conclusion: n the factor 2 and the closure property of integers. This result would not hold for non-integer x; for example, if x = 0.5, P = 7 cm, which is odd, but x must be an integer >1.
结论：周长为 (6x + 4) cm。我已证明当 x 为大于 1 的整数时，该表达式始终给出偶数。该推理依赖于因数 2 和整数的封闭性。此结果对于非整数 x 不成立；例如，若 x = 0.5，则 P = 7 cm，为奇数，但题目要求 x 为大于 1 的整数。
10. Sample 2: Investigating a Number Pattern | 范文二：探究数字模式
Problem: Investigate the difference between the squares of two consecutive numbers. Is there a pattern? Prove your conjecture.
题目：探究两个连续整数平方的差。是否存在规律？证明你的猜想。
Introduction: I will explore what happens when I subtract the square of one number from the square of the next integer. I will start by calculating some examples, then form a conjecture, and finally prove it algebraically.
引言：我将探究从一个整数的平方减去下一个整数的平方会发生什么。我会先计算几个例子，然后形成一个猜想，最后用代数方法证明它。
Body: Let the two consecutive numbers be n and n+1. The difference D = (n+1)² − n².
正文：设这两个连续整数为 n 和 n+1。差 D = (n+1)² − n²。
I generated a table of values:
我生成了一个数值表格：
| n | (n+1)² | n² | D |
|---|---|---|---|
| 1 | 4 | 1 | 3 |
| 2 | 9 | 4 | 5 |
| 3 | 16 | 9 | 7 |
| 4 | 25 | 16 | 9 |
5, etc.). So my conjecture is: The difference between the squares of consecutive integers n+1 and n is 2n + 1.
从表中可见，D = 3, 5, 7, 9, ... 这些是连续的奇数。实际上，这个差似乎是 2n + 1（因为当 n=1 时，D=2×1+1=3；n=2 时，D=2×2+1=5，依此类推）。因此我的猜想是：连续整数 n+1 与 n 的平方差为 2n + 1。
Now I will prove this algebraically:
现在我进行代数证明：
D = (n+1)² − n² = (n² + 2n + 1) − n² = 2n + 1.
D = (n+1)² − n² = (n² + 2n + 1) − n² = 2n + 1.
Since n is an integer, 2n is even, so 2n+1 is odd. Thus, the difference is always an odd number, specifically the (n+1)th odd number starting from 3.
由于 n 是整数，2n 是偶数，所以 2n+1 是奇数。因此，差值总是一个奇数，具体来说是从 3 开始的第 (n+1) 个奇数。
Conclusion: xplains why the differences in my table were consecutive odd numbers. The proof is valid for all real numbers n as well, but the parity observation only holds for integers. This pattern could be extended to investigate differences of squares with gaps larger than 1.
结论：我的探究证实，对于所有整数 n，(n+1)² − n² = 2n + 1。这就解释了为什么我表格中的差值都是连续奇数。该证明对所有实数 n 也成立，但奇偶性观察只对整数成立。这个规律可以拓展到探究间隔大于 1 的平方差。
11. Common Mistakes and Improvement Tips | 常见错误与改进技巧
ample (e.g., 3² − 2² = 5) is not a proof for all cases. Always generalise using variables.
常见错误是学生跳过了逻辑步骤，或将举例与证明混淆。单个数值举例（如 3² − 2² = 5）并不能作为所有情况的证明。一定要用变量进行一般化推导。
Another common error is poor notation. Writing '2x+3' without parentheses when substituting can lead to mistakes: (2x+3)² is not 2x²+9. Use brackets carefully.
另一个常见错误是符号使用不当。代入时缺少括号可能导致错误：(2x+3)² 并非 2x²+9。要谨慎使用括号。
Some students forget to define the domain. If your proof relies on the property that a variable is an integer, state it. Otherwise, your reasoning might be considered incomplete.
有些学生忘记定义取值范围。如果你的证明依赖于变量是整数这一性质，就要明确说明。否则推理可能被认为不完整。
To improve, always draft your solution on scrap paper first. Then rewrite it neatly, adding explanations. Read your work aloud – if it sounds confusing, revise it. Peer review can also help: ask a friend to follow your reasoning.
为了提高，可以先将解题过程写在草稿纸上。然后整齐地重新抄写，并加入解释。大声朗读你的作品——如果听起来很混乱，就修改它。同伴互评也有帮助：请一位朋友跟随你的推理思路。
12. Self-assessment Checklist | 自我评估清单
Before submitting your mathematical writing, go through this checklist:
在提交你的数学写作之前，请逐项检查以下清单：
- Have I restated the problem? 我重述问题了吗？
- Are all variables defined? 所有变量都定义了吗？
- Are assumptions stated? 假设陈述清楚了吗？
- Is the logical flow clear with connectives? 逻辑流程是否通过连接词清晰展现？
- Are all steps justified? 所有步骤都有理有据吗？
- Have I included diagrams or tables where helpful? 我在有帮助的地方使用了图表或表格吗？
- Is the notation correct and consistent? 符号是否正确且前后一致？
- Does the conclusion answer the question and interpret the result? 结论是否回答了问题并解读了结果？
- Are there any spelling or grammar errors? 是否有拼写或语法错误？
- Have I checked that an example is not used as a proof? 我是否确保没有用例子代替证明？
Using this checklist regularly will build a strong habit of thorough mathematical writing, preparing you for more advanced work in later years.
经常使用这份清单将培养你严谨数学写作的习惯，为今后更高阶的学业做好准备。
Published by TutorHao | Advanced Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图