📚 Year 7 WJEC Maths: Writing a Mathematical Essay – Framework and Model Answer | Year 7 WJEC 数学：论文写作框架与范文
In Year 7 WJEC Mathematics, pupils are often asked to write short structured explanations or 'maths essays' to demonstrate deeper understanding. These tasks are not about creative writing; they require logical reasoning, precise vocabulary and clear step-by-step arguments. Mastering the framework for such responses early will strengthen your problem-solving and examination skills throughout secondary school.
在 WJEC 的 Year 7 数学课程中，学生经常需要撰写简短、结构化的解释或"数学小论文"来展示更深层次的理解。这类任务并非创意写作，而是要求逻辑推理、精准的词汇和清晰的逐步论证。尽早掌握这类回答的框架，将强化你在整个中学阶段解决问题的能力和应考技巧。
1. What Is a Maths Essay? | 什么是数学论文？
A maths essay in Year 7 is a short piece of writing, usually between 80 and 150 words, that explains a concept, proves a statement or justifies a method. It is marked on clarity, accuracy and the use of mathematical language. Unlike an English essay, it relies heavily on definitions, algebraic expressions and numbered steps rather than descriptive paragraphs.
Year 7 的数学论文通常是一篇 80 到 150 词的短文，用于解释一个概念、证明一个论断或论证某种方法的合理性。评分依据是清晰度、准确性和数学语言的使用。与英文作文不同，数学论文更依赖定义、代数表达式和编号的步骤，而非描述性的段落。
Typical prompts include 'Explain why the sum of two odd numbers is always even', 'Prove that the angles in a triangle add up to 180 degrees' or 'What is a prime number? Give examples and non-examples'. These encourage you to show, not just tell.
典型的题目有"为什么两个奇数之和永远是偶数"，"证明三角形内角和为 180 度"或"什么是质数？请举例并给出非例"。这类题目鼓励你展示推理，而不仅仅是陈述结论。
2. The Basic Structure | 基本结构
Every strong maths essay follows a three-part structure: an opening statement that introduces the claim or definition, a reasoning section that builds the argument with logical steps and examples, and a concluding sentence that links back to the question. Use PEEL (Point, Evidence, Explanation, Link) or a similar model adapted for mathematics.
每一篇优秀的数学论文都遵循三部分结构：开篇陈述介绍论断或定义；推理部分通过逻辑步骤和例子构建论证；结尾句回到问题本身。可以使用 PEEL（观点、证据、解释、联系）模型，或针对数学稍加调整的类似模型。
Keeping this structure in mind helps you stay focused. Examiners look for a clear flow: say what you are going to prove, prove it rigorously and then restate why it matters.
牢记这个结构有助于保持思路集中。阅卷老师看重清晰的脉络：先说清你要证明什么，然后严谨地证明，最后重申这一结果的意义。
3. Writing a Strong Opening Statement | 写出强有力的开篇陈述
Your first sentence must directly address the question and set up the mathematical context. For a proof about odd numbers, start with: 'All odd numbers can be written in the form 2n + 1, where n is a whole number.' Avoid vague introductions such as 'This essay will talk about odd numbers.'
你的第一句话必须直接回应题目并建立数学语境。以证明奇数性质为例，可以这样开篇："所有奇数都可以写成 2n + 1 的形式，其中 n 是整数。" 避免使用模糊的引入，如"本文将讨论奇数"。
The opening should include key definitions or assumptions. If the question asks about a rule, state the rule in precise terms right away. This shows the examiner you understand the mathematical vocabulary.
开篇应包含关键定义或假设。若问题涉及某条法则，应立即用精确的语言陈述该法则。这能让阅卷老师看出你对数学词汇的掌握。
4. Building Your Argument with Reasoning | 用推理构建论证
After the opening, move step by step through the logic. Use linking words such as 'therefore', 'since', 'this means that' and 'consequently'. Each line should follow from the previous one. Write each step on a new line or use clear connectives to guide the reader through your algebraic manipulation or geometric reasoning.
开篇之后，应一步步展开逻辑。使用诸如"因此""由于""这意味着""从而"等连接词。每一行都应从上一步推导而来。可以将每一步单独成行，或使用清晰的连接词引领读者完成代数变形或几何推理。
2. Taking out the factor of 2 gives 2(a + b + 1). Since a + b + 1 is an integer, the sum is a multiple of 2, which is the definition of an even number. This chain of reasoning is what earns marks.
例如：设两个奇数为 2a + 1 和 2b + 1，其中 a 和 b 为整数。它们的和是 (2a + 1) + (2b + 1) = 2a + 2b + 2。提取因子 2 得到 2(a + b + 1)。由于 a + b + 1 是整数，和是 2 的倍数，这正是偶数的定义。这一推理链就是得分点。
5. Using Examples and Counterexamples | 使用例子与反例
re the general proof, or as a check at the end. For instance, show that 3 + 5 = 8 and 7 + 11 = 18, then explain why the pattern always holds using algebra.
仅靠举例无法证明一般性论断，但例子有助于说明推理。在数学论文中，你可以在一般性证明之前先用一个例子热身，或在结尾用例子验证。比如先展示 3 + 5 = 8 以及 7 + 11 = 18，再借助代数解释为什么该模式永远成立。
Counterexamples are powerful when you need to disprove a statement. If the claim is 'All prime numbers are odd', one counterexample — the number 2 — is enough. State the counterexample clearly and explain why it disproves the claim.
当需要反驳一个论断时，反例是很有力的工具。如果论断是"所有质数都是奇数"，一个反例——数字 2——就已足够。须清晰给出反例，并说明它如何推翻原来论断。
6. Mathematical Vocabulary and Notation | 数学词汇与符号
Using accurate terms is essential. Instead of saying 'the answer comes out even', write 'the result is an even integer'. Use words like 'consecutive', 'multiple', 'factor', 'integer', 'sum', 'product', 'divisible' and 'remainder' appropriately. This raises the quality of your communication and meets the WJEC Assessment Objectives for mathematical language.
使用准确的术语至关重要。不要写"答案出来是偶数"，而应写"结果为偶整数"。恰当地使用诸如"连续""倍数""因数""整数""和""积""可整除""余数"等数学词汇。这会提升表达质量，并满足 WJEC 评估目标中对数学语言的要求。
Notation must be clear: letters stand for numbers, and you must state what type of numbers they represent (e.g. 'n is an integer' or 'x is a positive whole number'). Use alignments and bullet points if they make the argument easier to follow. Do not overuse symbols at the expense of words; a sentence that connects symbols is still needed.
符号必须清晰：字母代表数字，且需说明它们属于哪一类数（如"n 为整数"或"x 为正整数"）。如果项目符号或对齐能使论证更易理解，可以使用。但不要为了符号而牺牲文字；仍需要句子将符号串联起来。
7. Linking Ideas with Connectives | 用连接词串联思路
Mathematical writing depends on logical connectives. Words like 'if … then …', 'given that', 'therefore', 'hence', 'thus', 'since' and 'because' show the direction of reasoning. Using a variety of these words prevents repetition and makes the essay more professional.
数学写作离不开逻辑连接词。"如果……那么……""鉴于""因此""由此""从而""由于""因为"等词语可以体现推理的方向。变换使用这些词可以避免重复，让论文更显专业。
Structuring your sentences to show cause and effect helps the reader follow your argument. For example: 'Because a and b are integers, a + b + 1 is also an integer. Therefore, 2(a + b + 1) is an even number.' This two-sentence pattern – reason then conclusion – is a building block of maths essays.
将句子组织成因果关系有助于读者跟上你的论证。例如："由于 a 和 b 是整数，a + b + 1 也是整数。因此，2(a + b + 1) 是偶数。" 这种先给出理由再得出结论的两句模式，是数学论文的基本构件。
8. Concluding Your Essay | 为论文写结论
A strong conclusion restates the original statement in light of the proof provided. It should not introduce new information. Frame it as: 'Therefore, the sum of any two odd numbers is always even' or 'This proves that the pattern holds for all integers.' Do not end with a weak phrase such as 'That is why odd numbers act like that.'
扎实的结论会基于给出的证明再次申明原论断，而不应引入新信息。可以表述为："因此，任意两个奇数之和永远是偶数" 或 "这证明该规律对所有整数都成立"。不要以"这就是奇数行为如此的原因"之类软弱无力的句子收尾。
If the essay was about explaining a concept, your conclusion can summarise the key property or link to another topic. Keep it concise and make sure the main idea echoes the opening.
如果论文是解释某个概念，结论可以总结关键性质或联系到另一个主题。要简洁，并确保主要思想与开篇呼应。
9. Model Answer: Why the Sum of Two Odd Numbers Is Even | 范文：为什么两个奇数之和是偶数
Below is a full model answer that uses the framework we have described. The prompt is: 'Explain why the sum of any two odd numbers is always an even number.' Read the English version first, then the Chinese translation, paying attention to the structure and phrasing.
以下是根据我们描述的框架写出的完整范文。题目是："解释为什么任意两个奇数之和永远是偶数。" 请先读英文版本，再看中文翻译，注意结构与措辞。
Opening statement: All odd numbers can be expressed in the form 2n + 1, where n is an integer.
开篇陈述：所有奇数都可以表示为 2n + 1 的形式，其中 n 为整数。
Setting up variables: Let the first odd number be 2a + 1 and the second odd number be 2b + 1, with a and b being integers.
设定变量：设第一个奇数为 2a + 1，第二个奇数为 2b + 1，其中 a 与 b 是整数。
Sum = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1)
Reasoning: Since a and b are integers, a + b + 1 is also an integer. The expression 2(a + b + 1) is clearly a multiple of 2. By definition, any multiple of 2 is an even number. Therefore, the sum of the two odd numbers is even.
推理：因为 a 和 b 是整数，a + b + 1 也是整数。表达式 2(a + b + 1) 显然是 2 的倍数。根据定义，任何 2 的倍数都是偶数。因此，这两个奇数的和为偶数。
Conclusion: This argument works for any odd numbers we choose because a and b can represent any integers. Thus, the sum of any two odd numbers is always even.
结论：由于 a 和 b 可以代表任何整数，这一论证对我们选择的任意奇数都成立。因此，任意两个奇数之和永远是偶数。
This model shows how definitions, algebra and logical connectives work together. Practise writing similar answers for other statements, such as 'the product of two even numbers is a multiple of 4' or 'the sum of three consecutive integers is a multiple of 3'.
这篇范文展示了定义、代数和逻辑连接词如何协同作用。请针对其他论断练习撰写类似的回答，例如"两个偶数的积是 4 的倍数"或"三个连续整数之和是 3 的倍数"。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Many pupils lose marks by relying only on examples without a general proof. A single example, or even ten examples, does not prove a statement for all cases. Always follow up illustrations with algebraic or logical reasoning that covers every possibility.
很多学生因仅依靠例子而缺乏一般性证明而丢分。一个例子，甚至十个例子，都无法证明一个对所有情况都成立的陈述。在展示实例之后，一定要用代数或逻辑推理覆盖所有可能的状况。
Another mistake is using letters without stating what kind of numbers they represent. Always define your variables: 'Let x be an integer' or 'Let n be a positive whole number'. Vague language such as 'it works because it just does' will not gain marks. Finally, check that your conclusion directly answers the question and matches your opening statement.
另一个错误是使用字母却不说明它们代表哪一类数。务必定义变量："设 x 为整数"或"设 n 为正整数"。诸如"它成立因为它就是成立"这种含糊的语言无法得分。最后，检查结论是否直接回答了题目并与开篇陈述一致。
11. Self-Assessment Checklist | 自我评估清单
Before submitting your work, run through this checklist: Have I included an opening sentence that defines key terms? Did I use variables and state their types? Are the steps in a logical order, with connectives? Have I used correct mathematical vocabulary? Did I include a concluding sentence that answers the question? Is the spelling of terms like 'integer', 'multiple' and 'remainder' accurate?
提交作业前，请对照这份清单检查：我是否写了定义关键术语的开篇句？是否使用了变量并说明了类型？各步骤是否有逻辑顺序且用了连接词？是否用了正确的数学词汇？是否有回答问题的结论句？"integer""multiple""remainder"等术语的拼写是否准确？
If you can tick all five boxes, your essay is likely to be clear and rigorous. Regular self-assessment will also prepare you for the more complex reasoning tasks in Years 8 and 9.
若能在这五点上全部打勾，你的论文很可能既清晰又严谨。定期自我评估还能为 Year 8 和 Year 9 更复杂的推理任务做好准备。
12. Where to Find More Practice | 更多练习资源
Your teacher may provide past WJEC-style prompts, but you can also create your own. Turn any statement from your textbook into a question: 'Prove that the sum of two even numbers is even', 'Show that a triangle cannot have two right angles' or 'Explain what a factor is and give examples'. Swap answers with a partner and check each other's structure.
你的老师可能会提供 WJEC 风格的旧题，但你也可以自己出题。将课本中的任意陈述转化成问题："证明两个偶数之和为偶数""证明一个三角形不可能有两个直角"或"解释什么是因数并举例"。与同伴交换答案，互相检查结构。
Remember that the skill of writing a mathematical explanation will strengthen your overall mathematical thinking. Use the framework in this article as a template until the process becomes natural. With consistent practice, you will be able to produce clear, concise and convincing maths essays.
请记住，撰写数学解释这一技能会增强你的整体数学思维。不断使用本文介绍的框架作为模板，直到这一过程变得自然而然。通过持续练习，你将能够写出清晰、简洁且具有说服力的数学论文。
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