📚 Year 10 Cambridge Mathematics: A Writing Framework and Model Paper | Year 10 Cambridge 数学：论文写作框架与范文
Writing a mathematical paper or investigation report is an essential skill for Year 10 Cambridge students. It allows you to explore a mathematical topic in depth, apply problem-solving strategies, and communicate your findings clearly. This guide provides a structured framework and a model paper to help you succeed.
撰写数学论文或探究报告是 Year 10 Cambridge 学生的一项重要技能。它让你深入探讨一个数学主题，运用解题策略，并清晰地表达你的发现。本指南提供了一个结构化框架和一篇范文，助你成功。
1. Why Write a Mathematical Paper? | 为什么要写数学论文？
Mathematical papers develop critical thinking and the ability to justify reasoning. They also prepare you for further study where extended writing and investigation are required.
数学论文培养批判性思维和论证推理能力，也为后续需要拓展写作和研究的学习做好准备。
For Cambridge IGCSE, while the examination focuses on problem-solving, many schools incorporate internal investigations to assess AO3 (analysis and communication). Writing a full paper helps you practise organising ideas logically and using mathematical notation correctly.
对 Cambridge IGCSE 而言，虽然考试侧重于解题，但许多学校会结合内部探究来评估 AO3（分析与交流）目标。撰写完整的论文有助于你练习逻辑地组织观点并正确使用数学符号。
2. Understanding the Task | 理解题目要求
Read the investigation prompt carefully. Identify whether the task involves statistical analysis, geometric proof, modelling, or historical exploration. Clarify the expected length, format, and assessment criteria.
仔细阅读探究题目。判断任务涉及统计分析、几何证明、数学建模还是数学史探索。明确预期的篇幅、格式和评分标准。
Common Year 10 tasks include: 'Investigate the relationship between two variables', 'Prove a theorem using geometry', or 'Explore the Fibonacci sequence in nature'. Always define your own clear research question.
常见的 Year 10 任务包括：“调查两个变量之间的关系”、“用几何证明定理”或“探究自然界中的斐波那契数列”。务必提出自己清晰的研究问题。
3. General Structure of an Investigative Paper | 探究报告的基本结构
A well-organised mathematical paper follows a consistent structure. The table below outlines the typical sections and their purposes.
一篇结构良好的数学论文遵循一致的框架。下表概述了典型的部分及其目的。
| Section | Purpose |
|---|---|
| Title and Abstract | A concise summary of the investigation |
| Introduction | Background and research question |
| Methodology | How data was collected or reasoning planned |
| Analysis | Calculations, diagrams, tables, proofs |
| Conclusion | Answer to the research question and reflection |
这个结构确保你的报告逻辑清晰、易于评估。每个部分都有明确的功能。
4. Part 1: Title and Abstract | 第一部分：标题与摘要
The title should clearly state the investigation's focus. An abstract, if required, summarises the aim, method, key results and conclusion in about 100 words.
标题应清晰地说明探究的焦点。摘要（如果需要）用约 100 字概括目的、方法、主要结果和结论。
Example title: 'Investigating the relationship between daily screen time and percentage score in a memory test among Year 10 students.'
示例标题：“探究 Year 10 学生每日屏幕时间与记忆测试得分百分比之间的关系”。
Write the abstract after completing the paper, ensuring it accurately reflects the content.
在完成论文后再撰写摘要，确保它准确反映内容。
5. Part 2: Introduction and Research Question | 第二部分：引言与研究问题
The introduction provides background information and explains why the topic is interesting. It must include a clearly stated research question and, if appropriate, a hypothesis.
引言提供背景信息，解释为何该课题有趣。它必须包含一个清晰陈述的研究问题，并在适当时给出假设。
For a statistical investigation, you might write: 'I hypothesise that there is a negative correlation between screen time and memory test score, because prolonged screen exposure may reduce concentration.'
对于统计调查，你可以写：“我假设屏幕时间与记忆测试分数之间存在负相关，因为长时间屏幕暴露可能降低注意力。”
Define any key terms and mention the mathematical techniques you plan to use, such as scatter graphs, mean, and correlation coefficient.
定义关键术语，并提及你将使用的数学方法，如散点图、平均数和相关系数。
6. Part 3: Methodology and Data Collection | 第三部分：方法与数据收集
Describe exactly how you collected or generated data. For a survey, state sample size, selection method, and any controls (e.g. same test conditions). For a geometric proof, list the axioms and theorems you will rely on.
准确描述你如何收集或生成数据。对于问卷调查，说明样本量、选择方法和任何控制措施（如同样的测试条件）。对于几何证明，列出你将依赖的公理和定理。
Ensure your method is ethical and feasible. In a school setting, you might ask classmates to complete a reaction test after varying sleep durations.
确保方法合乎伦理且可行。在学校环境中，你可以请同学们在不同睡眠时长后完成反应测试。
Present raw data in a clear, labelled table. This demonstrates organisation and makes verification possible.
用清晰、带标签的表格展示原始数据。这体现了条理性，并使验证成为可能。
7. Part 4: Mathematical Analysis | 第四部分：数学分析
This is the core of your paper. Perform calculations step by step, showing all working. Use appropriate measures such as mean, median, range, and standard deviation if relevant.
这是你论文的核心。逐步进行计算，展示所有过程。使用恰当的统计量，如平均数、中位数、极差，以及（若相关）标准差。
For instance, to calculate the mean reaction time:
Mean = (Σ xᵢ) / n
例如，计算平均反应时间：
平均数 = (Σ xᵢ) / n
Create accurate graphs by hand or using software. Label axes, include a title, and plot points carefully. For a linear correlation, draw a line of best fit and discuss its equation.
用手绘或软件创建精确的图表。标注坐标轴，添加标题，并小心描点。对于线性相关，画出最佳拟合线并讨论其方程。
If the investigation involves proof, present a logical chain of statements with justifications. Use symbols like ∴ (therefore) and ∵ (because) to keep reasoning concise.
如果探究涉及证明，展示一个有合理依据的逻辑陈述链。使用符号如 ∴（所以）和 ∵（因为）保持推理简洁。
8. Part 5: Results and Interpretation | 第五部分：结果与解释
Summarise what the calculations and graphs reveal. Do not simply repeat numbers; interpret them. For a correlation, comment on its strength and direction.
总结计算和图表所揭示的内容。不要仅仅重复数字；要解读它们。对于相关性，评论其强度和方向。
Compare findings with your initial hypothesis. State whether the data supports it, and suggest possible reasons for any unexpected results.
将发现与最初假设进行比较。陈述数据是否支持假设，并对任何意外结果提出可能的原因。
Use phrases like 'The scatter graph shows a moderate positive correlation, suggesting that as screen time increases, the memory score tends to decrease.'
使用诸如“散点图显示中等正相关，表明随着屏幕时间增加，记忆分数趋于降低”之类的表述。
9. Part 6: Conclusion and Reflection | 第六部分：结论与反思
Your conclusion should directly answer the research question. Summarise the key numerical findings and their implications.
你的结论应直接回答研究问题。总结关键数值发现及其影响。
Reflect on limitations: Was the sample size too small? Were there confounding variables? Discuss how the investigation could be improved or extended.
反思局限性：样本量是否太小？是否存在混淆变量？讨论如何改进或拓展你的研究。
A strong conclusion demonstrates critical evaluation and mathematical communication, which are valued in Cambridge assessments.
有力的结论体现了批判性评估和数学交流能力，这在 Cambridge 评价中很受重视。
10. Model Paper: Sleep and Reaction Time Investigation | 范文：睡眠与反应时间调查
The following model paper illustrates the framework in action. It investigates the question: 'Does the number of hours of sleep affect Year 10 students' reaction times?' Note the structure, use of tables, calculations, and formal tone.
下面的范文展示了该框架的应用。它探究的问题是：“睡眠时长是否影响 Year 10 学生的反应时间？”请注意结构、表格使用、计算和正式语气。
Title: Investigating the Relationship Between Sleep Duration and Reaction Time
标题：探究睡眠时长与反应时间的关系
Abstract: This investigation examines whether sleep duration affects reaction time in 20 Year 10 students. Reaction times were measured after participants recorded their previous night's sleep. The mean reaction time was 0.302 s for those with less than 7 hours' sleep and 0.267 s for those with 7 or more hours. A scatter graph revealed a weak negative correlation, suggesting that more sleep may slightly improve reaction speed. However, variability was high, and sample size limited firm conclusions.
摘要：本探究调查睡眠时长是否影响 20 名 Year 10 学生的反应时间。参与者在记录前一晚睡眠时长后测量了反应时间。睡眠不足 7 小时者的平均反应时间为 0.302 s，而睡眠达到 7 小时及以上者为 0.267 s。散点图显示弱负相关，表明较多睡眠可能略微提升反应速度。然而，变异性较大，且样本量限制了得出确切结论。
Introduction: Reaction time is crucial in daily activities and sports. I wanted to discover if sleep, which is known to affect cognitive function, has a measurable effect on simple reaction time in my peer group. My research question: 'Is there a correlation between hours of sleep and reaction time in Year 10 students?' I hypothesised a negative correlation – more sleep leads to faster (shorter) reaction times.
引言：反应时间在日常活动和体育运动中至关重要。我想了解已知会影响认知功能的睡眠，是否对同龄人的简单反应时间有可测量的影响。我的研究问题：“Year 10 学生的睡眠时长与反应时间之间是否存在相关性？”我假设存在负相关——睡眠越多，反应时间越快（越短）。
Methodology: Twenty Year 10 students (10 male, 10 female) volunteered. Each reported their hours of sleep the previous night. Using an online reaction test (red-green go/no-go), each performed 5 trials, and the median time was recorded to reduce outlier effects. Testing occurred at the same time of day to control for fatigue.
方法：20 名 Year 10 学生（10 男 10 女）自愿参与。每人报告前一晚的睡眠时长（小时）。使用在线反应测试（红绿灯反应），每人进行 5 次试验，记录中位数时间以减少异常值影响。测试在同一时段进行，以控制疲劳因素。
Raw Data:
原始数据：
| Student | Sleep (h) | Reaction time (s) |
|---|---|---|
| A | 6.5 | 0.321 |
| B | 8.0 | 0.254 |
| C | 5.5 | 0.345 |
| ... | ... | ... |
| T | 7.5 | 0.280 |
(Full table of 20 entries was used in the actual analysis.)
（实际分析中使用了完整的 20 条数据表格。）
Analysis:
分析：
10) mean = (0.254+0.280+...+0.264)/10 = 0.267 s. The difference of 0.035 s represents a 13% faster reaction in the high-sleep group.
按睡眠时长分组：低睡眠组（< 7 小时，n=10）平均反应时间 = (0.321+0.345+...+0.298)/10 = 0.302 s。高睡眠组（≥ 7 小时，n=10）平均 = (0.254+0.280+...+0.264)/10 = 0.267 s。差值 0.035 s 代表高睡眠组反应快了 13%。
sleep hours) + 0.388. The correlation coefficient r was calculated as -0.32, indicating a weak negative correlation.
绘制了以睡眠时长为 x 轴、反应时间为 y 轴的散点图。最佳拟合线方程为：反应时间 ≈ -0.015 ×（睡眠时长）+ 0.388。计算得到相关系数 r = -0.32，表明存在弱负相关。
r = [nΣxy – (Σx)(Σy)] / √{[nΣx² – (Σx)²][nΣy² – (Σy)²]} = -0.32
r = [nΣxy – (Σx)(Σy)] / √{[nΣx² – (Σx)²][nΣy² – (Σy)²]} = -0.32
Conclusion: The hypothesis is weakly supported. While the high-sleep group had a lower mean, the correlation is weak and individual variations are large. The investigation suggests sleep may have a small influence, but other factors like alertness and practice effect could not be ruled out. Limitations include a small, non-random sample and only one night's sleep record.
结论：假设得到弱支持。虽然高睡眠组有较低的平均值，但相关性较弱，且个体差异很大。本探究表明睡眠可能影响微小，但不能排除警觉性和练习效应等其他因素。局限性包括样本量小、非随机，且仅记录了一晚的睡眠数据。
Reflection: To improve, I would increase the sample size, measure sleep over multiple nights, and use a more precise reaction time instrument. This investigation reinforced the importance of controlling variables in experimental design.
反思：为了改进，我会增大样本量，测量多晚睡眠数据，并使用更精确的反应时间仪器。本次探究强化了实验设计中控制变量的重要性。
11. Common Mistakes to Avoid | 常见错误避免
Don't confuse correlation with causation. A mathematical paper should recognise that a relationship does not prove one variable causes the other.
不要混淆相关性与因果关系。数学论文应认识到，一种关系并不能证明一个变量导致了另一个变量。
Avoid presenting graphs without titles, labelled axes, or units. Sloppy presentation undermines your analysis.
避免呈现没有标题、坐标轴标签或单位的图表。草率的呈现会削弱你的分析。
Many students forget to explain their mathematical steps. Always show working and justify choices—for example, why you chose median over mean for skewed data.
许多学生忘记解释数学步骤。始终展示过程并说明理由——例如，对于偏态数据，为何选用中位数而非平均数。
12. Final Tips and Submission Checklist | 最终建议与提交清单
Before submission, use this checklist:
提交前，请使用此清单：
- Is the research question clearly stated? | 研究问题是否清晰陈述？
- Are all calculations shown and correct? | 所有计算是否展示且正确？
- Do graphs have titles, labels, and appropriate scales? | 图表是否有标题、标签和合适的刻度？
- Does the conclusion directly answer the question? | 结论是否直接回答了问题？
- Have you reflected on limitations and improvements? | 你是否反思了局限性和改进方向？
- Is the paper formatted consistently, with proper mathematical notation? | 论文格式是否一致，数学符号是否正确？
Follow this framework, and your mathematical writing will become clearer and more persuasive. Practise with different types of investigation to build confidence.
遵循本框架，你的数学写作将变得更清晰、更有说服力。通过不同类型的探究进行练习，以建立自信。
Published by TutorHao | Mathematics 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