📚 Writing Frame for Year 11 SQA Advanced Mathematics Investigation | 11年级SQA进阶数学论文写作框架
In Year 11 SQA Advanced Mathematics, the investigation component challenges you to explore a mathematical idea in depth and present your findings as a formal report. This article provides a clear writing frame and a sample passage to help you structure your paper effectively and meet examiner expectations.
在11年级SQA进阶数学课程中，探究项目要求你深入探索一个数学课题，并以正式报告的形式呈现研究成果。本文提供清晰的写作框架和范文节选，帮助你高效构建论文结构，满足评分标准。
1. Understanding the SQA Mark Scheme | 理解SQA评分方案
The investigation is assessed against criteria such as mathematical content, reasoning, structure, clarity, and reflection. Achieving top marks requires you to demonstrate a logical progression of ideas and use precise mathematical language throughout.
探究报告评分标准涵盖数学内容、推理过程、结构条理、表达清晰度和反思评价。要获得高分，你必须展示出清晰的思路推进，并全程使用准确的数学语言。
Each section must connect seamlessly, showing how your research question leads to analysis, results, and a critical evaluation. Examiners value originality, but a rigorous methodology matters more than a complex topic.
各节之间需无缝衔接，体现研究问题如何导向分析、结果和批判性评价。考官看重原创性，但严谨的方法论比花哨的选题更为重要。
2. Choosing a Suitable Topic | 选择合适的选题
A strong investigation begins with a topic that is neither too broad nor too trivial. Consider areas such as population growth models, projectile motion, statistical hypothesis testing, or the mathematics of music. Ensure the topic allows for genuine mathematical exploration beyond standard textbook exercises.
一个好的选题从不过于宽泛或过于浅显。可考虑人口增长模型、抛体运动、统计假设检验或音乐中的数学等领域。确保该选题能够支持超越常规课本练习的真正数学探索。
The topic must generate data or rely on theoretical models you can test. For instance, modelling the decay of a capacitor’s charge or the Fibonacci sequence in nature provides both practical and theoretical depth.
选题需要能生成数据或依赖可检验的理论模型。例如，模拟电容器电荷衰减或自然界中的斐波那契数列，既有实践价值也有理论深度。
3. Overview of the Writing Frame | 写作框架总览
Your report should follow the conventional structure of a mathematical paper: Title, Abstract, Introduction, Methodology, Analysis/Results, Discussion, Conclusion, and References. Sticking to this layout ensures a professional presentation.
你的报告应遵循数学论文的常规结构：标题、摘要、引言、方法、分析/结果、讨论、结论和参考文献。遵循这一布局能确保专业的呈现方式。
Use headings and subheadings consistently. All figures, tables, and equations must be labelled and referred to in the text. A clear framework allows the examiner to trace your mathematical argument effortlessly.
标题和副标题使用要统一。所有图表和方程必须编号并在正文中引用。清晰的框架能让考官毫不费力地追踪你的数学论证过程。
4. Title and Abstract | 标题与摘要
The title should be concise yet descriptive, such as ‘Modelling the Spread of a Rumour Using the Logistic Equation’. The abstract is a 150–200 word summary covering the aim, method, key findings, and main conclusion.
标题应简洁而具体，例如“利用逻辑斯谛方程建模谣言传播”。摘要是一段150–200词的概述，涵盖目的、方法、主要发现和重要结论。
Avoid vague phrases; use specific terms like ‘exponential growth’ or ‘correlation coefficient r = 0.92’. Write the abstract last, so it accurately reflects the completed investigation.
避免模糊措辞，使用诸如“指数增长”或“相关系数 r = 0.92”等具体术语。摘要在最后撰写，以便准确反映完成后的探究内容。
5. Writing the Introduction | 撰写引言
The introduction sets the context, states the research question, and explains why the topic is worth investigating. It should also include a brief outline of the report’s structure and mention any software or tools used, like Desmos or Excel.
引言部分搭建背景，阐明研究问题，并解释为什么该课题值得探究。还应简要概述报告结构，并提及使用的软件或工具，如 Desmos 或 Excel。
For example: ‘This investigation aims to determine whether the logistic model accurately predicts daily confirmed cases during the early phase of an epidemic, using publicly available data.’
例如：“本研究旨在利用公开数据，判断逻辑斯谛模型是否能准确预测流行病早期的每日确诊病例数。”
6. Methodology and Assumptions | 方法论与假设
Describe precisely how you collected or generated data, defined variables, and built your mathematical model. State all assumptions clearly, such as ‘the population is closed’ or ‘the gravitational acceleration is constant at 9.8 m/s²’.
准确描述如何收集或生成数据、定义变量和建立数学模型。清晰陈述所有假设，例如“人口为封闭系统”或“重力加速度恒为 9.8 m/s²”。
Include the rationale for choosing a particular regression or differential equation. The examiner needs to see that your choices are mathematically justified, not arbitrary.
说明选择特定回归或微分方程的理由。考卷需要看到你的选择有数学依据，而非随意为之。
7. Data Presentation and Analysis | 数据展示与分析
Present raw data in tidy tables with units and source references. Use visualisations like scatter plots with trendlines to reveal patterns. Every graph must have labelled axes, a title, and a legend if necessary.
用整洁的表格展示原始数据，注明单位和来源。使用散点图加趋势线等可视化手段揭示模式。每张图必须标注轴标签、标题，必要时添加图例。
| Day (t) | Infected I(t) |
| 0 | 1 |
| 2 | 2 |
| 4 | 5 |
Analysis should move from descriptive statistics to inferential or analytical methods. Show step-by-step calculations for derived quantities like the growth rate k or the root mean square error.
分析应从描述性统计过渡到推断性或解析方法。逐步展示推导量的计算，如增长率 k 或均方根误差。
8. Mathematical Rigour and Notation | 数学严谨性与符号
Use correct mathematical notation throughout. For example, write the logistic model as:
全程使用正确的数学符号。例如，逻辑斯谛模型写作：
dP/dt = kP(1 – P/N)
Define every symbol when it first appears: P is the population, k is the intrinsic growth rate, and N is the carrying capacity. Avoid shortcuts; justify transformations like natural log linearisations.
每个符号首次出现时需定义：P 为种群数量，k 为内禀增长率，N 为环境容纳量。避免走捷径；论证变换（如自然对数线性化）的理由。
When solving differential equations, outline the method (e.g., separation of variables, integrating factor) and check boundary conditions. Precision communicates a deep understanding.
解微分方程时，概述所用方法（如分离变量法、积分因子法）并检查边界条件。精确性体现深刻理解。
9. Discussion and Interpretation | 讨论与解释
Interpret the results in the context of the original problem. Compare predicted values with observed data, quantify errors, and comment on the goodness of fit. If residuals show a pattern, discuss possible model limitations.
结合原始问题解释结果。将预测值与观测数据对比，量化误差，评价拟合优度。若残差呈现规律，讨论可能的模型局限。
For instance, ‘The logistic model underestimated infections after day 10, suggesting that a time-dependent transmission rate might improve accuracy.’
例如，“逻辑斯谛模型在10天后低估了感染人数，表明时变传播率可能提高精确度。”
10. Conclusion and Evaluation | 结论与评价
The conclusion succinctly answers the research question and summarises the main findings. It should then critically evaluate the investigation, acknowledging limitations and proposing realistic improvements or extensions.
结论应简要回答研究问题并总结主要发现。随后需批判性评价探究过程，承认局限性，并提出可行的改进或拓展方向。
Do not introduce new data or calculations here. A strong evaluation might mention the impact of sample size, measurement error, or the failure of an assumption.
此处不可引入新数据或新计算。强有力的评价会提及样本量影响、测量误差，或某个假设不成立的情况。
11. Referencing and Academic Honesty | 参考文献与学术诚信
Cite all data sources, images, and theoretical frameworks using a consistent style (e.g., APA or Harvard). Include a reference list at the end of the report. Plagiarism, even unintentional, is severely penalised.
使用统一格式（如APA或哈佛格式）引用所有数据来源、图像和理论框架。在报告末尾附上参考文献列表。任何抄袭，即使无心，也会被严厉扣分。
Paraphrase ideas from textbooks or websites and always credit the original author. Using citation tools in Word or Zotero helps maintain accuracy.
改写教科书或网站中的观点，并始终注明原作者。使用 Word 或 Zotero 中的引用工具有助保持准确性。
12. Sample Investigation Excerpt | 范文节选
Below is an excerpt from an investigation titled ‘Modelling the Cooling of a Hot Beverage Using Newton’s Law’. It demonstrates how an introduction and early analysis can be structured according to the writing frame.
以下是一篇题为“用牛顿冷却定律建模热饮降温过程”的探究节选。它展示了如何按写作框架构建引言和初步分析。
Introduction
Newton’s law of cooling states that the rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings. This investigation tests the law by recording the temperature T of a cup of coffee cooling in a room at 21 °C. The aim is to determine the cooling constant k and assess whether an exponential decay model fits the data adequately.
引言
牛顿冷却定律指出，物体热量散失的速率与物体和环境的温差成正比。本研究通过记录一杯咖啡在21 °C房间内的温度 T，验证该定律，旨在确定冷却常数 k 并评估指数衰减模型是否充分拟合数据。
Analysis
The differential equation is dT/dt = –k(T – Ts), where Ts = 21 °C. Separation of variables yields ln(T – 21) = –kt + C. A graph of ln(T – 21) against time t gave an approximately straight line with slope –0.043, hence k ≈ 0.043 min⁻¹. The high linear correlation (r = –0.997) confirms the exponential behaviour. However, the model overestimated temperature after 30 minutes, possibly due to evaporative cooling not accounted for by Newton’s law.
分析
微分方程为 dT/dt = –k(T – Ts)，其中 Ts = 21 °C。使用分离变量法得到 ln(T – 21) = –kt + C。以 ln(T – 21) 对时间 t 作图得近似直线，斜率为 –0.043，因此 k ≈ 0.043 分⁻¹。高度的线性相关 (r = –0.997) 证实了指数变化趋势。然而，该模型在30分钟后高估了温度，这可能是由于蒸发冷却效应超出了牛顿定律的范畴。
This excerpt shows how theory, mathematical manipulation, graphical evidence, and critical reflection are woven together seamlessly.
此节选展示了如何将理论、数学推导、图形证据和批判性反思无缝地编织在一起。
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