📚 Year 12 SQA Mathematics: Project Writing Framework & Exemplar | SQA 12年级数学：论文写作框架与范文
Writing a mathematics investigation for SQA assignments is a skill that blends logical reasoning with clear communication. Whether you are tackling an Advanced Higher project or a Higher Applications of Mathematics task, your report must demonstrate structured thinking, accurate use of mathematical language, and the ability to reflect on your own methods. This guide provides a step‑by‑step framework and a worked exemplar to help you achieve top marks.
为 SQA 课程作业撰写数学探究报告，是一项将逻辑推理与清晰表达相结合的重要能力。无论你面对的是 Advanced Higher 的研究项目，还是 Higher Applications of Mathematics 的写作任务，报告都必须展现出结构化的思维、准确使用数学术语的能力，以及反思自己方法的意识。本指南提供逐步的结构框架和一篇典型范文，帮助你在评估中获得高分。
1. Understanding the Assessment Standards | 理解评分标准
Before you start writing, examine the SQA marking criteria carefully. Most mathematics assignments are graded on four pillars: planning and preparation, mathematical processing, interpretation, and evaluation. Each pillar requires you to demonstrate specific competencies — for example, selecting appropriate software or algebraic techniques counts towards processing, while critically discussing the limitations of your model falls under evaluation.
动笔之前，请仔细研读 SQA 评分标准。大多数数学作业根据四大支柱评分：规划与准备、数学处理、解读和评估。每个支柱都要求你展现具体能力——例如，选择合适的软件或代数方法属于处理范畴，而批判性地讨论模型局限性则属于评估范畴。
- Planning & preparation: Outline aims, variables, and data sources. | 规划与准备：概述目标、变量及数据来源。
- Processing: Carry out calculations, graphs, or statistical tests correctly. | 处理：正确进行计算、作图或统计检验。
- Interpretation: Explain what your results mean in context. | 解读：结合情境解释你的结果意味着什么。
- Evaluation: Reflect on reliability, improvements, and alternative approaches. | 评估：反思可靠性、改进空间和替代方法。
2. Choosing a Suitable Topic | 选择合适的题目
A focused, well‑defined topic is half the success. Choose an area that genuinely interests you — population growth, sports performance, optimisation of a design — and ensure it allows for a range of mathematical techniques. Avoid overly broad themes such as “climate change”; instead narrow it down to “modelling urban temperature rise in Glasgow using linear regression and integrals”.
一个聚焦且界限清晰的题目是成功的一半。选择你真正感兴趣的领域——人口增长、运动表现、设计优化——并确保题目能够容纳多种数学技术。避免“气候变化”这类过于宽泛的主题；不妨收窄为“使用线性回归和积分对格拉斯哥城市升温趋势进行建模”。
3. Structuring the Introduction and Aim | 构建引言与目标
The introduction sets the scene. Begin with a short background — why is this topic worth investigating? Then state your aim precisely, using phrases like “The aim of this project is to model…”, and list two to three objectives such as “to collect 30 data points”, “to compare exponential and polynomial curves”, and “to evaluate the best‑fit model using the coefficient of determination”.
引言部分为全文定下基调。先简要交代背景——为什么这个题目值得探究？然后准确陈述目标，使用“本研究旨在对……进行建模”等句式，并列出两到三个小目标，例如“收集30个数据点”“比较指数曲线与多项式曲线”“用决定系数评估最佳拟合模型”。
4. Methodology: Showing Your Mathematical Toolkit | 方法：展示你的数学工具箱
Describe each mathematical tool you intend to use, not just by name but by justifying why it fits the problem. If you are using differentiation to find maximum profit, explain that the first derivative reveals stationary points and the second confirms a maximum. Keep a clear log of algebraic processes, software packages (Excel, GeoGebra, Python), and any statistical tests such as the Chi‑squared test.
逐一介绍你计划使用的数学工具，不能只提名称，还要解释为什么适合该问题。如果利用微分求最大利润，就要说明一阶导数揭示稳定点、二阶导数确认极大值。清晰记录代数过程、软件包（Excel, GeoGebra, Python）以及卡方检验等统计检验方法。
Turning point test: if f'(a)=0 and f”(a) < 0, then x=a is a local maximum.
驻点判别：若 f'(a)=0 且 f”(a) < 0，则 x=a 为局部极大值点。
5. Data Collection and Presentation | 数据收集与呈现
Detail how you gathered your data — from a published dataset, a survey, or a simulation — and discuss its reliability. Include a sample of raw data in a table, making sure every column is labelled with units. Visual clarity matters: use scatter plots to show trends before modelling, and always number your figures and tables for easy reference.
详细说明数据来源——来自公开数据集、问卷调查还是模拟——并讨论其可靠性。用表格呈现原始数据样本，确保每列都标明单位。视觉清晰度很重要：建模前用散点图展示趋势，并始终为图表编号以便引用。
| Time (months) | 时间（月） | Population (thousands) | 人口（千） |
|---|---|
| 0 | 12.4 |
| 6 | 18.1 |
| 12 | 26.3 |
6. Carrying Out the Mathematical Processing | 执行数学处理
This is the core of your project. Show every step — rearranging equations, performing matrix multiplications, or coding a loop — with clear annotations. If you use technology, paste screenshots of command lines and the output, then interpret the results. For a regression model, state the equation, the correlation coefficient r, and the meaning of each parameter in context.
这是项目的核心部分。展示每一个步骤——移项、矩阵乘法或编写循环代码——并附上清晰的注解。如果借助技术，请粘贴命令行截图及输出结果，然后解读。对于回归模型，要给出方程、相关系数 r 以及每个参数的实际含义。
Least squares line: y = 2.34x + 11.2, r = 0.987
最小二乘直线：y = 2.34x + 11.2，r = 0.987
7. Interpreting Findings in Real‑World Terms | 用现实语言解读发现
An answer of ‘x = 3.7’ is meaningless without context. Translate your mathematics into everyday language: ‘The optimal selling price is £3.70, generating a maximum weekly profit of £1820.’ Connect your findings back to the original aim and gently highlight any patterns or anomalies you see.
一个“x = 3.7”的答案若没有情境便毫无意义。将数学语言转换成日常表达：“最优售价为3.70英镑，可实现最高周利润1820英镑。”将发现与最初目标联系起来，并自然地指出你观察到的任何规律或异常。
8. Critical Evaluation and Limitations | 批判性评估与局限性
High marks are awarded for honest reflection. Discuss what went well, what surprised you, and where your model breaks down. Mention sample size, measurement errors, assumptions (e.g. ‘growth is unbounded’), and external factors. Suggest at least two concrete improvements, such as collecting seasonal data or testing a logistic model instead.
诚实的反思能赢得高分。讨论进展顺利之处、令你惊讶之处以及模型在何种情况失效。提及样本大小、测量误差、假设条件（如“增长不受限制”）和外部因素。提出至少两条切实的改进建议，例如收集季节性数据或改用逻辑斯蒂模型。
9. Conclusion: Tying Everything Together | 结论：融会贯通
A short, punchy conclusion revisits the aim, summarises key results, and states your final judgement. Avoid introducing new material. End with a forward‑looking statement: ‘Further work could extend this model by incorporating temperature gradients.’
简短有力的结语将重温研究目标、总结关键结果并阐述最终判断。不要引入新内容。以展望式语句收尾：“进一步的工作可将温度梯度纳入模型，以扩展其适用范围。”
10. Referencing and Academic Integrity | 参考文献与学术诚信
Use a consistent referencing style (Harvard is common) for all sources, including datasets, textbooks, and websites. Cite the SQA data booklet if you used formulae from it, and list any AI tools only if permitted by your centre. Academic integrity is non‑negotiable.
所有来源——包括数据集、教材和网页——应使用统一的引用格式（哈佛格式较为通用）。若采用了 SQA 资料手册中的公式，也需注明。只有在中心允许的情况下才列出 AI 工具。学术诚信不容妥协。
11. Exemplar Extract with Commentary | 范文摘录与评注
Below is a brief extract from a model report on population dynamics. Notice how the writing is precise, every symbol is defined, and the interpretation immediately follows the mathematics.
以下是一篇人口动力学范例报告的节选。留意其行文如何做到精确、每个符号均加以定义，并且解读紧跟在数学处理之后。
Exemplar: “The exponential model P(t) = 12.4e^(0.23t) was fitted using the least squares method after log‑transforming the data. The average percentage error remained below 4% for the first 18 months, indicating a strong fit. However, extrapolation beyond 24 months gave unrealistic predictions, because the model assumes unlimited resources. This limitation led us to explore a logistic function.”
范文：“将数据取对数后，用最小二乘法拟合得到指数模型 P(t) = 12.4e^(0.23t)。前18个月的平均百分比误差低于4%，表明拟合良好。但在24个月之后进行外推却给出不切实际的预测值，原因是该模型假设资源无限。这一局限性促使我们进而探讨逻辑斯蒂函数。”
12. Final Presentation and Submission Checklist | 最终展示与提交核对清单
Before uploading, check formatting: consistent font and margins, labelled axes on graphs, numbered pages, and an embedded contents table. Reread your report aloud to catch awkward phrasing. Confirm that your file follows SQA naming conventions and that all embedded images are clear. A polished presentation reflects respect for your own work.
上传前检查格式：统一的字体和页边距、图表坐标轴标注、页码以及嵌入式目录。大声重读报告，找出别扭的措辞。确认文件名符合 SQA 命名规范，所有嵌入图像都清晰可辨。精心的排版体现了对自己作品的尊重。
- ✔ Graphs include title, axis labels with units. | 图表包含标题、带单位的坐标轴标签。
- ✔ All variables defined on first use. | 所有变量首次使用即予定义。
- ✔ Technical vocabulary (gradient, asymptote) used accurately. | 专业术语（梯度、渐近线）使用准确。
- ✔ File exported as PDF unless specified otherwise. | 除非另有规定，文件导出为 PDF。
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