📚 Year 10 SQA Further Mathematics: Essay Writing Framework and Exemplar | SQA 进阶数学论文写作框架与范文
In the SQA Further Mathematics curriculum, students are often required to produce extended written pieces or investigative reports that demonstrate deep understanding of mathematical concepts and their applications. This guide provides a structured approach to writing such essays, complete with a fully worked exemplar on modelling a basketball shot.
在 SQA 进阶数学课程中，学生常需撰写长篇论文或探究报告，以展现对数学概念及其应用的深刻理解。本指南提供结构化写作方法，并附上一篇篮球投篮建模的完整范文。
1. Why Writing Matters in Further Mathematics | 进阶数学中写作的重要性
Extended writing in mathematics goes beyond routine problem-solving. It allows you to explore real-world scenarios, formulate hypotheses, test models, and communicate findings clearly. This skill is crucial for Advanced Higher projects and university-level study.
数学中的拓展写作超越了常规解题。它能让你探索现实场景、提出假设、检验模型并清晰地传达发现。这一技能对 Advanced Higher 项目及大学阶段学习至关重要。
Unlike short-answer questions, a mathematics essay demands a logical narrative: you must explain why a method was chosen, how calculations were performed, and what the implications of the results are. This fosters critical thinking and academic rigour.
与简答题不同，数学论文要求逻辑叙述：你必须解释为何选择某种方法、如何进行演算，以及结果意味着什么。这能培养批判性思维和学术严谨性。
2. Understanding Assessment Objectives | 理解评估目标
SQA assessment criteria typically emphasise three areas: knowledge of mathematical techniques, application and reasoning, and communication. Your essay should demonstrate all three by presenting accurate mathematics, justifying decisions, and delivering a well-structured argument.
SQA 的评分标准通常强调三个方面：数学技巧的掌握、应用与推理，以及沟通表达。你的论文应通过呈现准确的数学、论证决策并给出结构良好的论述来展示这三方面。
| Objective | 目标 | What it means | 含义 |
|---|---|---|---|
| Knowledge | 知识 | Correct use of formulas, theorems, and notation | 正确使用公式、定理和符号 |
| Application | 应用 | Linking mathematics to a real context, testing models | 将数学与现实情境联系，检验模型 |
| Communication | 表达 | Clear structure, appropriate language, and proper referencing | 结构清晰、语言得体、参考文献规范 |
3. Selecting an Engaging Topic | 选择有趣的主题
Choose a topic that genuinely interests you and aligns with the Further Mathematics syllabus, such as calculus, vectors, matrices, or statistics. A good topic has a clear research question, measurable variables, and scope for mathematical modelling.
选择你真正感兴趣且符合进阶数学大纲的主题，例如微积分、向量、矩阵或统计。一个好主题应有清晰的研究问题、可衡量的变量以及数学建模的空间。
Examples of suitable topics include: modelling the spread of a rumour using differential equations, analysing the optimal angle for a projectile, or applying Markov chains to predict weather patterns. Avoid topics that are too broad or rely entirely on descriptive statistics.
合适的主题示例包括：用微分方程模拟谣言传播、分析抛射物的最优角度，或应用马尔可夫链预测天气。避免过于宽泛或完全依赖描述性统计的主题。
4. The IMRaD Structure | IMRaD 结构
Most mathematics investigations follow the IMRaD structure: Introduction, Methods, Results, and Discussion. This framework ensures a logical flow and helps examiners follow your reasoning. Occasionally, a separate Conclusion section is added.
大多数数学探究遵循 IMRaD 结构：引言（Introduction）、方法（Methods）、结果（Results）和讨论（Discussion）。该框架确保逻辑流畅，方便考官理解你的推理。有时会额外添加独立的结论部分。
- Introduction – presents the research question, background, and aims. / 引言 – 提出研究问题、背景和目标。
- Methods – explains the mathematical techniques, data collection, and model assumptions. / 方法 – 解释数学技巧、数据收集和模型假设。
- Results – displays calculations, graphs, and key findings. / 结果 – 展示演算、图表和关键发现。
- Discussion – interprets results, evaluates the model, and suggests improvements. / 讨论 – 解读结果、评价模型并提出改进建议。
5. Crafting the Introduction | 撰写引言
The introduction should hook the reader and set the scene. Begin with a real-world context or a historical note, then state your research question clearly. Outline the scope of your investigation and briefly mention the mathematical tools you intend to use.
引言应吸引读者并铺垫背景。从现实情境或历史注记入手，然后清晰陈述研究问题。概述探究范围，并简要提及你将使用的数学工具。
For instance, in a modelling essay: ‘This investigation aims to determine the optimal release angle for a basketball free throw by constructing a quadratic model from video data and comparing it with projectile motion equations derived from Newtonian mechanics.’
例如，在建模论文中：“本研究旨在通过从视频数据构建二次模型并与其基于牛顿力学推导的抛射体运动方程比较，确定篮球罚球的最佳出手角度。”
6. Explaining Methodology | 解释方法
This section is the core of your essay. Document every step: how you collected data, which assumptions were made (e.g. air resistance neglected, constant gravity), and what mathematical procedures were applied. Include derivations of key formulas.
这部分是论文的核心。记录每一步：如何收集数据、作出哪些假设（如忽略空气阻力、重力恒定）、应用了何种数学步骤。包含关键公式的推导。
Use proper mathematical notation: for a quadratic model, you might write the general form as y = ax² + bx + c and then show how to find coefficients using three data points. If calculus is involved, demonstrate differentiation to find the vertex.
使用正确的数学符号：对于二次模型，你可以写出一般式 y = ax² + bx + c，然后演示如何利用三个数据点求系数。若涉及微积分，可展示求导找顶点的方法。
7. Presenting Data and Analysis | 展示数据与分析
Results should be presented clearly with tables, graphs, and annotated calculations. Label axes, provide units, and reference any software used (e.g. Excel, GeoGebra). Avoid repeating raw data; summarise with descriptive statistics where appropriate.
结果应通过表格、图表及带注释的计算清晰呈现。标注坐标轴，提供单位，并注明所用软件（如 Excel、GeoGebra）。避免重复原始数据，适当用描述性统计总结。
When displaying a trajectory graph, you might include both the modelled curve and the experimental data points. A table comparing calculated maximum height and range with actual measurements strengthens the analysis.
展示轨迹图时，可同时包含模型曲线和实验数据点。比较计算的最大高度和射程与实际测量值的表格能强化分析。
8. Discussion and Conclusion | 讨论与结论
Discuss the meaning of your results. Did the model fit the data well? What were the sources of error? Compare your findings with theoretical predictions. Acknowledge limitations and suggest how the model could be refined (e.g. adding air resistance or spin).
讨论你结果的意义。模型与数据拟合得好吗？误差来源有哪些？将你的发现与理论预测进行比较。承认局限性，并提出如何改进模型（例如加入空气阻力或旋转）。
Conclude by answering the original research question concisely. For a basketball trajectory essay, you might state: ‘The optimal release angle was found to be approximately 52°, which is higher than the theoretical 45° due to the release point being above the ground.’
最后简明扼要地回答最初的研究问题。对于篮球轨迹论文，你可以陈述：“研究发现最优出手角约为 52°，这高于理论上的 45°，原因是出手点高于地面。”
9. Referencing and Formatting | 参考文献与格式
Every source must be cited using a consistent style (e.g. APA or Harvard). Include both in-text citations and a reference list at the end. Proper referencing demonstrates academic integrity and allows readers to verify your claims.
每个来源须用统一格式引用（如 APA 或哈佛格式）。包含文内引用和文末参考文献列表。规范参考文献显示学术诚信，也让读者能验证你的论述。
Format your paper with clear section headings, page numbers, and captions for all figures and tables. A well-presented essay creates a professional impression and may earn higher marks for communication.
使用清晰的章节标题、页码以及所有图表的标题来排版论文。一份美观的论文给人专业印象，并可能在表达方面获得更高分数。
10. Exemplar: Quadratic Modelling of a Basketball Free Throw | 范文：篮球罚球的二次函数建模
Title: Modelling the Trajectory of a Basketball Free Throw Using Quadratic Functions
题目：利用二次函数建模篮球罚球轨迹
Abstract
This investigation uses quadratic regression and projectile motion equations to model the trajectory of a basketball free throw. Video analysis provided coordinate data, from which a parabolic model y = -0.23x² + 1.15x + 2.10 was derived. The model predicted a maximum height of 3.54 m at a horizontal distance of 2.50 m and a launch angle of 51°. The results closely matched theoretical predictions and highlighted the importance of release point height in sports mechanics.
摘要
本探究使用二次回归和抛射体运动方程对篮球罚球轨迹进行建模。通过视频分析获得坐标数据，推导出抛物线模型 y = -0.23x² + 1.15x + 2.10。模型预测在水平距离 2.50 m 处达到最大高度 3.54 m，出手角为 51°。结果与理论预测高度吻合，凸显了出手点高度在运动力学中的重要性。
1. Introduction
The free throw is a critical scoring opportunity in basketball, yet its success depends heavily on release angle, speed, and height. The aim is to construct a quadratic model from recorded throws and determine the optimal launch angle for a player of height 1.85 m releasing the ball at 2.10 m above the floor.
1. 引言
罚球是篮球中至关重要的得分机会，但其成功与否很大程度上取决于出手角度、速度和高度。本研究的目的是通过录制的投篮构建二次模型，并为身高 1.85 m、出手高度 2.10 m 的球员确定最佳出手角。
2. Methodology
A basketball free throw was filmed at 60 fps. Three key points were extracted: release (0, 2.10), the highest point (2.50, 3.54), and entry into the hoop (4.60, 3.05). A quadratic model y = ax² + bx + c was assumed. Using the three points, a system of equations was solved to find a = -0.23, b = 1.15, c = 2.10. To verify, the projectile motion range formula was applied: R = (v₀² sin 2θ)/g, with g = 9.81 m s⁻². The launch speed v₀ was estimated as 6.8 m s⁻¹ from frame analysis.
2. 方法
以 60 fps 拍摄一次罚球。提取三个关键点：出手 (0, 2.10)、最高点 (2.50, 3.54) 和入筐 (4.60, 3.05)。假设二次模型 y = ax² + bx + c。利用三点解方程组得 a = -0.23, b = 1.15, c = 2.10。为验证，应用抛射体水平射程公式：R = (v₀² sin 2θ)/g，其中 g = 9.81 m s⁻²。通过逐帧分析估算出手速度 v₀ 为 6.8 m s⁻¹。
Vertex x-coordinate = -b/(2a) = -1.15/(2 × -0.23) = 2.50 m
顶点 x 坐标 = -b/(2a) = -1.15/(2 × -0.23) = 2.50 m
Maximum height y = a(2.50)² + b(2.50) + c = 3.54 m
最大高度 y = a(2.50)² + b(2.50) + c = 3.54 m
The launch angle θ was found from the derivative at x=0: dy/dx = 2ax + b, so at x=0, slope = b = 1.15. Thus tan θ = 1.15, giving θ = tan⁻¹(1.15) ≈ 49°. However, using the complete projectile equation accounting for release height, the optimal angle was recalculated as 51°, consistent with literature.
出手角 θ 由 x=0 处的导数求得：dy/dx = 2ax + b，x=0 时斜率为 b = 1.15。因此 tan θ = 1.15，θ = tan⁻¹(1.15) ≈ 49°。但使用考虑出手高度的完整抛射体方程重新计算后，最优角为 51°，与文献一致。
3. Results
The quadratic model produced an excellent fit (R² = 0.998). The trajectory peaked at 3.54 m, well above the 3.05 m rim height. The calculated launch angle of 51° matched the player’s observed technique. The model also predicted that for a release height of 2.10 m, the minimum required speed to reach the hoop was 5.9 m s⁻¹.
3. 结果
二次模型拟合极佳（R² = 0.998）。轨迹最高点为 3.54 m，远高于 3.05 m 的篮筐高度。计算出的 51° 出手角与球员实际动作相符。模型还预测，在 2.10 m 出手高度下，球到达篮筐所需的最低速度为 5.9 m s⁻¹。
4. Discussion
The slight discrepancy between the derivative-derived angle (49°) and the corrected optimum (51°) arises because the quadratic model ignores the initial velocity vector decomposition. Air resistance and ball spin were neglected, which could explain minor deviations. Nevertheless, the model provides a valid first approximation and could be extended with a parametric approach using trig functions.
4. 讨论
由导数得出的角度（49°）与校正后最优角（51°）之间的微小差异，源于二次模型忽略了初速度矢量分解。忽略空气阻力和球自转可能导致轻微偏差。不过，该模型提供了一个有效的初级近似，可通过三角函数的参数化方法进行拓展。
5. Conclusion
The optimal release angle for a basketball free throw with release height 2.10 m is approximately 51°, not the textbook 45°. This finding emphasises the role of launch height in projectile sports and demonstrates how quadratic modelling can yield practical insights.
5. 结论
出手高度为 2.10 m 时，篮球罚球的最佳出手角约为 51°，而非教科书上的 45°。这一发现凸显了出手高度在抛射运动中的作用，并展示二次建模如何提供实用洞见。
References / 参考文献
Brancazio, P. J. (1981). Physics of basketball. American Journal of Physics, 49(4), 356-365.
Hamilton, G. R. & Reinschmidt, C. (1997). Optimal trajectory for the basketball free throw. Journal of Sports Sciences, 15(5), 491-504.
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