📚 SQA Mathematics: Investigation Writing Framework & Exemplar | SQA 数学：探究报告写作框架与范文
Writing a successful mathematical investigation report for SQA qualifications—whether for Higher or Advanced Higher—demands a clear, logical structure and rigorous mathematical communication. This guide provides a comprehensive framework for planning and presenting your work, along with a fully worked exemplar investigation on optimising the volume of a cone. You will learn how to structure each section, present data effectively, justify your reasoning, and reflect critically on your findings. Mastering this format will not only help you meet assessment criteria but also build essential skills for university-level mathematics and beyond.
为 SQA 资格（无论是 Higher 还是 Advanced Higher）撰写一份成功的数学探究报告，需要清晰、逻辑严密的结构和严谨的数学表达。本指南提供了一个全面的框架，用于规划和展示你的研究，并附上了一份关于优化圆锥体积的完整范文。你将学习如何组织每个部分、有效地呈现数据、论证你的推理过程，并批判性地反思你的发现。掌握这一格式不仅有助于满足评分标准，还能为你培养大学数学及更高阶段所需的基本技能。
1. Understanding SQA Investigation Requirements | 理解 SQA 探究要求
In SQA Mathematics, an investigation is a piece of extended work where you explore a mathematical problem, formulate a strategy, collect or generate data, apply techniques, and interpret results. The aim is to demonstrate your ability to think independently, use appropriate notation, and communicate mathematical ideas clearly. At Advanced Higher level, the investigation often carries significant weighting and requires evidence of advanced algebraic manipulation, calculus, or statistical testing. Your report must show not only the final answer but the entire journey—including false starts, refinements, and justifications.
在 SQA 数学课程中，探究是一份拓展性作业，你需要探索一个数学问题，制定策略，收集或生成数据，应用技巧并解释结果。目的是展示你独立思考、使用恰当符号以及清晰表达数学思想的能力。在 Advanced Higher 级别，探究通常占很大比重，并要求体现高级代数运算、微积分或统计检验的证据。你的报告不仅要呈现最终答案，还要展示完整的过程——包括错误尝试、修正和论证。
Examiners look for a well-defined aim, a structured plan, correct use of mathematical language, and critical evaluation. The investigation is not a simple exercise; it should contain an element of discovery. You might explore a relationship that is not immediately obvious, test a conjecture, or generalise from specific cases. Keeping a log of your working as you go will help you write the final report efficiently.
考官看重明确的目标、结构化的计划、数学语言的正确使用以及批判性评价。探究不是简单的练习，它应当包含发现的元素。你可以探索一个并非一目了然的关系，检验一个猜想，或者从具体例子中归纳推广。在探究过程中随时记录你的演算，将有助于你高效地完成最终报告。
2. Overall Report Structure | 报告整体结构
A well-organised report follows a standard sequence, much like a scientific paper: Title, Abstract, Introduction, Methodology, Analysis/Results, Discussion, Conclusion, References, and Appendices. This structure helps the reader follow your logic without confusion. While the exact headings can be adapted to your investigation, the flow should be linear—each section building on the previous one. Avoid placing large blocks of raw data in the main body; instead, summarise key findings and refer to appendices for full datasets.
一份组织良好的报告遵循类似于科学论文的标准顺序：标题、摘要、引言、方法、分析/结果、讨论、结论、参考文献和附录。这种结构有助于读者毫无障碍地理解你的逻辑。虽然具体标题可根据探究内容进行调整，但整体流程应是线性的——每一节都建立在上一节的基础上。避免在正文中放置大段原始数据，而应总结关键发现，将完整数据集放在附录中。
Aim for a length that is proportionate to the complexity of your investigation; typically, an Advanced Higher report might be between 1500 and 3000 words, excluding appendices. Each section should be clearly labelled, and you must use consistent notation throughout. Remember that your report is a formal piece of mathematical writing—so avoid informal language, and always define variables when they first appear.
报告的篇幅应与探究的复杂程度相称；通常，一份 Advanced Higher 报告的词数在 1500 到 3000 之间（不含附录）。每个部分都应有清晰的标题，并且全文必须使用一致的符号。请记住，你的报告是一份正式的数学写作——因此要避免非正式语言，并在变量首次出现时对其进行定义。
3. Title and Abstract | 标题与摘要
The title should be concise yet descriptive, capturing the essence of your investigation. For example, “Optimising the Volume of a Cone under Fixed Lateral Surface Area” tells the reader exactly what to expect. Avoid vague titles like “Maths Investigation”. The abstract is a short paragraph (around 100–150 words) that summarises the entire investigation: the problem, method, key findings, and main conclusion. Write it last, even though it appears first, so that it accurately reflects your completed work.
标题应简洁而又具描述性，能抓住探究的核心。例如，“固定侧面积下圆锥体积的优化”就能让读者确切了解报告的内容。避免使用“数学探究”之类的模糊标题。摘要是一小段（约 100 到 150 词）概括整个探究的文字：问题、方法、关键发现和主要结论。尽管摘要位于报告最前面，但应在最后撰写，这样才能准确反映你已完成的工作。
A strong abstract does not contain references, figures, or unexplained jargon. It should be self-contained and understandable without reading the full report. Use it to highlight the mathematical significance of your work—for instance, whether you derived a general formula or discovered an interesting ratio. This paragraph sets the tone for the rest of the report.
一份优秀的摘要不应包含参考文献、图表或未解释的术语。它应当独立且无需阅读全文即可理解。用它来突出你研究的数学意义——例如，你是否推导出一个通用公式，或发现了一个有趣的比值。这一段将为报告的其余部分定下基调。
4. Introduction: Setting the Scene | 引言：设定背景
The introduction provides the context and motivation for your investigation. Begin by explaining the broader mathematical area—perhaps optimisation, sequences, or statistical modelling. Then narrow down to your specific research question. State your aim clearly: “This investigation aims to determine the dimensions that maximise the volume of a right circular cone given a fixed lateral surface area.” You should also outline the structure of the report, so the reader knows the path you will take.
引言为你的探究提供背景和动机。首先解释更广泛的数学领域——可能是优化、数列或统计建模。然后收窄到你的具体研究问题。清晰地陈述你的目标：“本探究旨在确定在固定侧面积下，使正圆锥体积最大化的尺寸。”你还应概述报告的结构，让读者了解你将展开的路径。
It is often helpful to include a brief review of relevant known results or formulas—for example, the formulas for the volume and surface area of a cone. However, do not turn the introduction into a lengthy textbook section. Keep it focused and end with a clear hypothesis or conjecture if appropriate. This shows you have thought critically from the start.
简要回顾相关的已知结果或公式通常会很有帮助——例如，圆锥体积和侧面积的公式。但是，不要把引言变成冗长的教科书章节。保持重点突出，并在适当时以一个清晰的假设或猜想结束。这表明你从一开始就进行了批判性思考。
5. Methodology: Describing Your Approach | 方法：描述你的研究途径
In this section, you explain how you carried out the investigation. For a purely theoretical investigation, this might involve the choice of variables, the algebraic approach, and the calculus techniques you planned to use. If the investigation includes a practical or numerical element, describe how you generated data, what software (if any) you used, and how you ensured accuracy. Be specific: rather than saying “I used differentiation,” state the function you differentiated and the condition for optimisation.
在这一节中，你需说明自己是如何开展探究的。对于纯理论性探究，这可能涉及变量的选择、代数方法以及你计划使用的微积分技巧。如果探究包含实践或数值元素，则描述你如何生成数据、使用了什么软件（如有），以及你如何确保准确性。要具体：不要说“我使用了微分”，而是说明你所微分的函数以及最优化的条件。
Ethical considerations are rarely needed in pure mathematics, but if you collected data from surveys or experiments, mention how you maintained integrity. Also discuss any assumptions you made—for instance, that the cone is a perfect geometric shape, or that the material has negligible thickness. Stating assumptions early prevents confusion later when results are interpreted.
纯数学探究很少需要考虑伦理问题，但如果你通过调查或实验收集数据，应提一下你是如何保持诚信的。此外，讨论你做出的所有假设——例如，假设圆锥是完美的几何形状，或者材料的厚度可以忽略不计。尽早说明假设可以避免在后期解读结果时出现混淆。
6. Data Presentation and Analysis | 数据展示与分析
Even in a theoretical investigation, you will generate results that need to be presented clearly. Use tables to summarise numerical checks, and graphs to visualise relationships. For an optimisation problem, a table might show volume values for various radii, while a graph of V against r can illustrate the maximum point before calculus is applied. Always label axes, include units where applicable, and provide a caption beneath each figure or table.
即使在理论性探究中，你也会产生需要清晰呈现的结果。使用表格总结数值检验，使用图表可视化关系。对于一个优化问题，表格可以展示不同半径下的体积值，而 V 关于 r 的图表则可以在应用微积分之前直观地显示最大值点。务必标注坐标轴，在适用时包含单位，并在每个图或表下方提供标题。
When presenting algebraic work, do not simply paste a jumble of symbols. Break the derivation into logical steps, and align equations neatly. Use centred displays for key formulas. For instance, to express the constraint, you could write:
S₀ = π r √(r² + h²)
当展示代数过程时，不要简单地把一堆符号粘贴上去。将推导过程分解为逻辑步骤，并整齐地对齐方程。对关键公式使用居中显示。例如，表达约束条件时可以写：
S₀ = π r √(r² + h²)
After each manipulation, comment on why you took that step. This demonstrates understanding and helps the examiner follow your reasoning. Numerical evidence can also be used to verify your analytical solution, strengthening the report’s credibility.
每一步运算之后，解释你为何采取这一步骤。这展示了你的理解，并帮助考官跟上你的推理过程。数值证据也可用于验证你的解析解，从而增强报告的可信度。
7. Mathematical Reasoning and Proof | 数学推理与证明
This is the core of your investigation, where you present the rigorous mathematical working. Start from the constraint equation, eliminate one variable, and express the quantity to be optimised as a function of a single variable. Then differentiate, find critical points, and verify the nature of the extremum using the first or second derivative test. All symbols must be defined, and each line should follow logically from the previous one.
这是你探究的核心部分，你需要在此展示严谨的数学演算。从约束方程开始，消去一个变量，将要优化的量表示成单一变量的函数。然后进行微分，找出临界点，并使用一阶或二阶导数检验验证极值的性质。所有符号都必须定义，每一行推导都应与前一行逻辑紧密衔接。
For the cone problem, substituting the constraint into the volume formula gives:
V(r) = (1/3)π r² √(S₀²/(π² r²) – r²)
在此圆锥问题中，将约束代入体积公式得到：
V(r) = (1/3)π r² √(S₀²/(π² r²) – r²)
Simplify the expression by squaring V (which preserves the location of the maximum) and differentiating. Show the full derivative, set it equal to zero, and solve for r. Finally, interpret the solution in terms of the original dimensions. By obtaining r:l = 1:√3 or an equivalent ratio, you demonstrate a concise, elegant result.
通过对 V 进行平方（这可以保持最大值点的位置不变）并微分来简化表达式。展示完整的导数，令其等于零，解出 r。最后，根据原始维度解释所得的解。通过得出 r:l = 1:√3 或与之等价的比值，你可以展示一个简洁而优美的结果。
8. Discussion and Interpretation | 讨论与解读
Once the mathematical result is obtained, step back and discuss what it means in the context of the original problem. Why does this ratio maximise the volume? Is the result intuitive or surprising? Connect the analytical outcome back to the numerical data you presented earlier—do the table and graph confirm the optimal dimensions? If any unexpected behaviour occurred, such as a singularity near r = 0, explain it mathematically.
一旦得出数学结果，退一步讨论它在原始问题背景下的含义。为什么这个比值能使体积最大化？结果是直观的还是令人惊讶的？将解析结果与你之前展示的数值数据联系起——表格和图表是否印证了最佳尺寸？如果出现了任何意外行为，例如在 r = 0 附近存在奇点，请从数学上加以解释。
Discuss any limitations of your model. For example, the assumption of a perfect right circular cone without thickness may not hold in real-world container design. Also, acknowledge if a different choice of constraint (e.g. total surface area instead of lateral area) would change the outcome. Reflective comments like these show higher-order thinking and can earn additional marks for evaluation.
讨论模型的各项局限性。例如，假设圆锥是完美的正圆锥且没有厚度，这在实际容器设计中可能不成立。同时，也要承认如果选择不同的约束条件（例如总表面积而非侧面积），结果将会改变。诸如此类的反思性评论展现了高阶思维能力，并能为评估赢得额外分数。
9. Conclusion and Evaluation | 结论与评估
The conclusion should succinctly answer the initial research question and state whether the aim was achieved. Summarise the key mathematical finding—for instance, that the volume of a cone with fixed lateral surface area is maximised when the slant height is √3 times the radius. Do not introduce new information here; it is a summary of what has already been established.
结论应简明扼要地回答最初的研究问题，并陈述目标是否达成。总结关键的数学发现——例如，在固定侧面积的条件下，当斜高为半径的 √3 倍时，圆锥的体积最大。不要在此处引入新信息；这是对已有内容的总结。
Evaluate the effectiveness of your methodology. Did the algebraic approach work smoothly? Would a numerical or graphical method have been equally valid? Reflect on what you would do differently if you repeated the investigation, and suggest possible extensions—such as investigating other solids or considering different optimisation criteria. This demonstrates a mature mathematical perspective.
评估你的方法论的有效性。代数方法进展顺利吗？数值或图形方法是否同样有效？反思如果重复此探究你会做哪些不同的处理，并提出可能的拓展方向——例如研究其他立体图形或考虑不同的优化标准。这展现了一种成熟的数学视野。
10. References and Appendices | 参考文献与附录
Any sources you used, such as textbooks, websites, or software, must be acknowledged in a references section. Use a consistent citation style (e.g., Harvard or APA). In SQA investigations, this is usually a short list, but it is essential for academic integrity. At a minimum, cite the formula sheet or textbook that provided the standard volume and surface area formulas.
你使用的任何来源，如教科书、网站或软件，都必须在参考文献部分加以标明。使用一致的引用风格（例如 Harvard 或 APA）。在 SQA 探究中，这通常是一个简短的列表，但对于学术诚信至关重要。至少，要引用提供标准体积和面积公式的公式表或教科书。
Appendices contain material that supports the main text but would interrupt the flow if included—detailed algebraic expansions, large tables of values, or screenshots of software output. Label each appendix clearly (Appendix A, Appendix B) and refer to them in the body of the report. Ensure that all appended material is relevant and neatly presented; untidy scribbles are not acceptable.
附录包含支撑正文但若置于正文中会打断行文的材料——详细的代数展开、大型数值表格或软件输出的截图。清晰标记每个附录（附录 A、附录 B），并在报告正文中提及它们。确保所有附录材料都是相关的，并且整齐呈现；潦草的涂鸦是不被接受的。
11. Exemplar Investigation: Optimising a Cone’s Volume | 范文探究：优化圆锥体积
Title: Optimising the Volume of a Right Circular Cone with Fixed Lateral Surface Area
标题：固定侧面积下正圆锥体积的优化
Abstract: This investigation explores the geometric problem of maximising the volume V of a right circular cone when the lateral surface area S₀ is held constant. By expressing V solely in terms of the base radius r, using calculus to find the stationary point, and verifying the maximum, the optimal ratio of slant height l to radius r is derived. The result l = √3 r yields a maximum volume, confirmed by both algebraic verification and numerical analysis.
摘要：本探究探讨在侧面积 S₀ 保持恒定的条件下，使正圆锥体积 V 最大化的几何问题。通过将 V 仅用底半径 r 表示，利用微积分求驻点并验证最大值，推导出斜高 l 与半径 r 的最佳比值。结果 l = √3 r 产生最大体积，该结论通过代数验证和数值分析得到确认。
Introduction: Cones appear frequently in engineering and packaging, where material efficiency is crucial. The lateral surface area of a cone is given by S = π r l, with l = √(r² + h²). The volume is V = ⅓ π r² h. The problem is to maximise V subject to constant S₀. Intuition suggests a very tall, thin cone might maximise volume, but preliminary numerical trials indicated an optimum at a moderate aspect ratio. This report systematically investigates that optimum.
引言：圆锥在工程和包装中频繁出现，这些领域对材料效率要求苛刻。圆锥的侧面积公式为 S = π r l，其中 l = √(r² + h²)。体积公式为 V = ⅓ π r² h。问题是在 S₀ 恒定的条件下最大化 V。直觉上，一个非常高、非常细的圆锥可能会使体积最大，但初步数值试验表明，最优值出现在一个中等的高宽比处。本报告系统地研究了这个最优值。
Methodology: An algebraic approach was chosen. The constraint S₀ = π r √(r² + h²) was rearranged to express h² in terms of r, and substituted into V. To simplify differentiation, V² was used because squaring preserves the location of positive maxima. The derivative d(V²)/dr was set to zero, and the critical r was found. The second derivative test then confirmed a maximum. All work was checked using a spreadsheet to calculate V for a range of r values.
方法：选择了代数方法。将约束条件 S₀ = π r √(r² + h²) 重新整理，用 r 表示 h²，并代入 V。为简化微分，使用了 V²，因为平方运算不会改变正值极大值点的位置。令导数 d(V²)/dr = 0，求出临界半径 r。然后使用二阶导数检验确认是极大值。所有计算均利用电子表格，针对一系列 r 值计算 V 进行了核对。
Analysis and Results: From the constraint, h² = S₀²/(π² r²) – r². Substituting into V = (1/3)π r² h gave V = (1/3)π r² √(S₀²/(π² r²) – r²). Squaring: V² = (1/9)π² r⁴ (S₀²/(π² r²) – r²) = (1/9)(S₀² r² – π² r⁶). Differentiating with respect to r: d(V²)/dr = (1/9)(2 S₀² r – 6 π² r⁵). Setting this to zero yields 2 S₀² r = 6 π² r⁵, hence r⁴ = S₀²/(3 π²), so r = (S₀/(π √3))^{½}. At this radius, l = S₀/(π r) = S₀/(π (S₀/(π √3))^{½}) = √(S₀√3/π). Simplifying the ratio l/r eventually gives l = √3 r. The maximum volume is V_max = (S₀/ (3√3π))^{½} S₀/(3√3)? (a simplified form). Numerical example: for S₀ = 100 cm², r ≈ 4.29 cm, h ≈ 6.62 cm, and V ≈ 127.3 cm³, which is larger than neighbouring values.
分析与结果：由约束条件，h² = S₀²/(π² r²) – r²。代入 V = (1/3)π r² h 得到 V = (1/3)π r² √(S₀²/(π² r²) – r²)。平方得：V² = (1/9)π² r⁴ (S₀²/(π² r²) – r²) = (1/9)(S₀² r² – π² r⁶)。关于 r 求导：d(V²)/dr = (1/9)(2 S₀² r – 6 π² r⁵)。令其为零，得到 2 S₀² r = 6 π² r⁵，因此 r⁴ = S₀²/(3 π²)，从而 r = (S₀/(π √3))^{½}。在该半径下，l = S₀/(π r) = S₀/(π (S₀/(π √3))^{½}) = √(S₀√3/π)。简化比值 l/r 最终得到 l = √3 r。最大体积为 V_max = (S₀/ (3√3π))^{½} S₀/(3√3)（简化形式）。数值实例：当 S₀ = 100 cm² 时，r ≈ 4.29 cm，h ≈ 6.62 cm，V ≈ 127.3 cm³，大于邻近值的体积。
Discussion: The result l = √3 r implies a specific angular configuration. At the maximum, the semi-vertical angle θ satisfies sin θ = r/l = 1/√3, giving θ ≈ 35.3°. This provides a clear geometric interpretation. Numerical checks confirmed the critical point was indeed a maximum, and the graph of V against r showed a smooth peak. The assumption of zero thickness and perfect shape is a limitation; real cones with material joints might deviate. An extension could fix total surface area (including base) and yield a different optimal ratio.
讨论：结果 l = √3 r 意味着一个特定的角度配置。在最大值处，半顶角 θ 满足 sin θ = r/l = 1/√3，即 θ ≈ 35.3°。这提供了一个清晰的几何解释。数值检验证实了临界点确实是最大值点，且 V 关于 r 的图形显示出一个平滑的峰值。零厚度和完美形状的假设是一个局限；带有材料接缝的真实圆锥可能会出现偏差。一个拓展方向可以是固定总表面积（包括底面），这将产生一个不同的最优比例。
Conclusion: The investigation successfully determined that for a fixed lateral surface area, the volume of a right circular cone is maximised when the slant height is √3 times the base radius. This elegant proportion can inform material-efficient cone design. The analytical method proved effective, verified by numerical modelling, and can be extended to related solids or constraints.
结论：本探究成功确定：对于固定的侧面积，当
Published by TutorHao | Year 12 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导