📚 Year 9 Cambridge Additional Mathematics: Investigation Writing Framework and Worked Example | 剑桥 Year 9 进阶数学：论文写作框架与范文
In the Cambridge Additional Mathematics course, students are often asked to conduct a mathematical investigation and present their findings in a structured report. This type of task goes beyond routine problem-solving; it requires clear communication, logical reasoning, and the ability to analyse patterns. A well-written investigation paper not only demonstrates your understanding of mathematical concepts but also develops skills valuable for further study. This article provides a step-by-step framework for writing such a paper and includes a complete worked example to illustrate how each section should be crafted.
在剑桥进阶数学课程中，学生经常需要进行数学探究并以结构化的报告呈现研究结果。这类任务超出了常规解题的范畴，它要求清晰的表达、逻辑推理以及分析规律的能力。一篇优秀的探究论文不仅能展现你对数学概念的理解，还能培养对今后学习十分宝贵的技能。本文提供了一个撰写此类论文的分步框架，并包含一篇完整的范文，以具体说明每个部分应如何撰写。
1. Why Write a Mathematical Investigation? | 为什么要撰写数学探究论文？
Writing an investigation encourages you to think like a mathematician. Instead of simply applying a formula, you explore a problem, make conjectures, test them, and draw conclusions. This process deepens your understanding of topics such as functions, graphs, sequences and trigonometry. Moreover, the written report is a key component of internal assessment and can help you stand out in your coursework.
撰写探究论文能促使你像数学家一样思考。你不再是简单地套用公式，而是探索问题、提出猜想、加以验证并得出结论。这个过程能加深你对函数、图像、数列和三角学等课题的理解。此外，书面报告是内部评估的重要组成部分，能帮助你在课程作业中脱颖而出。
2. Overall Structure of the Report | 报告的整体结构
A standard mathematical investigation report follows a logical flow: Title, Abstract, Introduction, Methodology, Results & Analysis, Discussion, Conclusion, and References. Each section has a distinct purpose. Keeping this structure in mind helps you organise your thoughts and ensures the reader can follow your reasoning from start to finish.
一份标准的数学探究报告遵循以下逻辑顺序：标题、摘要、引言、方法、结果与分析、讨论、结论和参考文献。每个部分都有明确的目的。牢记这一结构有助于整理你的思路，并确保读者能从头到尾理解你的推理过程。
3. Title and Abstract | 标题与摘要
The title should be precise and informative, for example ‘Investigating the Effect of a and b on the Graph of y = ax + b’. The abstract is a brief summary of the whole investigation — what you set out to do, how you did it, and your main findings. Keep it under 150 words and avoid using symbols only; write in full sentences.
标题应当准确明了，例如“探究 a 和 b 对 y = ax + b 图像的影响”。摘要则是对整个探究的简要概述——包括你的研究目的、方法和主要发现。摘要字数控制在150词以内，避免只使用符号，要用完整的句子表述。
4. Introduction: Setting the Scene | 引言：背景铺垫
The introduction explains the context of the investigation. Why is this topic interesting or important? Define key terms, state your aim clearly, and possibly list the research questions you intend to answer. For a Year 9 Additional Mathematics investigation, link the topic to the syllabus, such as linear functions or quadratic graphs.
引言部分说明探究的背景。为什么这个课题有趣或重要？定义关键术语，清晰地陈述你的目的，并尽可能列出你打算回答的研究问题。对于 Year 9 进阶数学探究，要将主题与课程大纲联系起来，例如一次函数或二次函数图像。
5. Methodology: How You Conducted the Investigation | 方法：你是如何进行探究的
Describe the tools and techniques you used. Did you plot graphs by hand, use graphing software like Desmos or GeoGebra, or write a small program? Explain how you controlled variables and collected data. This section should be precise enough for someone else to replicate your work.
描述你所使用的工具和技术。你是手绘图像，使用 Desmos 或 GeoGebra 等绘图软件，还是编写了一个小程序？解释你如何控制变量并收集数据。这部分应足够精确，以便他人能够重现你的探究过程。
6. Results and Analysis | 结果与分析
Present your findings clearly. Use tables to organise numerical data and refer to any graphs you have generated. For example, a table might show values of y for different values of a while b is fixed. Analyse the patterns you observe without yet explaining why they happen. Include calculations of slopes, intercepts, or other relevant measures.
清晰地展示你的发现。使用表格整理数值数据，并提及你所绘制的任何图像。例如，可以制作一张表格，显示在固定 b 值的情况下，不同 a 值所对应的 y 值。分析你所观察到的规律，但先不要解释为什么会发生这些规律。包括斜率、截距或其他相关度量的计算过程。
7. Discussion: Interpreting Your Findings | 讨论：解释你的发现
Now explain the mathematics behind the patterns. Why does changing a affect the steepness of the line y = ax + b? What does a negative a do? How does b translate the graph vertically? Relate your observations to algebraic rules and generalisations. If any results were unexpected, discuss possible reasons.
现在解释规律背后的数学原理。为什么改变 a 会影响直线 y = ax + b 的倾斜程度？a 为负数时会产生什么效果？b 如何使图像垂直平移？将你的观察与代数法则和推广结论联系起来。如果出现意外结果，讨论可能的原因。
8. Conclusion and References | 结论与参考文献
The conclusion summarises the main findings and answers the research questions posed in the introduction. Do not introduce new information here. You may also suggest further investigations. Always include a list of references if you used any books, websites, or software that influenced your work. Use a consistent citation style.
结论部分总结主要发现，并回答引言中提出的研究问题。不要在此处引入新信息。你也可以提出进一步的探究建议。如果你使用的任何书籍、网站或软件对你的探究产生了影响，请务必列出参考文献。引用格式要保持一致。
9. Worked Example: Investigating the Effect of a and b on y = ax + b | 完整范文：探究 y = ax + b 中 a 和 b 的影响
Below is a complete investigation report following the framework above. Each section is presented with the original English text followed immediately by its Chinese translation for bilingual learners.
以下是一篇遵循上述框架的完整探究报告。每个部分都先呈现英文原文，紧接着给出中文翻译，以便双语学习者参考。
Title: Investigating the Effect of a and b on the Graph of y = ax + b
标题：探究 a 和 b 对 y = ax + b 图像的影响
Abstract
This investigation explores how the parameters a and b in the linear equation y = ax + b influence the straight-line graph. By plotting several graphs with different values of a while keeping b constant, and vice versa, I observed that a controls the gradient (steepness and direction) of the line, while b determines the point where the line crosses the y-axis. The findings confirm that y = ax + b can represent any non-vertical straight line, with a = 0 giving a horizontal line.
摘要
本探究探讨线性方程 y = ax + b 中的参数 a 和 b 如何影响直线图像。通过固定 b 而改变 a 值，以及固定 a 而改变 b 值，绘制多幅图像，我观察到 a 控制着直线的斜率（倾斜程度和方向），而 b 决定了直线与 y 轴的交点。研究结果证实 y = ax + b 可以表示任何不垂直于 x 轴的直线，其中 a = 0 时得到水平直线。
Introduction
Linear functions are the simplest type of function studied in Additional Mathematics. The general form y = ax + b is often called the slope-intercept form. I wanted to discover exactly how the values of a and b change the appearance of the graph. My research question was: ‘What is the geometric meaning of a and b?’ Understanding this would help me sketch graphs quickly and solve coordinate geometry problems more efficiently.
引言
一次函数是进阶数学中最简单的一类函数。一般形式 y = ax + b 通常被称为斜截式。我希望确切地了解 a 和 b 的值如何改变图像的形状。我的研究问题是：“a 和 b 的几何意义是什么？”理解这一点将有助于我快速绘制图像并更有效地解决坐标几何问题。
Methodology
I used the graphing software Desmos to plot the lines. First, I set b = 0 and took a = -3, -2, -1, 0, 1, 2, 3. For each value, I recorded the coordinates of two points and the angle the line makes with the positive x-axis. Second, I set a = 2 and varied b from -3 to 3 in steps of 1. I tabulated the y-intercepts and observed the vertical shift. All graphs were plotted on the same coordinate grid for comparison.
方法
我使用绘图软件 Desmos 来绘制直线。首先，设定 b = 0，取 a = -3, -2, -1, 0, 1, 2, 3。对于每个 a 值，我记录了两个点的坐标以及直线与 x 轴正方向所成的角度。其次，设定 a = 2，使 b 从 -3 到 3 以 1 为步长变化。我将 y 截距制成表格，观察图像的垂直平移情况。所有图像都绘制在同一个坐标格纸上以便比较。
Results and Analysis
Table 1 shows the effect of a when b = 0.
表 1 展示了当 b = 0 时 a 的影响。
| a | Equation | Gradient | Line direction |
|---|---|---|---|
| -3 | y = -3x | -3 | Steep downwards |
| -1 | y = -x | -1 | Downwards at 135° |
| 0 | y = 0 | 0 | Horizontal |
| 1 | y = x | 1 | Upwards at 45° |
| 3 | y = 3x | 3 | Steep upwards |
The table reveals a clear pattern: the value of a is exactly the gradient of the line. When a > 0, the line slopes upward; when a < 0, it slopes downward. The magnitude of a determines steepness.
表格揭示了一个清晰的规律：a 的值恰好等于直线的斜率。当 a > 0 时，直线向上倾斜；当 a < 0 时，直线向下倾斜。a 的绝对值决定了倾斜的陡峭程度。
Table 2 shows the effect of b when a = 2.
表 2 展示了当 a = 2 时 b 的影响。
| b | Equation | y-intercept | Observation |
|---|---|---|---|
| -3 | y = 2x – 3 | (0, -3) | Shifted 3 units down |
| 0 | y = 2x | (0, 0) | Passes through origin |
| 2 | y = 2x + 2 | (0, 2) | Shifted 2 units up |
These results show that b simply moves the line up or down without changing its slope. The y-intercept is exactly (0, b).
这些结果表明，b 只是将直线上下平移，而不会改变其斜率。y 轴截距恰好为 (0, b)。
Discussion
The experiment confirms that in the equation y = ax + b, a represents the gradient, and b represents the y-intercept. The gradient a can be interpreted as the rate of change: for every 1 unit increase in x, y increases by a units. A negative gradient means y decreases as x increases. The constant b is the value of y when x = 0. Combining both parameters allows us to describe any non-vertical line. This matches the algebraic definition and provides a powerful tool for solving simultaneous equations graphically: the intersection point of two lines can be understood in terms of their gradients and intercepts.
讨论
实验证实，在方程 y = ax + b 中，a 代表斜率，b 代表 y 轴截距。斜率 a 可以理解为变化率：x 每增加 1 个单位，y 增加 a 个单位。斜率为负意味着 y 随 x 的增大而减小。常数 b 是 x = 0 时 y 的值。将两个参数结合起来，我们可以描述任何不垂直于 x 轴的直线。这与代数定义一致，并为我们用图像法解联立方程提供了一个强大的工具：两条直线的交点可以通过它们的斜率和截距来理解。
Conclusion
I found that a determines the gradient and direction of the line y = ax + b, while b determines its vertical position. This investigation helped me visualise linear functions and improved my graphing skills. A possible extension would be to explore the effect of similar parameters in quadratic functions, such as y = ax² + bx + c.
结论
我发现 a 决定了直线 y = ax + b 的斜率和方向，而 b 则决定了它的垂直位置。这次探究帮助我直观地认识了一次函数，并提升了我的绘制图像能力。一个可能的拓展是探究二次函数中类似参数的影响，例如 y = ax² + bx + c。
References
Desmos Graphing Calculator, available at https://www.desmos.com/calculator
Cambridge IGCSE Additional Mathematics Coursebook, Cambridge University Press.
参考文献
Desmos 图形计算器，网址 https://www.desmos.com/calculator
《剑桥 IGCSE 进阶数学教材》，剑桥大学出版社。
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