📚 IB AQA Maths: Experimental Operations Guide | IB AQA 数学：实验操作指南
Mathematical experimentation lies at the heart of both the IB Internal Assessment and AQA’s investigative coursework. Whether you are modelling the trajectory of a projectile, analysing the spread of a disease, or exploring geometric patterns, a systematic experimental approach transforms raw curiosity into rigorous mathematical evidence. This guide walks you through every stage of an experimental investigation, from formulating a sharp research question to presenting a polished final report, with practical advice for avoiding common pitfalls.
数学实验是 IB 内部评估和 AQA 探究式课程作业的核心。无论你是在建模抛体轨迹、分析疾病传播，还是探究几何图案，系统化的实验方法都能将原始的好奇心转化为严谨的数学证据。本指南将带你走过实验探究的每一阶段，从制定锐利的研究问题到呈现精炼的最终报告，并提供避免常见陷阱的实用建议。
1. Understanding the Experimental Task | 理解实验任务
In IB Mathematics, the Internal Assessment (IA) requires you to engage with an area of mathematics that interests you, develop a personal exploration involving data collection or theoretical modelling, and reflect on the process. AQA’s specifications similarly value statistical investigations and mathematical modelling tasks where you must gather, process, and interpret data. The key is to demonstrate mathematical thinking rather than just computing results.
在 IB 数学中，内部评估（IA）要求你涉足一个感兴趣的数学领域，开展涉及数据收集或理论建模的个人探索，并反思整个过程。AQA 的考试大纲同样重视统计调查和数学建模任务，需要你收集、处理并解读数据。关键在于展现数学思维，而不仅仅是计算结果。
Before launching into an experiment, clarify the task requirements: IB IA has a page limit (typically 12-20 pages) and strict criteria on presentation, mathematical communication, and personal engagement. AQA may have different mark schemes, but both reward clear rationale, appropriate mathematics, and critical evaluation. Read the assessment criteria first—this will shape every decision you make.
在着手实验之前，先厘清任务要求：IB IA 有页数限制（通常 12-20 页）以及关于展示、数学沟通和个人投入的严格标准。AQA 可能有不同的评分方案，但二者都奖励清晰的论证、合适的数学和批判性评价。首先要阅读评估标准——这将影响你做出的每一个决定。
2. Formulating a Research Question | 制定研究问题
A well-defined research question is narrow, measurable, and mathematically rich. Avoid vague topics like ‘Is there a relationship between height and basketball performance?’ Instead, refine it: ‘To what extent does height predict free-throw accuracy among high-school players, and can a linear regression model quantify this relationship?’ The question should imply the experiment you will conduct.
一个定义清晰的研究问题应该是范围窄小、可测量且数学丰富的。避免诸如“身高与篮球表现之间有关系吗？”这类模糊的主题。要将其细化：“身高在多大程度上能预测高中球员的罚球准确率，线性回归模型能否量化这一关系？”问题本身应暗示你将进行的实验。
For IB, your question must allow for personal engagement—choose something you genuinely care about. For AQA, the question often stems from a given dataset or a practical scenario. In both cases, ensure the mathematics required is at Higher or Standard Level appropriate, and that you can collect at least 30 data points to support statistical validity.
对 IB 而言，你的问题必须能体现个人投入——要选择你真正关心的事物。对 AQA 而言，问题往往源于给定的数据集或实际情境。无论是哪种情况，都要确保所需的数学内容适合高水平或标准水平，并且你能够收集至少 30 个数据点以支撑统计效度。
3. Designing the Experiment | 设计实验
Design is the blueprint of your investigation. Identify the independent variable (what you change), the dependent variable (what you measure), and control variables (what you keep constant). For instance, if investigating pendulum period, independent variable = length of string, dependent = time for 10 oscillations, controls = mass of bob, angle of release.
设计是你探究的蓝图。明确自变量（你改变什么）、因变量（你测量什么）和控制变量（你保持恒定什么）。例如，若研究单摆周期，自变量 = 摆线长度，因变量 = 10 次振荡的时间，控制变量 = 摆锤质量、释放角度。
Plan how to randomise trials and replicate measurements. Randomisation reduces systematic bias; replication allows you to estimate experimental uncertainty. Decide on the range and increments of the independent variable—enough levels to reveal a trend, but achievable within time limits. A pilot run can reveal flaws before you commit.
规划如何将试验随机化并重复测量。随机化能减少系统偏差；重复则能估算实验的不确定性。决定自变量的范围和步长——足够多的水平以显示趋势，同时能在时限内完成。在正式投入之前进行一次预实验，可以暴露其中的缺陷。
4. Data Collection Techniques | 数据收集技术
Generate primary data through hands-on experiments, surveys, or simulations. When measuring physical quantities, use the most precise instrument available and record raw data directly into a structured table. For surveys, craft unbiased questions and pilot them. For simulations, run enough iterations to stabilise outcomes.
通过动手实验、调查或模拟来产生主要数据。在测量物理量时，使用可获取的最精密仪器，并将原始数据直接记录到结构化的表格中。对于调查问卷，要精心设计无偏的问题并进行预测试。对于模拟，要运行足够多的迭代次数以使结果稳定。
Secondary data from reputable sources (government databases, scientific journals) can supplement your investigation, especially for AQA tasks where large datasets are provided. Always cite sources and comment on their reliability. Photographs of your experimental setup add authenticity and show personal engagement.
来自信誉良好的来源（政府数据库、科学期刊）的二手数据可以补充你的探究，特别是对于提供了大型数据集的 AQA 任务。始终注明出处并评论其可靠性。实验装置的照片可以增加真实感，并展示个人投入。
5. Organising and Presenting Data | 组织与呈现数据
Raw data belongs in clearly labelled tables with units. Processed data—means, standard deviations, or transformed values—should appear in separate tables. Use visual aids: scatter plots for raw data, line graphs for models, and bar charts for categorical comparisons. Every graph must have a title, labelled axes, and a legend if needed.
原始数据应放在标注清楚、带单位的表格中。处理后数据——平均值、标准差或转换值——应出现在单独的表格中。使用视觉辅助工具：散点图展示原始数据，折线图展示模型，条形图进行类别比较。每张图必须有标题、带标注的坐标轴，必要时还要有图例。
Consider appropriate scales: avoid distorted axes that exaggerate trends. Include error bars where uncertainty is known. For the IB IA, demonstrating excellent use of technology earns high marks in criterion E (use of mathematics). Software like GeoGebra, Desmos, or Excel can produce professional graphs.
考虑合适的比例尺：避免扭曲轴线而夸大趋势。在已知不确定度的地方加入误差棒。对于 IB IA，展示对技术的优秀运用可以在标准 E（数学的运用）中获得高分。像 GeoGebra、Desmos 或 Excel 这样的软件可以制作专业的图表。
6. Mathematical Modelling | 数学建模
The core of the investigation is fitting a mathematical model to your data. Common models include linear (y = mx + c), quadratic (y = ax² + bx + c), exponential (y = aeᵏˣ), and trigonometric functions. Choose a model based on theoretical grounds or the shape of the scatter plot. Justify your choice with reasoning, not just ‘it looks like it fits’.
探究的核心是将数学模型拟合到你的数据。常见的模型包括线性（y = mx + c）、二次（y = ax² + bx + c）、指数（y = aeᵏˣ）和三角函数。基于理论依据或散点图的形状来选择模型。用推理来证明你的选择，而不仅仅是“看起来拟合”。
Use regression tools to find parameters, and always display the equation and the coefficient of determination R². A high R² (close to 1) suggests a good fit, but check residuals: plot residuals against the independent variable; they should be randomly scattered, not forming a pattern. This validates the model’s appropriateness.
使用回归工具寻找参数，并始终显示方程和决定系数 R²。较高的 R²（接近 1）表明拟合良好，但还要检查残差：画出残差对自变量的图；残差应随机分布，不形成模式。这验证了模型的适当性。
7. Statistical Analysis and Hypothesis Testing | 统计分析与假设检验
Beyond modelling, statistical tests add rigour. For correlation, compute Pearson’s r and test its significance using a t-test or consult a critical value table. For comparing two groups, use a two-sample t-test or Mann–Whitney U test if data is non-normal. Clearly state null and alternative hypotheses, significance level (α = 0.05 is standard), and interpret the p-value in context.
除建模之外，统计检验增加了严谨性。对于相关性，计算皮尔逊 r 并用 t 检验测试其显著性，或查阅临界值表。对于两组比较，若数据非正态则使用双样本 t 检验或 Mann–Whitney U 检验。清晰地陈述原假设和备择假设、显著性水平（α = 0.05 为标准值），并在情境中解读 p 值。
Chi-squared tests are excellent for categorical data, such as testing whether a die is fair. For IB, you can include these tests even if they exceed the syllabus—just ensure you explain them. AQA often expects students to perform hypothesis tests as part of the statistics component, linking experimental data to formal inference.
卡方检验非常适合分类数据，例如检验骰子是否均匀。对于 IB，即使这些检验超出教学大纲，你也可以纳入——只要确保进行解释。AQA 通常期望学生进行假设检验，作为统计部分的要求，将实验数据与正式推断联系起来。
8. Error Analysis and Uncertainty | 误差分析与不确定性
Every measurement carries uncertainty. Distinguish between random errors (reduced by repetition) and systematic errors (e.g. a miscalibrated ruler). Calculate absolute and percentage uncertainties for direct measurements. For derived quantities, propagate uncertainties using these rules:
每一次测量都带有不确定性。区分随机误差（通过重复减少）和系统误差（如未校准的尺子）。计算直接测量的绝对和百分比不确定性。对于导出量，按下列规则传递不确定性：
- Addition/subtraction: add absolute uncertainties
- Multiplication/division: add percentage uncertainties
- Powers: multiply percentage uncertainty by the power
- 加法/减法：相加绝对不确定性
- 乘法/除法：相加百分比不确定性
- 幂：百分比不确定性乘以指数
Discuss how uncertainties affect your model parameters and final conclusion. Acknowledging limitations shows critical reflection. For example, ‘The ±0.1 s timing uncertainty contributed to a 5% error in the gradient, which may account for the deviation from the theoretical value.’
讨论不确定性如何影响你的模型参数和最终结论。承认局限性显示了批判性反思。例如：“±0.1 秒的计时不确定性导致了梯度上 5% 的误差，这或许能解释与理论值的偏差。”
9. Using Technology: Graphing Calculators and Software | 使用技术：图形计算器与软件
Technology streamlines analysis and enhances presentation. TI-Nspire or Casio graphing calculators can perform regression, statistical tests, and matrix operations. However, software like GeoGebra, Desmos, and Excel is often more flexible for experimenting with different models and creating publication-quality graphs.
技术能简化分析并提升展示效果。TI-Nspire 或卡西欧图形计算器可以执行回归、统计检验和矩阵运算。然而，像 GeoGebra、Desmos 和 Excel 这样的软件在尝试不同模型和创建出版级质量的图表方面往往更灵活。
Python or R can be used for advanced analysis, but keep the focus on the mathematics, not the coding. For IB, you should explain what the technology does, not just press buttons. Include screenshots of key steps. For AQA, familiarity with statistical software is advantageous for handling large datasets.
Python 或 R 可用于高级分析，但重点要保持在数学上，而非编码上。对于 IB，你应该解释技术做了什么，而不仅仅是按按钮。附上关键步骤的截图。对于 AQA，熟悉统计软件有助于处理大型数据集。
10. Writing the Report: Structure and Assessment Criteria | 撰写报告：结构与评估标准
A logical structure is essential. Begin with an introduction stating your research question, rationale, and a brief outline. Then describe your methodology, present data and analysis, discuss your model and its evaluation, and conclude with a summary and reflection. Use headings and subheadings to guide the reader.
逻辑结构至关重要。以引言开篇，陈述研究问题、理由和简要大纲。然后描述你的方法，展示数据和分析，讨论模型及其评价，最后以总结和反思收尾。使用标题和子标题来引导读者。
Map your draft onto the assessment criteria. For IB IA, check criterion A (presentation), B (mathematical communication), C (personal engagement), D (reflection), and E (use of mathematics). For AQA, align with the specified mark scheme sections. Peer review can reveal gaps before final submission.
将你的草稿与评估标准对照。对于 IB IA，检查标准 A（展示）、B（数学沟通）、C（个人投入）、D（反思）和 E（数学运用）。对于 AQA，要与指定的评分方案部分对齐。在最终提交前进行同伴评审可以发现缺漏。
| Criterion | Key Focus |
|---|---|
| A: Presentation | Organisation, readability, tables, graphs |
| B: Mathematical Communication | Notation, terminology, reasoned narrative |
| C: Personal Engagement | Originality, genuine interest, independence |
| D: Reflection | Critical evaluation of results and process |
| E: Use of Mathematics | Precision, sophistication, appropriate level |
标准：A 展示（组织、可读性、表格、图表），B 数学沟通（符号、术语、推理叙述），C 个人投入（原创性、真正兴趣、独立性），D 反思（对结果和过程的批判性评价），E 数学运用（精确性、深度、适当水平）。
11. Common Pitfalls and How to Avoid Them | 常见陷阱及如何避免
One major pitfall is choosing a superficial question that requires only descriptive statistics. Avoid this by ensuring your question demands modelling or inferential testing. Another is gathering insufficient data—fewer than 20 points leads to unreliable models. Plan to collect at least 30 data points, or more if natural variability is high.
一个主要陷阱是选择一个肤浅的问题，仅需描述性统计。通过确保你的问题需要建模或推断检验来避免。另一个是收集的数据不足——少于 20 个数据点会导致模型不可靠。计划收集至少 30 个数据点，如果自然变异性高则要更多。
Overfitting models is equally dangerous: a polynomial of degree 7 can pass through every point but has no predictive power. Stick to the simplest model that adequately describes the trend. Also, never ignore outliers without justification; investigate them—they might reveal systematic errors or interesting exceptions.
过度拟合模型同样危险：7 次多项式可以穿过每个点，但毫无预测能力。坚持使用能充分描述趋势的最简模型。此外，绝不要在无正当理由的情况下忽略异常值；要对其进行调查——它们可能揭示系统误差或有趣的例外。
12. Final Tips for Success | 成功要诀
Start early—experimental work takes time to refine. Maintain a logbook of all decisions, failed attempts, and insights; this demonstrates personal engagement and reflection. Rehearse discussions about your mathematics with peers or teachers, as verbalising your understanding deepens it.
尽早开始——实验工作需要时间来打磨。维护一本日志，记录所有决定、失败尝试和洞见；这能展现个人投入和反思。与同伴或老师排练关于你所用数学的讨论，因为口头表达能加深理解。
Finally, enjoy the process. The best investigations come from genuine curiosity. When you care about the question, your writing becomes more compelling, and the mathematical journey becomes a story worth telling. Good luck with your experimental exploration!
最后，享受这个过程。最好的探究源于真正的好奇心。当你在乎你的问题时，你的写作会变得更有说服力，而数学之旅就成为了一个值得讲述的故事。祝你的实验探索好运！
y = 2.3x + 0.5, R² = 0.98, uncertainty Δx = ±0.1
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