IB Maths: Guide to Experimental Investigations | IB 数学：实验操作指南

📚 IB Maths: Guide to Experimental Investigations | IB 数学：实验操作指南

The Internal Assessment (IA) in IB Mathematics is a unique opportunity to conduct your own mathematical exploration. Often centred on experimental data, it demands careful design, precise measurements, and rigorous statistical analysis. Understanding how to turn a real-world question into a mathematically sound investigation is key to achieving high marks.

IB 数学的内部评估（IA）是一次独特的数学探索机会，往往围绕实验数据展开，要求精心的实验设计、精确的测量和严谨的统计分析。学会如何将现实问题转化为具有数学深度的调查，是获得高分的关键。

1. Understanding Experimental Exploration in IB Maths IA | 理解 IB 数学 IA 中的实验探索

In the IB Mathematics IA, an experimental exploration involves collecting primary data through controlled trials, surveys, or physical experiments. The aim is to model, test hypotheses, or discover patterns using mathematical tools. This process mirrors the scientific method but is evaluated on mathematical criteria such as precision, use of mathematics, and personal engagement.

在 IB 数学 IA 中，实验探索指的是通过控制试验、调查或物理实验收集第一手数据。其目的是利用数学工具建模、检验假设或发现规律。这一过程类似科学方法，但评估着重于数学的精确性、数学运用程度以及个人参与。

Your exploration does not need to be groundbreaking; it should, however, demonstrate a clear connection between the experiment and a well-defined mathematical goal. Whether investigating projectile motion, the cooling of coffee, or the spread of a rumour, the mathematical lens you apply matters more than the complexity of the topic.

你的探索不必追求突破性，但必须清晰展示实验与明确数学目标之间的联系。无论是研究抛体运动、咖啡冷却还是谣言传播，你所运用的数学视角远比主题本身的复杂程度更重要。

2. Choosing a Suitable Experimental Topic | 选择合适的实验主题

The topic should be personally engaging and allow for rich mathematical analysis. Avoid overly simplistic experiments that yield only a straight line – examiners look for refinement of models, error analysis, and consideration of underlying assumptions. Consider areas like physics-based motion, biological growth, or statistical surveys where you can apply regression, calculus, or probability distributions.

主题应具有个人吸引力并且允许丰富的数学分析。避免过于简单的实验，仅得出一条直线——考官期望看到模型的改进、误差分析以及对基本假设的考量。可以考虑物理运动、生物生长或统计调查等领域，这些地方可以应用回归、微积分或概率分布。

For example, instead of just measuring the bounce height of a ball, you could explore the mathematical relationship with drop height using quadratic or exponential models and discuss energy loss. Alternatively, design a survey to test the Normal distribution of heights or reaction times among classmates.

例如，与其仅测量球的反弹高度，不如利用二次或指数模型探究其与下落高度之间的数学关系，并讨论能量损失。或者设计调查来检验同学身高或反应时间是否符合正态分布。

Always ensure the data you need can be collected ethically and with the available equipment. Confirm that the topic provides enough variables to manipulate and mathematical techniques to showcase at the level of your course, whether AA (Analysis and Approaches) or AI (Applications and Interpretation).

始终确保所需数据可以合乎道德地收集，并且使用现有设备。确认该主题能够提供足够多的变量以供操控，并有足够多的数学技巧来展示你所学课程（AA 或 AI）的水平。

3. Designing a Rigorous Experiment | 设计严谨的实验

A robust experiment begins with a clearly stated research question and a testable hypothesis. Identify independent and dependent variables, and decide which factors must be controlled. Write a step-by-step procedure so that the experiment can be replicated. This not only demonstrates good scientific practice but also strengthens the ‘Reflection’ criterion in your IA.

严谨的实验始于明确陈述的研究问题和可检验的假设。识别自变量和因变量，并确定必须控制的因素。写下逐步操作程序，使实验可被重复。这不仅展示了良好的科学实践，也加强了 IA 评估标准中的“反思”部分。

When collecting data, take repeated measurements at each setting to calculate a mean and to quantify uncertainty. For example, if you are timing the period of a pendulum, record at least three trials for each length. Use a table to organise raw data, noting any anomalies immediately. This habit makes the mathematics smoother later on.

收集数据时，在每个设定点进行多次测量以计算平均值并量化不确定度。例如，若在测量单摆周期，对每个摆长至少记录三次试验数据。用表格组织原始数据，立即记录任何异常值。这一习惯会使后续的数学处理更加顺畅。

Uncertainty = (half-range) or (instrument precision)

不确定度 = (半极差) 或 (仪器精度)

4. Mathematical Modelling and Assumptions | 数学建模与假设

Once the data is collected, the core of your IA lies in modelling. Begin by plotting the data points and visually assessing a trend. Propose a simple model first – a linear, quadratic, exponential, or power function – and use regression (by GDC or software) to find the parameters. State all assumptions clearly: no air resistance, constant room temperature, ideal gas behaviour, etc.

收集数据后，IA 的核心在于建模。首先绘制数据点，直观评估趋势。先提出一个简单模型——线性、二次、指数或幂函数——然后利用回归（通过 GDC 或软件）求出参数。清晰陈述所有假设：无空气阻力、恒定室温、理想气体行为等等。

If the initial model does not fit well, refine it by considering extra factors such as friction, cooling rate change, or logistic limitations. For instance, a population growth experiment may start with an exponential model P = P₀eᵏᵗ, but you might later switch to a logistic model P = K / (1 + Ae⁻ʳᵗ) if carrying capacity becomes relevant. Explain why each refinement is mathematically justified.

如果初始模型拟合不佳，通过考虑额外因素（如摩擦、冷却速率变化或逻辑限制）进行改进。例如，一个种群增长实验可以从指数模型 P = P₀eᵏᵗ 开始，但如果环境容量变得相关，你可能会转向逻辑模型 P = K / (1 + Ae⁻ʳᵗ)。解释每次改进为何在数学上是合理的。

5. Data Processing and Statistical Analysis | 数据处理与统计分析

Descriptive statistics such as mean, standard deviation, and quartiles provide an overview of your data. Process raw data into a form suitable for modelling: calculate averages, convert units, and compute derived quantities like acceleration or gradient. Always present processed data in clear tables, with columns labelled in mathematical notation where appropriate.

描述性统计量如均值、标准差和四分位数能提供数据概览。将原始数据处理成适合建模的形式：计算平均值、转换单位，并算出衍生量如加速度或斜率。始终以清晰的表格呈现处理后数据，适时用数学符号标记列名。

Inferential statistics, such as the Pearson correlation coefficient r or a χ² goodness-of-fit test, can assess the strength and significance of your model. For the correlation coefficient, use the formula:

推断性统计，如皮尔逊相关系数 r 或 χ² 拟合优度检验，可以评估模型的强度和显著性。对于相关系数，使用公式：

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √( Σ(xᵢ − x̄)² Σ(yᵢ − ȳ)² )

A value of r close to 1 or -1 indicates a strong linear relationship, but always verify with the coefficient of determination R² and residual plots. Non-linear models may require transformation (e.g., log-log plots) to simplify analysis.

r 值接近 1 或 -1 表明强线性关系，但始终要用决定系数 R² 和残差图进行验证。非线性模型可能需要变换（如双对数图）以简化分析。

6. Using Technology Tools Effectively | 有效使用技术工具

Graphical display calculators (GDCs), spreadsheets like Excel, and software such as GeoGebra or Desmos are essential for handling data and performing regressions. Show screen captures or computer outputs as evidence, but do not rely on them alone – you must interpret every graph and table mathematically.

图形计算器（GDC）、Excel 等电子表格以及 GeoGebra 或 Desmos 等软件对于处理数据和执行回归至关重要。展示屏幕截图或计算机输出作为证据，但不要仅依赖它们——必须用数学语言解释每一项图表和表格。

When using a GDC to find a regression line, always note the command used (e.g., LinReg(a+bx), ExpReg) and the resulting equation with correct variable names. For advanced explorations, you can write simple code in Python or R to simulate random processes or perform Monte Carlo trials, which adds depth to the mathematical sophistication.

使用 GDC 求回归线时，务必注明所用命令（如 LinReg(a+bx)、ExpReg），并给出使用正确变量名的方程。对于更高级的探索，可以编写 Python 或 R 的简单代码来模拟随机过程或进行蒙特卡洛试验，这会增加数学的深度。

7. Discussing Errors, Limitations, and Extensions | 讨论误差、局限性与扩展

No experiment is perfect. Distinguish between systematic errors (e.g., incorrectly calibrated equipment) and random errors (e.g., human reaction time). Quantify error propagation if possible, using formulas such as:

没有完美的实验。区分系统误差（如设备校准不当）和随机误差（如人类反应时间）。如果可能，量化误差传播，使用公式如：

For y = a + b: Δy = Δa + Δb

For y = a × b: Δy/y = Δa/a + Δb/b

Discuss how these uncertainties affect your model and conclusions. Suggest concrete improvements, such as using light gates instead of a stopwatch, or increasing sample size to satisfy the central limit theorem. This reflective thinking is highly rewarded under criterion E.

讨论这些不确定度如何影响模型和结论。提出具体改进措施，如使用光门代替秒表，或增加样本量以满足中心极限定理。这种反思性思考在标准 E 中得分很高。

8. Structuring the Written Report | 组织书面报告的结构

A well-structured IA follows a logical flow: Introduction, Aim, Rationale, Procedure, Data (raw and processed), Mathematical Analysis, Conclusion, and Reflection. Use clear headings and numbered sections. Write in a formal, yet personal tone – the exploration should be your own journey. All mathematical work must be typeset clearly, and pages should not be overcrowded.

结构良好的 IA 遵循逻辑流程：引言、目标、理由、步骤、数据（原始和处理后）、数学分析、结论和反思。使用清晰的标题和编号章节。采用正式但个人化的语气——探索应是属于你自己的旅程。所有数学工作必须排版清晰，页面不应过于拥挤。

In the ‘Aim’, state exactly what you hope to achieve mathematically, e.g., ‘To determine whether the rate of cooling of water follows Newton’s law of cooling, modelled by T(t)=Tₑ+T₀e⁻ᵏᵗ.’ Keep the report concise; the IB recommends around 12–20 pages, but quality matters much more than length.

在“目标”中，精确陈述你希望用数学实现什么，例如“确定水的冷却速率是否遵循牛顿冷却定律，其模型为 T(t)=Tₑ+T₀e⁻ᵏᵗ”。保持报告简洁；IB 推荐大约 12-20 页，但质量远比长度重要。

9. Meeting the Assessment Criteria | 满足评估标准

The IA is assessed on five criteria: Presentation (A), Mathematical Communication (B), Personal Engagement (C), Reflection (D), and Use of Mathematics (E). For experimental explorations, criteria B and E are particularly important. Show rigorous calculations, justify all steps, and demonstrate understanding beyond the textbook.

IA 按五项标准评估：呈现（A）、数学交流（B）、个人参与（C）、反思（D）和数学运用（E）。对于实验探索，标准 B 和 E 尤为重要。展示严谨的计算，证明所有步骤，并体现出超越课本的理解。

To score highly on personal engagement, explain why the topic intrigued you, show creative data collection, or devise your own experimental setup. Use first-person pronouns where appropriate: ‘I noticed that…’, ‘I was curious about…’. This genuine connection must shine through.

要在个人参与上得高分，解释为何该主题吸引你，展示富有创意的数据收集方式，或自行设计实验装置。适当使用第一人称：“我注意到……”，“我好奇……”。这种真实的联系必须清晰体现。

10. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法

Many students lose marks by presenting raw data without processing, using overly simplistic mathematics, or failing to link their conclusion back to the hypothesis. Another frequent mistake is ignoring the reflection criterion altogether – a brief paragraph at the end is not enough. Reflection must be woven throughout the exploration.

许多学生因未处理原始数据、数学过于简单，或结论未能回调假设而失分。另一个常见错误是完全忽略反思标准——文末的简短段落是不够的，反思必须贯穿整个探索过程。

Also, avoid treating the IA as a lab report for science; the focus must remain on the mathematics, not on the experimental apparatus. Every graph, equation, and statistical test should serve a mathematical purpose. Finally, carefully cite all sources and ensure academic honesty – this includes properly referencing any model you adapt.

此外，避免把 IA 写成一份科学实验报告；重点必须始终放在数学上，而非实验设备。每张图、每个方程和每次统计检验都应有数学目的。最后，仔细引用所有来源并确保学术诚信——这包括正确标注所改编的任何模型。

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IB Maths: Guide to Experimental Investigations | IB 数学：实验操作指南