📚 Experimental Operation Guide for IB & WJEC Mathematics | IB与WJEC数学实验操作指南
Mathematical experiments bring abstract concepts to life. Whether you are working on an IB Mathematics Internal Assessment (IA) or a WJEC Statistics coursework project, hands-on experimentation – including data collection, simulation, and statistical testing – forms the backbone of genuine mathematical inquiry. This guide provides step-by-step operational advice for planning, executing, and presenting a successful mathematical investigation, tailored to both IB and WJEC specifications. Follow these guidelines to strengthen your analytical skills and produce high-quality work that meets exam board criteria.
数学实验让抽象概念变得具体。不论你是在做 IB 数学内部评估（IA）还是 WJEC 统计课程作业，动手实验——包括数据收集、模拟和统计检验——都是真实数学探究的核心。本指南按照 IB 与 WJEC 的考纲要求，逐步为你提供规划、实施和展示数学调查的操作建议。遵循这些指引，你将提升分析能力，并提交出评分标准下的高质量作品。
1. Understanding the Role of Experiments | 理解实验在数学中的角色
In both IB and WJEC mathematics, an experiment goes beyond textbook exercises. It involves formulating a research question, designing a method to gather real or simulated data, and applying mathematical tools to draw conclusions. The IB IA expects a personal engagement with a topic, while WJEC coursework often focuses on statistical investigations, such as hypothesis testing or correlation analysis. Recognising this distinction will help you select an appropriate topic and employ suitable techniques.
无论是 IB 还是 WJEC 数学，实验都不只是做课本习题。它需要提出研究问题、设计方法收集真实或模拟数据，并运用数学工具得出结论。IB 内部评估强调对主题的个人投入，而 WJEC 课程作业通常聚焦统计调查，例如假设检验或相关性分析。认清这一区别，将有助于你选出合适的课题并采用恰当的方法。
2. Choosing a Feasible Research Question | 选择可行的研究问题
Start by brainstorming topics that genuinely interest you — sports statistics, population growth, or the physics of a bouncing ball. Narrow your focus to ensure the experiment can be completed with available resources. An ideal question should be clear, measurable, and rich enough to apply at least two mathematical processes (e.g., function modelling and regression). For IB, the question must allow for “personal engagement”; for WJEC, it should be suitable for a structured statistical cycle.
首先，头脑风暴你真正感兴趣的议题——体育统计、人口增长或弹跳球的物理学。缩小焦点，确保实验能用现有资源完成。理想的问题应清晰、可量化，并且足以运用至少两种数学过程（如函数建模与回归）。对 IB 而言，问题必须体现“个人投入”；对 WJEC，则需适合结构化统计循环。
3. Planning the Experimental Design | 实验设计规划
A sound experimental design outlines the variables, data collection method, and sampling strategy. Identify independent and dependent variables clearly. Decide whether you will carry out a controlled physical experiment, conduct a survey, or use a computer simulation. For WJEC coursework, you must state the population, sampling frame, and sample size in advance. In IB, you should also discuss potential confounding factors and how you intend to minimise bias. Document every decision in a logbook or draft.
良好的实验设计要阐明变量、数据收集方法和抽样策略。明确识别自变量与因变量。决定是进行受控的物理实验、开展问卷调查，还是使用计算机模拟。对 WJEC 课程作业，须提前声明总体、抽样框和样本量。在 IB 中，你还应讨论潜在的混杂因素以及如何减少偏差。将每个决策记录在日志或草稿中。
Key equation for sample size (z-based): n = (z² × p × (1−p)) / e²
样本量公式（基于 z 值）：n = (z² × p × (1−p)) / e²
4. Collecting and Recording Data Reliably | 可靠地收集与记录数据
Consistency is paramount during data collection. If you measure reaction times, use the same device and procedure for every trial. For surveys, ensure questions are unbiased and pilot-tested. Record raw data in a spreadsheet with clear headings and units. Both IB and WJEC moderators value transparency: photograph your experimental setup, attach blank questionnaires, and note any anomalies. Never fabricate or alter data — the mathematical integrity of your investigation depends on honesty.
在数据收集阶段，一致性至关重要。若测量反应时间，每次试验都使用相同的设备与步骤。对问卷来说，要保证问题无偏向性并经过预测试。将原始数据录入电子表格，标明清晰的表头和单位。IB 和 WJEC 评审人都看重透明度：拍摄实验装置的照片，附上空白的问卷，并记下任何异常。切勿编造或篡改数据——你研究的数学诚信依赖于诚实。
5. Using Technology for Simulation and Analysis | 运用技术进行模拟与分析
Modern mathematical experiments frequently involve technology. Graphing calculators (such as TI-84 or Casio fx-CG50), GeoGebra, Desmos, Excel, and Python libraries open up powerful simulation opportunities. For example, you can model projectile motion using parametric equations or run a Monte Carlo simulation for a probability investigation. Ensure you explain the code or calculator steps in your write-up, and always reference the software used. WJEC project work often requires candidates to demonstrate competence with statistical functions on a calculator or spreadsheet.
现代数学实验常常借助技术手段。图形计算器（如 TI-84 或 Casio fx-CG50）、GeoGebra、Desmos、Excel 和 Python 库提供了强大的模拟机会。例如，你可以利用参数方程模拟抛体运动，或者为概率问题运行蒙特卡罗模拟。确保在报告中解释代码或计算器步骤，并始终注明所用软件。WJEC 项目工作常要求考生展示使用计算器或电子表格进行统计函数运算的能力。
| English Tool | 中文工具 |
|---|---|
| GeoGebra for geometry & algebra simulation | GeoGebra 用于几何与代数模拟 |
| Excel / Google Sheets for regression and hypothesis tests | Excel / Google 表格用于回归与假设检验 |
| Python (with pandas, scipy) for advanced modelling | Python（配合 pandas、scipy）进行高级建模 |
| Graphing calculator for on-the-spot statistical plots | 图形计算器用于即时统计图形 |
6. Performing Statistical Calculations | 执行统计计算
Once you have raw data, compute descriptive statistics: mean, median, standard deviation, and quartiles. Construct box plots, histograms, or scatter diagrams to visualise distributions. The core inferential step often involves correlation coefficients (Pearson’s r or Spearman’s rank) and linear regression. For IB, you might go further by curve fitting exponential, sine, or logistic models. For WJEC, you need to calculate p-values and connect them to significance levels. Always show your working, even if technology performed the computation.
拿到原始数据后，计算描述统计量：均值、中位数、标准差和四分位数。绘制箱线图、直方图或散点图来可视化分布。核心的推断步骤通常涉及相关系数（Pearson r 或 Spearman 秩）和线性回归。对 IB，你可能要进一步拟合指数、正弦或逻辑模型。对 WJEC，需计算 p 值并将其与显著性水平联系起来。即使计算由技术完成，也始终展示你的运算过程。
Regression line: ŷ = a + bx, where b = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)², a = ȳ − bx̄
回归线：ŷ = a + bx，其中 b = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²，a = ȳ − bx̄
7. Hypothesis Testing Step by Step | 逐步进行假设检验
Hypothesis testing is central to WJEC coursework and also valuable for IB IAS. State the null hypothesis H₀ and alternative hypothesis H₁. Choose a test statistic (e.g., t-test, chi-squared test for independence) and a significance level, typically 5%. Calculate the test statistic and the p-value, then interpret the result in the context of your research question. Remember: a failure to reject H₀ is not proof H₀ is true; it simply means the evidence is insufficient.
假设检验是 WJEC 课程作业的核心，对 IB 内部评估也很有价值。陈述原假设 H₀ 和备择假设 H₁。选择检验统计量（如 t 检验、独立性卡方检验）和显著性水平（通常为 5%）。计算检验统计量和 p 值，然后结合研究问题解释结果。谨记：未能拒绝 H₀ 并不证明 H₀ 为真，仅表示证据不足。
8. Communicating Findings Through Mathematical Writing | 通过数学写作传达发现
Your report is the final product that examiners see. Structure it with an introduction, methodology, analysis, conclusion, and evaluation. Use clear mathematical language and define all symbols the first time they appear. In IB, insert reflective commentary explaining why you made each decision. In WJEC, follow the prescribed structure for statistical investigations: hypothesis, data collection, analysis, conclusion. Consistency, precision, and critical reflection will boost your marks substantially.
报告是评审人看到的最终成果。其结构应包括引言、方法、分析、结论与评价。使用清晰的数学语言，并在符号首次出现时加以定义。在 IB 中，插入反思性评论，解释每个决策的缘由。在 WJEC 中，遵循统计调查的预设结构：假设、数据收集、分析、结论。一致性、精确性和批判性反思将大幅提升分数。
9. Overcoming Common Pitfalls | 克服常见误区
Many students fall into the trap of choosing an overly broad topic, collecting too little data, or relying solely on automated software outputs without understanding the underlying mathematics. Another frequent mistake is ignoring ethical considerations or failing to reference secondary sources. Additionally, do not treat correlation as causation without further evidence. Plan a realistic timeline and seek feedback from your teacher at each stage. Early pilot experiments can reveal flaws before it is too late.
许多学生容易陷入选题过宽、数据量过少或仅依赖自动软件产出而不理解基本数学的陷阱。另一个常见错误是忽视伦理考量或未引用二手资料。此外，若无进一步证据，切勿将相关关系视为因果关系。制定切实可行的时间表，并于每个阶段向老师寻求反馈。早期预实验能在为时已晚之前揭示缺陷。
10. Final Checklist Before Submission | 提交前的最终检查清单
| English Checklist Item | 中文检查项 |
|---|---|
| Is the research question precise and mathematically meaningful? | 研究问题是否精确且具有数学意义？ |
| All variables are defined and units stated | 所有变量均已定义且注明单位 |
| Raw data is present with clear organisation | 原始数据呈现清晰、组织有序 |
| Calculations are shown stepwise (even if technology was used) | 计算过程逐步展示（即使使用了技术） |
| All graphs are labelled with titles and axis descriptions | 所有图形都带有标题和轴标签 |
| Hypothesis test conclusions are correctly interpreted | 假设检验结论解释正确 |
| Limitations of the experiment are honestly discussed | 诚实地讨论了实验的局限性 |
| Formatting follows IB or WJEC submission guidelines | 排版遵循 IB 或 WJEC 提交指南 |
11. Example: Exploring Projectile Motion Through Regression | 示例：通过回归探索抛体运动
ata points. Model the relationship between height (h) and range (R) by a power function R = a hᵇ. Linearise by taking logs: ln R = ln a + b ln h. Perform linear regression on (ln h, ln R) to obtain b and a. Analyse the residuals to assess fit. This demonstration applies curve fitting, logarithm transformation, and technology skills required by both boards.
设想一个实验：将球从坡道推出，测量不同初始高度下的水平射程。记录至少 20 个数据点。用幂函数 R = a hᵇ 对高度 h 和射程 R 的关系建模。通过取对数线性化：ln R = ln a + b ln h。对 (ln h, ln R) 进行线性回归，求出 b 和 a。分析残差以评估拟合优度。这一示例综合运用了曲线拟合、对数变换和技术技能，贴合两个考试委员会的要求。
Linearised model: ln R = ln a + b · ln h
线性化模型：ln R = ln a + b · ln h
12. Reflection and Continuous Improvement | 反思与持续改进
A strong investigation does not end with the conclusion. Spend time reflecting on what you would do differently next time. How could the sampling method be improved? What alternative mathematical models could be tested? For IB, the 'Evaluation' criterion rewards genuine reflection. For WJEC, a discussion of reliability and validity strengthens the final grade. Use this reflective process to deepen your understanding and prepare for higher-level mathematical thinking.
出色的研究不因结论而止步。花时间反思下一次你会有何不同做法。抽样方法如何改进？还有哪些替代数学模型可以检验？对 IB 而言，“评价”标准看重真诚的反思。对 WJEC，关于信度和效度的讨论能增强最终成绩。利用这一反思过程加深理解，为更高层次的数学思维做准备。
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