📚 IGCSE AQA Maths: A Guide to Experimental Investigations | IGCSE AQA 数学：实验操作指南
Mathematics is more than just abstract symbols and equations. For IGCSE AQA students, experimental investigations offer a hands-on way to explore probability, statistics, and data handling. This guide walks you through planning, conducting, and analysing your own maths experiments, helping you develop key practical skills assessed in the course.
数学不仅仅是抽象的符号和方程。对于 IGCSE AQA 学生来说，实验探究是一种亲手探索概率、统计和数据处理的方法。本指南将带你一步步规划、实施和分析自己的数学实验，帮助你培养课程评估中的关键实用技能。
1. Understanding Mathematical Experiments | 理解数学实验
A mathematical experiment involves collecting real-world or simulated data to test a hypothesis or explore a mathematical concept. It can range from tossing coins to investigate theoretical probability to measuring heights in class for statistical analysis.
数学实验涉及收集真实世界或模拟的数据，以检验假设或探索数学概念。它的范围可以从抛硬币研究理论概率，到测量班级同学的身高进行统计分析。
The core aim is to connect theory with practice. You will formulate a research question, gather data systematically, present it clearly, and interpret your findings using mathematical reasoning.
其核心目标是将理论与实践联系起来。你将提出一个研究问题，系统地收集数据，清晰地展示数据，并运用数学推理来解释你的发现。
2. Planning Your Experiment | 规划你的实验
Begin by defining a clear hypothesis. For example: ‘A fair dice is equally likely to land on each face’ or ‘Taller students tend to have larger hand spans’. Your hypothesis must be testable using data you can collect.
首先定义一个明确的假设。例如：“一个公平的骰子每个面朝上的可能性相等” 或 “个子较高的学生手长往往更大”。你的假设必须能够用你能收集到的数据进行检验。
Next, consider what data is needed. Decide on the sample size — larger samples give more reliable results but require more time. A sample size of at least 30 is often recommended for statistical experiments.
接下来，考虑需要什么数据。确定样本量 —— 样本量越大，结果越可靠，但需要更多时间。对于统计实验，通常建议样本量至少为 30。
Prepare data recording sheets in advance. Clearly label columns for each variable. If you are conducting a survey, design unbiased questions that do not lead respondents.
提前准备数据记录表。为每个变量清晰地标记列。如果你在进行调查，设计无偏见的、不引导受访者的问题。
3. Collecting Data Reliably | 可靠地收集数据
Consistency is key. If you are measuring, use the same equipment and standard units (cm, grams, seconds). When using a random process like rolling dice, ensure conditions remain unchanged — roll on a flat surface, from the same height.
一致性是关键。如果你在测量，请使用相同的设备和标准单位（厘米、克、秒）。当进行如掷骰子这样的随机过程时，确保条件不变 —— 在平坦的表面上，从相同高度掷出。
Record data immediately and accurately. Avoid rounding too early. If a measurement lies between markings on a ruler, estimate to the nearest half division to maintain precision.
立即准确地记录数据。避免过早四舍五入。如果测量值位于尺子刻度线之间，请估读到最近的半个刻度，以保持精度。
For probability experiments, repeat trials many times. The more repetitions, the closer the experimental probability gets to the theoretical probability (the Law of Large Numbers).
对于概率实验，多次重复试验。重复次数越多，实验概率就越接近理论概率（大数定律）。
4. Organising Raw Data | 整理原始数据
Raw data is messy. Use frequency tables to group data. For discrete data, list each possible value alongside its tally and frequency. For continuous data, create intervals (class intervals) ensuring no gaps or overlaps.
原始数据是杂乱的。使用频率表对数据进行分组。对于离散数据，列出每个可能的值及其计数和频率。对于连续数据，创建区间（组距），确保没有间隙或重叠。
| Height (cm) | Tally | Frequency |
|---|---|---|
| 150 ≤ h < 155 | IIII | 4 |
| 155 ≤ h < 160 | IIII I | 6 |
| 160 ≤ h < 165 | IIII II | 7 |
| 165 ≤ h < 170 | III | 3 |
Make sure the total frequency matches the number of data points you collected. This simple check prevents many errors.
确保总频率与你收集的数据点数量相匹配。这个简单的检查可以防止许多错误。
5. Presenting Data Graphically | 图形化呈现数据
Graphs make patterns visible. For discrete data, use bar charts with equal-width bars and clear labels on both axes. For continuous grouped data, histograms are suitable — remember that frequency is represented by bar area, but with equal class intervals, height shows frequency.
图表使规律可见。对于离散数据，使用条形图，条宽相等，两轴清洁标记。对于连续分组数据，直方图是合适的 —— 请记住，频率由直方条的面积表示，但在组距相等的情况下，高度表示频率。
Pie charts are excellent for showing proportions of a whole. Calculate each sector angle: (frequency ÷ total frequency) × 360°. Plot carefully with a protractor.
饼图非常适合显示整体中各部分的比例。计算每个扇形的角度：（频率 ÷ 总频率）× 360°。用半圆仪仔细绘制。
Scatter graphs help investigate correlation between two variables. Plot points precisely and draw a line of best fit if the relationship appears linear. Do not force the line through the origin unless data suggests it.
散点图有助于研究两个变量之间的相关性。准确描点，如果关系看起来是线性的，画出最佳拟合线。除非数据表明，否则不要强行让线通过原点。
6. Probability Experiments | 概率实验
Probability experiments test theoretical predictions. If you flip a fair coin 100 times, you expect about 50 heads. In reality, you might get 53 heads. The experimental probability is 53/100 = 0.53.
概率实验可以检验理论预测。如果你抛一枚公平的硬币 100 次，你预计约 50 次正面。实际上，你可能会得到 53 次正面。实验概率是 53/100 = 0.53。
Use the formula: Experimental probability = Number of successful outcomes / Total number of trials. As trials increase, this converges to theoretical probability.
使用公式：实验概率 = 成功结果的次数 / 试验总次数。随着试验次数增加，它会收敛到理论概率。
Combined events can be explored with two-step experiments. Use a two-way table or sample space diagram to list all outcomes before comparing with experimental frequencies.
组合事件可以通过两步实验来探索。在与实验频率进行比较之前，使用双向表或样本空间图列出所有可能结果。
7. Simulations in Maths | 数学模拟
When real experiments are impractical, simulations can model random processes. For example, you can use a random number generator or a spreadsheet to simulate rolling dice thousands of times.
当真实实验不切实际时，模拟可以对随机过程进行建模。例如，你可以使用随机数生成器或电子表格来模拟成千上万次掷骰子。
To simulate a coin flip using random numbers, assign 0-4 to heads and 5-9 to tails if using single digits 0-9. Clearly state your simulation rules in your report.
要使用随机数模拟抛硬币，如果使用个位数 0-9，则将 0-4 分配给正面，5-9 分配给反面。在报告中明确说明你的模拟规则。
Simulations allow you to explore probabilities of complex systems, like estimating π by dropping ‘matchsticks’ on lined paper (Buffon’s needle). Though not in the core syllabus, understanding simulation broadens experimental thinking.
模拟使你可以探索复杂系统的概率，比如通过向划线纸上投掷“火柴棒”来估算 π（蒲丰投针问题）。虽然不在核心大纲中，但理解模拟可以拓宽实验思维。
8. Sampling Techniques | 抽样技术
In larger investigations, you often need a sample from a population. A simple random sample gives every member an equal chance of selection. Use a random number generator or ‘names out of a hat’.
在更大规模的调查中，你经常需要从总体中抽取样本。简单的随机抽样使每个成员被选中的机会均等。可以使用随机数生成器或“抽签”的方法。
Stratified sampling ensures subgroups are proportionally represented. If 40% of your school are boys, your sample should be 40% boys. Calculate number from each stratum: (group size ÷ total population) × sample size.
分层抽样确保子组按比例代表。如果学校 40% 是男孩，你的样本中应有 40% 是男孩。计算每个层的样本数：（组大小 ÷ 总体大小）× 样本量。
Avoid convenience sampling, like only asking friends, as it introduces bias. Always discuss limitations of your sampling method in your evaluation.
避免便利抽样，比如只询问朋友，因为这会引入偏差。务必在评价中讨论你所用抽样方法的局限性。
9. Analysing Results | 结果分析
After collecting and presenting data, calculate measures of central tendency — mean, median, mode. For the mean, use: Mean = Σx / n (sum of all values divided by count).
在收集和呈现数据之后，计算集中趋势的度量 —— 平均数、中位数、众数。对于平均数，使用：平均数 = Σx / n（所有数值之和除以个数）。
Measure spread with range (max − min) or interquartile range (IQR = Q₃ − Q₁). Larger spread indicates more variability in your data.
用极差（最大值 − 最小值）或四分位距（IQR = Q₃ − Q₁）来衡量离散程度。较大的离散程度表明数据变异性更大。
If you have a scatter graph, calculate Spearman’s rank or simply describe correlation strength. Note potential outliers and whether they could be errors or genuine interesting points.
如果你有散点图，可以计算斯皮尔曼等级相关系数，或简单地描述相关性的强度。注意潜在的异常值，判断它们是误差还是真正有趣的数据点。
10. Drawing Conclusions and Evaluation | 得出结论与评价
Return to your original hypothesis. Does the evidence support it? Avoid claiming ‘proof’. Instead, say ‘the data suggests…’ or ‘the results are consistent with…’. Use your analysed statistics to justify your conclusion.
回到你最初的假设。证据是否支持它？避免宣称“证明”。相反，要说“数据表明……”或“结果与……相一致”。使用你分析过的统计数据来论证你的结论。
Evaluate your experiment honestly. Discuss limitations, such as small sample size, measurement errors, or biased sampling. Suggest improvements if you were to repeat the investigation.
诚实地评价你的实验。讨论局限性，例如样本量小、测量误差或抽样偏差。如果重复调查，提出改进建议。
Your final report should have sections: Introduction, Method, Results (tables and graphs), Analysis, Conclusion, Evaluation. A well-structured report demonstrates full experimental thinking.
你的最终报告应包括以下部分：引言、方法、结果（表格和图表）、分析、结论、评价。结构良好的报告展示了完整的实验思维。
11. Example Investigation: Dice Fairness | 示例探究：骰子的公平性
Hypothesis: A six-sided dice is fair, so each number 1-6 has probability 1/6. Roll the dice 120 times, recording each outcome. Organise frequencies in a table.
假设：一个六面骰子是公平的，因此每个数字 1-6 的概率为 1/6。将骰子投掷 120 次，记录每次结果。用表格整理频率。
Expected frequency for each face is 120/6 = 20. Compare observed frequencies. Use a bar chart to visualise deviations. If the dice shows 35 ones and 10 threes, it may be biased.
每个面的期望频率是 120/6 = 20。比较观察频率与期望频率。使用条形图可视化偏差。如果骰子显示 35 次 1 点和 10 次 3 点，可能是有偏的。
Calculate experimental probability for each number. Discuss possible reasons for unfairness (weight imbalance). Evaluate by suggesting more rolls or a different dice.
计算每个数字的实验概率。讨论不公平的可能原因（重量不平衡）。通过建议更多投掷次数或更换骰子进行评价。
12. Common Pitfalls and How to Avoid Them | 常见错误与避免方法
Not controlling variables: Ensure only the intended variable changes. In a coin flip, using different hands or surfaces can affect outcomes. Standardise your procedure.
没有控制变量：确保只有预期的变量在变化。在抛硬币时，使用不同的手或表面会影响结果。将你的操作标准化。
Misleading graphs: An axis that doesn’t start at zero can exaggerate differences. Always label axes and use appropriate scales. For histograms, use frequency density if class intervals are unequal.
误导性图表：不从零开始的坐标轴会夸大差异。始终标记坐标轴并使用合适的刻度。对于直方图，如果组距不等，使用频率密度。
Overlooking sample size: Conclusions from only 10 trials are unreliable. Aim for enough data to see patterns. Record any anomalies transparently.
忽视样本量：仅从 10 次试验中得出结论是不可靠的。目标是获取足够的数据来发现规律。透明地记录任何异常现象。
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