📚 GCSE CIE Mathematics: Practical Investigation Guide | GCSE CIE 数学：实验操作指南
In GCSE CIE Mathematics, practical investigations are not conducted in a laboratory with test tubes, but through carefully designed mathematical experiments. These activities involve collecting data, using geometrical instruments, running probability simulations, and analysing statistical information. A clear experimental guide helps you move from a mathematical question to a reliable conclusion using evidence, logic, and the correct tools.
在 GCSE CIE 数学课程中，实验探究并不是在实验室里使用试管，而是通过精心设计的数学实验来完成的。这些活动包括收集数据、使用几何工具、运行概率模拟以及分析统计信息。一份清晰的实验操作指南可以帮助你从数学问题出发，凭借证据、逻辑和正确工具得出可靠结论。
1. Understanding Mathematical Experiments | 理解数学实验
A mathematical experiment is a structured process of exploration that tests a hypothesis, uncovers a pattern, or estimates a probability. It might involve rolling dice a hundred times, measuring angles in a polygon, or surveying a sample of students. The goal is to use repeated observation, careful measurement, and tabulation to produce results that can be verified.
数学实验是一个结构化的探索过程，用于检验假设、揭示模式或估计概率。它可能涉及掷一百次骰子、测量多边形的角度，或调查一组学生样本。其目标是利用重复观察、仔细测量和表格化来产生可验证的结果。
Unlike physics or chemistry, mathematical experiments often rely on simulated randomness or geometric construction. You are expected to plan the procedure, select appropriate apparatus, record data systematically, and interpret the outcome within the context of the problem.
与物理或化学不同，数学实验通常依赖于模拟随机性或几何作图。你需要规划步骤、选择合适的工具、系统记录数据，并在问题背景下解释结果。
Recognise that every experiment has limitations: measurement errors, sampling bias, and the difference between theoretical and experimental probability. Acknowledge these in your evaluation.
要认识到每个实验都有局限性：测量误差、抽样偏差，以及理论概率与实验概率之间的差异。在评估中应说明这些局限。
2. Essential Apparatus and Their Correct Use | 基本工具及其正确使用
For geometric investigations, you need a ruler (cm and mm), a protractor, a pair of compasses, a pencil, a sharpener, and an eraser. A ruler must be placed so that the zero mark aligns with one end of the segment, not the edge of the ruler. Avoid parallax error by looking straight down at the scale.
进行几何探究时，需要一把刻度为厘米和毫米的直尺、一个量角器、一副圆规、一支铅笔、一个削笔刀和一块橡皮。直尺的放置应使零刻线与线段的一端对齐，而不是对齐尺子的边缘。直视刻度线以避免视差错误。
When using a protractor, always place the centre spot exactly at the vertex of the angle. Align the baseline with one arm of the angle, then read the measurement from the appropriate scale (inner or outer). Practise drawing and measuring angles of 30°, 45°, 95°, and 210° to build fluency.
使用量角器时，务必将中心点精确放置在角的顶点上。将量角器的底线与角的一条边对齐，然后从相应的刻度（内圈或外圈）读取角度值。练习绘制和测量 30°、45°、95° 和 210° 等角度以培养熟练度。
Compasses are used for constructing perpendicular bisectors, angle bisectors, circles, and arcs. Keep the compass tight enough so the radius does not change during drawing. Always mark the centre point clearly and draw arcs lightly for neatness. For accurate measurement, use a ruler with a sharp pencil tip.
圆规用于作垂直平分线、角平分线、圆和弧。保持圆规紧度适当，使得绘制过程中半径不会改变。始终清晰地标记圆心，并轻绘弧线以保持整洁。为了准确测量，使用削尖的铅笔配合直尺。
For statistical experiments, you may use dice, coins, spinners, or random number generators. Record outcomes in real time using tally marks. Organise data in frequency tables or spreadsheets.
对于统计实验，你可以使用骰子、硬币、转盘或随机数生成器。使用记数符号实时记录结果。将数据整理在频数表或电子表格中。
3. Designing a Geometric Experiment | 设计几何实验
Geometric experiments often test properties of shapes: the sum of interior angles of a polygon, the relationship between sides and angles in a triangle, or the constant of π (pi) through measuring circumference and diameter. Begin with a clear aim: for instance, ‘Investigate the sum of exterior angles of any convex polygon.’
几何实验通常检验图形的性质：多边形的内角和、三角形中边与角的关系，或通过测量周长和直径探究 π 的常量。从一个明确的目标开始：例如，“探究任何凸多边形的外角和”。
Draw several polygons — triangle, quadrilateral, pentagon, hexagon — accurately using ruler and protractor. Measure each exterior angle as shown below, where an exterior angle is formed by extending one side of the polygon. Record the measurements in a table.
使用直尺和量角器精确绘制多个多边形——三角形、四边形、五边形、六边形。测量每个外角，如图所示，外角是由延长多边形的一条边而形成的。将测量结果记录在表格中。
For the triangle, after measuring exterior angles a, b, c, calculate a + b + c. Repeat for the quadrilateral and pentagon. You should observe that the sum is always close to 360°, with small variations due to measurement errors.
对于三角形，测量外角 a、b、c 后，计算 a + b + c。对四边形和五边形重复实验。你应该观察到，除去测量误差造成的小幅波动外，总和始终接近 360°。
| Polygon | Number of sides n | Measured exterior angles sum (°) | Expected sum 360°? |
|---|---|---|---|
| Triangle | 3 | 359.5 | Yes (within error) |
| Quadrilateral | 4 | 360.8 | Yes |
| Pentagon | 5 | 359.2 | Yes |
Conclude that the sum of exterior angles is 360° regardless of the number of sides, illustrating a key geometric invariant.
得出结论：不论边数多少，外角和总是 360°，这证明了一个关键的几何不变量。
4. Measurement, Accuracy, and Error Analysis | 测量、精度与误差分析
Every measurement has an inherent uncertainty. In CIE Mathematics, you are expected to record lengths to the nearest millimetre and angles to the nearest degree. The maximum error for a single reading is half of the smallest division on the instrument. For a ruler marked in millimetres, the absolute error is ±0.5 mm.
每次测量都存在固有不确定性。在 CIE 数学中，要求将长度记录到最接近的毫米，将角度记录到最接近的度。单次读数的最大误差是仪器最小刻度的一半。对于标记毫米的直尺，绝对误差为 ±0.5 mm。
When combining measurements, errors propagate. For example, if you calculate the area of a rectangle by measuring length L and width W, the relative error in area can be estimated as (ΔL/L + ΔW/W). Always compute possible upper and lower bounds.
当合并测量值时，误差会传播。例如，如果通过测量长 L 和宽 W 计算矩形面积，面积中的相对误差可估算为 (ΔL/L + ΔW/W)。始终计算可能的上下界。
If an experiment asks you to measure the diameter d of a circle and compute the circumference C = πd, compare your measured circumference (using a string) with the theoretical value. Calculate percentage error:
如果实验要求测量圆的直径 d 并计算周长 C = πd，将用绳子测得的周长与理论值进行比较。计算百分比误差：
Percentage error = (|Measured value − Theoretical value| / Theoretical value) × 100%
Understanding error helps you evaluate the reliability of your experiment and justify discrepancies.
理解误差有助于评估实验的可靠性，并为差异提供合理理由。
5. Probability Experiments: Frequency and Estimation | 概率实验：频率与估计
Probability experiments aim to estimate the likelihood of an event by repeated trials. The experimental probability (relative frequency) is given by:
概率实验旨在通过重复试验估计事件发生的可能性。实验概率（相对频数）由下式给出：
Experimental probability = Number of successful trials / Total number of trials
Carry out an experiment with a biased coin. Flip it 50 times, recording heads and tails. Then compute the relative frequency of heads. The more trials, the closer the experimental probability tends to the theoretical probability (Law of Large Numbers).
用一枚不均匀的硬币进行实验。抛掷 50 次，记录正面和反面朝上的次数。然后计算正面的相对频数。试验次数越多，实验概率越接近理论概率（大数定律）。
Design a spinner with sectors labelled 1 to 5. The theoretical probability of landing on 3 is 1/5 = 0.2 if sectors are equal. Spin 100 times and record outcomes. Compare the experimental probability with 0.2. Is there evidence the spinner is fair?
设计一个分为 1 至 5 扇形的转盘。若各扇形面积相等，则转到 3 的理论概率为 1/5 = 0.2。旋转 100 次并记录结果。将实验概率与 0.2 进行比较。是否有证据表明转盘是公平的？
Use tally charts and bar charts to visualise the distribution. Large enough samples reduce random fluctuation. Explain that unexpected results might occur in small samples; for instance, flipping a coin 10 times could yield 7 heads and 3 tails, but that does not mean the coin is biased.
使用计数表和条形图将分布可视化。足够大的样本可以减少随机波动。解释在小型样本中可能出现意外结果；例如，抛硬币 10 次可能出现 7 次正面、3 次反面，但这并不表示硬币有偏差。
6. Simulation Using Random Digits | 使用随机数字进行模拟
Some real‑world problems cannot be physically repeated easily. Mathematical simulation uses random numbers to model these situations. For instance, simulate the gender distribution of 20 babies, assuming P(boy) = 0.5. You can assign digits: 0–4 represent boy, 5–9 represent girl. Read off two‑digit blocks from a random number table or use a calculator’s random function.
有些现实问题不容易实际重复。数学模拟使用随机数对这些情况进行建模。例如，模拟 20 个婴儿的性别分布，假设 P(男孩) = 0.5。可以分配数字：0–4 代表男孩，5–9 代表女孩。从随机数表中读取两位数块，或使用计算器的随机函数。
To simulate the outcome of a tennis match where the probability of Player A winning a point is 0.6, let digits 1–6 indicate a point for A, and 7–9, 0 indicate a point for B. Simulate points until one player reaches 4 points with a margin of 2. Run the simulation 10 times and estimate the probability that Player A wins the match.
要模拟网球比赛，其中选手 A 赢得一分的概率为 0.6，设数字 1–6 表示 A 得分，7–9、0 表示 B 得分。模拟得分过程直至一方先得 4 分且净胜 2 分。运行模拟 10 次，估计选手 A 赢得比赛的概率。
Recording simulation data in a tree diagram or table helps you verify outcomes. This method connects probability theory with practical investigation.
在树形图或表格中记录模拟数据有助于验证结果。这一方法将概率理论与实际探究联系起来。
7. Statistical Data Collection: Surveys and Sampling | 统计数据收集：调查与抽样
Many mathematical experiments involve collecting real‑world data via questionnaires or measurements. Define the population and choose a sampling method: random, stratified, or systematic. For example, to investigate the average time students spend on homework, you could draw a stratified sample based on year group, ensuring proportional representation.
许多数学实验涉及通过问卷或测量收集真实数据。定义总体并选择抽样方法：随机抽样、分层抽样或系统抽样。例如，要调查学生花在家庭作业上的平均时间，你可以根据年级进行分层抽样，确保比例代表性。
Avoid bias: do not only survey your friends or only those who are easy to reach. Frame questions clearly, without leading language. Pilot the questionnaire on a small group to identify ambiguous wording.
避免偏差：不要只调查你的朋友或那些容易接触到的人。设计的题目要清晰，不要使用引导性语言。先在一小群人中试调查，以找出含糊不清的表述。
Record responses as numerical data where possible, e.g. using a scale from 1 to 5 for agreement. Prepare a tally sheet and summarise data in frequency tables, calculating mean, median, mode, and range.
尽可能将回答记录为数值数据，例如使用 1 到 5 的同意程度量表。准备一张计数表，并将数据汇总在频数表中，计算平均数、中位数、众数和极差。
8. Organising and Displaying Experimental Data | 组织与展示实验数据
Once data is collected, it must be presented clearly. For discrete data with few values, a bar chart is appropriate. For continuous data, use a histogram or a frequency polygon. Draw axes with uniform scales and label them. The title of the chart must describe what is being shown.
收集到数据后，必须清晰地展示。对于数值较少的离散数据，条形图是合适的。对于连续数据，使用直方图或频数多边形。绘制标度一致的坐标轴并进行标注。图表的标题必须说明所展示的内容。
When comparing two distributions, a back‑to‑back stem‑and‑leaf plot or a comparative box‑and‑whisker plot is extremely useful. In CIE questions, you may need to construct these by hand, so practise finding median, quartiles, and interquartile range quickly. The box plot visualises spread and skew.
在比较两个分布时，背靠背茎叶图或比较箱线图极其有用。在 CIE 考试中，你可能需要手工绘制这些图形，因此要练习快速求中位数、四分位数和四分位距。箱线图直观展示了离散程度和偏态。
For probability experiments, a simple table of relative frequencies against number of trials demonstrates convergence. You can also plot a running relative frequency graph to illustrate the law of large numbers.
对于概率实验，一张关于试验次数和相对频数的简单表格可以展示收敛过程。你还可以绘制累积相对频率图来说明大数定律。
9. Analysing Results and Drawing Conclusions | 分析结果并得出结论
Analysis goes beyond stating the numbers. In a geometric experiment, compare measured properties with known theoretical values. Discuss any outliers or unexpected patterns. For example, a measured circle circumference might differ from πd because the string slipped or the ruler was misaligned.
分析不能只陈述数字。在几何实验中，将测量到的性质与已知理论值进行比较。讨论任何异常值或意外模式。例如，测得的圆的周长可能与 πd 不同，因为绳子滑动或直尺未对齐。
In a statistical experiment, interpret the mean and median in the context of the question. If the mean homework time is 2.5 hours with a standard deviation of 0.8 hour, comment on variability. Evaluate whether the sample is representative.
在统计实验中，结合问题情境解读平均数和中位数。如果平均作业时间是 2.5 小时，标准差为 0.8 小时，对变异性做出评论。评估样本是否具有代表性。
Probability experiments produce experimental relative frequencies. Compare them with theoretical probabilities using a simple difference or percentage. If the experimental probability deviates by more than 0.05 for 200 trials, suspect bias in the apparatus or in the procedure.
概率实验产生实验相对频率。通过简单的差值或百分比与理论概率进行比较。如果在 200 次试验中，实验概率偏差超过 0.05，则怀疑仪器或过程中存在偏差。
Always write a concise conclusion that answers the original aim, and suggest improvements such as increasing the number of trials, using more precise instruments, or refining the sampling method.
始终撰写简洁的结论，回答最初的研究目的，并提出改进建议，例如增加试验次数、使用更精确的仪器或改进抽样方法。
10. Using Technology: Calculators and Spreadsheets | 使用科技：计算器和电子表格
CIE Mathematics allows the use of scientific calculators. Use the STAT mode to enter data and compute mean, standard deviation, and quartiles. The random number function (RAND or RAN#) generates numbers between 0 and 1; multiply and round to simulate dice or coin flips.
CIE 数学允许使用科学计算器。使用统计（STAT）模式输入数据，计算平均数、标准差和四分位数。随机数函数（RAND 或 RAN#）生成 0 到 1 之间的数字；相乘并取整后可以模拟掷骰子或抛硬币。
Spreadsheets such as Excel or Google Sheets are powerful for simulations. Use formulas like =RANDBETWEEN(1,6) to simulate a die roll. Fill down for 1000 trials, then summarise with COUNTIF. Plot histograms quickly to see the distribution shape.
Excel 或 Google Sheets 等电子表格在模拟中功能强大。使用公式如 =RANDBETWEEN(1,6) 模拟掷骰子。向下填充 1000 次试验，然后用 COUNTIF 进行汇总。快速绘制直方图以观察分布形状。
For geometric experiments, dynamic geometry software (e.g., GeoGebra) lets you construct polygons, measure angles, and observe invariants instantly. While not required in the examination hall, it is an excellent tool for investigation during lessons.
对于几何实验，动态几何软件（例如 GeoGebra）可以让你构建多边形、测量角度并即时观察不变量。虽然考场内不作硬性要求，但它是课堂探究的绝佳工具。
When using technology, always cross‑check results with manual calculations to deepen understanding.
使用科技时，务必用手工计算进行核对，以加深理解。
11. Writing an Experimental Report | 撰写实验报告
A well‑structured report follows the AIM – METHOD – RESULTS – CONCLUSION – EVALUATION sequence. State the aim as a clear question or hypothesis. Describe the apparatus and step‑by‑step procedure so that another student could replicate it. Include a diagram if necessary.
一篇结构良好的实验报告遵循目的 – 方法 – 结果 – 结论 – 评估的顺序。将目的陈述为一个明确的问题或假设。描述仪器和逐步步骤，以便其他学生可以重复实验。如有必要，附上示意图。
In the results section, present raw data in tables, then processed data (e.g. averages, probabilities). Use charts to visualise key findings. The conclusion must directly answer the aim and refer to the data. For instance, ‘From the data, the sum of exterior angles is approximately 360°, supporting the theory.’
在结果部分，以表格形式呈现原始数据，然后是经过处理的数据（如平均数、概率）。利用图表将主要发现可视化。结论必须直接回答研究目的并引用数据。例如，“根据数据，外角和约为 360°，支持该理论。”
In the evaluation, identify sources of error, suggest improvements, and consider whether the experiment could be extended to further test the same concept. Acknowledge limitations honestly; it demonstrates critical thinking.
在评估中，识别误差来源、提出改进建议，并考虑该实验是否可以扩展以进一步检验同一概念。诚实地承认局限性体现了批判性思维。
12. Common Pitfalls and How to Avoid Them | 常见陷阱及其规避方法
Many students misuse instruments: not resetting the compass radius, reading the wrong protractor scale, or misaligning the ruler. Regularly check that your equipment is working and your technique is correct. When recording data, avoid rounding too early; keep intermediate values to at least one more significant figure than the final result requires.
许多学生滥用仪器：未重新设置圆规半径、读错量角器刻度，或直尺未对齐。定期检查设备是否正常、技术是否正确。记录数据时，避免过早舍入；中间值的有效数字至少要比最终结果要求的多保留一位。
In probability experiments, insufficient trials lead to misleading probabilities. If time is limited, combine results with classmates to increase total trials. When sampling, beware of voluntary response bias — people who choose to respond may not represent the whole population.
在概率实验中，试验次数不足会导致误导性概率。如果时间有限，可与同学合并结果以增加总试验次数。在抽样时，要警惕自愿响应偏差——选择响应的人可能不代表整个总体。
Never manipulate data to fit a preconceived result. Genuine anomalies can lead to deeper mathematical insight. Finally, present your work neatly; untidy tables or graphs can obscure patterns and lose marks in coursework.
切勿为了符合预设结论而捏造数据。真实的异常数据可以引发更深入的数学思考。最后，将作业呈现整洁；凌乱的表格或图表会掩盖规律并导致作业失分。
By following this guide, you will not only strengthen your investigative skills but also deepen your understanding of how mathematics models the real world — a core objective of the CIE GCSE Mathematics curriculum.
遵循本指南，你不仅能增强探究技能，还能深刻理解数学如何为现实世界建模——这是 CIE GCSE 数学课程的核心目标。
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