📚 KS3 Maths: A Guide to Experimental Investigations | KS3 数学：实验操作指南
In KS3 maths, experiments and hands-on investigations bring numbers, shapes, and data to life. This guide will help you plan, carry out, and evaluate mathematical experiments, whether you are measuring circles, testing probability, or exploring number patterns. By following the steps and ideas below, you will develop not only your practical skills but also a deeper understanding of key mathematical concepts.
在 KS3 数学中，实验和动手探究让数字、图形和数据栩栩如生。本指南将帮助你规划、实施和评估数学实验，无论你是在测量圆形、测试概率还是探索数字模式。通过遵循以下步骤和思路，你不仅能锻炼实践技能，还能加深对关键数学概念的理解。
1. What Are Mathematical Experiments? | 什么是数学实验？
A mathematical experiment is any structured activity where you gather data, test a prediction, or discover a rule through observation and measurement. Unlike pure calculation, experiments often involve physical objects such as coins, rulers, protractors, or computer simulations. For example, dropping a drawing pin to see how often it lands point-up is a probability experiment that generates real data.
数学实验是指任何通过观察和测量来收集数据、检验预测或发现规律的结构化活动。与纯计算不同，实验通常涉及实物，如硬币、直尺、量角器或计算机模拟。例如，抛掷图钉看它落地时钉尖朝上的频率，就是一个能产生真实数据的概率实验。
In KS3, experiments help you explore topics like geometry, statistics, and algebra in a concrete way. You might measure the angles of different triangles to discover that they always sum to 180°, or repeatedly roll dice to compare experimental and theoretical probabilities. These investigations allow you to see mathematics as a process of discovery rather than just a set of rules.
在 KS3，实验帮助你以具体的方式探索几何、统计和代数等主题。你可以测量不同三角形的内角，发现它们总和总是 180°，或者反复掷骰子来比较实验概率和理论概率。这些探究让你将数学视为一个发现的过程，而不仅仅是一套规则。
2. Planning Your Investigation | 规划你的探究
Every good experiment starts with a clear question or hypothesis. A hypothesis is a statement you can test, such as ‘The circumference of a circle is always about three times its diameter’ or ‘A fair coin lands on heads half the time.’ Write down your question before you begin, so you know exactly what you are trying to find out.
每个好的实验都从一个明确的问题或假设开始。假设是一个你可以检验的陈述，比如“圆的周长总是大约是其直径的三倍”或“一枚公平硬币正面朝上的概率是一半”。在开始之前写下你的问题，这样你就确切知道你要探究什么。
Next, decide what you will measure and which tools you need. If you are investigating the link between the radius of a circle and its area, you will need a compass, ruler, and squared paper. Make a list of materials and check that all measuring instruments are accurate. Also consider how many trials or measurements you will need – the more data you collect, the more reliable your conclusion is likely to be.
接下来，决定你要测量什么以及需要哪些工具。如果你在研究圆的半径与面积的关系，你将需要圆规、直尺和方格纸。列出材料清单并检查所有测量仪器是否准确。还要考虑你需要多少次试验或测量——收集的数据越多，你的结论可能越可靠。
Finally, think about variables. A variable is anything that can change. In a circle experiment, the diameter is the independent variable (the one you choose or control) and the circumference is the dependent variable (the one you measure). Try to keep other factors, like the measuring tape used, constant to make your test fair.
最后，思考变量。变量是指任何可以变化的东西。在圆形实验中，直径是自变量（你选择或控制的），周长是因变量（你测量的）。尽量保持其他因素不变，比如使用同一条卷尺，以确保测试公平。
3. Measuring and Drawing with Precision | 精确测量与绘图
Accurate measurement is vital for a successful experiment. Use a ruler marked in millimetres for lengths, and a protractor for angles. When measuring a line, always start at the zero mark, not the edge of the ruler, and read the measurement at eye level to avoid parallax errors. Record each measurement carefully, including the unit (mm, cm, m).
精确测量是实验成功的关键。用标有毫米的直尺测量长度，用量角器测量角度。测量线段时，始终从零刻度线（而非尺子边缘）开始，并在视线水平处读取刻度，以避免视差错误。仔细记录每次测量，包括单位（毫米、厘米、米）。
Drawing accurate diagrams is also part of mathematical experiments. For example, when constructing a triangle given three sides, use a sharp pencil and draw light construction lines. Label vertices clearly, and note the lengths and angles you have found. A well-drawn diagram can reveal patterns, such as the symmetry in an isosceles triangle, that you might otherwise miss.
绘制精确的图形也是数学实验的一部分。例如，给定三边画三角形时，使用削尖的铅笔并画出淡淡的辅助线。清楚地标记顶点，并记下你求出的边长和角度。一幅画得好的图形可以揭示你原本可能错过的模式，比如等腰三角形的对称性。
When using digital tools like dynamic geometry software, always check that the measurements shown on screen match your manual ones. Software can speed up investigations, but understanding how to measure by hand is a foundational skill.
当使用动态几何软件等数字工具时，始终检查屏幕上显示的测量值是否与手动测量值匹配。软件可以加快探究速度，但理解如何手动测量是一项基础技能。
4. Exploring Pi (π) Through Measurement | 通过测量探索圆周率（π）
One classic KS3 experiment is to discover the constant π. Gather several circular objects – lids, jars, cans – of different sizes. For each object, carefully measure the circumference (C) using a flexible tape measure or by rolling the object along a ruler. Then measure the diameter (d) across its centre. Record your results in a table.
一个经典的 KS3 实验是发现常数 π。收集几个不同大小的圆形物体——盖子、罐子、罐头。对于每个物体，用软卷尺或沿直尺滚动的方式仔细测量周长（C）。然后测量穿过圆心的直径（d）。将结果记录在表格中。
| Object | Circumference (C) in cm | Diameter (d) in cm | C ÷ d |
|---|---|---|---|
| Lid | 15.7 | 5.0 | 3.14 |
| Can | 22.0 | 7.0 | 3.14 |
| Jar lid | 9.4 | 3.0 | 3.13 |
For each object, calculate C ÷ d. You should find that the ratio is always just over 3, no matter the size. In fact, it approaches a constant value of approximately 3.14159…, which we represent by the Greek letter π. This experiment demonstrates that the circumference of any circle is about 3.14 times its diameter.
对于每个物体，计算 C ÷ d。你会发现，无论大小如何，这个比值总是略大于 3。实际上，它趋近于一个约 3.14159… 的常数，我们用希腊字母 π 表示。这个实验证明，任何圆的周长都大约是其直径的 3.14 倍。
C = π × d → π = C ÷ d
The more precise your measurements, the closer your results will be to the true value of π. If your ratios vary, discuss possible sources of error, such as taping problems or an off-centre diameter measurement. This kind of reflection is an important part of any investigation.
测量越精确，结果就越接近 π 的真实值。如果比值有偏差，讨论可能的误差来源，如卷尺问题或直径测量偏离中心。这种反思是任何探究中的重要部分。
5. Probability Experiments: Tossing Coins and Dice | 概率实验：抛硬币与掷骰子
Probability experiments help you see how chance behaves in the real world and how it compares to theoretical predictions. A simple experiment is to toss a fair coin 50 times and record the number of heads. According to theory, the probability of heads is ½, so you might expect 25 heads. However, in practice, your result may be slightly different due to randomness.
概率实验帮助你了解机会在现实世界中的表现，以及如何与理论预测进行比较。一个简单的实验是抛一枚公平硬币 50 次并记录正面朝上的次数。根据理论，正面的概率是 ½，所以你可能会期待 25 次正面。然而，在实际中，由于随机性，你的结果可能略有不同。
Write down your experimental probability using the formula:
使用下面的公式写下你的实验概率：
Experimental probability = Number of successful outcomes ÷ Total number of trials
For example, if you get 27 heads in 50 tosses, the experimental probability is 27/50 = 0.54. This is close to 0.5, but not exactly equal. As you increase the number of tosses to 100, 200, or more, you should see the experimental probability get closer to the theoretical value – this is called the Law of Large Numbers.
例如，如果你在 50 次抛掷中得到 27 次正面，实验概率是 27/50 = 0.54。这接近 0.5，但并不完全相等。当你把抛掷次数增加到 100、200 或更多时，你会发现实验概率越来越接近理论值——这被称为大数定律。
Dice experiments follow a similar pattern. Roll a fair six-sided die 60 times and record the frequency of each face. The theoretical probability of rolling a 3 is 1/6 ≈ 0.1667. Create a frequency table and compare your empirical frequencies with the expected frequency of 10 for each number. Such experiments build intuition about probability distributions.
骰子实验遵循类似的模式。掷一枚公平的六面骰子 60 次，记录每个面的频率。掷出 3 点的理论概率是 1/6 ≈ 0.1667。创建频率表，并将你的经验频率与每个数字的期望频率 10 进行比较。这类实验能建立关于概率分布的直觉。
6. Data Collection and Recording | 数据收集与记录
Good data collection starts with a structured recording system. Use a table with clear column headings, such as ‘Trial number’, ‘Measured value’, and ‘Notes’. For example, when measuring the angle of a ramp and the distance a toy car travels, your table might look like this:
好的数据收集始于结构化的记录系统。使用具有清晰列标题的表格，例如“试验编号”、“测量值”和“备注”。例如，当测量斜坡角度和玩具车行驶距离时，你的表格可能如下所示：
| Trial | Angle (°) | Distance (cm) |
|---|---|---|
| 1 | 15 | 42 |
| 2 | 15 | 44 |
| 3 | 15 | 43 |
Always note any unusual occurrences during a trial, such as the car hitting an obstacle. These annotations help explain unexpected results later. If you repeat measurements, take the mean of your values to improve reliability.
始终记下试验过程中任何不寻常的情况，例如汽车撞到了障碍物。这些注释有助于日后解释意外结果。如果你重复测量，取数值的平均值以提高可靠性。
In digital experiments or surveys, keep a copy of raw data in a spreadsheet. Sort and filter tools can help you spot trends, but always check for data entry errors before drawing conclusions. A well-kept logbook is a sign of a careful mathematician.
在数字实验或调查中，在电子表格中保存原始数据的副本。排序和筛选工具可以帮助你发现趋势，但在得出结论前，始终检查数据输入错误。一本保存完好的记录本是严谨的数学家的标志。
7. Creating Charts and Graphs | 创建图表
Visual representations make patterns easier to see. For continuous data, line graphs or scatter plots are often best. For example, if you investigated how the area of a square changes with its side length, plot side length (x-axis) against area (y-axis). Label axes clearly and include units.
视觉呈现让模式更容易被发现。对于连续数据，线状图或散点图通常是合适的。例如，如果你探究了正方形的面积如何随边长变化，则绘制边长（x 轴）与面积（y 轴）的图形。清楚地标记坐标轴并包括单位。
For categorical data, such as the frequency of colours in a bag of sweets, use a bar chart or pie chart. In a bar chart, the height of each bar represents the frequency. Ensure that the bars are of equal width and that there are spaces between them. A pie chart shows proportions: each sector angle is calculated as (frequency ÷ total) × 360°.
对于分类数据，例如一袋糖果中颜色的频率，使用条形图或饼图。在条形图中，每个条形的高度代表频率。确保条形宽度相等且它们之间有间隔。饼图显示比例：每个扇区的角度计算为（频率 ÷ 总数） × 360°。
When using digital tools, choose the graph type that fits your data. Avoid 3D effects that can distort perception. Always give your chart a title that explains what it shows, for instance ‘Mean distance travelled by toy car for different ramp angles’.
使用数字工具时，选择适合数据的图表类型。避免使用会扭曲视觉的 3D 效果。始终给图表添加一个能解释其内容的标题，例如“不同斜坡角度下玩具车行驶的平均距离”。
8. Averages and Spread: Finding Mean, Median, Mode, and Range | 平均值与离散度：寻找平均数、中位数、众数和范围
When you have a set of measurements, averages summarise the data, while the range tells you about spread. The mean is what many people call the average:
当你有一组测量值时，平均值可以概括数据，而范围则告诉你数据的离散情况。平均数就是许多人所说的平均值：
Mean = (Sum of all values) ÷ (Number of values)
For example, if five students spend (12, 15, 8, 20, 10) minutes on a puzzle, the mean is (12+15+8+20+10)/5 = 65/5 = 13 minutes. The mean is useful but can be affected by extreme values (outliers).
例如，如果五个学生花在一道谜题上的时间分别是 (12, 15, 8, 20, 10) 分钟，平均数是 (12+15+8+20+10)/5 = 65/5 = 13 分钟。平均数很有用，但可能会受极端值（离群值）的影响。
The median is the middle value when data is ordered. For the set 8, 10, 12, 15, 20, the median is 12. If there is an even number of values, take the mean of the two middle ones. The median is often a better measure when data is skewed. The mode is simply the value that occurs most often. The range is the difference between the largest and smallest values: 20 − 8 = 12 minutes here.
中位数是将数据排序后位于中间的值。对于 8, 10, 12, 15, 20，中位数是 12。如果有偶数个数值，则取中间两个的平均值。当数据偏斜时，中位数通常是更好的度量。众数就是出现最频繁的那个值。范围是最大值与最小值之差：这里 20 – 8 = 12 分钟。
In your experiment report, calculate all relevant averages and the range. Discuss why one average might be more representative than another. For instance, if one coin-toss session gave an unusually high number of heads, the median might be closer to the theoretical expectation than the mean.
在实验报告中，计算所有相关的平均值和范围。讨论为什么一个平均值可能比另一个更具代表性。例如，如果某次抛硬币得到了异常高的正面次数，中位数可能比平均数更接近理论期望。
9. Investigating Number Patterns and Sequences | 探究数字模式和序列
Algebraic thinking can be developed through pattern experiments. A classic KS3 activity is to generate a sequence of shapes, such as matchstick triangles, and count the number of sticks needed for each term.
代数思维可以通过模式实验来培养。一个经典的 KS3 活动是生成一系列形状，比如火柴棍三角形，并计算每项所需的火柴棍数量。
For example, to make a row of n triangles, you might need 2n+1 sticks. Set up a table to record the term number and the sticks counted. Then try to find the rule connecting the two. Writing the rule in words first, then in algebraic symbols, is a powerful way to bridge arithmetic and algebra.
例如，要搭一排 n 个三角形，你可能需要 2n+1 根火柴棍。建立一个表格来记录项数和数出的火柴棍数量。然后尝试找出连接两者的规律。先用文字写出规律，再用代数符号表示，是连接算术和代数的有力方式。
You could also explore sequences generated by a given rule, such as ‘start with 3, then add 5 each time’. List the first six terms and plot them on a graph. The resulting points should lie in a straight line, revealing a linear relationship. Experiments like this make the concept of ‘nth term’ concrete and visual.
你也可以探索由给定规则生成的序列，例如“从 3 开始，然后每次加 5”。列出前六项并在图上描点。得到的点应位于一条直线上，揭示了一个线性关系。像这样的实验使“第 n 项”的概念变得具体而直观。
10. Scale and Proportion: Model Building | 比例与比例：模型构建
Scale models are a hands‑on way to explore ratio and proportion. Choose an object, such as a classroom or a football pitch, and measure its real dimensions. Then decide on a scale, for instance 1 cm represents 1 m (a scale of 1:100). Use this ratio to calculate the model dimensions.
比例模型是探究比和比例的一种动手方式。选择一个物体，例如一间教室或一个足球场，并测量其实际尺寸。然后选定一个比例尺，例如 1 cm 代表 1 m（比例 1:100）。使用此比值计算模型尺寸。
Draw the scaled diagram on paper, carefully converting every length. If the real length is 6 m, the model length will be 6 cm. Check that all lengths have been reduced by the same factor; otherwise, the shape will be distorted. This activity reinforces the idea that similar shapes have proportional sides.
在纸上绘制比例图，仔细转换每个长度。如果实际长度为 6 m，则模型长度为 6 cm。检查所有长度是否按相同因子缩小；否则形状会失真。这个活动强化了相似形状具有成比例边的概念。
You can extend the investigation by calculating areas. If the scale is 1:100, the area scale factor is 1:10000, because area scales by the square of the linear factor. Build a simple physical model using card, and compare its weight or material cost to deduce scaling effects – a beautiful blend of geometry and real‑world application.
你可以通过计算面积来扩展探究。如果线性比例是 1:100，面积比例因子就是 1:10000，因为面积按线性因子的平方缩放。用卡片制作一个简单的实体模型，并比较其重量或材料成本以推断缩放效应——这是几何与现实应用的完美结合。
11. Presenting Your Findings and Evaluating | 展示你的发现与评估
After collecting and analysing data, you need to communicate your findings clearly. Write a short report that includes your initial hypothesis, the method you used, your results (tables and graphs), and your conclusion. State whether your results support the hypothesis or not. Use mathematical vocabulary like ‘experimental probability’, ‘mean’, ‘correlation’ where appropriate.
收集并分析数据后，你需要清晰地交流你的发现。写一份简短的报告，包括你最初的假设、使用的方法、结果（表格和图形）以及结论。陈述结果是否支持假设。在适当的地方使用数学词汇，如“实验概率”、“平均数”、“相关性”。
Evaluation is crucial: discuss any limitations or errors. Were your measurements precise enough? Did you have enough trials? If you repeated the experiment, would you do anything differently? For example, you might note that the scale on your protractor was hard to read, which could explain some variability in angle measurements.
评估至关重要：讨论任何局限性或误差。你的测量是否足够精确？你有足够的试验次数吗？如果重复实验，你会做哪些不同的尝试？例如，你可能会注意到量角器上的刻度难以读取，这可以解释角度测量中的某些变异性。
Finally, suggest how the experiment could be extended. Perhaps you could test more values, change a variable you kept constant, or use a computer simulation to run thousands of trials. An investigative mindset is at the heart of mathematical discovery.
最后，建议如何扩展实验。也许你可以测试更多数值，改变一个你原先保持不变的变量，或者使用计算机模拟运行成千上万次试验。探究式思维是数学发现的核心。
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