KS3 Advanced Maths: Experimental Investigation Guide | KS3 进阶数学：实验操作指南

📚 KS3 Advanced Maths: Experimental Investigation Guide | KS3 进阶数学：实验操作指南

Mathematics is not just about solving textbook problems; it is a living subject full of inquiry and discovery. In this guide, you will learn how to carry out hands-on experiments and investigations that deepen your understanding of KS3 advanced maths topics, from probability to geometry, patterns to data handling. Each section provides clear steps, tips, and recording methods to help you think like a mathematician.

数学不仅仅是解课本上的题目；它是一门充满探究与发现的活学问。在这份指南中，你将学习如何进行动手实验和探究活动，从而加深对 KS3 进阶数学主题的理解，范围涵盖概率、几何、规律模式以及数据处理。每一节都提供清晰的步骤、建议和记录方法，帮助你像数学家一样思考。

1. Setting Up Your Toolkit | 准备实验工具

Before starting any investigation, gather the essential tools. You will need a ruler, a protractor, a pair of compasses, a sharp pencil, squared paper, plain paper, and a scientific calculator. For probability experiments, have coins and dice ready. A notebook for recording observations and a laptop or tablet with dynamic geometry software (such as GeoGebra) can also be very useful.

在开始任何探究之前，先准备好基本工具。你需要一把直尺、一个量角器、一副圆规、一支尖铅笔、方格纸、白纸和一个科学计算器。进行概率实验时，要准备好硬币和骰子。还需要一本记录观察结果的笔记本，以及装有动态几何软件（如 GeoGebra）的笔记本电脑或平板电脑，这些也会非常有用。

Keep your workspace organised. Label each investigation clearly, and always note the date and the question you are trying to answer. A well-prepared toolkit allows you to focus on the mathematics rather than searching for equipment.

保持工作区域整洁。为每项探究清楚地标注标题，并始终记下日期和你试图回答的问题。准备充分的工具能让你专注于数学本身，而不是四处寻找器材。

2. Probability Experiments: Flipping Coins | 概率实验：抛硬币

Probability experiments help you understand the difference between theoretical and experimental probability. Start with a fair coin. Predict the probability of getting heads: theoretically it is ½. Flip the coin 50 times, recording each outcome in a tally chart. Calculate the experimental probability by dividing the number of heads by 50.

概率实验有助于你理解理论概率与实验概率之间的差异。从一枚均匀硬币开始。预测得到正面的概率：理论上它是 ½。抛硬币 50 次，用正字计数表记录每次的结果。用正面次数除以 50 计算实验概率。

Experimental P(heads) = (Number of heads) ÷ 50

实验 P(正面) = 正面次数 ÷ 50

Repeat the experiment with 100 flips. Notice how the experimental probability tends to get closer to 0.5 as the number of trials increases. This demonstrates the Law of Large Numbers. Record your results in a simple table.

用 100 次抛掷重复实验。观察随着试验次数的增加，实验概率如何趋向于接近 0.5。这就演示了大数定律。用一个简单的表格记录你的结果。

Number of flips (抛掷次数)	Number of heads (正面次数)	Experimental probability (实验概率)
50
100

3. Rolling Dice and Exploring Outcomes | 掷骰子与结果探究

When you roll a fair six-sided die, there are six equally likely outcomes: 1, 2, 3, 4, 5, 6. The theoretical probability of rolling a 3 is ₁/₆. To explore this, roll a die 60 times, tally the results, and draw a bar chart of frequencies. Compare the shape of your chart to a uniform distribution.

抛掷一枚均匀的六面骰子时，有六种等可能的结果：1、2、3、4、5、6。掷出 3 的理论概率是 ₁/₆。为了探究这一点，掷骰子 60 次，用正字计数表记录结果，并画一张频率条形图。将你图表的形状与均匀分布进行比较。

Now investigate the sum of two dice. List all 36 possible ordered pairs. Count how many ways you can obtain sums from 2 to 12. The sum of 7 has the highest theoretical probability (⁶/₃₆ = ₁/₆). Test this by rolling two dice 100 times and recording the sum each time. Compare your experimental probabilities with the theoretical values.

现在探究两颗骰子的和。列出所有 36 种可能的有序数对。统计得到从 2 到 12 各个和的方法数。和为 7 的理论概率最高（⁶/₃₆ = ₁/₆）。通过投掷两颗骰子 100 次并记录每次的和来检验这一结论。将你的实验概率与理论值进行比较。

P(sum = 7) = 6/36 = 1/6

P(和为 7) = 6/36 = 1/6

4. Collecting and Charting Data | 收集数据与绘制图表

Data collection is at the heart of many mathematical investigations. Choose a question, such as ‘How many books do students in my class read per month?’ Then design a simple data recording sheet or a digital form. Collect responses from at least 30 participants. Organise the raw data into a frequency table, grouping the data into intervals if necessary.

数据收集是许多数学探究的核心。选择一个研究问题，例如“我班上的学生每个月读多少本书？”然后设计一张简单的数据记录表或一份电子表单。从至少 30 位参与者那里收集回答。将原始数据整理成频率表，如有必要可将数据分组到区间。

Represent your data visually using a bar chart for discrete data or a histogram for continuous data. You can also draw a pie chart to show proportions. Always label axes, provide a title, and use consistent scales. Calculate the mean, median, and mode to summarise the data set. The mean is found by summing all values and dividing by the number of values.

使用条形图表示离散数据，或用直方图表示连续数据，将你的数据直观地展示出来。你也可以绘制饼图来展示比例。务必标注坐标轴、提供标题并使用一致的刻度。计算平均数、中位数和众数以概括数据组。平均数通过将所有数值相加再除以数值的个数来求得。

Mean = (Sum of all data values) ÷ (Number of data values)

平均数 = 所有数据值的总和 ÷ 数据值的个数

5. Geometric Constructions with Compass and Ruler | 几何构造：圆规与直尺

Precise geometric constructions are a practical way to explore properties of shapes. Begin with constructing the perpendicular bisector of a line segment. Draw a segment AB. Open your compass to more than half its length, draw arcs from A and B that intersect above and below. Join the intersection points; this line bisects AB at a right angle.

精确的几何作图是探索图形性质的一种实践方式。从作一条线段的垂直平分线开始。画一条线段 AB。将圆规张开至大于线段长度的一半，分别以 A 和 B 为圆心画弧，使两弧在上方和下方相交。连接两个交点；这条直线垂直平分 AB。

Next, construct the angle bisector. Draw an angle ∠ABC. With the compass point on the vertex B, draw an arc cutting both arms at points P and Q. Without changing the compass width, draw arcs from P and Q that intersect at X. The line BX bisects the angle. Always leave your construction arcs visible as evidence of your method.

接下来，作角平分线。画一个角 ∠ABC。将圆规的针尖放在顶点 B 上，画一条弧，与角的两边分别交于点 P 和 Q。保持圆规张开宽度不变，分别以 P 和 Q 为圆心画弧，两弧相交于点 X。直线 BX 就是该角的平分线。务必将作图弧线保留清晰，作为作图方法的证据。

6. Investigating Triangle Angle Sum | 探究三角形内角和

A fundamental property of all triangles is that the sum of their interior angles is always 180°. You can verify this by drawing several different triangles on paper, measuring each angle carefully with a protractor, and adding them up. Record your measurements in a table, and note any small errors that occur due to measurement inaccuracy.

所有三角形的一个基本性质是其内角和总是 180°。你可以通过在纸上画几个不同的三角形，用量角器仔细测量每个角，再将它们相加来验证这一性质。将测量结果记录在表格中，并留意由于测量不精确而产生的微小误差。

Another investigative method is to tear off the three corners of a paper triangle and place them around a point. They will fit together to form a straight line, confirming the 180° sum. This hands-on approach provides a concrete visual proof of the angle sum theorem.

另一种探究方法是将纸三角形的三个角撕下，并把它们拼在同一个点周围。它们会拼合成一条直线，从而证实 180° 的和。这种动手操作的方法为内角和定理提供了一个具体的直观证明。

∠A + ∠B + ∠C = 180°

7. Exploring Pythagoras’ Theorem through Area | 通过面积探索勾股定理

Pythagoras’ theorem states that in a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². To explore this, draw a right‑angled triangle with legs of 3 cm and 4 cm. Measure the hypotenuse; it should be 5 cm. Then construct squares on each side and calculate their areas. You will find that 9 cm² + 16 cm² = 25 cm².

勾股定理指出，在直角三角形中，斜边的平方等于两条直角边的平方和：a² + b² = c²。为了探究这一定理，画一个直角边长为 3 cm 和 4 cm 的直角三角形。测量斜边；它应该是 5 cm。然后在每条边上作正方形并计算它们的面积。你会发现 9 cm² + 16 cm² = 25 cm²。

Try this with other integer side lengths, such as 5-12-13 or 6-8-10 triangles. Cut out the squares and show that the two smaller squares can be rearranged to cover the largest square exactly. This dissection experiment offers a powerful visual justification of the theorem.

用其他整数边长，如 5-12-13 或 6-8-10 的三角形，也这样试试看。把正方形剪下来，展示两个较小的正方形经过重新排列后可以恰好覆盖最大的正方形。这种分割实验为定理提供了有力的视觉论证。

8. Patterns and Sequences | 模式与数列

Exploring number patterns leads to deep algebraic understanding. Start with the sequence of square numbers: 1, 4, 9, 16, 25… Build these geometrically using counters or dot diagrams. Each square number can be expressed as the sum of consecutive odd numbers: 1, 1+3=4, 1+3+5=9, and so on. Write down the nth term rule for square numbers: T(n) = n².

探索数字模式能带来深刻的代数理解。从平方数数列开始：1, 4, 9, 16, 25… 用计数圆片或点图来几何地构建它们。每个平方数都可以表示为连续奇数的和：1，1+3=4，1+3+5=9，依此类推。写下平方数的第 n 项规则：T(n) = n²。

Investigate triangular numbers: 1, 3, 6, 10, 15… Arrange counters in triangular formations. Find the formula T(n) = n(n+1)/2. Test it for n = 6: T(6) = 6×7÷2 = 21. Compare the visual arrangement of two identical triangular numbers to see why they form a rectangle, which helps derive the formula.

探究三角形数：1, 3, 6, 10, 15… 将计数圆片排列成三角形阵型。找出公式 T(n) = n(n+1)/2。用 n = 6 来检验：T(6) = 6×7÷2 = 21。将两个相同的三角形数进行视觉排列，看看它们为什么能拼成一个矩形，这有助于推导出该公式。

T(n) = 1 + 2 + 3 + … + n = n(n+1)/2

9. Dynamic Geometry Software Experiments | 动态几何软件实验

Dynamic geometry software, such as GeoGebra, allows you to create precise constructions and then drag points to observe how properties remain invariant. Construct a triangle with three medians. Notice that the medians always intersect at a single point, the centroid, regardless of how you reshape the triangle. Measure the segments on each median; you will see that the centroid divides each median in a 2:1 ratio.

动态几何软件，如 GeoGebra，能让你创建精确的作图，然后拖动点来观察性质如何保持不变。画一个拥有三条中线的三角形。注意，无论你如何改变三角形的形状，中线总是相交于同一点，即重心。测量每条中线上被分成的线段；你会看到重心将每条中线分成 2:1 的两部分。

Use the software to explore circle theorems. Draw a chord and the angle at the centre versus the angle at the circumference subtended by the same chord. Measure both angles and observe that the centre angle is always twice the circumference angle. These experiments provide instant feedback and deepen your understanding of geometric relationships.

使用该软件探索圆定理。画一条弦，并画出圆心角以及由同一条弦所对的圆周角。测量这两个角，观察圆心角总是圆周角的两倍。这些实验能提供即时的反馈，加深你对几何关系的理解。

10. Writing Up Your Investigation | 撰写实验报告

A well-structured investigation report helps you communicate your findings clearly. Start with a title and a brief introduction stating the aim of your investigation. Then describe your method step by step, including the equipment used. Present your results using tables, charts, and written explanations. When interpreting results, link back to your original prediction or hypothesis.

一份结构良好的探究报告有助于你清晰地传达自己的发现。从标题和简要的引言开始，陈述探究的目的。然后逐步描述你的方法，包括所使用的器材。用表格、图表和文字说明来呈现你的结果。在解释结果时，要回扣你最初的预测或假设。

Include a section for evaluation where you discuss any limitations or unexpected outcomes. Suggest how you could improve the investigation if you were to repeat it. Finally, write a conclusion that summarises what you have learned and possibly raises new questions for further inquiry. Always use precise mathematical language.

要包含一个评估部分，在其中讨论任何局限或意外结果。提出如果你要重复这项探究可以如何改进。最后，写一个结论来总结你学到了什么，并可能提出供进一步探究的新问题。务必使用准确的数学语言。

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KS3 Advanced Maths: Experimental Investigation Guide | KS3 进阶数学：实验操作指南