📚 IGCSE Mathematics: Mathematical Requirements | IGCSE 数学：数学能力要求
In IGCSE Mathematics, the syllabus is built around a clear set of mathematical requirements. These are the core skills and ways of thinking that examiners expect you to demonstrate, no matter which topic is being tested. Understanding these requirements helps you move beyond memorising formulas and focus on applying mathematics with accuracy, reasoning and confidence.
在 IGCSE 数学中，课程大纲围绕一套明确的数学能力要求构建。这些是考官希望你在任何考查主题中都能展现的核心技能与思维方式。理解这些要求能帮助你超越死记公式，专注于准确、有条理、有信心地应用数学。
1. Number and Calculation Skills | 数字与计算技能
You must be able to work confidently with integers, fractions, decimals, percentages, ratio and proportion. This includes converting between forms, finding a fraction of a quantity, and solving problems involving percentage increase or decrease.
你必须能够熟练处理整数、分数、小数、百分数、比和比例。这包括不同形式之间的转换、求一个量的几分之几，以及解决涉及百分数增减的问题。
- Use the four operations with positive and negative numbers.
- Round numbers to a given number of decimal places or significant figures.
- Apply order of operations correctly, including brackets and indices.
- 对正数和负数进行加减乘除四则运算。
- 将数字四舍五入到指定的小数位数或有效数字。
- 正确运用运算顺序，包括括号和指数。
37% of 240 = 0.37 × 240 = 88.8
240 的 37% = 0.37 × 240 = 88.8
2. Algebraic Manipulation | 代数运算
Algebra is the language of generalised number. You need to collect like terms, expand brackets, factorise expressions, and rearrange formulae to change the subject.
代数是广义数字的语言。你需要合并同类项、展开括号、因式分解表达式，以及变换公式的主项。
- Expand expressions such as 3(x + 2) and (x + 3)(x − 4).
- Factorise by taking out a common factor or by using a quadratic method.
- Solve linear equations, quadratic equations, and simple simultaneous equations.
- 展开如 3(x + 2) 和 (x + 3)(x − 4) 的表达式。
- 通过提取公因式或二次三项式方法进行因式分解。
- 解一次方程、二次方程和简单的联立方程。
3(x + 2) − 2(x − 1) = 3x + 6 − 2x + 2 = x + 8
3(x + 2) − 2(x − 1) = 3x + 6 − 2x + 2 = x + 8
3. Graphs and Coordinate Geometry | 图形与坐标几何
You should be able to plot points, draw straight-line graphs, and interpret real-life graphs such as distance–time and speed–time graphs. You also need to understand gradient and intercept.
你应当能够描点、绘制直线图，并解读距离—时间和速度—时间等实际生活图形。你还需要理解斜率和截距。
- Find the gradient of a line from two points.
- Write the equation of a straight line in the form y = mx + c.
- Identify parallel lines from their gradients.
- 由两点求直线的斜率。
- 写出 y = mx + c 形式的直线方程。
- 根据斜率判断平行线。
gradient m = (y₂ − y₁) ÷ (x₂ − x₁)
斜率 m = (y₂ − y₁) ÷ (x₂ − x₁)
4. Geometry and Angle Facts | 几何与角度知识
You must know angle properties on a straight line, around a point, in triangles and in polygons. You should also use symmetry and properties of circles where they appear in the syllabus.
你必须掌握直线上的角、点周围的角、三角形和多边形中的角的性质。你还要在考纲范围内运用对称性和圆的性质。
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Sum of interior angles of an n-sided polygon is (n − 2) × 180°.
- 直线上的角之和为 180°。
- 点周围的角之和为 360°。
- n 边形内角之和为 (n − 2) × 180°。
Interior angle sum of a hexagon = (6 − 2) × 180° = 720°
六边形内角和 = (6 − 2) × 180° = 720°
5. Trigonometry and Pythagoras | 三角学与毕达哥拉斯定理
In right-angled triangles, you need to apply Pythagoras' theorem and the sine, cosine and tangent ratios. You must also choose the correct ratio based on the sides and angle given.
在直角三角形中，你需要应用毕达哥拉斯定理以及正弦、余弦和正切比值。你还必须根据已知边和角选择正确的比值。
- Pythagoras: a² + b² = c² for the hypotenuse c.
- sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
- Use inverse trig functions to find an unknown angle.
- 毕达哥拉斯定理：斜边为 c 时，a² + b² = c²。
- sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。
- 使用反三角函数求未知角。
If a = 3 and b = 4, then c = √3² + 4² = √25 = 5
若 a = 3，b = 4，则 c = √3² + 4² = √25 = 5
6. Mensuration and Units | 测量与单位
You need to calculate perimeter, area and volume for common shapes and solids. You must also convert between units of length, area, volume, mass and time, including compound units such as speed and density.
你需要计算常见图形和立体的周长、面积和体积。你还必须转换长度、面积、体积、质量和时间单位，包括速度和密度等复合单位。
- Area of triangle = ½ × base × height.
- Area of circle = πr²; circumference = 2πr.
- Volume of prism = area of cross-section × length.
- 三角形面积 = ½ × 底 × 高。
- 圆的面积 = πr²；周长 = 2πr。
- 棱柱体积 = 横截面积 × 长。
Speed = distance ÷ time, so 60 km in 1.5 h gives 40 km/h
速度 = 路程 ÷ 时间，因此 1.5 小时行驶 60 km，速度为 40 km/h
7. Statistics and Data Handling | 统计与数据处理
You should collect, organise and display data in charts and diagrams. You must calculate averages and measures of spread, and interpret results in context.
你应当收集、整理数据并用图表展示。你必须计算平均数和离散程度的度量，并结合实际情况解释结果。
- Mean = sum of values ÷ number of values.
- Median is the middle value when data are ordered.
- Mode is the most frequent value; range = highest − lowest.
- 平均数 = 所有数值之和 ÷ 数值个数。
- 中位数是数据排序后位于中间的值。
- 众数是出现次数最多的值；极差 = 最大值 − 最小值。
For 2, 3, 5, 7, 8: mean = 25 ÷ 5 = 5, median = 5, range = 8 − 2 = 6
对于 2, 3, 5, 7, 8：平均数 = 25 ÷ 5 = 5，中位数 = 5，极差 = 8 − 2 = 6
8. Probability | 概率
You need to understand the probability scale from 0 to 1, calculate simple probabilities, and use tree diagrams or Venn diagrams for combined events where required.
你需要理解从 0 到 1 的概率标度，计算简单概率，并在需要时使用树状图或维恩图处理组合事件。
- P(event) = number of favourable outcomes ÷ total number of outcomes.
- For mutually exclusive events, P(A or B) = P(A) + P(B).
- For independent events, P(A and B) = P(A) × P(B).
- P(事件) = 有利结果数 ÷ 总结果数。
- 对于互斥事件，P(A 或 B) = P(A) + P(B)。
- 对于独立事件，P(A 且 B) = P(A) × P(B)。
P(two heads) = ½ × ½ = ¼
P(两次正面) = ½ × ½ = ¼
9. Ratio, Proportion and Rates of Change | 比、比例与变化率
You must solve problems involving direct and inverse proportion, scale factors, and rates of change. This includes dividing a quantity in a given ratio and finding best buys.
你必须解决涉及正比例、反比例、比例因子和变化率的问题。这包括按给定比例分配数量和寻找最划算的购买方案。
- Divide £60 in the ratio 2 : 3 by finding one part first.
- Use a scale factor to enlarge or reduce shapes.
- Compare unit prices to identify the best value.
- 通过先求出一份，将 60 英镑按 2 : 3 分配。
- 使用比例因子放大或缩小图形。
- 比较单价以确定最划算的方案。
2 : 3 gives 5 parts, so one part = 60 ÷ 5 = 12; amounts are 24 and 36
2 : 3 共 5 份，所以一份 = 60 ÷ 5 = 12；结果分别为 24 和 36
10. Problem Solving and Mathematical Reasoning | 问题解决与数学推理
IGCSE questions often combine several topics. You need to identify what is being asked, choose a strategy, and communicate your method clearly. Estimation and checking for reasonableness are essential.
IGCSE 题目常常综合多个知识点。你需要识别题目要求、选择策略，并清晰地表达解题方法。估算和检查答案是否合理是必不可少的。
- Translate a written problem into mathematical expressions or equations.
- Break a multi-step problem into smaller, manageable parts.
- Check units, signs and whether the final answer makes sense.
- 将文字问题转化为数学表达式或方程。
- 将多步骤问题分解为更小、更易处理的部分。
- 检查单位、符号以及最终答案是否合理。
If a shop offers 20% off, a £45 item costs 80% of £45 = 0.8 × 45 = £36
如果商店打八折，原价 45 英镑的商品现价为 45 的 80% = 0.8 × 45 = 36 英镑
11. Use of Calculator and Estimation | 计算器使用与估算
You must know when and how to use a calculator efficiently, but also be able to estimate to check calculator results. This includes using brackets, memory, and trigonometric or statistical functions correctly.
你必须知道何时以及如何高效使用计算器，同时能够通过估算检查计算器结果。这包括正确使用括号、存储功能以及三角函数或统计函数。
- Estimate before calculating to avoid large errors.
- Round answers only at the end of a calculation.
- Use the calculator's fraction, square, cube and root functions accurately.
- 计算前先估算，以避免出现大的错误。
- 只在计算结束时才对答案进行四舍五入。
- 准确使用计算器的分数、平方、立方和开方功能。
√50 ≈ 7.07, so 3√50 ≈ 3 × 7.07 ≈ 21.2
√50 ≈ 7.07，因此 3√50 ≈ 3 × 7.07 ≈ 21.2
12. Communication and Presentation of Method | 方法表达与书写规范
Examiners award marks for clear working, not just the final answer. You should show each step, label diagrams where helpful, and use correct mathematical notation.
考官不仅根据最终答案给分，还会为清晰的解题过程给分。你应当展示每一步，必要时标注图形，并使用正确的数学符号。
- Write down the formula before substituting values.
- Show substitution clearly and evaluate step by step.
- State units and round answers as requested.
- 先写出公式，再代入数值。
- 清楚展示代入过程，并逐步计算。
- 写明单位，并按要求对答案进行四舍五入。
Area = ½ × base × height = ½ × 8 × 5 = 20 cm²
面积 = ½ × 底 × 高 = ½ × 8 × 5 = 20 cm²
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