IGCSE Mathematics Core Topics Overview | IGCSE数学核心考点梳理

📚 IGCSE Mathematics Core Topics Overview | IGCSE数学核心考点梳理

This article provides a structured review of the core topics in the IGCSE Mathematics syllabus. It is designed to help you identify key formulas, understand essential concepts, and avoid common mistakes in your revision.

本文旨在系统梳理 IGCSE 数学考纲中的核心考点，帮助你明确重点公式、理解关键概念，并在复习过程中避开常见误区。

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1. Number and Arithmetic | 数与算术

Number forms the foundation of the entire IGCSE Mathematics course. You must be comfortable with integers, fractions, decimals, percentages, ratios, and standard form.

数与算术是整个 IGCSE 数学课程的基础。你必须熟练掌握整数、分数、小数、百分数、比与比例，以及科学计数法（标准形式）。

• Understand the order of operations: brackets, indices, division and multiplication, addition and subtraction.

  理解运算顺序：括号、指数、乘除、加减。

• Use prime factorisation to find the highest common factor (HCF) and lowest common multiple (LCM).

  使用质因数分解求最大公因数（HCF）和最小公倍数（LCM）。

• Express numbers in standard form, such as (3.2 times 10⁵) — wait, no LaTeX is allowed. Use 3.2 × 10⁵.

  用标准形式表示数，例如 3.2 × 10⁵。

Percentage change = (new value − original value) ÷ original value × 100%

百分变化量 =（新值 − 原值）÷ 原值 × 100%

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2. Algebra: Expressions and Equations | 代数：表达式与方程

Algebra requires you to manipulate symbols, simplify expressions, and solve linear and quadratic equations.

代数部分要求你操作符号、化简表达式，并求解线性方程与二次方程。

• Collect like terms, expand brackets, and factorise expressions.

  合并同类项、展开括号、分解因式。

• Solve linear equations including those with fractions and unknown on both sides.

  解线性方程，包括含分数以及未知数在等号两边的方程。

• Use the quadratic formula: x = (−b ± √b² − 4ac) ÷ (2a).

  使用二次方程求根公式：x = (−b ± √b² − 4ac) ÷ (2a)。

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

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3. Sequences and Functions | 数列与函数

Sequences and functions help you recognise patterns and describe relationships using algebra.

数列与函数帮助你识别模式，并用代数语言描述数量关系。

• Find the nth term of a linear sequence: nth term = dn + c, where d is the common difference and c is a constant.

  求线性数列的第 n 项：第 n 项 = dn + c，其中 d 为公差，c 为常数。

• Recognise quadratic sequences by second differences.

  通过二阶差分识别二次数列。

• Evaluate functions using f(x) notation and find inverse functions.

  使用 f(x) 记号求函数值，并求反函数。

nth term of a linear sequence = first term + (n − 1) × common difference

线性数列第 n 项 = 首项 + (n − 1) × 公差

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4. Coordinate Geometry | 坐标几何

Coordinate geometry connects algebra and geometry. You need to work with straight lines, gradients, and midpoints.

坐标几何将代数与几何联系起来。你需要掌握直线、斜率和中点等问题。

• Gradient = (change in y) ÷ (change in x) = (y₂ − y₁) ÷ (x₂ − x₁).

  斜率 = y 的变化量 ÷ x 的变化量 = (y₂ − y₁) ÷ (x₂ − x₁)。

• Equation of a straight line: y = mx + c, where m is the gradient and c is the y-intercept.

  直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。

• Midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

  两点 (x₁, y₁) 和 (x₂, y₂) 的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。

Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1.

平行线斜率相等；垂直线斜率相乘等于 −1。

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5. Mensuration: Area and Volume | 测量：面积与体积

Mensuration involves calculating lengths, areas, and volumes of two-dimensional and three-dimensional shapes.

测量涉及计算二维图形和三维图形的长度、面积与体积。

• Area of a circle: A = πr². Circumference: C = 2πr.

  圆面积：A = πr²。圆周长：C = 2πr。

• Volume of a prism: V = cross-sectional area × length.

  棱柱体积：V = 横截面积 × 长度。

• Volume of a cylinder: V = πr²h. Surface area of a cylinder: 2πrh + 2πr².

  圆柱体积：V = πr²h。圆柱表面积：2πrh + 2πr²。

Volume of a sphere = 4⁄3 πr³; Volume of a cone = 1⁄3 πr²h

球体积 = 4⁄3 πr³；圆锥体积 = 1⁄3 πr²h

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6. Trigonometry | 三角学

Trigonometry is used to solve problems involving right-angled triangles and to find angles and side lengths.

三角学用于解决直角三角形相关问题，以及求角度和边长。

• Pythagoras' theorem: a² + b² = c², where c is the hypotenuse.

  勾股定理：a² + b² = c²，其中 c 为斜边。

• sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent.

  sin θ = 对边 ÷ 斜边；cos θ = 邻边 ÷ 斜边；tan θ = 对边 ÷ 邻边。

• For non-right-angled triangles, use the sine rule and cosine rule.

  对于非直角三角形，使用正弦定理和余弦定理。

Sine rule: a⁄sin A = b⁄sin B = c⁄sin C

正弦定理：a⁄sin A = b⁄sin B = c⁄sin C

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7. Statistics and Probability | 统计与概率

Statistics and probability require you to interpret data, calculate averages, and determine likelihoods.

统计与概率要求你解读数据、计算平均值，并判断事件发生的可能性。

• Mean, median, mode, and range are the key measures of central tendency and spread.

  平均数、中位数、众数和极差是描述集中趋势和离散程度的关键统计量。

• Probability of an event = number of favourable outcomes ÷ total number of outcomes.

  事件概率 = 有利结果数 ÷ 总结果数。

• For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).

  对于独立事件，P(A 且 B) = P(A) × P(B)。对于互斥事件，P(A 或 B) = P(A) + P(B)。

Relative frequency = frequency of an outcome ÷ total number of trials

相对频率 = 某结果出现次数 ÷ 总试验次数

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8. Geometry: Angles and Properties of Shapes | 几何：角与图形性质

Geometry includes angle facts, triangle properties, circle theorems, and congruence or similarity.

几何部分包括角度性质、三角形性质、圆定理，以及全等与相似。

• Angles in a triangle sum to 180°. Angles on a straight line sum to 180°.

  三角形内角和为 180°。直线上的邻角之和为 180°。

• Angle at the centre of a circle is twice the angle at the circumference.

  圆心角是圆周角的两倍。

• Similar shapes have proportional side lengths; congruent shapes are identical in size and shape.

  相似图形对应边成比例；全等图形大小和形状完全相同。

For a regular polygon, each interior angle = (n − 2) × 180° ÷ n

正多边形每个内角 = (n − 2) × 180° ÷ n

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9. Vectors and Transformations | 向量与变换

Vectors describe translations and directions, while transformations include reflection, rotation, enlargement, and translation.

向量描述平移和方向，变换包括反射、旋转、放大（伸缩）和平移。

• Add and subtract vectors, and multiply a vector by a scalar.

  进行向量的加减运算，以及向量与标量的乘法。

• Find the magnitude of a vector using Pythagoras' theorem.

  利用勾股定理求向量的模（大小）。

• Describe transformations carefully, including centre of rotation, angle, and scale factor.

  准确描述变换，包括旋转中心、旋转角度和缩放比例。

If a vector is (x, y), its magnitude is √x² + y²

若向量为 (x, y)，其模为 √x² + y²

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10. Graphs and Inequalities | 图像与不等式

Graphs include linear, quadratic, cubic, reciprocal, and exponential functions. Inequalities extend equation solving to ranges of values.

图像包括线性、二次、三次、反比例和指数函数。不等式将方程求解扩展为取值范围。

• Plot graphs from equations, and find the gradient and intercepts.

  根据方程绘制图像，并求斜率和截距。

• Solve quadratic equations graphically by finding the x-intercepts.

  通过找图像与 x 轴交点，用图像法解二次方程。

• Represent inequalities on a number line and shade regions on a coordinate grid.

  在数轴上表示不等式，并在坐标平面中画出阴影区域。

The solution of an inequality is written with an open circle for < or >, and a closed circle for ≤ or ≥.

不等式解集在数轴上用空心圆表示 < 或 >，用实心圆表示 ≤ 或 ≥。

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11. Sets and Number Patterns | 集合与数值规律

Set notation appears in more advanced IGCSE papers, but the core ideas are simple and logical.

集合符号在更高级的 IGCSE 试卷中出现，但核心概念简单且富有逻辑。

• Understand union (A ∪ B), intersection (A ∩ B), and complement (A').

  理解并集（A ∪ B）、交集（A ∩ B）和补集（A′）。

• Use Venn diagrams to solve problems involving two or three sets.

  使用维恩图解决涉及两个或三个集合的问题。

• Recognise square numbers, cube numbers, and Fibonacci-type sequences.

  识别平方数、立方数以及斐波那契型数列。

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

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12. Exam Strategies and Common Mistakes | 考试策略与常见错误

Knowing the content is not enough; you must also avoid common pitfalls and manage your time efficiently.

仅仅知道知识点还不够；你还必须避免常见陷阱，并高效管理时间。

• Read the question carefully and underline key words such as "estimate", "exact", "give your answer in standard form".

  仔细读题，并在关键词如“估算”“精确值”“用标准形式表示答案”下方划线。

• Show all working steps. In IGCSE, method marks are often awarded even if the final answer is wrong.

  展示所有解题步骤。在 IGCSE 中，即使最终答案错误，也常常可以获得方法分。

• Check units, especially when converting between cm² and m², or cm³ and litres.

  检查单位，尤其是在 cm² 与 m²，或 cm³ 与升之间进行换算时。

Always substitute your answer back into the original equation to verify it.

始终将答案代回原方程进行验证。

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