📚 IGCSE Maths Teacher's Guide: 100 Core Skills Review | IGCSE 数学教师用书：100 项核心技能复习
This article collects 100 essential IGCSE Mathematics points that teachers and students commonly revisit when using the teacher's guide. It is organised as a rapid revision map for the core and extended tiers.
本文整理 IGCSE 数学教师用书中高频复习的 100 个核心要点，按数与代数、几何、统计与概率等主线编排，适合核心与扩展层级的学生快速查漏补缺。
1. Number Types and Set Notation | 数系与集合符号
The IGCSE number system includes natural numbers (1, 2, 3, ...), integers (positive, negative and zero), rational numbers (fractions such as 3/4 and terminating or recurring decimals), irrational numbers (such as √2 and π) and real numbers, which include all of the above.
IGCSE 数系包括自然数（1、2、3……）、整数（正数、负数和零）、有理数（如 3/4 以及有限小数或循环小数）、无理数（如 √2 和 π）以及实数——实数包含上述所有数。
Set language is used to describe groups of numbers or outcomes. A ∩ B means the intersection (elements in both sets), A ∪ B means the union (elements in either set), and A' means the complement (elements not in A).
集合语言用来描述一组数或结果。A ∩ B 表示交集（同时属于两个集合的元素），A ∪ B 表示并集（属于任一集合的元素），A' 表示补集（不在 A 中的元素）。
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
This counting rule subtracts the overlap once, because it has been counted twice when adding n(A) and n(B).
这个计数法要减去重叠部分一次，因为重叠部分在 n(A) 与 n(B) 相加时被计算了两次。
Venn diagrams are the standard way to show these relationships, and n(A) simply counts the number of elements in set A.
维恩图是表示这些关系的标准方法，而 n(A) 只表示集合 A 中元素的个数。
2. Fractions, Decimals and Percentage Conversions | 分数、小数和百分数互化
To convert a fraction to a decimal, divide the numerator by the denominator; to convert a decimal to a percentage, multiply by 100. Recurring decimals such as 0.333... can be written as 1/3 using algebraic or place-value methods.
分数化小数用分子除以分母；小数化为百分数乘以 100。循环小数如 0.333... 可以用代数或位值法写成 1/3。
Reverse percentage questions should return to the original amount by dividing by the multiplier. For example, a price after a 20% increase is 120% of the original, so divide by 1.2.
逆向百分数问题应通过除以乘数回到原量。例如，增加 20% 后的价格是原价的 120%，因此除以 1.2 即可。
percentage change = (new − original) ÷ original × 100%
This formula gives a positive value for an increase and a negative value for a decrease.
这个公式在增加时给出正值，在减少时给出负值。
3. Ratio, Speed and Direct Proportion | 比、速度与正比例
Ratios compare quantities and are simplified by dividing by the highest common factor (HCF). To split a quantity in the ratio a : b, divide it into a + b parts.
比用来比较数量，并通过除以最大公因数（HCF）化简。按 a : b 分配一个量时，应把它分成 a + b 份。
speed = distance ÷ time, density = mass ÷ volume
These compound measures can be rearranged using the same balancing method as linear equations.
这些复合单位可以像一元一次方程一样通过平衡移项进行变形。
or decrease at the same rate, written as y = kx. Inverse proportion is written as y = k/x.
正比例意味着两个量以相同速度增减，写作 y = kx；反比例写作 y = k/x。
4. Algebraic Expansion and Factorisation | 代数展开与因式分解
e law: a(b + c) = ab + ac. Expanding two brackets (x + a)(x + b) gives x² + (a + b)x + ab.
展开单项括号使用分配律：a(b + c) = ab + ac。展开两个括号 (x + a)(x + b) 得到 x² + (a + b)x + ab。
(a + b)(a − b) = a² − b²
This is called the difference of two squares and appears very often in IGCSE factorisation questions.
这个恒等式叫做平方差公式，在 IGCSE 因式分解题中出现频率很高。
Factorising reverses this process: look for the highest common factor, then check for the difference of two squares. Rearranging a formula uses the same balancing moves as solving equations.
因式分解是逆过程：先提取最大公因式，再检查平方差。变换公式使用与解方程相同的平衡移项方法。
5. Solving Linear Equations and Inequalities | 解一元一次方程与不等式
To solve a linear equation, use inverse operations to isolate the unknown. If the equation has brackets or fractions, expand or multiply by the lowest common denominator first.
解一元一次方程要用逆运算把未知数分离。若方程含括号或分数，先展开或乘以最小公分母。
If ax + b = c, then x = (c − b) ÷ a
Always check your solution by substituting the value back into the original equation.
一定要把求得的数值代回原方程检验。
lutions on a number line with open circles for < or > and closed circles for ≤ or ≥.
对于不等式，解法与方程类似，但乘以或除以负数时要改变不等号方向。在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
6. Coordinate Geometry and Graphs | 坐标几何与函数图像
s m = (y₂ − y₁) ÷ (x₂ − x₁). The midpoint of two points is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
直线的斜率是 m = (y₂ − y₁) ÷ (x₂ − x₁)。两点的中点是 ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)。
y = mx + c and for perpendicular gradients m₁m₂ = −1
that multiply to −1. Solving f(x) = g(x) can be read from the intersection of two graphs.
方程 y = mx + c 中，m 为斜率，c 为 y 轴截距。平行线斜率相等；互相垂直的直线斜率之积为 −1。求 f(x) = g(x) 可以从两图像的交点读出。
7. Quadratic Functions and the Parabola | 二次函数与抛物线
A quadratic expression in the form ax² + bx + c can be solved by factorising, completing the square, or the quadratic formula.
形如 ax² + bx + c 的二次式可以用因式分解、配方法或二次求根公式求解。
x = (−b ±
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