📚 IGCSE Mathematics: Core Revision Guide | IGCSE数学：核心复习指南
This guide covers the essential topics and problem-solving strategies you need to master for the IGCSE Mathematics examination. Each section focuses on high-yield concepts, common question patterns, and the precise mathematical techniques that earn full marks.
本指南涵盖IGCSE数学考试中必须掌握的核心主题与解题策略。每一节聚焦高频考点、常见题型模式，以及能够获得满分的精确数学技巧。
1. Number and Ratio | 数与比例
Understanding the hierarchy of numbers is fundamental. You must be comfortable with natural numbers, integers, rational numbers, and irrational numbers, and be able to identify which set a given number belongs to.
理解数的层级体系是基础。你必须熟悉自然数、整数、有理数和无理数，并能够判断一个给定数字归属于哪个数集。
Prime numbers have exactly two distinct factors: 1 and themselves. For example, 17 is prime, but 1 is not prime by definition.
质数恰好有两个不同的因数：1和它本身。例如，17是质数，但根据定义1不是质数。
ion. For 24 = 2³ × 3 and 36 = 2² × 3², the HCF is 2² × 3 = 12 and the LCM is 2³ × 3² = 72.
最大公因数（HCF）和最小公倍数（LCM）最好通过质因数分解来求。对于24 = 2³ × 3和36 = 2² × 3²，HCF为2² × 3 = 12，LCM为2³ × 3² = 72。
Ratios can be simplified in the same way as fractions. If a sum of money is divided in the ratio 2 : 5, the total is 7 parts, so one part equals the total divided by 7.
比例与分数一样可以化简。如果一笔钱按2 : 5的比例分配，总共有7份，那么每份等于总数除以7。
Percentage change = (new value − original value) ÷ original value × 100%
Percentages are used constantly in IGCSE questions. A 20% increase followed by a 20% decrease does not return to the original value, because the second percentage is applied to the increased amount.
百分比在IGCSE题目中频繁出现。先增加20%再减少20%并不会回到原值，因为第二个百分比是作用在增加后的数值上的。
2. Algebra: Manipulation | 代数：运算技巧
Algebraic manipulation is the backbone of the IGCSE curriculum. You must be able to expand brackets, factorise expressions, and simplify algebraic fractions with confidence.
代数运算是IGCSE课程体系的支柱。你必须能够熟练地展开括号、分解因式以及化简代数分式。
le brackets requires the distributive law: (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.
展开双重括号需要运用分配律：(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6。
cess. The expression x² + 5x + 6 factors to (x + 2)(x + 3), since 2 + 3 = 5 and 2 × 3 = 6.
因式分解是逆向过程。表达式x² + 5x + 6分解为(x + 2)(x + 3)，因为2 + 3 = 5且2 × 3 = 6。
Always look for a common factor first before attempting any other factorisation technique.
在尝试任何其他分解技巧之前，务必先寻找公因数。
b² = (a + b)(a − b). This appears frequently in both straightforward and disguised forms.
平方差公式是一个值得牢记的特殊情形：a² − b² = (a + b)(a − b)。这一公式在直接和变形的题目中都很常见。
a² − b² = (a + b)(a − b)
3. Linear Equations and Inequalities | 线性方程与不等式
Solving linear equations requires performing the same operation on both sides. The goal is always to isolate the unknown variable on one side of the equation.
解线性方程需要在等式两边执行相同的运算。目标始终是将未知变量隔离在等式的一侧。
ample, solving (x/2) + 3 = (x/3) + 5: multiply all terms by 6 to obtain 3x + 18 = 2x + 30.
对于含分数的方程，先将每一项乘以公分母。例如，解(x/2) + 3 = (x/3) + 5：将所有项乘以6得3x + 18 = 2x + 30。
When solving inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign.
解不等式时，记住两边同时乘以或除以负数会改变不等号的方向。
Always check your solution by substituting it back into the original equation.
始终将解代回原方程进行验证。
Inequalities can be represented on number lines. An open circle means the endpoint is not included, while a closed circle means it is included.
不等式可以在数轴上表示。空心圆表示端点不包含，实心圆表示端点包含。
−3 ≤ x < 5
This notation means x is greater than or equal to −3 and strictly less than 5.
此记号表示x大于或等于−3且严格小于5。
4. Simultaneous Equations | 联立方程组
Simultaneous equations involve finding values of two unknowns that satisfy both equations at the same time. The elimination method is generally the most efficient for linear systems.
联立方程组要求找到同时满足两个方程的两个未知数的值。对于线性方程组，消元法通常是最有效的方法。
2x + 3y = 7
3x − y = 5
e second equation by 3: 9x − 3y = 15. Now add the two equations to get 11x = 22, so x = 2.
为了消去y，将第二个方程乘以3：9x − 3y = 15。现在将两个方程相加得11x = 22，所以x = 2。
Substitute x = 2 into the first equation: 2(2) + 3y = 7, which gives 3y = 3, so y = 1.
将x = 2代入第一个方程：2(2) + 3y = 7，得3y = 3，所以y = 1。
The solution is (2, 1). Always verify in both original equations.
解为(2, 1)。始终在两个原方程中验证。
Graphically, the solution to a pair of simultaneous linear equations is the point where the two lines intersect.
从图像上看，一对联立线性方程组的解就是两条直线的交点。
5. Quadratic Equations | 二次方程
There are three main methods for solving quadratic equations: factorisation, the quadratic formula, and completing the square. You should be fluent in all three.
解二次方程主要有三种方法：因式分解、求根公式和配方法。你应该对这三种方法都熟练掌握。
ion factorises cleanly. For x² − 5x + 6 = 0, we get (x − 2)(x − 3) = 0, so x = 2 or x = 3.
当方程能干净地分解因式时，因式分解是最快的方法。对于x² − 5x + 6 = 0，得到(x − 2)(x − 3) = 0，所以x = 2或x = 3。
The quadratic formula works for any quadratic equation, including those that do not factorise.
求根公式适用于任何二次方程，包括那些无法因式分解的方程。
Completing the square is valuable because it directly reveals the vertex of the parabola.
配方法很有价值，因为它直接揭示抛物线的顶点。
x = (−b ± √b² − 4ac) ÷ 2a
The discriminant, b² − 4ac, determines the number of real roots. If it is positive, there are two distinct roots; if it is zero, there is one repeated root; if it is negative, there are no real roots.
判别式b² − 4ac决定实数根的数量。如果为正，有两个不同的根；如果为零，有一个重根；如果为负，则没有实数根。
6. Coordinate Geometry | 坐标几何
Coordinate geometry connects algebra and geometry through the Cartesian plane. You must be able to find gradients, midpoints, and lengths of line segments.
坐标几何通过笛卡尔平面将代数和几何联系起来。你必须能够求斜率、中点和线段长度。
The gradient of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated as (y₂ − y₁) ÷ (x₂ − x₁).
经过两点(x₁, y₁)和(x₂, y₂)的直线斜率为(y₂ − y₁) ÷ (x₂ − x₁)。
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to give −1, meaning they are negative reciprocals.
平行线的斜率相等。垂直线的斜率相乘等于−1，即它们互为负倒数。
The midpoint of a segment is found by averaging the x-coordinates and the y-coordinates separately.
线段中点通过分别对x坐标和y坐标取平均值来求得。
Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
x + c, where m represents the gradient and c represents the y-intercept. You should be able to rewrite equations into this standard form.
直线方程为y = mx + c，其中m表示梯度，c表示y截距。你应该能够将方程改写为此标准形式。
7. Mensuration: Area and Volume | 测量学：面积与体积
Mensuration deals with the measurement of geometric figures. You must memorise the standard formulas for area and volume and know when to apply each one.
测量学研究几何图形的度量。你必须记住面积和体积的标准公式，并知道何时应用每一个。
| Shape | 图形 | Formula | 公式 |
| Rectangle 矩形 | Area = base × height | 面积 = 底 × 高 |
| Triangle 三角形 | Area = ½ × base × height | 面积 = ½ × 底 × 高 |
| Circle 圆 | Area = πr² | 面积 = πr² |
| Cylinder 圆柱 | Volume = πr²h | 体积 = πr²h |
| Sphere 球 | Volume = (4/3)πr³ | 体积 = (4/3)πr³ |
When calculating the area of a compound shape, divide it into simpler shapes, calculate each area separately, and then add or subtract as required.
计算复合图形面积时，将其分割为简单图形，分别计算每个面积，然后根据需要进行加法或减法。
Circumference = 2πr
Remember that the circumference of a circle is the distance around it, and π is approximately 3.14159.
记住圆的周长是绕圆一周的距离，π约等于3.14159。
8. Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
Trigonometry allows us to relate angles and side lengths in right-angled triangles. The three primary ratios are sine, cosine, and tangent.
三角学使我们能够建立直角三角形中角度和边长的关系。三个基本比率是正弦、余弦和正切。
ne = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
SOH CAH TOA是标准的助记符：正弦 = 对边 ÷ 斜边，余弦 = 邻边 ÷ 斜边，正切 = 对边 ÷ 邻边。
The angle of elevation is measured upward from the horizontal line, while the angle of depression is measured downward from the horizontal.
仰角是从水平线向上测量的角度，而俯角是从水平线向下测量的角度。
Always identify which side of the triangle is opposite, which is adjacent, and which is the hypotenuse before selecting the correct ratio.
在选择正确的比率之前，务必先确定三角形中哪条边是对边、哪条是邻边、哪条是斜边。
To find an angle when you know two sides, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, and tan⁻¹.
当已知两边求角度时，使用反三角函数：sin⁻¹、cos⁻¹和tan⁻¹。
tan θ = opposite ÷ adjacent
These techniques extend naturally to calculating distances and heights that cannot be measured directly.
这些技巧自然延伸到计算无法直接测量的距离和高度。
9. Statistics: Mean, Median, Mode, Range | 统计学：均值、中位数、众数、极差
Descriptive statistics summarise data sets using single representative values. Each measure has its own strength and is appropriate in different contexts.
描述性统计使用单一代表值概括数据集。每种度量各有其优势，适用于不同的情境。
The mean is the sum of all data values divided by the number of values. It is the most commonly used average but can be skewed by extreme outliers.
均值是所有数据值之和除以值的个数。它是最常用的平均数，但可能被极端异常值歪曲。
The median is the middle value when the data is arranged in ascending order. For an even number of values, take the average of the two middle numbers.
中位数是数据按升序排列后的中间值。对于偶数个数值，取中间两个数的平均值。
The mode is the most frequently occurring value. A data set can have no mode, one mode, or multiple modes.
众数是出现频率最高的值。一个数据集可以没有众数、有一个众数或多个众数。
The range measures the spread of the data and is calculated as the maximum value minus the minimum value.
极差衡量数据的离散程度，其计算方式为最大值减去最小值。
A grouped frequency table requires you to estimate the mean using the midpoint of each class interval as a representative value.
对于分组频数表，需要使用每个组距的中点作为代表值来估算均值。
10. Probability | 概率
Probability quantifies the likelihood of an event occurring and is always expressed as a number between 0 and 1 inclusive.
概率量化事件发生的可能性，其取值始终是0到1之间（包含0和1）。
utcomes, P(event) = (number of favourable outcomes) ÷ (total number of possible outcomes).
对于等可能的结果，P(事件) = 有利结果数 ÷ 所有可能结果总数。
The complement rule states that P(event not occurring) = 1 − P(event occurring).
补事件规则指出，P(事件不发生) = 1 − P(事件发生)。
If two events are mutually exclusive, the probability of either event occurring is the sum of their individual probabilities.
如果两事件互斥，其中任一事件发生的概率等于它们各自概率之和。
P(A or B) = P(A) + P(B) − P(A and B)
For independent events, the probability of both occurring is the product of their individual probabilities.
对于独立事件，两者同时发生的概率等于它们各自概率的乘积。
Tree diagrams are particularly useful for multi-stage probability problems, especially when the probabilities change at each branch as in the case of drawing without replacement.
树形图对于多阶段概率问题特别有用，尤其是在不放回抽取等情境中，每个分支的概率会发生变化。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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