📚 IGCSE Maths: G-1 Student Book 500 Core Revision | IGCSE 数学：G-1 学生用书 500 核心复习
This revision guide is designed for IGCSE Mathematics students working through the G-1 Student Book, page 500 checkpoint and beyond. It brings together the core skills most often tested in Paper 1 and Paper 2, including number, algebra, graphs, geometry, statistics and probability. Each section below pairs an English explanation with a Chinese explanation so you can study confidently in both languages.
本复习指南专为使用 G-1 学生用书第 500 页及后续内容的 IGCSE 数学学生设计。它汇集了试卷一和试卷二中最常考查的核心技能，包括数、代数、图像、几何、统计与概率。以下每个小节都同时提供英文和中文解释，帮助你自信地进行双语学习。
1. Number Types and Operations | 数的类型与运算
IGCSE candidates frequently meet questions on classifying numbers, prime factorisation, highest common factor (HCF) and lowest common multiple (LCM).
IGCSE 考生经常遇到数字分类、质因数分解、最大公约数（HCF）和最小公倍数（LCM）的题目。
A key result states that the product of two positive integers equals the product of their HCF and LCM.
一个关键结论是：两个正整数的乘积等于它们的 HCF 与 LCM 的乘积。
a × b = HCF(a, b) × LCM(a, b)
For example, 24 and 36 have HCF 12 and LCM 72, so 24 × 36 = 864 and 12 × 72 = 864.
例如，24 和 36 的 HCF 为 12，LCM 为 72，所以 24 × 36 = 864，且 12 × 72 = 864。
- Prime factorisation: 24 = 2³ × 3 | 质因数分解：24 = 2³ × 3
- Rational numbers include fractions and terminating or recurring decimals | 有理数包括分数和有限小数或循环小数
- Irrational numbers cannot be written as simple fractions, such as √2 and π | 无理数不能写成简单分数，例如 √2 和 π
2. Fractions, Decimals and Percentages | 分数、小数与百分比
You must be able to convert freely between fractions, decimals and percentages. For example, 3/8 = 0.375 = 37.5%.
你必须能够在分数、小数和百分比之间自由转换。例如，3/8 = 0.375 = 37.5%。
Percentage increase and decrease questions often use the multiplier method. A 15% increase uses the multiplier 1.15, while a 15% decrease uses 0.85.
百分比增加和减少题常使用乘数法。增加 15% 使用乘数 1.15，减少 15% 使用乘数 0.85。
New value = Original value × (1 ± r/100)
Compound interest follows the same idea over several time periods.
复利在多个时间段内遵循相同的思路。
- Recurring decimal: 1/3 = 0.333… = 0.3̇ | 循环小数：1/3 = 0.333… = 0.3̇
- Reverse percentage: divide by the multiplier to find the original amount | 反向百分比：除以乘数以求出原值
3. Ratio and Proportion | 比与比例
Ratio problems require you to divide a quantity into parts according to a given ratio. If the ratio is 3 : 5, the total number of parts is 8.
比的问题要求你按照给定比例将一个量分成若干份。如果比为 3 : 5，总份数就是 8。
Direct proportion means that as one quantity increases, the other increases at the same rate. Inverse proportion means that as one quantity increases, the other decreases.
正比例意味着一个量增加时，另一个量以相同速率增加。反比例意味着一个量增加时，另一个量减少。
y = kx (direct proportion) and y = k/x (inverse proportion)
Always simplify ratios to their lowest terms and use the same units for both quantities.
始终将比化为最简形式，并确保两个量的单位相同。
4. Algebraic Expressions and Identities | 代数表达式与恒等式
Expanding brackets and factorising are essential algebra skills. You should know the standard identities below.
展开括号和因式分解是代数中的基本技能。你应该熟悉以下标准恒等式。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
For example, x² − 9 factorises as (x + 3)(x − 3), and x² + 6x + 9 factorises as (x + 3)².
例如，x² − 9 可以因式分解为 (x + 3)(x − 3)，x² + 6x + 9 可以因式分解为 (x + 3)²。
- Collect like terms before solving or factorising | 在求解或因式分解前先合并同类项
- Always check your expansion by substituting a simple value | 总是通过代入一个简单数值来检验展开结果
5. Linear Equations and Inequalities | 一次方程与不等式
A linear equation can be solved by rearranging to isolate the variable. For example, 3x − 5 = 7 becomes 3x = 12, so x = 4.
一次方程可以通过移项来求解。例如，3x − 5 = 7 变成 3x = 12，所以 x = 4。
When solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality sign.
解不等式时，请记住：乘以或除以负数时，不等号方向要反转。
−2x < 6 → x > −3
Solutions to inequalities are often shown on a number line or using interval notation.
不等式的解常用数轴或区间表示法来表示。
6. Simultaneous Equations | 联立方程
Simultaneous linear equations can be solved by substitution or elimination. The elimination method is usually faster when both equations are in the form ax + by = c.
联立一次方程可以用代入法或消元法求解。当两个方程都是 ax + by = c 的形式时，消元法通常更快。
For example, solve: 2x + y = 7 and x − y = 2. Adding the equations gives 3x = 9, so x = 3. Substituting back gives y = 1.
例如，解方程组：2x + y = 7 和 x − y = 2。将两式相加得 3x = 9，所以 x = 3。代回得 y = 1。
Always check your solution in both original equations.
一定要把解代入两个原方程进行检验。
7. Quadratic Equations | 二次方程
Quadratic equations often appear in IGCSE exams. They can be solved by factorising, completing the square, or using the quadratic formula.
二次方程在 IGCSE 考试中经常出现。它们可以通过因式分解、配方法或二次公式求解。
For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / 2a
The expression b² − 4ac is called the discriminant. It tells you how many real roots the quadratic has.
表达式 b² − 4ac 称为判别式。它告诉你二次方程有多少个实根。
- If b² − 4ac > 0, there are two distinct real roots | 如果 b² − 4ac > 0，有两个不同实根
- If b² − 4ac = 0, there is one repeated real root | 如果 b² − 4ac = 0，有一个重实根
- If b² − 4ac < 0, there are no real roots | 如果 b² − 4ac < 0，没有实根
8. Functions and Graphs | 函数与图像
You need to recognise the shapes of linear, quadratic, cubic and reciprocal graphs. A linear graph has equation y = mx + c, where m is the gradient and c is the y-intercept.
你需要识别一次、二次、三次和反比例函数的图像形状。一次函数图像方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
For a quadratic graph y = ax² + bx + c, the turning point and roots are key features. If a > 0 the graph is U-shaped; if a < 0 it is an upside-down U.
对于二次函数图像 y = ax² + bx + c，顶点和根是关键特征。如果 a > 0，图像呈 U 形；如果 a < 0，图像呈倒 U 形。
- Linear graph: straight line | 一次函数图像：直线
- Quadratic graph: parabola | 二次函数图像：抛物线
- Reciprocal graph: hyperbola with asymptotes | 反比例函数图像：具有渐近线的双曲线
9. Geometry: Pythagoras and Trigonometry | 几何：勾股定理与三角
In a right-angled triangle, Pythagoras’ theorem links the side lengths: the square of the hypotenuse equals the sum of the squares of the other two sides.
在直角三角形中，勾股定理描述了三条边的关系：斜边的平方等于另外两条边的平方和。
a² + b² = c²
The basic trigonometric ratios are defined as follows for a right-angled triangle:
对于直角三角形，基本三角比定义如下：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ when you need to find an angle.
当你需要求角度时，使用反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。
10. Statistics and Probability | 统计与概率
For a data set, the mean is the sum of values divided by the number of values. The median is the middle value when data are ordered, and the mode is the most frequent value.
对于一组数据，平均数是所有值之和除以值的个数。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。
Mean = ∑ x / n
Probability measures the chance of an event occurring. It is always between 0 and 1 inclusive.
概率衡量事件发生的可能性。它始终在 0 到 1 之间（含 0 和 1）。
P(A) = number of favourable outcomes / total number of possible outcomes
For independent events A and B, the probability that both occur is P(A and B) = P(A) × P(B).
对于独立事件 A 和 B，两者同时发生的概率为 P(A 且 B) = P(A) × P(B)。
- Range = largest value − smallest value | 极差 = 最大值 − 最小值
- Mutually exclusive events cannot happen at the same time | 互斥事件不能同时发生
- P(A or B) = P(A) + P(B) for mutually exclusive events | 对于互斥事件，P(A 或 B) = P(A) + P(B)
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