📚 IGCSE Mathematics G-4 Workbook 100: Comprehensive Revision Guide | IGCSE数学G-4练习册100：全面复习指南
Welcome to your dedicated revision guide for the G-4 Workbook 100. This resource covers core IGCSE mathematics topics from algebra and graphs to geometry and statistics, with step-by-step explanations and worked examples that mirror the style of exam questions.
欢迎阅读G-4练习册100的专属复习指南。本资料覆盖IGCSE数学的核心主题，从代数、图像到几何与统计，配有循序渐进的讲解和与考试题型一致的例题。
1. Understanding the G-4 Workbook Structure | 了解G-4练习册结构
The G-4 workbook is designed to help you consolidate essential skills. Each problem set, including #100, focuses on applying concepts in unfamiliar contexts. Start by scanning the questions and grouping them by topic, then allocate time according to your weaker areas.
G-4练习册旨在帮助你巩固核心技能。每个练习集（包括第100题）都注重在陌生情境中应用概念。先浏览所有题目并按主题分组，然后根据薄弱环节分配时间。
- Identify topic keywords in each question | 识别每道题中的主题关键词
- Estimate the difficulty level before solving | 解题前预估难度
- Check your answers against the mark scheme | 对照评分标准检查答案
2. Core Algebra: Expanding and Factorising | 核心代数：展开与因式分解
Expanding and factorising are inverse processes. You must be fluent with products like (a + b)(c + d), and recognise common identities such as the difference of two squares.
展开与因式分解是互逆过程。你必须熟练计算(a + b)(c + d)这类乘积，并识别常见的恒等式，例如平方差公式。
(a + b)² = a² + 2ab + b²
For factorising quadratics, look for two numbers that multiply to give the constant term and add to give the coefficient of x.
因式分解二次式时，寻找两个数，使它们的乘积等于常数项，和等于x的系数。
- Always expand before simplifying expressions | 化简表达式前先展开
- Check factorisation by expanding again | 通过再次展开来检查因式分解
3. Linear Equations and Inequalities | 线性方程与不等式
Solving a linear equation requires you to isolate the variable using inverse operations. Inequalities follow the same rules, except that multiplying or dividing by a negative number reverses the inequality sign.
解线性方程需要通过逆运算分离变量。不等式遵循同样的规则，但乘以或除以负数时要反转不等号。
3x − 7 = 8 → 3x = 15 → x = 5
solutions on a number line, use an open circle for < or >, and a closed circle for ≤ or ≥.
在数轴上表示解集时，< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆圈。
4. Simultaneous Equations | 联立方程
Simultaneous equations can be solved by elimination or substitution. The elimination method is efficient when coefficients align well, while substitution works best when one equation has a variable already isolated.
联立方程可通过消元法或代入法求解。当系数匹配较好时消元法高效；当一个方程中变量已经被隔离时，代入法最适用。
2x + y = 7 (1)
x − y = 2 (2)
x = 3. Substituting back gives y = 1. Always test the solution in both original equations.
将方程(1)和(2)相加得3x = 9，所以x = 3。代回得y = 1。始终在两个原方程中检验解。
5. Quadratic Equations | 二次方程
Quadratics can be solved by factorising, completing the square, or using the quadratic formula. The formula works for any quadratic, even when the roots are not rational.
二次方程可通过因式分解、配方法或求根公式求解。求根公式适用于任何二次方程，即使根不是有理数。
x = (−b ± √b² − 4ac) / 2a
For example, x² + 5x + 6 = 0 factorises as (x + 2)(x + 3) = 0, giving x = −2 or x = −3.
例如，x² + 5x + 6 = 0 可因式分解为(x + 2)(x + 3) = 0，得到x = −2或x = −3。
- Check the discriminant b² − 4ac to determine the nature of roots | 检查判别式b² − 4ac以判断根的性质
- For word problems, reject solutions that do not make sense | 对于应用题，舍弃不合理的解
6. Straight Line Graphs | 直线图像
t and c is the y-intercept. You need to be able to sketch graphs from an equation and find the equation from a graph.
直线方程写作y = mx + c，其中m是斜率，c是y轴截距。你需要能根据方程画草图，并能根据图像求方程。
m = (y₂ − y₁) / (x₂ − x₁)
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1.
平行线斜率相同。垂直线斜率之积为−1。
| Equation form | 方程形式 | Key feature | 关键特征 |
| y = mx + c | m gradient, c intercept | m斜率，c截距 |
| ax + by = d | standard form | 标准形式 |
7. Quadratic Graphs and Transformations | 二次函数图像与变换
re (h, k) is the vertex. The sign of a determines whether the parabola opens upwards or downwards.
二次函数图像是抛物线。其方程可写为顶点式y = a(x − h)² + k，其中(h, k)为顶点。a的符号决定抛物线开口向上或向下。
y = x² → y = (x − 3)² + 2
This translation shifts the graph right by 3 units and up by 2 units.
这个平移将图像向右移动3个单位，向上移动2个单位。
- The axis of symmetry is x = h | 对称轴为x = h
- To find x-intercepts, set y = 0 | 求x截距时令y = 0
- To find y-intercept, set x = 0 | 求y截距时令x = 0
8. Geometry: Angles and Circle Theorems | 几何：角度与圆定理
Angle reasoning is essential in IGCSE geometry. You should know that angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.
角度推理在IGCSE几何中至关重要。你应该知道直线上的角之和为180°，周角之和为360°，对顶角相等。
For circles, the following theorems are frequently tested: the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, and the angle in a semicircle is 90°.
关于圆，以下定理经常考查：圆心角是圆周角的2倍，同弧上的圆周角相等，半圆内的圆周角为90°。
| Theorem | 定理 | Statement | 表述 |
| Angle in semicircle | 半圆角 | Always 90° | 恒为90° |
| Tangent and radius | 切线与半径 | Perpendicular | 垂直 |
9. Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
OH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
在直角三角形中，正弦、余弦和正切比率将边长与锐角联系起来。记住SOH CAH TOA：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。
sin θ = opposite / hypotenuse
Use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle given two sides.
已知两边时，使用反函数（sin⁻¹、cos⁻¹、tan⁻¹）求角度。
- Label the sides relative to the given angle | 根据已知角标记各边
- Make sure your calculator is in degree mode | 确保计算器处于角度模式
10. Statistics and Probability | 统计与概率
Statistics and probability together often appear in the final section of the G-4 workbook. For a data set you should be able to calculate the mean, median, mode and range, and interpret them appropriately.
统计和概率通常出现在G-4练习册的最后部分。对于数据集，你应该能够计算均值、中位数、众数和极差，并合理分析。
mean = sum of all values / total number of values
e outcomes is 1. For independent events, multiply probabilities: P(A and B) = P(A) × P(B).
概率在0到1的范围内衡量。所有互斥结果的概率之和为1。对于独立事件，概率相乘：P(A且B) = P(A) × P(B)。
11. Exam Tips for G-4 Workbook 100 | G-4练习册第100题考试技巧
Problem 100 is designed to be challenging. It often combines multiple topics in a multi-part question. Write all working clearly, because method marks are awarded even if the accuracy mark is lost.
第100题通常具有挑战性。它往往以多小问的形式结合多个主题。请清晰写出所有解题步骤，因为即使正确性分数丢失，方法分仍会获得。
- Read the question twice and underline key values | 读题两遍并划出关键数据
- State any formulas or equations you use | 写出所使用的公式或方程
- Check the units and rounding instructions | 检查单位和小数位数要求
- Use the final 5 minutes to revisit calculations | 用最后5分钟重新检查计算
Consistent practice with previous G-4 workbook problems will build speed and confidence. Use this guide alongside your notes, and attempt 2–3 unguided problems each day.
持续练习以前的G-4练习册题目可以提升速度和信心。请将本指南与笔记结合使用，每天尝试2–3道不参考提示的题目。
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