IGCSE Mathematics G-3 Workbook: Complete Revision Collection | IGCSE数学G-3练习册内容合集

📚 IGCSE Mathematics G-3 Workbook: Complete Revision Collection | IGCSE数学G-3练习册内容合集

This comprehensive revision collection is designed for IGCSE Mathematics students working through the G-3 workbook series. It brings together the essential topics, worked examples, and exam-style practice that you need to master the syllabus. Each section below pairs an English explanation with its Chinese translation, allowing you to study bilingually and reinforce your understanding of core mathematical concepts.

本合集专为正在学习 G-3 练习册系列的 IGCSE 数学学生编写，汇集了考纲中的核心主题、典型例题和考试风格练习。以下每一节均配有英文讲解和中文翻译，帮助你通过双语学习巩固对数学核心概念的理解。

1. Number and Ratio | 数与比例

The G-3 workbook begins with number work and proportional reasoning. You must be confident in working with integers, fractions, decimals, percentages, indices, and standard form. A strong grasp of these skills is essential because they appear in almost every other topic.

G-3 练习册从数与比例推理开始。你必须熟练掌握整数、分数、小数、百分数、指数和科学记数法。这些技能几乎出现在其他所有主题中，是后续学习的基础。

Percentages: Convert between fractions, decimals, and percentages. Calculate percentage increase and decrease, and reverse percentages.
百分数：在分数、小数和百分数之间互化；计算百分数增减以及反向百分数。
Ratio and proportion: Divide quantities in a given ratio, solve problems involving direct and inverse proportion.
比与比例：按给定比分配数量，解决正比例和反比例问题。
Standard form: Write very large and very small numbers as a × 10ⁿ, where 1 ≤ a < 10.
科学记数法：将极大或极小的数写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10。

Example: Increase 80 by 15% → 80 × 1.15 = 92

例：80 增加 15% → 80 × 1.15 = 92

When solving proportion problems, set up a ratio table or use the unitary method. For inverse proportion, remember that as one quantity doubles, the other halves.

解决比例问题时，可以列比例表或使用单位法。对于反比例，记住当一个量翻倍时，另一个量减半。

2. Algebra: Manipulation and Equations | 代数：运算与方程

Algebra is a major part of the IGCSE G-3 workbook. You need to expand brackets, factorise expressions, and solve linear, quadratic, and simultaneous equations. Accuracy with algebraic manipulation will save you time in exams.

代数是 IGCSE G-3 练习册的重要组成部分。你需要展开括号、因式分解表达式，并解线性方程、二次方程和联立方程。熟练的代数运算能力能在考试中为你节省宝贵时间。

Expanding brackets: Use FOIL or the distributive law: (x + 3)(x – 2) = x² + x – 6.
展开括号：使用分配律（首项、外项、内项、末项）：(x + 3)(x – 2) = x² + x – 6。
Factorising quadratics: Find two numbers that multiply to give c and add to give b in x² + bx + c.
因式分解二次式：对于 x² + bx + c，找到两个数使乘积为 c、和为 b。
Simultaneous equations: Solve by elimination or substitution. For example, 2x + y = 7 and x – y = 2.
联立方程：用消元法或代入法求解。例如 2x + y = 7 和 x – y = 2。
Quadratic formula: If ax² + bx + c = 0, then x = (-b ± √(b² – 4ac)) / (2a).
二次公式：若 ax² + bx + c = 0，则 x = (-b ± √(b² – 4ac)) / (2a)。

Solve x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 or x = 3

解 x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 或 x = 3

Always check your solutions by substituting back into the original equation. This is a simple but powerful error-checking habit.

始终将解代回原方程进行检验。这是一个简单却有效的检查错误习惯。

3. Functions and Graphs | 函数与图像

The workbook introduces function notation f(x), composite functions, and inverse functions. You must also be able to sketch and interpret graphs of linear, quadratic, and cubic functions.

练习册引入函数记号 f(x)、复合函数和反函数。你还必须能够绘制并解读线性函数、二次函数和三次函数的图像。

Function notation: If f(x) = 2x + 1, then f(3) = 7.
函数记号：若 f(x) = 2x + 1，则 f(3) = 7。
Composite functions: fg(x) means apply g first, then f. For f(x) = x + 1 and g(x) = 2x, fg(3) = f(6) = 7.
复合函数：fg(x) 表示先求 g，再求 f。例如 f(x) = x + 1，g(x) = 2x，则 fg(3) = f(6) = 7。
Inverse functions: To find f⁻¹(x), rearrange y = f(x) to make x the subject, then swap x and y.
反函数：求 f⁻¹(x) 时，将 y = f(x) 重新排列为以 x 为未知量，然后交换 x 与 y。
Graphs of quadratics: The vertex of y = ax² + bx + c lies on the axis of symmetry x = -b/(2a).
二次函数图像：y = ax² + bx + c 的顶点位于对称轴 x = -b/(2a) 上。

Practise drawing graphs by plotting key points: intercepts, turning points, and roots. For the graph of y = ax² + bx + c, the y-intercept is c, and the roots are found by setting y = 0.

通过描点来练习画图：截距、顶点和根。对于 y = ax² + bx + c，y 截距为 c，根通过令 y = 0 求得。

4. Coordinate Geometry | 坐标几何

Coordinate geometry connects algebra with geometry. In the G-3 workbook, you calculate gradients, midpoints, lengths, and equations of straight lines. You should also understand parallel and perpendicular lines.

坐标几何将代数与几何联系在一起。在 G-3 练习册中，你需要计算斜率、中点、长度和直线方程，还要理解平行线与垂直线。

Gradient: m = (y₂ – y₁) / (x₂ – x₁).
斜率：m = (y₂ – y₁) / (x₂ – x₁)。
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。
Equation of a line: y – y₁ = m(x – x₁), or y = mx + c.
直线方程：y – y₁ = m(x – x₁)，或 y = mx + c。
Parallel and perpendicular: Parallel lines have m₁ = m₂; perpendicular lines have m₁ × m₂ = -1.
平行与垂直：平行线斜率相等 m₁ = m₂；垂直线斜率满足 m₁ × m₂ = -1。

Line through (1, 2) and (3, 8): gradient = 3, equation: y – 2 = 3(x – 1) → y = 3x – 1

过 (1, 2) 和 (3, 8) 的直线：斜率 = 3，方程：y – 2 = 3(x – 1) → y = 3x – 1

Always label your axes and use graph paper when working on coordinate geometry questions in the workbook. This makes it easier to verify your answers visually.

在练习册中做坐标几何题时，始终标好坐标轴并使用坐标纸，这样能更容易直观验证答案。

5. Geometry and Mensuration | 几何与测量

Geometry in the G-3 workbook covers angles, triangles, polygons, circles, and three-dimensional shapes. Mensuration requires calculating perimeter, area, surface area, and volume using standard formulae.

G-3 练习册中的几何内容包括角、三角形、多边形、圆和三维图形。测量部分需要使用标准公式计算周长、面积、表面积和体积。

Angles in a polygon: Sum of interior angles = (n – 2) × 180°, where n is the number of sides.
多边形内角：内角和 = (n – 2) × 180°，其中 n 为边数。
Circle formulae: Circumference = 2πr; area = πr².
圆公式：周长 = 2πr；面积 = πr²。
Volume of a prism: Volume = cross-sectional area × length.
棱柱体积：体积 = 截面面积 × 长度。
Surface area of a cylinder: 2πrh + 2πr².
圆柱表面积：2πrh + 2πr²。

Area of a sector = (θ/360°) × πr², where θ is the sector angle

扇形面积 = (θ/360°) × πr²，其中 θ 为扇形圆心角

For compound shapes, split them into simpler shapes. In the workbook, always write the formula first, then substitute the numbers. This earns method marks even if the final answer is wrong.

对于组合图形，将其拆分为简单图形。在练习册中，务必先写公式，再代入数值。即使最终答案错误，也能获得方法分。

6. Trigonometry | 三角学

Trigonometry is a central topic in the IGCSE G-3 workbook. You need to know the three basic ratios, the sine rule, the cosine rule, and the formula for the area of a triangle.

三角学是 IGCSE G-3 练习册的核心主题。你需要掌握三个基本三角比、正弦定理、余弦定理和三角形面积公式。

Right-angled triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
直角三角形：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Sine rule: a/sin A = b/sin B = c/sin C.
正弦定理：a/sin A = b/sin B = c/sin C。
Cosine rule: a² = b² + c² – 2bc cos A.
余弦定理：a² = b² + c² – 2bc cos A。
Area of a triangle: Area = ½ab sin C.
三角形面积：面积 = ½ab sin C。

In a triangle with sides a = 7, b = 9 and angle C = 35°, Area = ½ × 7 × 9 × sin 35°

在三角形中，边 a = 7，b = 9，角 C = 35°，面积 = ½ × 7 × 9 × sin 35°

When using the sine rule, be careful with the ambiguous case. If the given angle is acute, there may be two possible triangles.

使用正弦定理时注意多解情况。若已知角为锐角，可能存在两个不同的三角形。

7. Vectors and Transformations | 向量与变换

Vectors and transformations are important for understanding movement and mapping in the plane. In the G-3 workbook, you work with column vectors, vector addition, and scalar multiplication. You also study reflections, rotations, translations, and enlargements.

向量与变换对于理解平面中的移动和映射很重要。在 G-3 练习册中，你要处理列向量、向量加法和数乘，还要学习反射、旋转、平移和放大。

Column vector: A vector from (x₁, y₁) to (x₂, y₂) is written as (x₂ – x₁, y₂ – y₁)ᵀ.
列向量：从 (x₁, y₁) 到 (x₂, y₂) 的向量记为 (x₂ – x₁, y₂ – y₁)ᵀ。
Vector addition: Add components separately: (3, 4) + (1, -2) = (4, 2).
向量加法：对应分量相加：(3, 4) + (1, -2) = (4, 2)。
Scalar multiplication: Multiply each component: 2(3, -1) = (6, -2).
数乘：每个分量乘以标量：2(3, -1) = (6, -2)。
Enlargement: Multiply the position vectors of vertices by the scale factor from the centre of enlargement.
放大：以放大中心为基准，将各顶点的位置向量乘以缩放因子。

When describing a transformation, always give the full description: type, line of reflection, angle and direction of rotation, centre, or scale factor and centre of enlargement.

描述变换时，务必给出完整信息：变换类型、反射轴、旋转角度与方向、旋转中心，或缩放因子与放大中心。

8. Probability | 概率

Probability in the G-3 workbook involves calculating the likelihood of events using sample spaces, tree diagrams, and the laws of probability. You must understand mutually exclusive and independent events.

G-3 练习册中的概率部分涉及使用样本空间、树形图和概率法则计算事件发生的可能性。你必须理解互斥事件和独立事件。

Basic rule: P(event) = number of favourable outcomes / total number of outcomes.
基本规则：P(事件) = 有利结果数 / 总结果数。
Mutually exclusive events: P(A or B) = P(A) + P(B).
互斥事件：P(A 或 B) = P(A) + P(B)。
Independent events: P(A and B) = P(A) × P(B).
独立事件：P(A 且 B) = P(A) × P(B)。
Complement rule: P(not A) = 1 – P(A).
对立事件规则：P(非 A) = 1 – P(A)。

Two coins are tossed: P(both heads) = ½ × ½ = ¼

抛两枚硬币：P(两个正面) = ½ × ½ = ¼

Tree diagrams are especially useful for multi-stage experiments. Multiply along the branches and add between branches. Always check that the probabilities on any set of branches sum to 1.

树形图特别适合多阶段试验。沿分支相乘，分支之间相加。务必检查同一组分支的概率之和等于 1。

9. Statistics | 统计

Statistics in the G-3 workbook focuses on data handling: calculating averages, measures of spread, and representing data in charts and graphs. Interpretation of data is just as important as calculation.

G-3 练习册中的统计部分侧重于数据处理：计算平均数、离散程度度量，以及用图表表示数据。解读数据与计算同样重要。

Mean: Sum of all data values divided by the number of values.
平均数：所有数据值之和除以数据个数。
Median: The middle value when data are arranged in order.
中位数：数据按顺序排列后的中间值。
Mode: The value that occurs most often.
众数：出现次数最多的值。
Range: Maximum value – minimum value.
极差：最大值 – 最小值。
Interquartile range (IQR): Upper quartile – lower quartile; measures the spread of the middle 50%.
四分位距 (IQR)：上四分位数 – 下四分位数；衡量中间 50% 数据的离散程度。

For grouped data, use mid-interval values to estimate the mean. For histograms, remember that the area of each bar represents frequency, not the height alone.

对于分组数据，用区间中值估计平均数。对于直方图，记住每个长条的面积代表频数，而不仅是高度。

10. Problem-Solving Strategies | 解题策略

The final section of the G-3 workbook revision collection is about general problem-solving strategies. These skills help you apply all the mathematical content to unfamiliar exam questions.

G-3 练习册复习合集的最后一部分是通用解题策略。这些技能帮助你运用所有数学内容应对不熟悉的考试题目。

Read carefully: Identify what is given and what is asked. Underline key information and units.
仔细审题：明确已知条件与所求目标，划出关键信息和单位。
Choose a method: Decide whether the problem needs algebra, geometry, trigonometry, or statistics. Look for keywords such as ‘maximum’, ‘constant’, ‘intersect’.
选择方法：判断问题需要代数、几何、三角还是统计方法。注意 ‘最大’、’恒定’、’相交’ 等关键词。
Show working: Write every step clearly. This helps you track your thinking and earns method marks.
写出过程：清晰书写每一步，这有助于跟踪思路并获得方法分。
Check answers: Substitute results back, estimate approximate values, and make sure the answer is sensible in context.
检查答案：将结果代回原题，估算近似值，确保答案在实际情境中合理。

If a question gives a diagram, mark all known lengths and angles before you begin solving.

如果题目给出图形，解题前先标出所有已知长度和角度。

Practice with past papers and timed conditions. The G-3 workbook is designed to build both fluency and confidence. Use it consistently, and review your mistakes to avoid repeating them.

使用历年真题并限时练习。G-3 练习册旨在提升熟练度与自信心。坚持使用，并认真复习错题以避免重复犯错。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE Mathematics G-3 Workbook: Complete Revision Collection | IGCSE数学G-3练习册内容合集