📚 Year 10 CIE Maths: Core Knowledge Review | Year 10 CIE 数学：核心知识点梳理
In Year 10 of the Cambridge IGCSE Mathematics syllabus, you will consolidate and extend the fundamental skills needed for success at the extended or core level. This article provides a structured review of the key topics, including number operations, algebraic manipulation, coordinate geometry, trigonometry, mensuration, transformations, statistics, and probability. Each section pairs an essential idea with clear explanations in both English and Chinese, helping bilingual learners to master the subject efficiently.
在 Year 10 的剑桥 IGCSE 数学课程中，你将巩固并拓展基础技能，为后续的核心或扩展考试做好准备。本文系统梳理了核心知识点，涵盖数的运算、代数处理、坐标几何、三角学、测量、变换、统计和概率。每个要点均提供中英双语解释，帮助双语学生高效掌握数学。
1. Number Types and Operations | 数的类型与运算
Understand natural numbers, integers, rational and irrational numbers, as well as directed numbers and the order of operations (BIDMAS/BODMAS). Apply four operations to fractions, decimals, and percentages, and convert fluently between these forms. Use index notation and simple standard form (scientific notation) to represent very large or very small numbers.
理解自然数、整数、有理数和无理数，以及带符号数的运算顺序（BIDMAS/BODMAS）。熟练掌握分数、小数和百分数的四则运算与互化。使用指数记法和简单的标准形式（科学记数法）表示极大或极小的数。
- Example: 2/3 × 0.75 = 2/3 × 3/4 = 1/2
- 示例：2/3 × 0.75 = 2/3 × 3/4 = 1/2
2. Algebraic Fundamentals | 代数基础
Simplify expressions by collecting like terms, use the distributive law to expand brackets, and factorise by taking out common factors. Work with linear equations, including those with brackets and unknowns on both sides. Substitute values into formulae and rearrange simple formulae to change the subject.
通过合并同类项化简代数式，利用分配律展开括号，并提取公因式进行因式分解。解一元一次方程，包括含括号和未知数在两侧的方程。将数值代入公式，并能对简单公式进行变形以改变主项。
3(x – 2) = 2x + 4 → 3x – 6 = 2x + 4 → x = 10
3. Linear Graphs and Coordinate Geometry | 线性图像与坐标几何
n the Cartesian plane. Find the gradient of a straight line given two points using m = (y₂ – y₁)/(x₂ – (y-intercept). Determine the equation of a line parallel or perpendicular to a given line.
在直角坐标系中绘制点和直线。根据两点求直线斜率，m = (y₂ – y₁)/(x₂ – x₁)。认识 y = mx + c 的形式，理解 m（斜率）和 c（y轴截距）的意义。求与给定直线平行或垂直的直线的方程。
- Parallel lines → same gradient.
- 平行线 → 斜率相等。
- Perpendicular lines → product of gradients = –1.
- 垂直线 → 斜率乘积为 –1。
4. Sequences and Patterns | 数列与规律
Generate terms of a sequence from a term-to-term rule or a position-to-term rule (nth term). Recognise arithmetic sequences and find the nth term formula a + (n – 1)d. Explore quadratic sequences and deduce the second difference. Use sigma notation at a basic level if required by the extended curriculum.
根据递推规则或通项公式（第n项）生成数列的项。识别等差数列并写出通项公式 a + (n – 1)d。探究二次数列并推算二次差。若扩展课程要求，能简单使用求和符号。
For the sequence 3, 7, 11, 15, … the nth term is 4n – 1.
数列 3, 7, 11, 15, … 的通项公式为 4n – 1。
5. Ratio, Proportion and Rates | 比、比例与率
Simplify ratios and divide a quantity in a given ratio. Solve problems involving direct and inverse proportion, including real-life contexts such as recipes, maps, and foreign currency conversions. Calculate with speed, density, and other compound measures.
化简比并按给定比例分配数量。解决正比例和反比例问题，包括食谱、地图和外汇兑换等实际情景。运用速度、密度等复合量进行计算。
| Direct proportion: y = kx | 正比例：y = kx |
| Inverse proportion: y = k/x | 反比例：y = k/x |
6. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
Apply Pythagoras' es in right-angled triangles. Use sine, cosine, and tangent ratios (SOH CAH TOA) to solve for unknown sides and angles. Extend to angles of elevation and depression, and apply trigonometry in 3D contexts where appropriate.
运用勾股定理 a² + b² = c² 求直角三角形的未知边。使用正弦、余弦和正切比（SOH CAH TOA）求未知边和角。延伸到仰角和俯角问题，并在适当情况下应用于三维情景。
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边
7. Mensuration: Area, Volume and Surface Area | 测量：面积、体积与表面积
Recall and use formulae for the area of a triangle, parallelogram, trapezium, and circle. Calculate perimeter and arc length, area of sector. Find the volume and surface area of prisms, pyramids, cylinders, cones, and spheres. Know the relationship between cm³ and litres, and between area scale factor and volume scale factor of similar figures.
回忆并运用三角形、平行四边形、梯形和圆的面积公式。计算周长、弧长及扇形面积。求棱柱、棱锥、圆柱、圆锥和球体的体积与表面积。了解立方厘米与升的关系，以及相似形面积比与体积比的关系。
Area of a circle = πr², Circumference = 2πr, Volume of cylinder = πr²h
圆面积 = πr²，周长 = 2πr，圆柱体积 = πr²h
8. Transformations and Symmetry | 变换与对称
Describe and perform reflections, rotations, translations, and enlargements on the coordinate plane. Use a column vector to describe a translation. Find the centre of rotation and the positive or negative scale factor of an enlargement, including fractional scale factors. Identify lines of symmetry and rotational symmetry of 2D shapes.
在坐标平面上描述并进行反射、旋转、平移和放大（位似）变换。用列向量描述平移。确定旋转中心以及放大的比例因子（包括分数比例因子和负比例因子）。识别平面图形的对称轴和旋转对称。
nlargement with scale factor k: image length = k × original length; area multiplied by k².
比例因子为 k 的放大：像长 = k × 原长；面积乘以 k²。
9. Statistics: Data Handling and Averages | 统计：数据处理与平均数
Collect, classify, and represent data using bar charts, pie charts, histograms (for continuous data with equal class intervals), and cumulative frequency diagrams. Calculate mean, median, mode, and range for discrete and grouped data. Interpret scatter graphs and draw lines of best fit, understanding correlation.
使用条形图、饼图、直方图（等距分组连续数据）和累积频率图收集、分类并展示数据。计算离散数据和分组数据的平均数（均值）、中位数、众数和极差。解读散点图并画最佳拟合线，理解相关性。
Mean from a frequency table: Σ(fx) / Σf
频数表求均值：Σ(fx) / Σf
10. Probability | 概率
Calculate the probability of a single event using the scale 0 to 1. Understand that the sum of probabilities of all mutually exclusive outcomes is 1. Use sample space diagrams and tree diagrams to find probabilities of combined events. Distinguish between 'and' and 'or' rules, and apply the addition rule for mutually exclusive events and the multiplication rule for independent events.
使用 0 至 1 的概率尺度计算单一事件的概率。理解互斥结果概率之和为 1。利用样本空间图和树状图求组合事件的概率。区分“与”和“或”的法则，并应用互斥事件的加法法则及独立事件的乘法法则。
P(A) × P(B) for independent events; P(A or B) = P(A) + P(B) for mutually exclusive events.
独立事件：P(A 且 B) = P(A) × P(B)；互斥事件：P(A 或 B) = P(A) + P(B)。
11. Inequalities and Regions | 不等式与区域
Solve linear inequalities in one variable and represent the solution on a number line. Solve graphically by shading regions defined by one or more linear inequalities in two variables. Interpret strict and inclusive inequalities appropriately.
解一元一次不等式并在数轴上表示解集。通过绘制二元一次不等式所定义的区域来图解不等式组，并正确解释严格不等式与包含不等式。
y > 2x + 1 → dashed line; y ≥ 2x + 1 → solid line.
y > 2x + 1 → 虚线；y ≥ 2x + 1 → 实线。
12. Vectors (Extended Level Introduction) | 向量（扩展层级入门）
Understand a vector as a quantity with both magnitude and direction. Use column notation and perform addition, subtraction, and scalar multiplication of vectors. Find the magnitude of a vector using Pythagoras' theorem, and use vectors to describe geometric relationships such as midpoints and parallel lines.
理解向量是具有大小和方向的量。使用列向量记法，并进行向量的加法、减法和数乘运算。利用勾股定理求向量的模，并运用向量描述几何关系，如中点和平行线。
Vector AB = b – a, where a and b are position vectors of A and B.
向量 AB = b – a，其中 a 和 b 分别是点 A 和 B 的位置向量。
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