📚 Year 11 CIE Maths: Core Concepts Review | Year 11 CIE 数学：核心知识点梳理
This comprehensive guide covers the essential topics for the Cambridge IGCSE Mathematics syllabus (Year 11). Whether you are taking Core or Extended, these key areas form the foundation of your exam preparation. Each section presents the concepts in English followed by Chinese, with clear explanations, formulas, and practical examples to help you master the material.
这份全面的指南涵盖了剑桥 IGCSE 数学教学大纲（11 年级）的核心主题。无论你参加的是核心课程还是扩展课程，这些关键领域都是你备考的基础。每个部分先英文后中文介绍概念，并提供清晰的解释、公式和实际示例，帮助你掌握这些内容。
1. Number and Operations | 数与运算
Understanding number types and operations is fundamental. You should be comfortable with integers, fractions, decimals, percentages, and their conversions.
理解数字类型和运算是基础。你应该熟练掌握整数、分数、小数、百分比以及它们之间的转换。
Key operations include prime factorisation, highest common factor (HCF), and lowest common multiple (LCM). The product of prime factors is often written using index notation.
关键运算包括质因数分解、最大公因数 (HCF) 和最小公倍数 (LCM)。质因数的乘积通常使用指数表示法书写。
For any number, you can estimate by rounding to a given number of significant figures or decimal places. Also remember the rules of directed numbers (positive and negative) and the order of operations (BIDMAS/BODMAS).
对于任何数字，你可以通过四舍五入到指定有效数字或小数位数来估算。还要记住有向数字规则（正数和负数）以及运算顺序（BIDMAS/BODMAS）。
Order of operations: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right)
运算顺序：括号、指数、除/乘（从左到右）、加/减（从左到右）
2. Algebra: Expressions and Equations | 代数：表达式与方程
Algebra involves manipulating symbols and solving equations. You must be able to simplify expressions by collecting like terms, expanding brackets, and factorising.
代数涉及操作符号和求解方程。你必须能够通过合并同类项、展开括号和因式分解来简化表达式。
For linear equations, solve by isolating the variable on one side. For quadratic equations, you can factorise, complete the square, or use the quadratic formula.
对于线性方程，通过将变量隔离在一侧来求解。对于二次方程，你可以进行因式分解、配方法或使用求根公式。
Quadratic formula: x = [–b ± √(b² – 4ac)] / 2a
求根公式：x = [–b ± √(b² – 4ac)] / 2a
Remember that simultaneous equations can be solved by elimination or substitution. Inequalities are treated similarly to equations, but the sign reverses when multiplying or dividing by a negative number.
请记住，联立方程组可以通过消元法或代入法求解。不等式的处理方式与方程类似，但当乘以或除以负数时，不等号方向会反转。
3. Sequences | 数列
A sequence is an ordered list of numbers following a rule. You need to recognise arithmetic sequences (common difference) and geometric sequences (common ratio).
数列是按照一定规律排列的一列数。你需要识别等差数列（公差）和等比数列（公比）。
The nth term of an arithmetic sequence can be written as uₙ = a + (n–1)d, where a is the first term and d is the common difference.
等差数列的第 n 项可以写成 uₙ = a + (n–1)d，其中 a 是首项，d 是公差。
For quadratic sequences, the second differences are constant. Use the method of differences to find the expression for the nth term, often in the form an² + bn + c.
对于二次数列，二阶差是常数。使用差分法找到第 n 项的表达式，通常形式为 an² + bn + c。
4. Ratio, Proportion and Rates | 比、比例和比率
Ratios compare quantities in the same unit. You can simplify ratios by dividing by common factors. Direct proportion means that as one quantity increases, the other increases at the same rate; inverse proportion means one increases as the other decreases.
比是同一单位中的量之间的比较。你可以通过除以公因数来化简比。正比意味着一个量增加时，另一个量以相同的速率增加；反比意味着一个量增加时另一个量减少。
Percentage increase or decrease is calculated using: [(new – original) / original] × 100%. Compound interest is an application of repeated percentage change.
百分比的增减计算公式为：[(新值 – 原值) / 原值] × 100%。复利是重复百分比变化的应用。
Compound interest: A = P(1 + r/100)^n
复利公式：A = P(1 + r/100)^n
Speed, density and other compound measures are rates. Use the unitary method for best value problems.
速度、密度和其他复合度量都是比率。使用归一法解决最优价值问题。
5. Geometry: Angles and Polygons | 几何：角与多边形
Understand angle properties on a straight line (sum to 180°), around a point (360°), vertically opposite angles, parallel lines (alternate, corresponding, co-interior).
理解直线上的角（和为 180°）、点周围的角（360°）、对顶角、平行线（内错角、同位角、同旁内角）的特性。
For polygons, the sum of interior angles of an n-sided polygon is (n–2) × 180°. The sum of exterior angles is always 360°.
对于多边形，n 边形内角和为 (n–2) × 180°。外角和总是 360°。
Congruence and similarity are key geometric concepts. Two triangles are similar if they have equal angles or proportional sides.
全等和相似是关键的几何概念。如果两个三角形角相等或边成比例，则它们相似。
6. Mensuration | 测量与求积
Mensuration involves calculating lengths, areas, and volumes of 2D and 3D shapes. These formulas must be memorised.
求积法涉及计算二维和三维图形的长度、面积和体积。这些公式必须牢记。
| Shape | Area / Volume |
|---|---|
| Triangle | ½ × base × height |
| Circle | π × radius² |
| Cylinder (volume) | π × r² × h |
| Sphere (surface area) | 4 × π × r² |
For composite shapes, break them down into simpler parts. Also be able to find arc lengths and sector areas using the fraction of the circle (θ/360).
对于复合图形，将其分解为更简单的部分。还要能够使用圆的分数 (θ/360) 求弧长和扇形面积。
7. Trigonometry | 三角学
Trigonometry relates the sides and angles of right-angled triangles. The basic ratios are sine, cosine, and tangent:
三角学将直角三角形的边与角联系起来。基本的比是正弦、余弦和正切：
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
Use the sine rule and cosine rule for non-right-angled triangles. The sine rule: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² – 2bc cos A.
对于非直角三角形，使用正弦定理和余弦定理。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。
You also need to know exact trigonometric values for 30°, 45°, 60°, and their related angles. Bearings are measured clockwise from north.
你还需要知道 30°、45°、60° 及其相关角度的精确三角函数值。方位角是从正北顺时针方向测量的。
8. Graphs of Functions | 函数图像
Plotting and interpreting graphs is a major skill. For a linear function y = mx + c, m is the gradient and c is the y-intercept.
绘制和解释图像是一项重要技能。对于线性函数 y = mx + c，m 是斜率，c 是 y 轴截距。
Quadratic graphs are parabolas; turning points can be found by completing the square. Recognise cubic, reciprocal, and exponential graphs. Understand how to solve equations graphically.
二次函数图像是抛物线；可通过配方法求顶点。识别三次函数、反比例函数和指数函数的图像。理解如何通过图像求解方程。
Distance–time and speed–time graphs give information about motion: the gradient of a distance–time graph gives speed; the area under a speed–time graph gives distance.
距离-时间图像和速度-时间图像提供运动信息：距离-时间图像的斜率表示速度；速度-时间图像下的面积表示距离。
9. Coordinate Geometry | 坐标几何
Coordinate geometry deals with points, lines, and their relationships on the Cartesian plane. The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).
坐标几何处理平面直角坐标系中的点、线及其关系。连接点 (x₁, y₁) 和 (x₂, y₂) 的线段中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The distance between two points is found using Pythagoras’ theorem: distance = √[(x₂–x₁)² + (y₂–y₁)²].
两点之间的距离使用勾股定理求得：距离 = √[(x₂–x₁)² + (y₂–y₁)²]。
The gradient of a line is (y₂–y₁) / (x₂–x₁). Parallel lines have equal gradients; perpendicular lines’ gradients multiply to –1.
直线的斜率是 (y₂–y₁) / (x₂–x₁)。平行线的斜率相等；垂直线的斜率乘积为 –1。
10. Statistics | 统计
Statistics involves collecting, representing, and analysing data. You should be able to construct and interpret pie charts, bar charts, histograms (with frequency density), and cumulative frequency diagrams.
统计学涉及收集、表示和分析数据。你应该能够构建和解释饼图、条形图、直方图（使用频率密度）以及累积频率图。
Measures of central tendency: mean (sum of values ÷ number of values), median (middle value), mode (most frequent). Measures of spread: range, interquartile range (IQR = Q₃ – Q₁), and standard deviation (if covered).
集中趋势的度量：平均数（总和÷数量）、中位数（中间值）、众数（出现次数最多的值）。离散程度的度量：极差、四分位距 (IQR = Q₃ – Q₁) 以及标准差（若涉及）。
Scatter diagrams help show correlation; a line of best fit can be used to estimate values. Beware of extrapolation.
散点图有助于显示相关性；最佳拟合线可用于估计值。注意外推法的局限性。
11. Probability | 概率
Probability measures the chance of an event, written as a number between 0 and 1 (or as a fraction, decimal, or percentage). The probability of an event not happening is 1 – P(event).
概率衡量事件发生的可能性，用 0 到 1 之间的数字（或分数、小数、百分比）表示。事件不发生的概率是 1 – P(事件)。
For combined events, use sample space diagrams, tree diagrams (multiply along branches, add for different outcomes). Conditional probability is the probability of an event given that another event has occurred.
对于组合事件，使用样本空间图、树状图（沿分支相乘，不同结果相加）。条件概率是在另一事件已发生的情况下某事件发生的概率。
P(A and B) = P(A) × P(B) for independent events. P(A or B) = P(A) + P(B) – P(A and B)
对于独立事件，P(A 且 B) = P(A) × P(B)。P(A 或 B) = P(A) + P(B) – P(A 且 B)
12. Transformations and Vectors | 变换与向量
Transformations change the position or size of a shape. The four main types are translation, reflection, rotation, and enlargement. An enlargement is defined by a centre and a scale factor.
变换改变图形的位置或大小。四种主要类型是平移、反射、旋转和放大。放大由中心和比例因子定义。
Vectors describe a translation. They are written as column vectors (x, y) and can be added, subtracted, and multiplied by a scalar. The magnitude of vector (x, y) is √(x² + y²).
向量描述了平移。它们写成列向量 (x, y)，可以进行加法、减法和标量乘法。向量 (x, y) 的模长是 √(x² + y²)。
Combining transformations: the order matters. Also, a combination of two reflections is equivalent to a rotation or translation.
组合变换的顺序很重要。此外，两次反射的组合相当于一次旋转或平移。
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