📚 Year 11 Cambridge Mathematics: Core Topic Summary | 核心知识点梳理
This article provides a structured revision of the essential topics in the Cambridge IGCSE Mathematics syllabus. Covering number, algebra, geometry, statistics and probability, the notes present each key concept in paired English and Chinese explanations to support bilingual learning. Key formulas, strategies and common pitfalls are highlighted throughout.
本文系统梳理了剑桥 IGCSE 数学课程中的核心知识点，涵盖数、代数、几何、统计与概率等模块。每个关键概念均以中英文对照的形式呈现，帮助双语学习者巩固理解。文中重点标注了必会公式、解题思路与常见易错点。
1. Number – Integers, Fractions, Decimals and Percentages | 数 – 整数、分数、小数与百分数
ng two numbers with the same sign gives a positive result, while different signs give a negative result, e.g., (-4) × (-3) = 12 but (-4) × 3 = -12.
整数是可以为正、负或零的整数。整数运算时，同号相乘除得正，异号相乘除得负，例如 (-4) × (-3) = 12，而 (-4) × 3 = -12。
de the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%. Recurring decimals can be written using dot notation, e.g., 0.3̇ = 1/3.
分数、小数和百分数可以互相转换。将分数化为小数用分子除以分母；将小数化为百分数乘以 100%。循环小数可用点标记表示，例如 0.3̇ = 1/3。
Percentage increase and decrease: new value = original × (1 ± percentage change). Reverse percentage problems require working backwards from the final amount by dividing by the multiplier, not simply multiplying.
百分比增减：新值 = 原值 × (1 ± 百分数)。逆向百分数问题需要从最终量反推，应除以乘数而不是直接乘。
n is an integer. Operations with standard form rely on handling powers of ten separately.
标准形式将极大或极小的数写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。标准形式的运算需分别处理 10 的幂次。
2. Sets and Venn Diagrams | 集合与维恩图
A set is a collection of distinct objects. The universal set is denoted by E or U, and the empty set by Ø. Notation: n(A) means the number of elements in set A; A' is the complement; A ∪ B is union; A ∩ B is intersection.
集合是一组互异的对象。全集用 E 或 U 表示，空集用 Ø 表示。记号：n(A) 表示集合 A 的元素个数；A' 表示补集；A ∪ B 表示并集；A ∩ B 表示交集。
hips between sets. The formula for the probability or number of elements in a union is n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Shading regions and interpreting three-set Venn diagrams are common exam tasks.
维恩图可用于直观展示集合间的关系。关于并集元素个数的公式为 n(A ∪ B) = n(A) + n(B) - n(A ∩ B)。考试中常考查三集合维恩图的区域涂色与解读。
3. Algebra – Expressions, Equations and Inequalities | 代数 – 表达式、方程与不等式
law: a(b + c) = ab + ac. Factorising reverses this process; e.g., x² - 9 = (x + 3)(x - 3).
代数变换包括去括号、因式分解与化简。乘法分配律：a(b + c) = ab + ac。因式分解是逆过程，例如 x² - 9 = (x + 3)(x - 3)。
Linear equations are solved by isolating the variable. Quadratic equations can be solved by factorising, completing the square, or using the quadratic formula: x = [-b ± √b² - 4ac] / (2a). Always check the discriminant b² - 4ac to determine the number of real roots.
线性方程通过隔离变量求解。二次方程可因式分解、配方法或用求根公式 x = [-b ± √b² - 4ac] / (2a) 求解。应先用判别式 b² - 4ac 判断实根个数。
Inequalities are solved similarly to equations, but when multiplying or dividing by a negative number, the inequality sign must be reversed. Number-line representation and set-builder notation are frequently tested.
不等式解法与方程类似，但当两边同乘除以负数时不等号方向需调转。数轴表示和集合构造式是常见考点。
4. Graphs and Functions | 图形与函数
efines a linear function. Composite functions fg(x) means applying g then f. Inverse functions f⁻¹(x) reverse the mapping.
函数 f(x) 将输入 x 映射到唯一输出。记号 f(x) = 2x + 3 定义了一个线性函数。复合函数 fg(x) 表示先执行 g 再执行 f。反函数 f⁻¹(x) 反转映射关系。
s y-intercept. Parallel lines share the same gradient; perpendicular lines have gradients that multiply to -1.
直线图形方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。平行的直线斜率相等；垂直的直线斜率乘积为 -1。
Quadratic graphs produce a parabola. Key features include the vertex and axis of symmetry. Exponential, reciprocal and trigonometric graphs each have characteristic shapes that you should recognise instantly.
二次函数图像为抛物线，关键特征包括顶点和对称轴。指数函数、反比例函数和三角函数图像各有特征形状，需能够快速辨认。
5. Sequences | 数列
A sequence is an ordered list of numbers. An arithmetic sequence has a constant difference d: the nth term is a + (n - 1)d. The sum of the first n terms is Sn = n/2 [2a + (n - 1)d].
数列是一组有顺序的数。等差数列有公差 d，第 n 项为 a + (n - 1)d，前 n 项和为 Sn = n/2 [2a + (n - 1)d]。
A geometric sequence has a constant ratio r: the nth term is arⁿ⁻¹. The sum of the first n terms (finite) is Sn = a(1 - rⁿ) / (1 - r), used when |r| < 1 for infinite sum.
等比数列有公比 r，第 n 项为 arⁿ⁻¹，有限前 n 项和为 Sn = a(1 - rⁿ) / (1 - r)；当 |r| < 1 时可用于无限项求和。
Other sequences include quadratic sequences where the second difference is constant, and cubic sequences. Finding the nth term formula requires identifying the pattern in differences.
其他数列如二次差为常数的二次数列、三次数列。求第 n 项公式需分析差分的模式。
6. Ratio, Proportion and Rates | 比、比例和速率
Ratios compare two or more quantities and are written as a:b or a:b:c. Equivalent ratios simplify by dividing by the highest common factor. Dividing a quantity in a given ratio uses the total number of parts.
比用于比较两个或多个数量，写作 a:b 或 a:b:c。等价比可以除以最大公因数简化。按给定比例分配数量要用总份数来计算。
x; as x increases, y decreases. Graphs and algebraic models help distinguish between them.
正比例：y = kx，x 增大时 y 以恒定速率增大。反比例：y = k/x，x 增大时 y 减小。可通过图像和代数模型区分二者。
asures. Convert between units carefully, e.g., km/h to m/s multiply by (1000/3600) = 5/18.
速度 = 距离 ÷ 时间、密度 = 质量 ÷ 体积等复合量是速率的例子。单位换算时需仔细，如 km/h 转为 m/s 乘以 (1000/3600) = 5/18。
7. Geometry – Angles, Triangles and Circles | 几何 – 角、三角形与圆
Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal. Alternate angles equal, corresponding angles equal and co-interior angles sum to 180° when parallel lines are cut by a transversal.
直线上的角之和为 180°，周角为 360°。对顶角相等。平行线被截线所截时，内错角相等，同位角相等，同旁内角互补成 180°。
In triangles, the sum of interior angles is 180°. The exterior angle equals the sum of the two opposite interior angles. Congruence and similarity determine when triangles are identical or proportionally identical.
三角形内角和为 180°。外角等于两不相邻内角之和。全等和相似用于判断两个三角形是否完全相同或成比例相同。
Circle theorems: the angle at the centre is twice the angle at the circumference; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; tangents from a point are equal in length.
圆定理：圆心角等于两倍圆周角；同弧上的圆周角相等；半圆上的圆周角为 90°；圆内接四边形对角互补；从同一点出发的两条切线长度相等。
8. Mensuration – Area, Volume and Surface Area | 测量 – 面积、体积与表面积
b)h, circle A = πr². For sectors, arc length = (θ/360) × 2πr, sector area = (θ/360) × πr².
常见面积公式：矩形 A = lw，三角形 A = ½ bh，梯形 A = ½ (a+b)h，圆 A = πr²。扇形的弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
olume = ⁴⁄₃ πr³, surface area = 4πr². Cylinder: volume = πr²h, curved surface area = 2πrh.
体积与表面积：棱柱体积 = 横截面积 × 长度；棱锥体积 = ⅓ × 底面积 × 高；圆锥体积 = ⅓ πr²h；球体积 = ⁴⁄₃ πr³，表面积 = 4πr²。圆柱：体积 = πr²h，侧面积 = 2πrh。
When solving measurement problems, use the same units throughout and leave answers in terms of π or to a stated degree of accuracy if required.
解测量问题时需统一单位，并按照要求保留 π 或精确到指定位数。
9. Trigonometry and Pythagoras' Theorem | 三角学与勾股定理
Pythagoras’ theorem applies to right-angled triangles: c² = a² + b², where c is the hypotenuse. It can be used to find missing sides or to check if a triangle is right-angled.
勾股定理适用于直角三角形：c² = a² + b²，c 为斜边。可用于求未知边或验证三角形是否为直角。
djacent. Use the mnemonic SOH CAH TOA. In right-angled triangles, sides and angles can be found using these ratios.
三个三角函数比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。记忆口诀 "SOH CAH TOA"。在直角三角形中可利用这些比求边和角。
+ c² - 2bc cos A. The area formula using two sides and included angle: Area = ½ ab sin C.
对于非直角三角形，可用正弦定理：a/sin A = b/sin B = c/sin C，和余弦定理：a² = b² + c² - 2bc cos A。已知两边及夹角求面积：面积 = ½ ab sin C。
10. Vectors and Transformations | 向量与变换
A vector has magnitude and direction, represented as a column vector (x, y) or by a directed line segment. Addition uses the triangle law; multiplication by a scalar changes magnitude but not direction (unless the scalar is negative).
向量有大小和方向，可用列向量 (x, y) 或有向线段表示。向量加法遵循三角形法则；标量乘法改变大小但不改变方向（标量为负时方向反转）。
Transformations include translation (sliding by a vector), reflection (mirroring in a line), rotation (turning around a centre by an angle), and enlargement (resizing from a centre with a scale factor). Negative scale factors produce an inverted image.
变换包括平移（按向量滑动）、反射（关于直线翻折）、旋转（绕中心转一定角度）和放大（从中心按比例因子缩放）。负的比例因子会产生倒置的图像。
rotation 90° clockwise about (0,0). Combining transformations requires attention to order.
描述变换需完整给出类型、参数与细节，例如关于直线 y=x 的反射，或绕 (0,0) 顺时针旋转 90°。组合变换时需要注意顺序。
11. Statistics – Averages, Charts and Cumulative Frequency | 统计 – 平均数、图表与累积频数
sum of values ÷ number of values; median is the middle value when ordered; mode is the most frequent value. The range measures spread: highest value minus lowest value.
集中量数：平均数 = 总和 ÷ 个数；中位数是排序后的中间值；众数是出现频次最高的值。极差衡量离散度：最大值减最小值。
ency density = frequency ÷ class width, so that the area of each bar represents frequency.
分组频数表需用估计平均数 = Σ(fx) / Σf，x 为组中点。直方图使用频数密度 = 频数 ÷ 组距，使得每个条形的面积代表频数。
Cumulative frequency diagrams help find medians, quartiles and interquartile range. Box plots (box-and-whisker) give a five-number summary: minimum, Q1, median, Q3, maximum. Scatter diagrams show correlation; a line of best fit can be drawn for prediction.
累积频数图用于求中位数、四分位数和四分位距。箱形图（盒须图）给出五数概括：最小值、Q1、中位数、Q3、最大值。散点图显示相关性，可画最佳拟合线进行预测。
12. Probability | 概率
nt). For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率是事件发生可能性的度量，范围从 0（不可能）到 1（必然）。事件不发生的概率为 1 - P(事件)。对于等可能结果，P(事件) = 有利结果数 / 总结果数。
Tree diagrams handle combined independent events by multiplying probabilities along branches; add probabilities for mutually exclusive combined outcomes. Remember to check that the sum of probabilities from a single point is 1.
树状图通过分支相乘处理独立事件的组合；互斥结果的概率相加。记住检查从同一个点出发的概率之和为 1。
Conditional probability deals with the probability of an event given that another has occurred, written P(A|B). Always adjust the sample space accordingly. Venn diagrams and two-way tables are useful tools.
条件概率处理在另一事件已发生的条件下某事件的概率，记作 P(A|B)。必须相应调整样本空间。维恩图和双向表是有效的工具。
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