📚 Core Topics for Year 11 Cambridge Mathematics | 剑桥Year 11 数学核心知识点梳理
Year 11 Cambridge Mathematics (IGCSE 0580) builds a strong foundation across number, algebra, geometry, statistics and problem‑solving. This revision guide consolidates the core topics, highlighting key formulae, common misconceptions and typical exam approaches — helping you revise efficiently and aim for high marks.
剑桥 Year 11 数学 (IGCSE 0580) 旨在为代数、几何、统计和问题解决打下坚实基础。本复习指南梳理核心知识点，突出关键公式、常见误区以及典型解题方法，帮助你高效复习，冲刺高分。
1. Number and Operations | 数与运算
Master the number system, including integers, fractions, decimals and percentages. Understand directed numbers and how to apply the four operations in the correct order (BIDMAS/BODMAS). Converting between fractions, decimals and percentages is essential for both calculator and non‑calculator papers.
掌握数的体系，包括整数、分数、小数和百分数。理解有向数，能正确运用四则运算的顺序（括号、指数、乘除、加减）。在可用和不可用计算器的试卷中，熟练进行分数、小数与百分数之间的转换都至关重要。
Be confident with ratios and proportions. You should be able to divide a quantity in a given ratio, find an unknown term in a proportion, and solve real‑life problems involving map scales, recipes and mixtures. Direct and inverse proportion, expressed as y ∝ x or y ∝ 1/x, often appears alongside algebraic manipulation.
对比和比例要有信心。应能将一个量按给定比例分配，求比例中的未知项，解决涉及地图比例尺、配方和混合物的实际问题。正比例和反比例（y ∝ x 或 y ∝ 1/x）常与代数运算结合出现。
Work accurately with standard form (scientific notation) a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This includes adding, subtracting, multiplying and dividing numbers in standard form. Also, understand upper and lower bounds: when measurements are rounded, the true value lies in an interval determined by half the unit of precision.
准确使用标准形式（科学记数法）a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。这包括对标准形式下的数进行加、减、乘、除运算。同时，理解上界和下界：当测量值被舍入时，真值位于由半个精度单位确定的区间内。
2. Algebra: Expressions, Equations and Formulae | 代数：表达式、方程与公式
Expand and factorise algebraic expressions fluently. This includes removing brackets such as (x + 2)(x – 5), factorising quadratics like x² + 7x + 10, and taking out common factors. Recognise the difference of two squares a² – b² = (a – b)(a + b) — a frequently tested pattern.
熟练地对代数式进行展开和因式分解。包括去掉括号如 (x + 2)(x – 5)，对二次式如 x² + 7x + 10 进行因式分解，以及提取公因式。识别平方差公式 a² – b² = (a – b)(a + b)，这是常考的模型。
Solve linear equations, including those with brackets and fractions. For simultaneous linear equations, you must be confident with both elimination and substitution methods. When representing graphically, the solution is the intersection point of the two lines.
解线性方程，包括含有括号和分数的方程。对于联立线性方程组，必须熟练掌握消元法和代入法。用图形表示时，解即为两条直线的交点坐标。
Rearrange formulae to change the subject. This involves applying inverse operations in the correct order, treating the variable you want as the unknown. Typical examples: making x the subject of y = ax + b, or making r the subject of A = πr².
变换公式的主项。这需要按正确顺序使用逆运算，将所需求的变量视为未知数。典例：将 y = ax + b 变为 x = (y – b)/a，或将 A = πr² 变为 r = √(A/π)。
3. Coordinate Geometry and Straight‑Line Graphs | 坐标几何与直线图像
The equation of a straight line is usually written as y = mx + c, where m is the gradient and c is the y‑intercept. You must be able to find the gradient from two points using m = (y₂ – y₁)/(x₂ – x₁). The intercept c is where the line cuts the y‑axis (x = 0).
直线的方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。必须会用两点求斜率：m = (y₂ – y₁)/(x₂ – x₁)。截距 c 是直线与 y 轴交点的纵坐标（x = 0）。
Parallel lines have the same gradient: m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = -1. These relationships are essential for solving problems involving line equations, such as finding the equation of a line parallel to a given line passing through a specific point.
平行线斜率相等：m₁ = m₂。垂直线满足 m₁ × m₂ = -1。这些关系在解决直线方程问题时至关重要，例如求过某点且平行于已知直线的直线方程。
You should also be able to calculate the midpoint and the length of a segment between two points. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance: √((x₂ – x₁)² + (y₂ – y₁)²). These formulas are the foundation of coordinate geometry.
还应会计算两点间的中点坐标和线段长度。中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。距离：√((x₂ – x₁)² + (y₂ – y₁)²)。这些公式是坐标几何的基础。
4. Quadratic Equations and Functions | 二次方程与函数
Quadratic equations take the form ax² + bx + c = 0. You can solve them by factorising, completing the square, or using the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a. The formula is especially useful when factorisation is not obvious or when answers are required to a given number of decimal places.
二次方程的形式为 ax² + bx + c = 0。可通过因式分解、配方法或二次公式求解：x = [-b ± √(b² – 4ac)] / 2a。当因式分解不显著或要求保留指定小数位数时，公式法尤为有用。
The discriminant Δ = b² – 4ac determines the nature of the roots: if Δ > 0 there are two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 means no real roots (the parabola does not intersect the x‑axis). Exam questions often ask you to find a condition for real roots using inequalities.
判别式 Δ = b² – 4ac 决定根的性质：若 Δ > 0，有两个相异实根；Δ = 0 给出一个重根；Δ < 0 意味着无实根（抛物线不与 x 轴相交）。试题常要求利用不等式求出存在实根的条件。
The graph of y = ax² + bx + c is a parabola. The sign of a determines its direction (positive a: ∪‑shape; negative a: ∩‑shape). The turning point can be found by completing the square or by using x = -b/(2a). The y‑intercept is at c, and the x‑intercepts (roots) are found by solving ax² + bx + c = 0.
y = ax² + bx + c 的图像是一条抛物线。a 的符号决定开口方向（a > 0：开口向上 ∪ 形；a < 0：开口向下 ∩ 形）。顶点坐标可通过配方法或 x = -b/(2a) 求得。y 轴截距为 c，x 轴截距（根）通过解 ax² + bx + c = 0 获得。
5. Inequalities | 不等式
Solve linear inequalities in a similar way to linear equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number. For example, -2x < 6 becomes x > -3. Compound inequalities like -3 ≤ 2x + 1 < 7 are solved by isolating the variable in the middle.
解线性不等式的方法与线性方程类似，但需注意：当两边同乘或除以负数时，不等号方向要改变。例如，-2x < 6 变为 x > -3。复合不等式如 -3 ≤ 2x + 1 < 7 可通过将变量分离在中间来解决。
Represent solutions on a number line using open or closed circles: open circle for strict inequalities (), closed circle for inclusive inequalities (≤, ≥). The arrow indicates the direction of the solution set.
在数轴上用空心圆或实心圆表示解集：严格不等式 () 用空心圆，包含等号的不等式 (≤, ≥) 用实心圆。箭头表示解集的方向。
Graphical inequalities involve shading unwanted regions. When dealing with two variables, draw the boundary lines (dashed for ; solid for ≤, ≥), then identify and shade the region that satisfies all constraints simultaneously. Label the unshaded region R as requested.
图形不等式涉及涂去不符合要求的区域。处理两个变量时，先画出边界线（ 用虚线；≤, ≥ 用实线），然后确定并涂去同时满足所有约束的区域。按题目要求将未涂区域标记为 R。
6. Geometry and Trigonometry | 几何与三角学
Angle properties are fundamental: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal. For parallel lines, know alternate angles, corresponding angles and interior (co‑interior) angles. In polygons, the sum of interior angles = (n – 2) × 180°, and the sum of exterior angles is always 360°.
角的性质是基础：直线上的邻角之和为 180°，同顶点的角之和为 360°，对顶角相等。对于平行线，掌握内错角、同位角和同旁内角。多边形中，内角和 = (n – 2) × 180°，外角和始终为 360°。
Pythagoras’ theorem applies to right‑angled triangles: a² + b² = c², where c is the hypotenuse. Use it to find missing sides or to prove a triangle is right‑angled. The three trigonometric ratios — sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent — are used to find angles and side lengths.
毕达哥拉斯定理适用于直角三角形：a² + b² = c²，其中 c 为斜边。用它来求未知边长或证明一个三角形是直角三角形。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——用于求角度和边长。
For non‑right‑angled triangles, the sine rule (a/sin A = b/sin B = c/sin C) and the cosine rule (a² = b² + c² – 2bc cos A) are required. The area of any triangle can be found using ½ ab sin C. These often appear in multi‑step problems combining bearings and geometry.
对于非直角三角形，需用正弦定理 (a/sin A = b/sin B = c/sin C) 和余弦定理 (a² = b² + c² – 2bc cos A)。任意三角形的面积可用 ½ ab sin C 求得。这些常出现在结合方位角与几何的多步骤问题中。
7. Mensuration: Perimeter, Area and Volume | 测量：周长、面积与体积
You must recall and apply formulas for the perimeter and area of common 2D shapes. For circles, circumference = 2πr and area = πr². When dealing with sectors, arc length = (θ/360) × 2πr, and sector area = (θ/360) × πr², where θ is the central angle in degrees.
必须记住并能应用常见二维图形的周长与面积公式。对于圆，周长 = 2πr，面积 = πr²。处理扇形时，弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²，其中 θ 为圆心角度数。
| Shape | Area Formula (English) | 面积公式 (中文) |
|---|---|---|
| Rectangle | A = l × w | 矩形面积 = 长 × 宽 |
| Triangle | A = ½ × base × height | 三角形面积 = ½ × 底 × 高 |
| Parallelogram | A = b × h | 平行四边形面积 = 底 × 高 |
| Trapezium | A = ½(a + b)h | 梯形面积 = ½(上底 + 下底) × 高 |
Surface area and volume of 3D solids: prism (volume = area of cross‑section × length), cylinder (volume = πr²h, curved surface = 2πrh), cone (volume = ⅓πr²h, curved surface = πrl, where l is slant height), and sphere (volume = ⁴⁄₃πr³, surface = 4πr²). Compound measures like density = mass/volume and speed = distance/time must be converted correctly.
三维体的表面积和体积：棱柱（体积 = 横截面积 × 长），圆柱（体积 = πr²h，侧面积 = 2πrh），圆锥（体积 = ⅓πr²h，侧面积 = πrl，l 为母线长），球体（体积 = ⁴⁄₃πr³，表面积 = 4πr²）。复合度量如密度 = 质量/体积、速率 = 路程/时间，须正确转换单位。
8. Statistics and Probability | 统计与概率
Represent data using bar charts, pie charts, histograms (for continuous data with equal or unequal class widths), frequency polygons and cumulative frequency graphs. Calculate and interpret the mean, median, mode and range. The interquartile range (IQR = Q₃ – Q₁) is a measure of spread often used with box‑and‑whisker plots.
使用条形图、饼图、直方图（用于等宽或不等宽组距的连续数据）、频数多边形和累积频数图表示数据。计算并解释平均数、中位数、众数和极差。四分位距 (IQR = Q₃ – Q₁) 是常与箱线图一起使用的离散度量。
Probability of an event = (number of favourable outcomes) / (total number of outcomes), provided all outcomes are equally likely. Relative frequency can estimate probability from experimental data. Combined events are tackled with tree diagrams (multiply along branches, add probabilities of mutually exclusive paths). Understand conditional probability: P(A given B) = P(A and B) / P(B).
在所有结果等可能的前提下，事件概率 = (有利结果数)/(总结果数)。相对频数可从实验数据中估计概率。处理组合事件使用树图（沿分支相乘，互斥路径的概率相加）。理解条件概率：P(A given B) = P(A and B) / P(B)。
Venn diagrams and two‑way tables are powerful tools for organising information and solving “and/or” probability problems. Always check that probabilities sum to 1 and be careful with replacement versus without replacement scenarios when drawing multiple items.
维恩图和双向表是组织信息并解决“且/或”概率问题的有力工具。始终检查概率之和是否为 1，并注意多次抽取物品时放回与不放回情形的区别。
9. Vectors and Transformations | 向量与变换
A vector quantity has both magnitude and direction, often written as a column vector or as a letter combination like AB. Vector addition and subtraction are performed component‑wise. Scalar multiplication changes the length without affecting the direction (unless the scalar is negative).
向量同时具有大小和方向，常写作列向量形式或像 AB 这样的字母组合。向量加减法按分量进行。标量乘法改变长度但不影响方向（除非标量为负）。
Be able to find the magnitude (length) of a vector using Pythagorean theorem. Parallel vectors are scalar multiples of each other. Position vectors and the concept of AB = b – a are vital for proving collinearity or finding midpoints in vector form.
会使用勾股定理求向量的大小（长度）。平行向量互为标量倍数。位置向量以及 AB = b – a 的概念对于用向量形式证明共线性或求中点至关重要。
Transformations: translation (moving every point by a given vector), reflection (mirroring in a given line), rotation (turning about a centre through an angle), and enlargement (scaling by a factor k from a centre). You must be able to describe a single transformation fully and use matrices to represent certain transformations in the Extended syllabus.
变换包括：平移（按给定向量移动各点）、反射（关于给定直线作镜像）、旋转（绕中心转过一个角度）和放大（以某点为中心按比例 k 缩放）。须能完整描述一个单一变换，并在扩展大纲中会用矩阵表示某些变换。
Combined transformations are applied in a given order; the final image depends on the sequence. In matrix terms, the transformation matrices are multiplied, with the first transformation on the right. Recognise that enlargement with a negative scale factor produces an image on the opposite side of the centre and is inverted.
组合变换按指定顺序进行；最终图像取决于次序。在矩阵表示中，变换矩阵相乘，先进行的变换位于右侧。识别负比例放大会使图像出现在中心的另一侧，且呈倒置状态。
10. Functions and Sequences | 函数与数列
A function is a rule that assigns exactly one output for each input. Notation: f(x) = 2x + 3. Domain is the set of allowed inputs, and range the set of possible outputs. Composite functions fg(x) = f(g(x)) mean applying g first, then f. Inverse function f⁻¹ reverses the mapping, provided f is one‑to‑one.
函数是一种为每个输入分配唯一输出的规则。记作：f(x) = 2x + 3。定义域是允许输入值的集合，值域是可能输出值的集合。复合函数 fg(x) = f(g(x)) 表示先作用 g，再作用 f。反函数 f⁻¹ 逆转映射，前提是 f 为一一映射。
Recognising and sketching graphs of basic functions — linear, quadratic, cubic, reciprocal and exponential — is helpful. Transformations of functions: f(x) + a (vertical translation), f(x + a) (horizontal translation), -f(x) (reflection in x‑axis) and af(x) (vertical stretch) modify the graph in predictable ways.
识别并绘制基本函数的图像——线性、二次、三次、倒数和指数函数——很有帮助。函数变换：f(x) + a（纵向平移），f(x + a)（横向平移），-f(x)（关于 x 轴反射）和 af(x)（纵向拉伸），它们以可预测的方式改变图像。
Sequences: linear (arithmetic) sequences have a constant difference d and their nth term is a + (n – 1)d, where a is the first term. Quadratic sequences have a constant second difference; their nth term is of the form An² + Bn + C. You should be able to determine the nth term formula and use it to find any term.
数列：线性（等差）数列有恒定公差 d，其第 n 项为 a + (n – 1)d，其中 a 为首项。二次数列有恒定的二级差分；其第 n 项形式为 An² + Bn + C。应能求出通项公式并用之求任意项。
Be prepared for problems linking sequences with patterns, diagrams, or practical contexts. Always check the first few terms to verify your formula, and ensure you present it in its simplest expanded form.
准备好处理将数列与图形、图表或实际情境相关联的问题。始终检查最初几项以验证公式，并确保以最简展开形式呈现通项。
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