📚 Year 11 CAIE Mathematics: Core Knowledge Points Review | 年11 CAIE 数学：核心知识点梳理
This article provides a comprehensive review of the most essential topics in the Year 11 CAIE IGCSE Mathematics syllabus. It covers number concepts, algebra, geometry, mensuration, trigonometry, vectors, transformations, probability and statistics – all presented with clear bilingual explanations and key formulas to support effective revision.
本文全面梳理年11 CAIE IGCSE 数学课程中最核心的知识点，涵盖数的概念、代数、几何、测量与求积、三角学、向量与变换、概率和统计，配以清晰的双语讲解与关键公式，为高效复习提供有力支持。
1. Number Concepts | 数的概念
The number system includes natural numbers ℕ, integers ℤ, rational numbers ℚ (fractions and recurring or terminating decimals), irrational numbers (non‑recurring, non‑terminating decimals like √2 or π) and real numbers ℝ.
数系包括自然数 ℕ、整数 ℤ、有理数 ℚ（分数以及循环或终止小数）、无理数（不循环无限小数，例如 √2 或 π）以及实数 ℝ。
A prime number has exactly two distinct factors: 1 and itself. Composite numbers have more than two factors. The numbers 0 and 1 are neither prime nor composite.
质数恰好有两个不同的因数：1 和自身。合数有两个以上的因数。0 和 1 既不是质数也不是合数。
To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM), write each number as a product of prime factors. HCF is the product of the smallest powers of common prime factors; LCM is the product of the highest powers of all prime factors present.
求最大公因数（HCF）和最小公倍数（LCM）时，先将每个数写成质因数乘积。HCF 是公共质因数的最小次幂之积；LCM 是所有出现质因数的最高次幂之积。
Standard form writes a number as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. It is used for very large or very small numbers, e.g., 4.2 × 10⁵ or 6.7 × 10⁻³.
标准形式将数字写作 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数，用于表示极大或极小的数，如 4.2 × 10⁵ 或 6.7 × 10⁻³。
When rounding to a given number of decimal places or significant figures, identify the decisive digit and look at the next digit. If it is 5 or more, round up; otherwise, round down. Upper and lower bounds follow from the accuracy of measurement.
将数值四舍五入到指定的小数位数或有效数字时，识别决定位并观察下一位数字。如果该位数字 ≥ 5，则进位，否则舍去。测量值的上界和下界取决于其精确度。
Ratios compare two or more quantities. Dividing a quantity in a given ratio involves finding the value of one part, then multiplying. Direct proportion: y = kx. Inverse proportion: y = k/x.
比用于比较两个或多个数量。按给定比例分配时，先求出一份的量再相乘。正比例关系：y = kx；反比例关系：y = k/x。
Percentage change is calculated as (change / original) × 100%. Successive percentage changes are applied step‑by‑step, not simply added.
百分比变化 = (变化量 ÷ 原值) × 100%。连续的百分比变化需逐步计算，不可直接相加。
2. Algebra Fundamentals | 代数基础
Expand brackets by multiplying each term inside: a(b + c) = ab + ac. Factorising is the reverse process: write an expression as a product of its factors, e.g., 6x² + 9x = 3x(2x + 3).
去括号时，将括号外的项乘以括号内的每一项：a(b + c) = ab + ac。因式分解是其逆过程：将表达式写成几个因式的乘积，如 6x² + 9x = 3x(2x + 3)。
To factorise a quadratic trinomial x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c with a ≠ 1, use the method of splitting the middle term or trial factors.
对于二次三项式 x² + bx + c，寻找两个数，其积为 c，和为 b。对于 a ≠ 1 的 ax² + bx + c，可使用拆分中项法或尝试因式组合。
The difference of two squares factors as a² − b² = (a + b)(a − b). Recognising this pattern saves time in both expansion and factorising.
平方差公式 a² − b² = (a + b)(a − b) 在展开和因式分解中都非常有用，识别该模式可节省时间。
Simplify algebraic fractions by cancelling common factors after factorising numerators and denominators. Always state restrictions where the denominator cannot be zero.
化简代数分式时，先将分子分母因式分解，然后约去公因式。需注明分母不为零的限制条件。
Changing the subject of a formula means rearranging it to isolate a target variable. Perform inverse operations in reverse order, treating the variable as the unknown.
公式变形是指将公式重新排列，使目标变量成为主体。应按照逆向运算顺序，将目标变量视为未知数处理。
3. Equations and Inequalities | 方程与不等式
Solve linear equations by performing the same operation on both sides, collecting like terms and isolating the variable. Always check your solution by substitution.
解一元一次方程时，等式两边同时进行相同运算，合并同类项后求出未知数。应将解代入原方程验算。
For quadratic equations ax² + bx + c = 0, we can factorise, use the quadratic formula, or complete the square. The formula is:
x = (−b ± √(b² − 4ac)) / (2a)
解二次方程 ax² + bx + c = 0 可用因式分解法、求根公式或配方法。求根公式为：
x = (−b ± √(b² − 4ac)) / (2a)
Simultaneous linear equations can be solved by elimination or substitution. When one equation is linear and the other is quadratic, substitute the linear expression into the quadratic and solve.
解二元一次联立方程可用消元法或代入法。若一个为一次方程、另一个为二次方程，则将一次式代入二次式求解。
Inequalities behave like equations, except multiplying or dividing by a negative number reverses the inequality sign. Represent solutions on a number line with open or closed circles.
解不等式与解方程类似，但当两边同乘或同除以一个负数时，不等号方向要改变。解集在数轴上用空心或实心圆点表示。
For quadratic inequalities, first find the critical values where the expression equals zero, then test intervals to determine where the inequality holds.
解二次不等式时，先求出表达式等于零的临界值，再通过区间检验确定满足不等式的区域。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is given by uₙ = a + (n − 1)d, where a is the first term.
等差数列相邻两项的差 d 为常数，第 n 项（通项）公式为 uₙ = a + (n − 1)d，其中 a 为首项。
A geometric sequence has a constant ratio r between consecutive terms. The nth term is uₙ = arⁿ⁻¹. Be careful with negative and fractional ratios.
等比数列相邻两项的比 r 为常数，第 n 项为 uₙ = arⁿ⁻¹。注意比值为负数或分数的情况。
To find whether a number is a term of a given sequence, set the nth term formula equal to that number and solve for n; n must be a positive integer.
判断某数是否为数列的项，可将通项公式令其等于该数，解出 n，若 n 为正整数则该数是数列的一项。
Finding the next terms or a missing term relies on identifying the pattern. Patterns may use linear, quadratic, or other rules.
根据已知项找后续项或缺失项，需要识别规律。规律可能是线性的、二次的或其他形式。
5. Functions and Graphs | 函数与图像
A function is a rule that assigns each input exactly one output. Notation: f(x) = expression. The domain is the set of allowed inputs; the range is the set of possible outputs.
函数是每个输入值对应唯一输出值的规则。记作 f(x)。定义域是允许的输入集，值域是可能得到的输出集。
Composite functions combine two functions: e.g., fg(x) means apply g first, then f. Inverse function f⁻¹(x) reverses the effect of f(x); it exists only if f is one‑to‑one.
复合函数将两个函数结合：fg(x) 表示先作用 g 再作用 f。反函数 f⁻¹(x) 逆转 f(x) 的作用，仅当 f 是一一对应时才存在。
Graphs of linear functions: y = mx + c, where m is the gradient and c is the y‑intercept. Sketch by plotting the intercept and using gradient rise/run.
线性函数的图像为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。作图时标出截距，并根据斜率 delta y/delta x 画出直线。
Quadratic graphs y = ax² + bx + c are parabolas. The sign of a determines upward (a > 0) or downward (a < 0) opening. Find the turning point by completing the square or using x = −b/(2a).
二次函数的图像 y = ax² + bx + c 是抛物线。a > 0 开口向上，a < 0 开口向下。可通过配方法或用 x = −b/(2a) 求顶点。
Gradients of curves change. Draw a tangent at a point and find its slope to estimate the gradient at that point. This also applies to distance‑time and speed‑time graphs.
曲线的斜率是变化的。在某点作切线，切线的斜率即为该点处曲线的瞬时变化率。此方法同样用于距离‑时间图和速度‑时间图。
6. Geometry and Angles | 几何与角度
Basic angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal.
基本角度知识：直线上的邻角之和为 180°，绕一点的周角之和为 360°，对顶角相等。
Parallel line rules: alternate angles are equal, corresponding angles are equal, interior (co‑interior) angles sum to 180°. Use these to find unknown angles and prove lines are parallel.
平行线的性质：内错角相等，同位角相等，同旁内角之和为 180°。可运用这些性质求未知角并证明两直线平行。
Interior angles of an n‑sided polygon sum to (n − 2) × 180°. For a regular polygon, each interior angle = (n − 2) × 180° / n, and each exterior angle = 360° / n.
n 边形的内角和为 (n − 2) × 180°。正多边形中，每个内角 = (n − 2) × 180° / n，每个外角 = 360° / n。
Circle theorems (extended): angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent‑radius perpendicularly; alternate segment theorem.
圆定理（扩展内容）：圆心角是圆周角的两倍；半圆上的圆周角为 90°；同弧上的圆周角相等；圆内接四边形对角互补；切线与过切点的半径垂直；弦切角定理（alternate segment theorem）。
7. Mensuration | 测量与求积
Perimeter and area of basic shapes must be memorised. Use appropriate units and convert between mm², cm², m² carefully.
需熟记基本平面图形的周长和面积公式。注意使用恰当单位，并在 mm², cm², m² 之间谨慎换算。
| Shape / 形状 | Area / 面积 | Perimeter / 周长 |
| Rectangle / 矩形 | length × width | 2(l + w) |
| Triangle / 三角形 | ½ × base × height | sum of sides |
| Parallelogram / 平行四边形 | base × perpendicular height | sum of sides |
| Trapezium / 梯形 | ½(a + b)h | sum of sides |
| Circle / 圆 | πr² | 2πr (circumference) |
For composite shapes, split them into simpler parts, calculate individual areas and sum them. Include subtraction of cut‑out areas.
对于复合图形，可将其拆分为简单图形，分别计算面积后再求和。挖空部分则要减去相应面积。
Volume and surface area of prisms and cylinders: volume = cross‑sectional area × length. For a cylinder, volume = πr²h, curved surface area = 2πrh, total surface area = 2πr² + 2πrh.
棱柱和圆柱的体积 = 横截面积 × 长度。圆柱体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πr² + 2πrh。
Pyramid, cone and sphere formulas (given in exams but must be used fluently): volume of pyramid/cone = ⅓ × base area × height; volume of sphere = ⁴⁄₃πr³; surface area of sphere = 4πr²; curved surface area of cone = πrl (l = slant height).
棱锥、圆锥和球的公式（考试会提供，但需熟练运用）：棱锥/圆锥体积 = ⅓ × 底面积 × 高；球体积 = ⁴⁄₃πr³；球表面积 = 4πr²；圆锥侧面积 = πrl（l 为母线长）。
8. Trigonometry | 三角学
In right‑angled triangles, SOHCAHTOA relates angles and sides:
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
在直角三角形中，通过 SOHCAHTOA 将角和边关联：
sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边
Use inverse trig ratios to find an angle: θ = sin⁻¹(opp/hyp), etc. Ensure the calculator is in degree mode.
利用反三角函数求角度：θ = sin⁻¹(对/斜) 等。注意计算器应处于角度模式。
For non‑right‑angled triangles (extended), use the sine rule: a/sin A = b/sin B = c/sin C, or alternatively sin A / a = sin B / b. Use cosine rule: a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²) / (2bc) when finding an angle.
对于非直角三角形（扩展内容），使用正弦定理：a/sin A = b/sin B = c/sin C，或 sin A / a = sin B / b。余弦定理：a² = b² + c² − 2bc cos A；求角时用 cos A = (b² + c² − a²) / (2bc)。
Area of any triangle can be found by ½ ab sin C, where a and b are two sides and C is the included angle.
任意三角形的面积可用公式 ½ ab sin C 计算，其中 a、b 为两边长，C 为这两边的夹角。
In 3D problems, identify the relevant right‑angled triangles within the solid and apply SOHCAHTOA or rules for lengths/diagonals.
在三维问题中，先在立体图形中找出相关的直角三角形，再运用 SOHCAHTOA 或边长与对角线之间的关系求解。
9. Vectors and Transformations | 向量与变换
A vector is represented by a column (xy) or by a directed line segment. It has magnitude (length) and direction. Magnitude of (xy) is √(x² + y²).
向量可用列向量 (x y) 或有向线段表示，具有大小和方向。向量 (x y) 的模为 √(x² + y²)。
Vector operations: addition and subtraction are done by adding/subtracting corresponding components. Scalar multiplication multiplies each component by the scalar. Use these for translations.
向量的运算：加减法为对应分量相加减；数乘是将每个分量乘以该标量。平移变换即为向量加法。
Transformations in the plane: reflection (mirror line), rotation (centre, angle, direction), translation (column vector), enlargement (centre, scale factor). Be precise in describing each.
平面内的变换：反射（指定对称轴）、旋转（中心、角度、方向）、平移（列向量）、位似（中心、比例因子）。描述变换时应精确说明各要素。
An enlargement with negative scale factor produces an image on the opposite side of the centre, inverted and resized. Fractional scale factors reduce the shape.
使用负比例因子的位似会在中心的另一侧生成一个旋转 180° 且大小变化的像。比例因子小于 1 则缩小图形。
Combinations of transformations: apply one after the other. The single transformation equivalent to a combination may be identified by comparing object and final image.
变换的组合：依次进行多个变换。可比较原像与最终像，确定与组合变换等效的单一变换。
10. Probability | 概率
Probability of an event = (number of favourable outcomes) / (total number of equally likely outcomes). It lies between 0 and 1.
事件的概率 = 有利结果数 ÷ 等可能结果总数，取值范围在 0 至 1 之间。
The sum of probabilities of all mutually exclusive outcomes is 1. For complementary events, P(A’) = 1 − P(A).
所有互斥结果的概率之和为 1。对于对立事件，有 P(A’) = 1 − P(A)。
Tree diagrams display sequences of events. Multiply along branches for combined probabilities; add probabilities from different branches for alternative outcomes.
树状图展示事件发生的顺序。沿分支相乘得到该路径的组合概率；将不同分支的概率相加则得到满足条件的总概率。
Conditional probability (extended) is the probability of an event B occurring given that A has already occurred: P(B|A) = P(A and B) / P(A). Use two‑way tables or tree diagrams.
条件概率（扩展内容）指在事件 A 已发生的条件下事件 B 发生的概率：P(B|A) = P(A∩B) / P(A)。可运用二维表或树状图分析。
11. Statistics | 统计
Measures of central tendency: mean = sum of values / number of values; median = middle value when ordered; mode = most frequent value. For grouped data, use the modal class and estimate the mean using midpoints.
集中趋势的度量：平均值 = 数据总和 ÷ 数据个数；中位数 = 排序后居中的值；众数 = 出现频率最高的值。对于分组数据，用组中值估计平均值，众数则使用众数组。
Measures of spread: range = maximum − minimum. Interquartile range (IQR) = upper quartile − lower quartile. The IQR shows the spread of the middle 50% of data.
离散程度的度量：极差 = 最大值 − 最小值。四分位距 (IQR) = 上四分位数 − 下四分位数。四分位距反映中间 50% 数据的离散情况。
Cumulative frequency graphs plot cumulative frequency against upper class boundaries. Use the graph to estimate medians, quartiles and percentiles.
累积频数图将累积频数对组距上界作图，可用于估计中位数、四分位数和百分位数。
Box‑and‑whisker plots show minimum, lower quartile, median, upper quartile and maximum. They are useful for comparing distributions.
箱线图（盒须图）显示最小值、下四分位数、中位数、上四分位数及最大值，便于比较数据分布。
Histograms display grouped continuous data where frequency is represented by bar area. Frequency density = frequency / class width. Use this to plot uneven class widths.
直方图用于展示分组连续数据，以矩形面积表示频数。频数密度 = 频数 ÷ 组距。组距不等时，必须用频数密度作图。
Scatter graphs show relationships between two variables. A line of best fit can be drawn to model linear correlation and make predictions, but avoid extrapolation far beyond the data range.
散点图展示两个变量之间的关系。可以画出最佳拟合线来模拟线性相关并进行预测，但应避免对远离数据范围的区域外推。
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