📚 Year 11 Edexcel Maths: Quick-Reference Formula & Theorem Handbook | Year 11 Edexcel 数学：公式定理速查手册
This quick-reference guide collects the essential formulas, rules and theorems you need to master for your Edexcel Year 11 mathematics course. All content is aligned with the Pearson Edexcel GCSE specification and covers both Foundation and Higher tier key facts. Use this handbook to support your revision, tackle exam questions with confidence and check your understanding of core mathematical relationships.
这本速查手册汇集了 Year 11 Edexcel 数学课程必须掌握的关键公式、法则和定理。内容完全对标 Pearson Edexcel GCSE 考纲，涵盖基础与高阶等级的重要知识点。借助本手册辅助复习，你将能够自信应对考题，并检验自己对核心数学关系的理解。
1. Algebra Essentials | 代数基础
Algebraic manipulation lies at the heart of the GCSE syllabus. Remember the basic rules for expanding, factorising and working with powers.
代数运算是 GCSE 教学大纲的核心。牢记展开、因式分解及幂运算的基本法则。
aw: a(b + c) = ab + ac. Always multiply the term outside the bracket by every term inside.
分配律：a(b + c) = ab + ac。务必用括号外的项乘以括号内的每一项。
squares: a² − b² = (a + b)(a − b). This is one of the most common factorisation patterns.
平方差公式：a² − b² = (a + b)(a − b)。这是最常见的因式分解模式之一。
ᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. These apply for all rational values of m and n at Higher tier.
指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。高阶等级中 m、n 为任意有理数时这些定律仍适用。
Factorising quadratics x² + bx + c: find two numbers that multiply to c and add to b.
二次三项式 x² + bx + c 的因式分解：寻找乘积为 c、和为 b 的两个数。
2. Quadratic Equations | 二次方程
Quadratics appear in many forms. You must be able to solve them by factorising, using the quadratic formula or completing the square.
二次方程以多种形式出现。你必须掌握因式分解法、公式法和配方法求解。
Standard form: ax² + bx + c = 0, where a ≠ 0.
标准式：ax² + bx + c = 0，其中 a ≠ 0。
Quadratic formula: x = [−b ± √b² − 4ac] / (2a). This is given on the Higher formula sheet but not always on Foundation — memorise it.
二次公式：x = [−b ± √b² − 4ac] / (2a)。高阶公式表中会提供此公式，但基础卷有时不提供，建议牢记。
real roots; if Δ = 0 there is exactly one repeated root; if Δ < 0 there are no real roots.
判别式 Δ = b² − 4ac：若 Δ > 0 则有两个不等实根；Δ = 0 则有一个重根；Δ < 0 则无实根。
+ b/2)² − (b/2)². This rewrites a quadratic in vertex form and reveals the turning point.
配方法：x² + bx = (x + b/2)² − (b/2)²。这可将二次式改写为顶点式并揭示图像的转折点。
3. Linear Graphs & Coordinate Geometry | 直线图像与坐标几何
The ability to interpret and manipulate linear equations is essential. Key concepts include gradient, intercept and parallel/perpendicular lines.
解读和处理一次方程的能力至关重要。核心概念包括斜率、截距以及平行线与垂直线。
Equation of a straight line: y = mx + c, where m is the gradient and c is the y‑intercept. Alternatively, y − y₁ = m(x − x₁).
直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。也可使用点斜式 y − y₁ = m(x − x₁)。
Gradient formula: m = (y₂ − y₁) / (x₂ − x₁). Rise over run.
斜率公式：m = (y₂ − y₁) / (x₂ − x₁)。纵向增量除以横向增量。
Parallel lines have the same gradient: m₁ = m₂.
平行直线斜率相等：m₁ = m₂。
Perpendicular lines satisfy m₁ × m₂ = −1 (provided neither is vertical).
垂直线满足 m₁ × m₂ = −1（假设均不垂直于 x 轴）。
Midpoint of (x₁, y₁) and (x₂, y₂): ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ).
两点中点公式：( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。
4. Sequences | 数列
Sequences are a key part of the algebra topic. You should be comfortable with both arithmetic and special sequences.
数列是代数主题的重要组成。你应熟练掌握等差数列及特殊的数列类型。
Arithmetic sequence: term‑to‑erm rule adds a constant difference d. nth term = a + (n − 1)d, where a is the first term.
等差数列：逐项增加固定差值 d。第 n 项公式：a + (n − 1)d，其中 a 为首项。
Recognising special sequences: square numbers (1, 4, 9, 16, …), cube numbers (1, 8, 27, 64, …), triangular numbers (1, 3, 6, 10, …), Fibonacci‑type sequences.
识别特殊数列：平方数（1, 4, 9, 16, …）、立方数（1, 8, 27, 64, …）、三角形数（1, 3, 6, 10, …）以及类斐波那契数列。
Quadratic sequences: the second difference is constant. The nth term takes the form an² + bn + c.
二次数列：二阶差为一常数。第 n 项形式为 an² + bn + c。
5. Pythagoras’ Theorem & Trigonometry (Right‑angled triangles) | 勾股定理与三角学（直角三角形）
These are foundational for solving side lengths and angles in right‑angled triangles. Learn the exact values of sin, cos and tan for key angles.
这些是求解直角三角形边长和角度的基础。务必熟记关键角度的正弦、余弦和正切的精确值。
orem: a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle).
勾股定理：a² + b² = c²，其中 c 为斜边（直角对边）。
os θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use SOH CAH TOA to recall them.
三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用 SOH CAH TOA 助记。
Exact values (must be memorised):
sin 30° = 1/2, sin 45° = 1/√2 = √2/2, sin 60° = √3/2;
cos 30° = √3/2, cos 45° = 1/√2 = √2/2, cos 60° = 1/2;
tan 30° = 1/√3 = √3/3, tan 45° = 1, tan 60° = √3.精确值（必须牢记）：
sin 30° = 1/2，sin 45° = 1/√2 = √2/2，sin 60° = √3/2；
cos 30° = √3/2，cos 45° = 1/√2 = √2/2，cos 60° = 1/2；
tan 30° = 1/√3 = √3/3，tan 45° = 1，tan 60° = √3。Angles of elevation and depression: measured from the horizontal. Draw a diagram and apply trigonometry inside the right‑angled triangle formed.
仰角与俯角：均以水平线为基准测量。画出示意图并在构成的直角三角形中应用三角学。
6. Non‑right‑angled Triangle Rules (Higher) | 非直角三角形法则（高阶）
For triangles that do not contain a right angle, you need the sine rule, cosine rule and the area formula using sine.
对于不含直角的三角形，需使用正弦定理、余弦定理以及含正弦的面积公式。
= sin B / b = sin C / c. Use when you know two angles and one side, or two sides and a non‑included angle.
正弦定理：a / sin A = b / sin B = c / sin C，或其倒数形式。用于已知两角一边或两边一对角的情形。
+ c² − 2bc cos A. Use when you have two sides and the included angle, or all three sides.
余弦定理：a² = b² + c² − 2bc cos A。用于已知两边及其夹角或已知三边的情形。
Area of a triangle: ½ ab sin C. This is the area formula when two sides and the included angle are known.
三角形面积公式：½ ab sin C。已知两边及其夹角时使用。
7. Perimeter, Area & Volume | 周长、面积与体积
Memorise standard geometric formulas for 2D shapes and 3D solids. Be ready to apply them in compound shapes and real‑world contexts.
熟记二维图形与三维立体的标准几何公式。能够在组合图形和实际情境中加以应用。
| Shape / Solid | Formula (English) | 公式 (中文) |
|---|---|---|
| Rectangle area | length × width | 长 × 宽 |
| Triangle area | ½ × base × perpendicular height | ½ × 底 × 垂直高 |
| Parallelogram area | base × perpendicular height | 底 × 垂直高 |
| Trapezium area | ½(a + b)h | ½(a + b)h |
| Circle circumference | 2πr (or πd) | 2πr（或 πd） |
| Circle area | πr² | πr² |
| Prism volume | area of cross‑section × length | 横截面积 × 长度 |
| Cylinder volume | πr²h | πr²h |
| Pyramid volume | ⅓ × base area × vertical height | ⅓ × 底面积 × 垂直高 |
| Cone volume (Higher) | ⅓πr²h | ⅓πr²h |
| Sphere volume (Higher) | ⁴⁄₃πr³ | ⁴⁄₃πr³ |
8. Angles & Circle Theorems | 角度与圆定理
Angle facts and circle theorems appear frequently in both calculator and non‑calculator papers. Know them thoroughly.
角度性质与圆定理在计算器与非计算器试卷中频繁出现，务必彻底掌握。
Angles on a straight line sum to 180°; angles around a point sum to 360°.
平角为 180°；周角为 360°。
Vertically opposite angles are equal.
对顶角相等。
Alternate angles, corresponding angles and co‑interior angles on parallel lines.
平行线中的内错角、同位角与同旁内角。
ngle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; radius is perpendicular to tangent at point of contact; the angle between a tangent and a chord equals the angle in the alternate segment.
圆定理：圆心角等于两倍圆周角；半圆上的圆周角为 90°；同弧上的圆周角相等；圆内接四边形对角和为 180°；半径垂直于切点处的切线；切线与弦的夹角等于交错弧上的圆周角。
9. Transformations & Vectors | 变换与向量
Transformations change the position or size of a shape. Vectors describe translations and directed movements.
变换会改变图形的位置或大小。向量用于描述平移及有向线段。
Four transformations: translation (vector), reflection (mirror line), rotation (centre, angle, direction), enlargement (centre, scale factor).
四种变换：平移（列向量）、反射（镜面线）、旋转（中心、角度、方向）、放大（中心、比例因子）。
For enlargements, if scale factor k > 1 the shape is enlarged; 0 < k < 1 gives a reduction; negative k also appears at Higher tier and includes rotation by 180°.
放大时，比例因子 k > 1 则放大；0 < k < 1 则缩小；高阶等级中还出现负比例因子（k < 0），此时图形同时旋转 180°。
Vector notation: column vector (x, y)ᵀ or bold a. The magnitude is √x² + y².
向量表示法：列向量 (x, y)ᵀ 或粗体 a。模长为 √x² + y²。
Vector addition and scalar multiplication: a + b corresponds to the diagonal of a parallelogram; k a stretches the vector by factor k.
向量加法与数乘：a + b 对应平行四边形对角线；k a 将向量拉伸 k 倍。
10. Probability | 概率
Probability rules help quantify chance. Learn to handle combined events using tree diagrams and the AND/OR rules.
概率法则用于量化可能性。需掌握通过树状图以及“与/或”规则处理复合事件。
Probability scale: 0 (impossible) to 1 (certain). For any event A, 0 ≤ P(A) ≤ 1.
概率尺度：0（不可能）至 1（必然）。对任意事件 A，满足 0 ≤ P(A) ≤ 1。
Sum of probabilities of all possible outcomes = 1. P(not A) = 1 − P(A).
所有可能结果的概率之和为 1。P(非 A) = 1 − P(A)。
Expected frequency = probability × number of trials.
期望频数 = 概率 × 实验次数。
For two independent events A and B: P(A and B) = P(A) × P(B).
两独立事件的交事件：P(A 且 B) = P(A) × P(B)。
For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
互斥事件的并事件：P(A 或 B) = P(A) + P(B)。
Tree diagrams: multiply along branches for "and" probabilities, add relevant end‑branch probabilities for "or". At Higher tier, conditional probability appears: P(B|A) = P(A and B) / P(A).
树状图：沿分支相乘求“与”概率，将相关末端分支概率相加求“或”概率。高阶等级要求条件概率：P(B|A) = P(A 且 B) / P(A)。
11. Statistics & Data Handling | 统计与数据处理
Statistics requires fluency with averages, measures of spread, and chart interpretation. Remember to check whether data is ungrouped or grouped.
统计学要求熟练运用平均数、离散度量以及图表解读。注意区分数据是未分组还是分组。
Measures of central tendency: mean (sum ÷ count), median (middle value), mode (most frequent).
集中趋势度量：平均数（总和 ÷ 个数）、中位数（中间值）、众数（出现频率最高）。
Range = maximum value − minimum value.
极差 = 最大值 − 最小值。
ange (IQR) = upper quartile − lower quartile. The IQR measures the spread of the middle 50% of data.
四分位距（IQR）= 上四分位数 − 下四分位数。IQR 衡量中间 50% 数据的离散程度。
d frequency, the mean is estimated using midpoints: estimated mean = Σ(f × midpoint) / Σf.
分组数据求平均数用组中值估算：估算平均数 = Σ(f × 组中值) / Σf。
Cumulative frequency graphs: the median and quartiles can be read from the graph. Use a smooth curve through points plotted at the upper class boundary.
累积频数图：可以从图上读取中位数与四分位数。在组上界处描点，并用平滑曲线连接。
Scatter graphs show correlation. A line of best fit can be drawn to predict values, but avoid extrapolation beyond the data range.
散点图显示相关性。可绘制最佳拟合线进行预测，但应避免超出数据范围的外推。
12. Compound Measures & Proportion | 复合单位与比例
Compound measures link two different quantities. Proportional reasoning is tested across many topics, from recipes to exchange rates.
复合单位将两个不同的量联系起来。比例推理在许多主题中都会涉及，从食谱到汇率无处不在。
o distance = speed × time, time = distance ÷ speed. Speed units are typically km/h or m/s.
速度 = 距离 ÷ 时间。变形为 距离 = 速度 × 时间，时间 = 距离 ÷ 速度。速度单位通常为 km/h 或 m/s。
Density = mass ÷ volume. Commonly g/cm³ or kg/m³.
密度 = 质量 ÷ 体积。常见单位 g/cm³ 或 kg/m³。
Pressure = force ÷ area. Unit: N/m² (pascal). Used in Higher tier.
压强 = 力 ÷ 面积。单位：N/m²（帕斯卡）。用于高阶等级。
, where k is the constant of proportionality. Graph is a straight line through the origin.
正比例：y = kx，k 为比例常数。图像为过原点的直线。
Inverse proportion: y = k / x. Graph is a rectangular hyperbola (Higher).
反比例：y = k / x。图像为等轴双曲线（高阶）。
Percentage change: % hange = (change / original value) × 100. Compound interest: final amount = P (1 + r/100)ⁿ.
百分比变化：变化百分比 = (变化量 / 原值) × 100。复利：终值 = 本金 × (1 + 利率/100)ⁿ。
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