📚 Year 8 CIE Mathematics Formula & Theorem Quick Reference | 八年级CIE数学公式定理速查手册
This quick reference guide collects the essential formulas, rules and theorems for the Year 8 CIE Mathematics curriculum. Keep it handy for homework, revision and exam preparation. Each section presents a clear statement of the rule in English, followed by a Chinese explanation, so you can reinforce your understanding in both languages.
这本速查手册汇集了八年级CIE数学课程的核心公式、法则与定理，方便你随时翻阅，用于作业、复习和备考。每个部分先用英文给出法则的准确描述，再用中文解释，帮助你在双语环境中巩固理解。
1. Number Operations & Index Laws | 数字运算与指数法则
When multiplying powers with the same base, add the exponents: am × an = am+n. For example, 2³ × 2² = 2⁵.
同底数幂相乘时，指数相加：am × an = am+n。例如 2³ × 2² = 2⁵。
When dividing powers with the same base, subtract the exponents: am ÷ an = am−n (a ≠ 0). For example, 5⁴ ÷ 5² = 5².
同底数幂相除时，指数相减：am ÷ an = am−n（a ≠ 0）。例如 5⁴ ÷ 5² = 5²。
Any non‑zero number raised to the power of zero equals 1: a⁰ = 1 (a ≠ 0). Negative exponents indicate reciprocals: a−n = 1 / an.
任何非零数的零次幂等于1：a⁰ = 1（a ≠ 0）。负指数表示倒数：a−n = 1 / an。
Raising a power to another power means multiplying the exponents: (am)n = am×n. Example: (3²)⁴ = 3⁸.
幂的乘方指数相乘：(am)n = am×n。例如 (3²)⁴ = 3⁸。
2. Algebraic Expressions & Expanding Brackets | 代数表达式与去括号
To expand a single bracket, multiply each term inside the bracket by the factor outside: a(b + c) = ab + ac. For example, 3(x + 4) = 3x + 12.
展开单个括号时，将括号外的因数与括号内的每一项相乘：a(b + c) = ab + ac。例如 3(x + 4) = 3x + 12。
To expand two brackets, multiply every term in the first bracket by every term in the second bracket (the distributive law): (a + b)(c + d) = ac + ad + bc + bd. In Year 8 this often appears as (x + p)(x + q) = x² + (p+q)x + pq.
展开两个括号时，将第一个括号中的每一项与第二个括号中的每一项相乘（分配律）：(a + b)(c + d) = ac + ad + bc + bd。在八年级，常见形式为 (x + p)(x + q) = x² + (p+q)x + pq。
Factorising reverses expanding. Look for the highest common factor (HCF) of the terms: e.g., 4x + 8 = 4(x + 2).
因式分解是展开的逆运算。寻找各项的最高公因式（HCF），例如 4x + 8 = 4(x + 2)。
3. Linear Equations & Inequalities | 线性方程与不等式
To solve a linear equation, isolate the variable on one side by performing the same operation on both sides. For example: 2x + 5 = 13 ⇒ 2x = 8 ⇒ x = 4.
解一元一次方程时，通过等号两边同时进行相同的运算来分离变量。例如 2x + 5 = 13 ⇒ 2x = 8 ⇒ x = 4。
When an equation contains brackets, expand first, then simplify. Always check your solution by substituting it back into the original equation.
若方程中含有括号，先去括号再化简。始终将解代回原方程检验。
Inequalities use signs like <, >, ≤ and ≥. Solving them is similar to equations, except that multiplying or dividing by a negative number reverses the inequality sign.
不等式使用 <、>、≤ 和 ≥ 等符号。解法与方程类似，但在两边同乘或同除以一个负数时，不等号方向要改变。
4. Angles & Parallel Lines | 角与平行线
Angles on a straight line add up to 180°. Vertically opposite angles are equal.
直线上的相邻角之和为180°（平角）。对顶角相等。
When a transversal cuts parallel lines, corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles sum to 180°.
当一条截线与两条平行线相交时，同位角相等，内错角相等，同旁内角互补（和为180°）。
These angle facts are used to find unknown angles in diagrams. Always give reasons for each step.
这些角的性质可用来求解图形中的未知角。每一步推理都需要写明理由。
5. Triangles, Polygons & Angle Sums | 三角形、多边形与内角和
The sum of the interior angles of a triangle is 180°. The sum of the interior angles of a quadrilateral is 360°.
三角形的内角和为180°。四边形的内角和为360°。
For any n‑sided polygon, the sum of interior angles = (n − 2) × 180°. The sum of exterior angles (one at each vertex) is always 360°.
对于任意n边形，内角和 = (n − 2) × 180°。任意多边形的外角和（每个顶点只取一个外角）始终为360°。
A triangle with two equal sides is called an isosceles triangle; the base angles are equal. An equilateral triangle has three equal sides and three 60° angles.
有两条边相等的三角形叫等腰三角形，其底角相等。等边三角形的三条边都相等，每个角都是60°。
6. Perimeter, Area & Volume | 周长、面积与体积
Perimeter is the total distance around a 2D shape. For a rectangle: P = 2(l + w). Circumference of a circle: C = 2πr or C = πd.
周长是二维形状一周的长度。矩形周长：P = 2(长+宽)。圆的周长：C = 2πr 或 C = πd。
Common area formulas:
常见面积公式：
| Rectangle | A = l × w |
| Triangle | A = ½ × base × height |
| Parallelogram | A = base × vertical height |
| Trapezium | A = ½ × (a + b) × height |
| Circle | A = πr² |
Volume of a cuboid: V = l × w × h. Surface area of a cuboid: SA = 2(lw + lh + wh).
长方体体积：V = 长 × 宽 × 高。长方体表面积：SA = 2(长×宽 + 长×高 + 宽×高)。
7. Fractions, Decimals & Percentages | 分数、小数与百分比
To convert a fraction to a percentage, divide numerator by denominator and multiply by 100. For example, 3/5 = 0.6 = 60%.
将分数化为百分数，用分子除以分母再乘100。例如 3/5 = 0.6 = 60%。
Percentage of an amount: multiply the amount by the percentage divided by 100. E.g., 20% of 50 = 50 × 0.2 = 10.
求一个量的百分之几：用该量乘百分比除以100。例如 50 的 20% = 50 × 0.2 = 10。
Increasing by a percentage: new value = original × (1 + percentage/100). Decreasing: new value = original × (1 − percentage/100).
增加一个百分比：新值 = 原值 × (1 + 百分数/100)。减少：新值 = 原值 × (1 − 百分数/100)。
8. Ratio & Proportion | 比与比例
A ratio compares two or more quantities in the same unit. Simplify a ratio by dividing each part by their highest common factor.
比用于比较两个或多个相同单位的量。将比的每一项除以它们的最大公因数即可化简。
To divide a quantity in a given ratio, first find the total number of parts, then work out the value of one part.
按给定比例分配一个量时，先算出总份数，再求出一份的值，最后按份分配。
Direct proportion means two quantities increase or decrease at the same rate. If y is directly proportional to x, then y = kx for a constant k.
正比关系是指两个量同速增减。若 y 与 x 成正比，则 y = kx，其中 k 为常数。
9. Statistics: Mean, Median, Mode & Range | 统计：平均数、中位数、众数与极差
The mean is the sum of all values divided by the number of values: Mean = Σx / n.
平均数是所有数值的总和除以数值的个数：平均数 = Σx / n。
The median is the middle value when the data are arranged in order. If there are two middle values, take their mean.
中位数是将数据排序后处于中间位置的值。若中间位置有两个数，则取它们的平均数。
The mode is the value that appears most often. The range is the difference between the largest and smallest values.
众数是出现频率最高的数值。极差是最大值与最小值之差。
10. Probability Basics | 概率基础
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
概率用来衡量事件发生的可能性，用0（不可能）到1（必然）之间的数表示。
For equally likely outcomes: Probability of an event = number of favourable outcomes / total number of outcomes.
对于等可能的结果：事件的概率 = 有利结果数 / 全部可能结果数。
The sum of the probabilities of all mutually exclusive outcomes of an experiment is 1.
一次试验中所有互斥结果的概率之和为1。
11. Coordinates & Straight‑line Graphs | 坐标与直线图像
Plot a point in the coordinate plane using an ordered pair (x, y). The x‑coordinate gives the horizontal position, the y‑coordinate the vertical position.
在坐标平面中用有序数对 (x, y) 描点。x 坐标表示水平位置，y 坐标表示垂直位置。
A linear relationship can be written in the form y = mx + c, where m is the gradient (slope) and c is the y‑intercept.
线性关系可以写成 y = mx + c 的形式，其中 m 是斜率，c 是直线在 y 轴上的截距。
To find the gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes upward; a negative gradient slopes downward.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率：m = (y₂ − y₁) / (x₂ − x₁)。斜率为正时直线向上倾斜，为负时向下倾斜。
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