📚 Year 8 Edexcel Mathematics: Formula & Theorem Quick Revision Guide | Year 8 Edexcel 数学：公式定理速查手册
This quick revision guide compiles the essential formulas, rules, and theorems you need for the Year 8 Edexcel Mathematics curriculum. Use it to check your understanding of key topics before tests and exams.
本速查手册汇总了 Year 8 Edexcel 数学课程中必须掌握的公式、法则和定理，供你在测验和考试前快速回顾和查漏补缺。
1. Number Properties and Operations | 数的性质与运算
Addition and multiplication are commutative: a + b = b + a and a × b = b × a.
加法和乘法满足交换律：a + b = b + a 且 a × b = b × a。
d multiplication are associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
加法和乘法满足结合律：(a + b) + c = a + (b + c) 且 (a × b) × c = a × (b × c)。
tion distributes over addition: a × (b + c) = a × b + a × c. This is the distributive law.
乘法对加法满足分配律：a × (b + c) = a × b + a × c。
When multiplying or dividing two numbers with the same sign, the answer is positive. When the signs differ, the answer is negative.
两个同号数相乘或相除，结果为正；异号数相乘或相除，结果为负。
2. Fractions, Decimals, and Percentages | 分数、小数和百分比
fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 0.375.
分数化小数：用分子除以分母。例如 3/8 = 0.375。
ion, write the decimal as the numerator over a power of 10 (e.g. 0.6 = 6/10) and simplify.
小数化分数：以 10 的幂为分母写出分数（如 0.6 = 6/10），再约分。
To increase a value by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 - r/100).
增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 - r/100)。
3. Laws of Indices | 指数定律
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
Any non-zero number raised to the power of 0 is 1: a⁰ = 1.
任何非零数的 0 次幂均为 1：a⁰ = 1。
A negative exponent indicates the reciprocal: a⁻ⁿ = 1 / aⁿ.
负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
4. Algebraic Expressions and Equations | 代数表达式与方程式
To expand a bracket, multiply the term outside by each term inside: a(b + c) = ab + ac.
去括号时，将括号外的项乘以括号内的每一项：a(b + c) = ab + ac。
Factorising is the reverse of expanding. Look for a common factor: ab + ac = a(b + c).
因式分解是展开的逆过程，提取公因式：ab + ac = a(b + c)。
c, perform inverse operations in reverse order. Keep the balance by doing the same to both sides.
解形如 ax + b = c 的一次方程，按照逆运算顺序进行，并在等号两边同时操作以保持平衡。
When solving inequalities, reverse the inequality sign if you multiply or divide by a negative number.
解不等式时，如果两边同时乘以或除以一个负数，不等号方向要反转。
5. Linear Graphs and Slope | 线性图形与斜率
f a straight line is y = mx + c, where m is the slope (gradient) and c is the y-intercept.
直线方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
pe m between two points (x₁, y₁) and (x₂, y₂) is calculated as: m = (y₂ − y₁) / (x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率公式：m = (y₂ − y₁) / (x₂ − x₁)。
Parallel lines have equal slopes.
平行线斜率相等。
y = mx + c
6. Angles and Polygons | 角度与多边形
Vertically opposite angles are equal. Angles on a straight line add up to 180°.
对顶角相等；平角（直线上的角）之和为 180°。
When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.
平行线被截线所截时，同位角相等、内错角相等、同旁内角互补（和为 180°）。
angle is 180°. For any polygon with n sides, the sum of interior angles is (n − 2) × 180°.
三角形内角和为 180°；n 边形内角和为 (n − 2) × 180°。
The sum of exterior angles of any convex polygon is always 360°.
任意凸多边形的外角和均为 360°。
7. Perimeter, Area, and Volume | 周长、面积和体积
Below are common area and perimeter formulas. For π, use 3.14 or the π button on your calculator.
以下是常用面积与周长公式。圆周率 π 取 3.14 或使用计算器上的 π 键。
| Shape 形状 | Perimeter / Circumference 周长 | Area 面积 |
|---|---|---|
| Rectangle 矩形 (l × w) | P = 2(l + w) | A = l × w |
| Triangle 三角形 (b, h) | Sum of sides | A = ½ × b × h |
| Parallelogram 平行四边形 (b, h) | Sum of sides | A = b × h |
| Trapezium 梯形 (a, b parallel, h) | Sum of sides | A = ½(a + b) × h |
| Circle 圆 (r) | C = 2πr or πd | A = πr² |
The volume of a prism is found by multiplying the area of its cross-section by its length (height).
棱柱体积 = 底面积 × 高（长度）。
Vprism = Area of cross‑section × length
8. Pythagoras' Theorem | 勾股定理
(c, the longest side) equals the sum of the squares of the other two sides: a² + b² = c².
在直角三角形中，斜边（最长边）的平方等于两条直角边的平方和：a² + b² = c²。
You can rearrange to find a shorter side: a = √c² − b². Always check that the triangle is right‑angled before applying the theorem.
可通过变形求直角边：a = √c² − b²。使用定理前务必确认三角形为直角三角形。
a² + b² = c²
9. Ratio and Proportion | 比例与比率
To simplify a ratio, divide all parts by their highest common factor. For example, 8:12 simplifies to 2:3.
化简比：将比的各项除以它们的最大公因数。例如 8:12 化简为 2:3。
When sharing a quantity in a given ratio, add the ratio parts to find the total number of shares, then divide accordingly.
按比例分配时，先求出比例的总份数，再用总量除以总份数，最后乘以各项份数。
portion if their ratio stays constant: y = kx, where k is the constant of proportionality.
两个量成正比例时，它们的比值保持不变：y = kx，其中 k 为比例常数。
10. Statistics and Probability | 统计与概率
n data are ordered. Mode is the most frequent value. Range = highest value − lowest value.
平均数（均值）是所有数据之和除以数据个数。中位数是排序后位于中间的数值。众数是出现次数最多的数值。极差 = 最大值 − 最小值。
The probability of an event is: Number of favourable outcomes / Total number of possible outcomes. Probabilities always lie between 0 (impossible) and 1 (certain).
事件发生的概率 = 有利结果数 / 所有可能结果数。概率始终在 0（不可能）到 1（必然）之间。
The sum of probabilities of all mutually exclusive outcomes of an experiment is 1.
互斥事件的概率之和为 1。
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