📚 Year 8 WJEC Maths: Quick Formula & Theorem Reference Handbook | Year 8 WJEC 数学：公式定理速查手册
This handbook brings together all the essential formulas, rules and theorems you will meet in Year 8 WJEC Mathematics. Keep it handy for quick revision, homework help and exam preparation. Each topic is presented with clear statements followed by step-by-step explanations in both English and Chinese to strengthen your understanding and memory.
本手册汇集了你在 Year 8 WJEC 数学中将遇到的所有重要公式、法则和定理。随时查阅，可用于快速复习、家庭作业和备考。每个主题都配有清晰的表述，并以中英双语逐步解释，帮助你加深理解和记忆。
1. Basic Arithmetic and Fractions | 基本算术与分数
All calculations follow the BIDMAS order: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
所有计算都遵循 BIDMAS 顺序：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
Example: 3 + 6 × (5 − 2) = 3 + 6 × 3 = 3 + 18 = 21
To add or subtract fractions, rewrite them with a common denominator. Multiply fractions by multiplying the numerators and denominators separately. Divide by flipping the second fraction and multiplying.
分数的加减：先通分，找到公分母。分数乘法：分子相乘，分母相乘。分数除法：将第二个分数颠倒后相乘。
a/b + c/d = (ad + bc) / bd a/b × c/d = ac / bd a/b ÷ c/d = a/b × d/c
The reciprocal of a number x is 1/x. The reciprocal of a fraction a/b is b/a. Multiplying a number by its reciprocal always gives 1.
一个数 x 的倒数是 1/x。分数 a/b 的倒数是 b/a。一个数乘以其倒数总是等于 1。
2. Percentages, Decimals and Ratio | 百分数、小数与比例
To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100. One whole is 100%, 1 or 1/1.
将百分数转换为小数，除以 100。将小数转换为百分数，乘以 100。一个整体是 100%、1 或 1/1。
To find a percentage of an amount, write the percentage as a fraction or decimal and multiply. To increase or decrease by a percentage, add or subtract the percentage to/from 100%, then multiply.
求一个数量的百分比：把百分数写成分数或小数，再相乘。要增加或减少一个百分比，先用 100% 加上或减去该百分比，再乘以原数。
Ratio compares two or more quantities. Simplify ratios by dividing all parts by their highest common factor. To share a quantity in a given ratio, find the total number of parts, divide the quantity by that total, then multiply by each part.
比例用于比较两个或多个量。化简比例：将各部分除以它们的最大公因数。按比例分配：先求总份数，总量除以总份数，再乘以各部份的份数。
3. Algebraic Expressions and Equations | 代数表达式与方程
letters and powers. Simplify by adding or subtracting the coefficients: 5x + 3x − 2x = 6x.
同类项具有完全相同的字母和指数。合并时，将系数相加或相减：5x + 3x − 2x = 6x。
+ c) = ab + ac. When expanding two brackets, use FOIL: (a + b)(c + d) = ac + ad + bc + bd.
展开括号：用括号外的项乘以括号内的每一项：a(b + c) = ab + ac。展开两个括号时，使用 FOIL 法则：(a + b)(c + d) = ac + ad + bc + bd。
To solve a linear equation, perform the same operations on both sides to isolate the unknown. Always keep the equation balanced.
解一元一次方程时，在等式两边进行相同的运算以分离未知数。始终保持等式平衡。
Example: 2x + 5 = 13 → 2x = 8 → x = 4
Substitution means replacing letters with given numbers. Always use brackets for negative values and follow BIDMAS.
代入法：用给定的数字替换字母。对于负数要加括号，并遵循 BIDMAS 顺序。
4. Sequences and Patterns | 序列与模式
A sequence is a list of numbers following a rule. The term-to-term rule tells you how to get from one term to the next. The position-to-term rule (nth term) lets you find any term directly.
序列是按照某种规则排列的一列数。逐项规则告诉你如何从一项得到下一项。项序规则（第 n 项）可以让你直接求出任意一项。
ally an + b, where a = d. Find b by using the first term: when n = 1, an + b = first term.
对于公差为 d 的线性序列，第 n 项通常表示为 an + b，其中 a = d。利用第一项求 b：当 n = 1 时，an + b = 第一项的值。
Sequence: 4, 7, 10, 13, … → nth term = 3n + 1
Square numbers: 1, 4, 9, 16, … nth term = n². Triangle numbers: 1, 3, 6, 10, … nth term = n(n + 1)/2.
平方数：1, 4, 9, 16, … 第 n 项 = n²。三角形数：1, 3, 6, 10, … 第 n 项 = n(n + 1)/2。
5. Geometry: Angles and Lines | 几何：角与线条
Angles are measured in degrees (°). Acute: 0° – 90°; Right: exactly 90°; Obtuse: 90° – 180°; Straight: exactly 180°; Reflex: 180° – 360°.
角以度 (°) 为单位。锐角：0° – 90°；直角：正好 90°；钝角：90° – 180°；平角：正好 180°；优角：180° – 360°。
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°。一点周围的角之和为 360°。对顶角相等。
When a transversal crosses parallel lines: corresponding angles are equal; alternate angles are equal; co-interior angles sum to 180°.
当一条截线与平行线相交：同位角相等；内错角相等；同旁内角互补（和为 180°）。
triangle is 180°. The sum of interior angles of a polygon with n sides is (n − 2) × 180°.
三角形内角和为 180°。n 边形的内角和为 (n − 2) × 180°。
6. Geometry: Area and Perimeter | 几何：面积与周长
ll side lengths together. For regular shapes, it is the number of sides × one side length.
周长是图形一周的总长度。将所有边长相加。对于规则图形，周长 = 边数 × 一条边的长度。
Rectangle: Area = length × width (A = lw)
矩形：面积 = 长 × 宽 (A = lw)
Triangle: Area = ½ × base × vertical height (A = ½ bh)
三角形：面积 = ½ × 底 × 垂直高 (A = ½ bh)
Parallelogram: Area = base × perpendicular height (A = bh)
平行四边形：面积 = 底 × 垂直高 (A = bh)
Trapezium: Area = ½ × (a + b) × height, where a and b are the parallel sides.
梯形：面积 = ½ × (a + b) × 高，其中 a 和 b 是平行边。
For compound shapes, split them into rectangles, triangles or parallelograms, find each area, then sum or subtract.
对于复合图形，将其分解为矩形、三角形或平行四边形，分别求出面积，然后相加或相减。
7. Geometry: Volume and Surface Area | 几何：体积与表面积
Volume measures the space inside a 3D solid, in cubic units. Surface area is the total area of all faces.
体积衡量立体图形内部的空间大小，单位为立方单位。表面积是所有面的总面积。
Cube: Volume = side³ (s³); Surface area = 6s²
立方体：体积 = 边长³ (s³)；表面积 = 6s²
Cuboid: Volume = length × width × height (V = lwh); Surface area = 2(lw + lh + wh)
长方体：体积 = 长 × 宽 × 高 (V = lwh)；表面积 = 2(lw + lh + wh)
Prism: Volume = area of cross-section × length
棱柱：体积 = 横截面积 × 长度
To find the surface area of a prism, calculate the area of each face and add them. The two ends are the cross-sections; the sides form rectangles.
要计算棱柱的表面积，计算每个面的面积并相加。两个端面是横截面；侧面构成矩形。
8. Pythagoras' Theorem | 勾股定理
In a right‑angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
在直角三角形中，斜边（直角所对的边）的平方等于另外两条边的平方和。
a² + b² = c² where c is the hypotenuse
a² + b² = c²，其中 c 为斜边
To find the hypotenuse, use c = √a² + b². To find a shorter side, rearrange to a = √c² − b². Always check your rounding if the answer is a decimal.
求斜边：c = √a² + b²。求一条直角边：a = √c² − b²。如果答案是小数，注意按要求取整。
Pythagoras’ theorem only works for right‑angled triangles. Label the sides carefully before substituting.
勾股定理只适用于直角三角形。代入前务必仔细标注各边。
9. Statistics: Mean, Median, Mode, Range | 统计：均值、中位数、众数、范围
The mean is the average value. Add all the numbers and divide by how many there are.
均值是平均数。将所有数值相加，再除以数值的个数。
Mean = sum of values ÷ number of values
均值 = 数值总和 ÷ 数值个数
The median is the middle value when the data is in order. If there are two middle values, average them. The mode is the value that occurs most often. The range is the difference between the largest and smallest values.
中位数是将数据排序后中间的值。如果有两个中间值，则取它们的平均数。众数是出现次数最多的值。范围 = 最大值 − 最小值。
When data is in a frequency table, multiply each value by its frequency, sum these, then divide by the total frequency to get the mean. The modal class is the group with the highest frequency.
当数据用频数表给出时，用每个值乘以其频数，求和，再除以总频数即得均值。众数类别是频数最高的组。
10. Introduction to Probability | 概率简介
Probability measures how likely an event is. It is a number between 0 (impossible) and 1 (certain), often written as a fraction, decimal or percentage.
概率衡量事件发生的可能性。它是一个介于 0（不可能）和 1（必然）之间的数，通常用分数、小数或百分数表示。
lity of an event = number of favourable outcomes / total number of equally likely outcomes
事件的概率 = 有利结果的个数 / 所有等可能结果的总数
The probabilities of all possible outcomes add up to 1. If the probability of an event is P, the probability it does not happen is 1 − P.
所有可能结果的概率之和为 1。如果一个事件发生的概率为 P，那么它不发生的概率为 1 − P。
For mutually exclusive events, you can add their probabilities to find the probability of either happening. Use sample space diagrams to list all outcomes for combined events.
对于互斥事件，可以将它们的概率相加来求二者之一发生的概率。使用样本空间图表列出组合事件的所有结果。
11. Coordinates and Straight-line Graphs | 坐标与直线图
Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. The origin is (0, 0).
坐标写作 (x, y)，其中 x 表示水平位置，y 表示垂直位置。原点是 (0, 0)。
The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
(steepness), calculated as rise / run. c is the y-intercept, where the line crosses the y‑axis.
直线图的方程为 y = mx + c。m 是斜率（倾斜度），计算公式为 垂直变化 / 水平变化。c 是 y 轴截距，即直线与 y 轴的交点。
To plot a straight line, choose three x values, calculate the corresponding y values, plot the points and join them with a ruler. Parallel lines have the same gradient.
要画出直线，选取三个 x 值，计算对应的 y 值，描点后用直尺连接。平行线具有相同的斜率。
12. Unit Conversion | 单位换算
Always write the units with your answer. Converting between metric units uses powers of 10. Learn these key relationships.
答题时务必写上单位。公制单位之间的换算使用 10 的幂。记住以下关键换算关系。
| Length | 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm | 长度: 1 千米 = 1000 米, 1 米 = 100 厘米, 1 厘米 = 10 毫米 |
| Mass | 1 tonne = 1000 kg, 1 kg = 1000 g, 1 g = 1000 mg | 质量: 1 吨 = 1000 千克, 1 千克 = 1000 克, 1 克 = 1000 毫克 |
| Capacity | 1 L = 1000 mL, 1 L = 1000 cm³ | 容量: 1 升 = 1000 毫升, 1 升 = 1000 立方厘米 |
| Time | 1 year = 365 days, 1 day = 24 h, 1 h = 60 min, 1 min = 60 s | 时间: 1 年 = 365 天, 1 天 = 24 小时, 1 小时 = 60 分, 1 分 = 60 秒 |
e conversion factor. For example, 1 m² = 100 × 100 = 10 000 cm², and 1 m³ = 1 000 000 cm³.
在换算平方或立方单位时，记得将换算因数也平方或立方。例如，1 m² = 100 × 100 = 10 000 cm²，1 m³ = 1 000 000 cm³。
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