📚 Year 10 WJEC Maths: Quick Reference Handbook of Formulas and Theorems | Year 10 WJEC 数学：公式定理速查手册
This handbook is your go-to revision companion for Year 10 WJEC Mathematics, covering every essential formula, theorem and key fact you need across Number, Algebra, Geometry, Statistics and Probability. Use it to check definitions, reinforce memory and build confidence before tests. Each section presents concepts in clear pairs of English and Chinese, so you can study bilingually.
本手册是 Year 10 WJEC 数学的随身复习宝典，涵盖数、代数、几何、统计与概率中所有关键公式、定理与重要知识点。可用作定义查询、记忆强化和考前信心训练。每个小节都以中英对照的形式呈现概念，方便双语学习。
1. Number – Types of Numbers and Operations | 数的类型与运算
Natural numbers are positive whole numbers starting from 1 (sometimes 0 is included). Integers include all whole numbers and their negatives. Rational numbers can be written as a fraction a/b where a and b are integers and b ≠ 0. Irrational numbers cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-repeating. Real numbers include all rational and irrational numbers.
自然数是从 1 开始的正整数（有时包含 0）。整数包含所有正整数、负整数和零。有理数可以写成分数 a/b 的形式，其中 a、b 为整数且 b ≠ 0。无理数不能用简单分数表示，小数部分无限不循环。实数包含所有有理数和无理数。
Order of operations: BIDMAS (Brackets, Indices, Division and Multiplication from left to right, Addition and Subtraction from left to right) determines the correct sequence for calculations.
运算顺序：BIDMAS（括号、指数/幂、除与乘按从左到右进行、加与减按从左到右进行）规定了计算的正规步骤。
x laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (for a ≠ 0). For square roots: √a × b = √a × √b.
指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0）。关于平方根：√a × b = √a × √b。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
To add or subtract fractions, write them with a common denominator, then add or subtract the numerators. To multiply fractions, multiply the numerators and multiply the denominators. To divide fractions, multiply by the reciprocal of the divisor.
分数加减法：先通分化为同分母，分子相加减。分数乘法：分子乘分子，分母乘分母。分数除法：乘以除数的倒数。
de the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%. To convert a percentage to a decimal, divide by 100. For example, 3/4 = 0.75 = 75%.
形式转换：分数化小数，用分子除以分母。小数化百分数，乘以 100%。百分数化小数，除以 100。例如，3/4 = 0.75 = 75%。
× (1 + percentage/100). Percentage decrease: new value = original × (1 − percentage/100).
百分比增长：新值 = 原值 × (1 + 百分数/100)。百分比减少：新值 = 原值 × (1 − 百分数/100)。
3. Algebraic Expressions and Equations | 代数表达式与方程
ouble brackets: (x + p)(x + q) = x² + (p + q)x + pq. Example: (x + 3)(x − 2) = x² + x − 6.
单重括号展开：a(b + c) = ab + ac。双重括号展开：(x + p)(x + q) = x² + (p + q)x + pq。例如，(x + 3)(x − 2) = x² + x − 6。
+ bx + c, find two numbers that multiply to c and add to b. x² + 5x + 6 = (x + 2)(x + 3).
因式分解是展开的逆运算。首先提取公因式，如 6x + 9 = 3(2x + 3)。对形如 x² + bx + c 的二次式分解，需找到乘积为 c、和为 b 的两数。x² + 5x + 6 = (x + 2)(x + 3)。
e the variable. For ax + b = c, x = (c − b)/a. Always check your solution by substitution.
解线性方程：运用逆运算逐步分离变量。对于 ax + b = c，x = (c − b)/a。务必代入原方程检验。
4. Linear Graphs and Gradients | 线性图与斜率
The gradient measures steepness: m = (change in y) ÷ (change in x) = (y₂ − y₁)/(x₂ − x₁).
直线方程为 y = mx + c，其中 m 表示斜率，c 是 y 轴截距。斜率衡量倾斜程度：m = (y 的变化) ÷ (x 的变化) = (y₂ − y₁)/(x₂ − x₁)。
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1, so if one gradient is m, the other is −1/m.
平行直线斜率相同。互相垂直的直线斜率乘积为 −1，若一条直线的斜率为 m，另一条为 −1/m。
se the gradient to move right 1 and up (or down) m. Draw a straight line through the points.
绘制 y = mx + c 图像时，从点 (0, c) 出发，利用斜率向右移动 1 个单位，纵向移动 m 个单位（上移或下移），过点作直线。
5. Simultaneous Equations | 联立方程组
Simultaneous equations are two (or more) equations with the same variables. The solution is the pair(s) of values that satisfy all equations. Graphically, it is the intersection point of two lines.
联立方程组是具有相同变量的两个（或多个）方程。其解是同时满足所有方程的一对值。几何上，就是两直线图像的交点坐标。
Elimination method: multiply equations if needed to make coefficients of one variable equal, then add or subtract the equations to eliminate that variable. Substitute back to find the other variable.
消元法：必要时将方程乘以适当倍数，使某一变量的系数相等，再通过相加或相减消去该变量。回代求得另一变量。
nd x − y = 2. Adding gives 3x = 9, so x = 3. Then y = 1. Check in both original equations.
示例：2x + y = 7 和 x − y = 2。两式相加得 3x = 9，x = 3，代入得 y = 1。务必回代检验。
6. Inequalities | 不等式
reater than or equal to). They are solved similarly to equations, but when multiplying or dividing by a negative number, the inequality sign must be reversed.
不等式用符号比较大小：<（小于），>（大于），≤（小于等于），≥（大于等于）。解法与方程类似，但当乘以或除以一个负数时，不等号方向必须反转。
r inequalities like 2 < x ≤ 5, x is any number greater than 2 and less than or equal to 5.
在数轴上表示不等式：< 或 > 用空心圆，≤ 或 ≥ 用实心圆，并涂上表示方向的阴影。例如 2 < x ≤ 5 表示 x 取所有大于 2 且小于等于 5 的数。
7. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
squares of the other two sides: a² + b² = c². This is used to find a missing side length.
直角三角形中，斜边（最长边）的平方等于两直角边的平方和：a² + b² = c²。该定理用于求未知边长。
cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. A common mnemonic is SOH CAH TOA.
直角三角形中的三角比（θ 为锐角）：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。常用口诀 SOH CAH TOA 帮助记忆。
pp/hyp), θ = cos⁻¹(adj/hyp), θ = tan⁻¹(opp/adj). Ensure your calculator is in degree mode.
求角度时使用反三角函数：θ = sin⁻¹(对/斜)，θ = cos⁻¹(邻/斜)，θ = tan⁻¹(对/邻)。计算前务必确认计算器处于角度（degree）模式。
8. Area and Volume | 面积与体积
Below is a quick-reference table of common 2D shapes and their area formulas, with bilingual entries.
以下为常见二维图形的面积公式速查表，含有中英对照。
| Shape | Area Formula (English) | 面积公式 (中文) |
|---|---|---|
| Rectangle | Area = length × width | 面积 = 长 × 宽 |
| Triangle | Area = ½ × base × height | 面积 = ½ × 底 × 高 |
| Parallelogram | Area = base × vertical height | 面积 = 底 × 垂直高 |
| Trapezium | Area = ½ × (a + b) × h | 面积 = ½ × (上底 a + 下底 b) × 高 h |
| Circle | Area = π × r² (r = radius) | 面积 = π × r²（r = 半径） |
r a cylinder, Volume = π r² h. Surface area of a solid is the total area of all its faces.
棱柱体积：体积 = 截面积 × 长度。对于长方体（立方体），体积 = 长 × 宽 × 高。圆柱体积 = π r² h。立体图形的表面积为所有面的面积之和。
9. Angles in Geometry | 几何中的角
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角（直线上角）之和为 180°。点周角之和为 360°。对顶角相等。
In any triangle, the interior angles add up to 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
任何三角形的内角和为 180°。三角形的一个外角等于与其不相邻的两个内角之和。
here n is the number of sides. The sum of exterior angles (one per vertex) is always 360°.
任意多边形的内角和 = (n − 2) × 180°，其中 n 为边数。外角和（每个顶点取一个外角）恒为 360°。
10. Data Handling – Averages and Range | 数据处理 – 平均数和极差
, take their mean. Mode is the most frequently occurring value. Range = largest value − smallest value.
平均数（算术平均）= 所有数据值的和 ÷ 数据个数。中位数是排序后位于中间的值；若居中值有两个，取两者的平均数。众数是出现次数最多的值。极差 = 最大值 − 最小值。
the frequency and the midpoint is the class centre. The modal class is the group with the highest frequency.
对于分组数据或频率表，平均数 ≈ Σ(f × 组中值) ÷ Σf，其中 f 为频数，组中值为组中点。众数类别是频数最高的组。
11. Probability | 概率
f an event = number of favourable outcomes ÷ total number of equally likely outcomes. Prob
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