📚 Year 9 WJEC Maths: Quick Reference Formula & Theorem Handbook | Year 9 WJEC 数学：公式定理速查手册
This handbook collects all the essential formulas, rules, and theorems you need for Year 9 WJEC Mathematics. Keep it close for quick revision, homework help, and exam preparation.
本手册汇集了 Year 9 WJEC 数学所需的所有关键公式、法则与定理。随身携带，便于快速复习、辅助作业和备考。
1. Number & Arithmetic | 数与算术
Order of operations follows BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction). Always calculate inside brackets first, then powers, then multiply/divide from left to right, finally add/subtract.
运算顺序遵循 BIDMAS（括号、指数、乘除、加减）。先算括号，再算指数，接着从左到右进行乘除，最后进行加减。
To add or subtract fractions, use a common denominator: a/b ± c/d = (ad ± bc) / bd. Multiply fractions straight across: a/b × c/d = ac/bd. To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.
分数的加减需通分：a/b ± c/d = (ad ± bc) / bd。乘法直接相乘：a/b × c/d = ac/bd。除以一个分数等于乘上它的倒数：a/b ÷ c/d = a/b × d/c。
Converting between percentages, decimals and fractions: percent to decimal – divide by 100 (e.g. 45% = 0.45); fraction to percent – multiply by 100; decimal to fraction – use place value. A percentage increase or decrease uses multiplier: increase by r% → multiply by (1 + r/100).
百分数、小数和分数互化：百分数化小数——除以100（如45% = 0.45）；分数化百分数——乘以100；小数化分数——根据位值转化。百分数增减使用乘数：增加r% → 乘以 (1 + r/100)。
Index laws are vital: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. Standard form writes large or small numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
指数法则至关重要：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。标准形式将大数或小数写为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。
Squares and square roots: √(a²) = a. The cube root of a is the number whose cube is a. Estimating roots between two integers is a common skill.
平方与平方根：√(a²) = a。a 的立方根是立方等于 a 的数。在两个整数之间估算平方根是一项常见技能。
2. Algebra & Equations | 代数与方程
Solving linear equations: perform the same operation to both sides to isolate the unknown. For ax + b = c, subtract b then divide by a: x = (c − b)/a. Always check your answer by substitution.
解一元一次方程：对方程两边进行相同的运算以分离未知数。对于 ax + b = c，先减去 b 再除以 a：x = (c − b)/a。务必将答案代回检验。
Expanding a single bracket: a(b + c) = ab + ac. With two brackets, use FOIL: (a + b)(c + d) = ac + ad + bc + bd. Remember to collect like terms afterwards.
单项式乘括号：a(b + c) = ab + ac。两项式相乘使用 FOIL 方法：(a + b)(c + d) = ac + ad + bc + bd。之后记得合并同类项。
Factorising is the reverse of expanding. Common factor: ab + ac = a(b + c). Difference of two squares: a² − b² = (a + b)(a − b). This is a key Year 9 identity.
因式分解是展开的逆运算。提取公因式：ab + ac = a(b + c)。平方差公式：a² − b² = (a + b)(a − b)。这是 Year 9 的一个重要恒等式。
Rearranging formulae: treat the subject variable as the unknown and use inverse operations. For example, to make t the subject of v = u + at, subtract u then divide by a: t = (v − u)/a.
公式变形：将要表示的量视为未知数，并使用逆运算。例如，将 v = u + at 改写为 t，先减去 u 再除以 a：t = (v − u)/a。
3. Sequences & Patterns | 数列与规律
An arithmetic sequence has a constant difference d between consecutive terms. The nth term formula is: nth term = a + (n − 1)d, where a is the first term.
等差数列的相邻两项之差 d 为常数。第 n 项公式为：第 n 项 = a + (n − 1)d，其中 a 是首项。
For sequences with a second common difference (quadratic patterns), Year 9 often covers finding the second difference and linking to n². But the linear nth term is the key focus.
对于二阶差不变（二次型规律）的数列，Year 9 通常会涉及求二阶差并联系 n²。不过重点是线性数列的第 n 项。
Fibonacci sequence: after the first two terms 1, 1, each term is the sum of the previous two. You may be asked to generate terms.
斐波那契数列：从 1, 1 开始，每一项是前两项之和。考题可能要求写出若干项。
4. Geometry: Angles & Lines | 几何：角与线
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°。绕一点的周角为 360°。对顶角相等。
Interior angles of a triangle sum to 180°. For a quadrilateral, sum = 360°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为 180°。四边形内角和为 360°。三角形的外角等于其两个不相邻的内角之和。
With parallel lines: corresponding angles are equal, alternate angles are equal, allied (co‑interior) angles sum to 180°.
平行线：同位角相等，内错角相等，同旁内角互补（和为 180°）。
Polygon interior angle sum = (n − 2) × 180°, where n is the number of sides. The sum of exterior angles of any convex polygon is 360°. For a regular polygon, each interior angle = 180° − (360°/n).
多边形内角和 = (n − 2) × 180°，n 为边数。任意凸多边形的外角和为 360°。在正多边形中，每个内角 = 180° − (360°/n)。
5. Perimeter, Area & Volume | 周长、面积与体积
Rectangle: perimeter P = 2(l + w), area A = lw. Triangle: area A = ½ × base × height. Parallelogram: A = b × h. Trapezium: A = ½(a + b)h.
矩形：周长 P = 2(长 + 宽)，面积 A = 长 × 宽。三角形：面积 A = ½ × 底 × 高。平行四边形：A = 底 × 高。梯形：A = ½(上底 + 下底)高。
Circle: circumference C = πd = 2πr, area A = πr². Use π ≈ 3.142 or the calculator’s π button.
圆：周长 C = πd = 2πr，面积 A = πr²。计算时使用 π ≈ 3.142 或计算器上的 π 键。
Volume of a prism = area of cross‑section × length. For a cuboid, V = l × w × h. For a cylinder, V = πr²h. Remember area is in square units (cm², m²) and volume in cubic units (cm³, m³).
棱柱的体积 = 横截面积 × 长度。长方体体积 V = 长 × 宽 × 高。圆柱体积 V = πr²h。注意面积单位为平方单位，体积单位为立方单位。
6. Transformations & Symmetry | 变换与对称
Reflection: a mirror line acts as a perpendicular bisector of the segment joining a point and its image. Rotation: specify centre, angle and direction. Translation: described by a column vector.
反射（轴对称）：对称轴是点与像点连线的垂直平分线。旋转：需指定旋转中心、旋转角度和方向。平移：用列向量描述。
Enlargement: scale factor k from a centre. If k > 1, the shape enlarges; if 0 < k < 1, it shrinks. Side lengths multiply by k; area multiplies by k².
放大：从中心以比例因子 k 放大。k > 1 时形状放大；0 < k < 1 时缩小。边长乘以 k；面积乘以 k²。
Symmetry: a line of symmetry divides a shape into two mirror‑image halves. The order of rotational symmetry tells how many times a shape looks the same in a full turn.
对称：对称轴将图形分为镜像的两半。旋转对称的阶数描述图形绕中心旋转一周时重合的次数。
7. Pythagoras’ Theorem | 勾股定理
In a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.
在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²，其中 c 为斜边。
To find the length of the hypotenuse, use c = √(a² + b²). To find a shorter side, rearrange: a = √(c² − b²). Always identify the hypotenuse first.
求斜边长：c = √(a² + b²)。求直角边：a = √(c² − b²)。解题时务必先确定斜边。
Converse of Pythagoras: if a² + b² = c² holds in a triangle with sides a, b, c, then the triangle is right‑angled.
勾股定理的逆定理：若三角形三边满足 a² + b² = c²，则该三角形为直角三角形。
8. Ratio & Proportion | 比例与比率
Simplify a ratio 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 number.
化简比例时，将所有项除以最大公因数。按比例分配：先求总份数，总量除以总份数得到一份的量，再乘以各项的份数。
Two quantities are in direct proportion if their ratio stays constant: y = kx, where k is the constant of proportionality. Recognising and using y ∝ x is a key skill.
若两个量的比值恒定，它们成正比例：y = kx，k 为比例常数。识别及运用 y ∝ x 是一项关键技能。
Best‑buy problems and recipe scaling are applications of ratio and proportion. Use unitary method to compare unit prices or adjust ingredient amounts.
最划算购买和配方调整是比例的应用。使用归一法比较单价或调整配料分量。
9. Statistics: Averages & Charts | 统计：平均数和图表
Mean = sum of all values ÷ number of values. Median = middle value when data is ordered. Mode = most frequent value. Range = maximum − minimum.
平均数（均值）= 所有数值之和 ÷ 数值个数。中位数 = 排序后位于中间的值。众数 = 出现次数最多的值。极差 = 最大值 − 最小值。
Bar charts display discrete or categorical data; pie charts show proportions of a whole. Line graphs are used for time series. Scatter graphs show relationship between two variables; correlation can be positive, negative or none.
条形图用于显示离散或分类数据；饼图显示各部分占整体的比例。线形图用于时间序列。散点图显示两个变量间的关系；相关性可以是正、负或无。
When interpreting charts, always read scales carefully and check for misleading representations. The sum of angles in a pie chart is 360°, and each sector angle = (frequency/total) × 360°.
解读图表时务必仔细阅读刻度，警惕误导性表达。饼图中角度总和为 360°，每个扇形的角度 = (频数/总数) × 360°。
10. Probability | 概率
Probability of an event = number of favourable outcomes / total number of possible outcomes, written as a fraction, decimal or percentage. Probability scale runs from 0 (impossible) to 1 (certain).
事件发生的概率 = 有利结果数 / 所有可能结果数，可用分数、小数或百分数表示。概率介于 0（不可能）和 1（必然）之间。
The probabilities of all mutually exclusive outcomes sum to 1. Expected frequency = probability × number of trials.
所有互斥结果的概率之和为 1。预期频数 = 概率 × 试验次数。
Sample space diagrams and two‑way tables help list all possible outcomes of combined events. Use them to find probabilities of “and” and “or” events systematically.
样本空间图与双向表有助于列出组合事件的所有可能结果，用来系统计算“且”和“或”事件的概率。
11. Measures & Units | 度量与单位
Metric conversions: length – 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. Mass – 1 tonne = 1000 kg, 1 kg = 1000 g. Capacity – 1 litre = 1000 ml, 1 cm³ = 1 ml.
公制换算：长度 – 1 千米 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米。质量 – 1 吨 = 1000 千克，1 千克 = 1000 克。容量 – 1 升 = 1000 毫升，1 立方厘米 = 1 毫升。
Time: 1 hour = 60 minutes, 1 minute = 60 seconds. Always use 24‑hour clock when appropriate and be careful when adding mixed units.
时间：1 小时 = 60 分，1 分 = 60 秒。必要时使用 24 小时制，混合单位运算时须格外小心。
Compound measure formulas: speed = distance ÷ time, density = mass ÷ volume. You can rearrange these using formula triangles if needed.
复合量公式：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积。如有需要可用公式三角形进行变形。
Area conversion: 1 m² = 10 000 cm² (not 100!). Volume conversion: 1 m³ = 1 000 000 cm³. Remember that units of area and volume scale with the square and cube of the length ratio.
面积换算：1 平方米 = 10 000 平方厘米（不是 100！）。体积换算：1 立方米 = 1 000 000 立方厘米。记住面积和体积单位随长度比的平方和立方缩放。
12. Graphs & Functions | 图形与函数
Coordinates are written as (x, y). The x‑axis is horizontal, the y‑axis vertical. Plot points carefully using the “along the corridor, up the stairs” rule.
坐标写为 (x, y)。x 轴为水平轴，y 轴为垂直轴。按照“先走横轴，再上竖轴”的规则仔细描点。
Linear graphs have equation y = mx + c. m is the gradient (steepness), calculated as rise/run. c is the y‑intercept, where the line crosses the y‑axis.
线性图像方程为 y = mx + c。m 为斜率（坡度），由垂直变化/水平变化算出。c 为 y 截距，即直线与 y 轴的交点。
To find the gradient of a straight line from two points (x₁, y₁) and (x₂, y₂), use m = (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes up, a negative gradient slopes down.
已知两点 (x₁, y₁) 和 (x₂, y₂) 求直线斜率：m = (y₂ − y₁) / (x₂ − x₁)。正斜率向上倾斜，负斜率向下倾斜。
Real‑life graphs: distance‑time graphs show journey; a horizontal line means stationary, a straight sloped line means constant speed. Speed is found from the gradient of a distance‑time graph.
实际场景图：距离‑时间图反映行程；水平线段表示静止，倾斜直线表示匀速运动。速度可由距离‑时间图的斜率求得。
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