📚 Year 9 WJEC Maths: Formulas and Theorems Quick Reference Handbook | 九年级 WJEC 数学：公式定理速查手册
This handbook brings together all the key mathematical formulas, theorems and essential knowledge needed for the Year 9 WJEC curriculum. Use it as a quick revision aid to memorise rules, understand relationships and check your work before tests. The content is organised by topic so you can find exactly what you need at a glance.
本手册汇总了九年级 WJEC 数学课程中所有关键的公式、定理和必备知识。你可以把它当作快速复习工具，用来记忆法则、理解关系并在测试前检查自己的作业。内容按主题编排，让你能一目了然地找到所需内容。
1. Number Properties | 数的性质
A prime number has exactly two distinct factors: 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. 2 is the only even prime.
质数只有两个不同的因数：1 和它本身。前十个质数是 2、3、5、7、11、13、17、19、23 和 29。2 是唯一的偶质数。
A factor divides exactly into a number. A multiple is the result of multiplying a number by an integer. The highest common factor (HCF) is the largest factor shared by two or more numbers; the lowest common multiple (LCM) is the smallest multiple they share.
因数可以整除一个数。倍数是一个数乘以整数的结果。最大公因数 (HCF) 是两个或多个数共有的最大因数；最小公倍数 (LCM) 是它们共有的最小倍数。
4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100. The square root of a number n, written √n he positive number that squares to give n. Cube numbers: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125.
平方数：1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100。一个数 n 的平方根记作 √n，是平方后得到 n 的正数。立方数：1³=1, 2³=8, 3³=27, 4³=64, 5³=125。
Index laws for positive integers: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n); (aᵐ)ⁿ = aᵐˣⁿ. Any number raised to the power 0 is 1 (as long as the base is not 0).
正整数指数律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n)；(aᵐ)ⁿ = aᵐˣⁿ。任何非零数的 0 次幂等于 1。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent fractions are made by multiplying or dividing numerator and denominator by the same number. To add or subtract fractions, write them with a common denominator, then add or subtract the numerators. Multiplication: multiply numerators and multiply denominators. Division: multiply by the reciprocal (swap the second fraction's numerator and denominator).
等值分数是通过将分子和分母乘以或除以同一个数得到的。加减分数时，先通分，再加减分子。乘法：分子乘分子，分母乘分母。除法：乘以倒数（交换第二个分数的分子与分母）。
Mixed numbers: convert to improper fractions before multiplying or dividing. An improper fraction has a numerator larger than its denominator. To convert a fraction to a decimal, divide the numerator by the denominator.
带分数：在乘除前先化为假分数。假分数的分子大于分母。将分数化为小数时，用分子除以分母。
A percentage means 'out of 100' ge increase = (increase ÷ original) × 100; percentage decrease = (decrease ÷ original) × 100.
百分数表示“占一百份中的多少”。分数化百分数，乘以 100。求一个量的百分之几，把百分数化成小数再相乘。增长百分数 = (增长量 ÷ 原值) × 100；减少百分数 = (减少量 ÷ 原值) × 100。
Common conversions to memorise:
需要记住的常见换算：
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33⅓% |
| 2/3 | 0.666... | 66⅔% |
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. The ratio a : b means for every a units of the first quantity there are b units of the second. To simplify a ratio, divide all parts by the highest common factor. Ratios can be written in the form 1 : n by dividing both sides by the first number.
比用来比较整体中的各个部分。比 a : b 表示每 a 个单位的第一量对应 b 个单位的第二量。化简比时，将所有项除以它们的最大公因数。通过两边同时除以第一项，可以将比写成 1 : n 的形式。
ne quantity changes in relation to another. If y is directly proportional to x, we write y ∝ oportionality. The ratio y : x stays constant. Unitary method: find the value for one unit first, then scale up or down.
比例描述一个量如何随另一个量变化。若 y 与 x 成正比，记作 y ∝ x 或 y = kx，其中 k 为比例常数。比值 y : x 保持不变。单位法：先求出一个单位的值，再进行放大或缩小。
4. Algebraic Expressions | 代数表达式
an be combined: 3a + 2a = 5a. Unlike terms such as 3a and 2b cannot be simplified further.
表达式包含数字、变量和运算符号。同类项拥有完全相同的变量部分，可以合并：3a + 2a = 5a。非同类型项如 3a 和 2b 不能进一步化简。
m in the first bracket multiplies each term in the second: (a + b)(c + d) = ac + ad + bc + bd.
展开括号使用分配律：a(b + c) = ab + ac。务必用括号外的项乘以括号内的每一项。对于双括号，第一个括号中的每一项乘以第二个括号中的每一项：(a + b)(c + d) = ac + ad + bc + bd。
nd place it outside the bracket. For example, 6x + 9 = 3(2x + 3). Check by expanding back.
因式分解是展开的逆运算。找出所有项的最高公因式 (HCF)，放在括号外面。例如 6x + 9 = 3(2x + 3)。可以通过重新展开来检验。
5. Linear Equations | 线性方程
An equation shows two expressions are equal. To solve, perform the same operation on both sides to isolate the variable. Always keep the equation balanced.
方程表示两个表达式相等。求解时，在等号两边同时进行相同的运算以分离变量。始终保持方程两边平衡。
ct b from both sides, then divide by a. If the variable appears on both sides, collect like terms to one side first. Always check your solution by substituting it back into the original equation.
对于线性方程 ax + b = c，先在两边减去 b，再除以 a。若变量出现在等号两边，先将含变量项移到一边。务必把解代回原方程进行检验。
es parallel to the x-axis are y = constant; lines parallel to the y-axis are x = constant.
直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率 = 纵距 ÷ 横距 = y 的变化量 ÷ x 的变化量。平行于 x 轴的直线为 y = 常数；平行于 y 轴的直线为 x = 常数。
6. Sequences | 序列
A number sequence is a list of numbers that follow a rule. In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference, d.
数字序列是一列按照规则排列的数。在等差数列中，相邻两项的差是常数。这个差值称为公差 d。
of an arithmetic sequence with first term a and common difference d is: nth term = a + (n – n use this formula to find any term or to check if a given number belongs to the sequence.
首项为 a、公差为 d 的等差数列的第 n 项公式为：第 n 项 = a + (n – 1)d。你可以用这个公式求任意一项，或检验某个数是否属于该序列。
Other sequences have different rules: geometric sequences multiply by a fixed ratio; square sequences 1, 4, 9, 16... have nth term n²; triangular numbers 1, 3, 6, 10... have nth term n(n+1)/2.
其他序列遵循不同的规则：等比数列乘以固定比值；平方序列 1, 4, 9, 16... 的第 n 项为 n²；三角形数 1, 3, 6, 10... 的第 n 项为 n(n+1)/2。
7. Coordinates and Graphs | 坐标与图像
Coordinates are written as (x, y). The x-axis runs horizontally, the y-axis runs vertically. The origin is (0, 0). Plotting points and joining them creates a graph.
坐标记作 (x, y)。x 轴为水平方向，y 轴为垂直方向。原点是 (0, 0)。描点并将它们连接起来就得到图像。
The midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between points can be found using Pythagoras' theorem.
两点 (x₁, y₁) 和 (x₂, y₂) 的中点是 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点之间的距离可以利用勾股定理求出。
or non-linear graphs such as y = x², create a table of values and plot the curve smoothly.
线性图像为一条直线。画 y = mx + c 的图像时，从 y 轴上的 (0, c) 出发，再利用斜率 m 找到其他点。对于 y = x² 这样的非线性图像，先列出数值表，再用光滑曲线连接各点。
8. Geometry: Angles and Shapes | 几何：角与图形
Angle facts: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. In a triangle, the sum of interior angles is 180°. In a quadrilateral, the sum is 360°.
角的事实：直线上的邻角之和为 180°；绕一点的周角之和为 360°；对顶角相等。三角形的内角和为 180°。四边形的内角和为 360°。
Parallel lines: corresponding angles are equal; alternate angles are equal; interior (co‑interior) angles sum to 180°. These facts help find unknown angles in diagrams with parallel lines.
平行线：同位角相等；内错角相等；同旁内角互补（和为 180°）。这些事实能帮助我们求出含平行线的图形中的未知角。
Properties of triangles: an equilateral triangle has three equal sides and three 60° angles. An isosceles triangle has two equal sides and two equal base angles. A right‑angled triangle contains a 90° angle. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形的性质：等边三角形三条边相等，三个角均为 60°。等腰三角形两条边相等，两个底角相等。直角三角形含有一个 90° 角。三角形的外角等于与其不相邻的两个内角之和。
Quadrilaterals named: square, rectangle, rhombus, parallelogram, trapezium, kite. Know their side and angle properties.
四边形的名称：正方形、矩形、菱形、平行四边形、梯形、筝形。应熟记它们的边角性质。
9. Pythagoras' Theorem | 勾股定理
In any right‑angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
在任何直角三角形中，斜边的平方等于两直角边的平方和。
c² = a² + b²
posite the right angle) and a and b are the shorter sides. To find the hypotenuse, use c = √a² + b². To find a shorter side, use a = √c² – b².
其中 c 为斜边（最长边，对应直角），a 和 b 为两直角边。求斜边用 c = √a² + b²。求直角边用 a = √c² – b²。
distance between two points on a coordinate grid and checking whether a triangle is right‑angled (if a² + b² = c² holds for the three sides).
应用包括求坐标网格上两点间的距离，以及检验三角形是否为直角三角形（若三边满足 a² + b² = c²）。
10. Perimeter, Area and Volume | 周长、面积与体积
= 2πr, where d is diameter and r is radius. Use π ≈ 3.14 or the π button on a calculator.
周长是图形一周的总长度。矩形周长 P = 2(l + w)，其中 l 为长，w 为宽。圆的周长：C = πd 或 C = 2πr，其中 d 为直径，r 为半径。π 取近似值 3.14 或使用计算器上的 π 键。
Area measures the space inside a shape. Key formulas:
面积衡量图形内部的区域大小。关键公式：
| Rectangle | A = l × w |
| Triangle | A = ½ × base × height |
| Parallelogram | A = base × vertical height |
| Trapezium | A = ½(a + b)h, where a and b are parallel sides, h is height |
| Circle | A = πr² |
ded. Volume of a cuboid: V = length × width × height. Volume of a prism: V = area of cross‑section × length.
组合图形可分割成简单图形，分别求出各部分的面积，再根据需要相加或相减。长方体的体积：V = 长 × 宽 × 高。棱柱的体积：V = 横截面积 × 长度。
11. Statistics | 统计
alue when data are ordered. The mode is the most frequent value. The range = largest value – smallest value, measuring spread.
平均数是算术平均值：平均数 = 所有数据值之和 ÷ 数据个数。中位数是将数据排序后处于中间位置的值。众数是出现次数最多的值。极差 = 最大值 – 最小值，用来衡量离散程度。
al frequency) × 360°. Tally charts and frequency tables organise raw data before plotting.
数据可以用条形图、象形图、折线图和饼图展示。在饼图中，扇形的角度 = (频数 ÷ 总频数) × 360°。在画图之前，用划记表和频数表整理原始数据。
Scatter graphs show relationships between two variables. Positive correlation means as one variable increases, the other tends to increase. A line of best fit can be drawn to make predictions.
散点图展示两个变量之间的关系。正相关意味着一个变量增大时，另一个也倾向于增大。可以画出最佳拟合线进行预测。
12. Probability | 概率
otal number of equally likely outcomes. It can be written as a fraction, decimal or percentage.
概率衡量事件发生的可能性大小。概率值总是在 0（不可能）和 1（必然）之间。概率 = 有利结果的数量 ÷ 所有等可能结果的总数。它可以用分数、小数或百分数表示。
(B). The sum of probabilities of all possible outcomes is 1, so P(event not happening) = 1 – ple space diagrams list all outcomes and help calculate probabilities for combined events.
对于互斥事件（不能同时发生），P(A 或 B) = P(A) + P(B)。所有可能结果的概率之和为 1，因此 P(事件不发生) = 1 – P(事件发生)。样本空间图列出所有结果，有助于计算组合事件的概率。
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