📚 Year 9 CCEA Mathematics: Formula & Theorem Quick Reference Guide | 九年级CCEA数学：公式定理速查手册
This reference guide brings together all the essential formulas, theorems and rules you need for Year 9 CCEA Mathematics. Use it as a fast recap before tests, homework checks or when you want to remind yourself of a key idea. Each section pairs an English explanation with a Chinese translation so you can study in the language you prefer or strengthen your bilingual mathematical vocabulary.
这份速查手册汇总了九年级 CCEA 数学中所有重要的公式、定理与规则。无论是测试前的快速复习、作业检查，还是想重温某个关键概念，都可以随时翻阅。每个知识点都配有英文说明和中文翻译，方便你用自己习惯的语言学习，同时也能强化双语数学词汇。
1. Number Facts: Operations, Negative Numbers and BIDMAS | 数系基础：运算、负数与运算顺序
When adding a negative number, the result is the same as subtracting its positive value: a + (−b) = a − b.
加上一个负数等于减去它的相反数：a + (−b) = a − b。
When subtracting a negative number, change the two minus signs into a plus: a − (−b) = a + b.
减去一个负数时，把两个负号变成加号：a − (−b) = a + b。
Multiplying or dividing two numbers with the same sign gives a positive answer; with different signs the answer is negative.
同号两数相乘或相除结果为正；异号相乘或相除结果为负。
The order of operations is remembered by BIDMAS (Brackets, Indices, Division & Multiplication, Addition & Subtraction). Work from left to right for operations of the same priority.
运算顺序用 BIDMAS 来记忆：括号、指数、乘除、加减。同一优先级的运算从左向右计算。
Example: 3 + 4 × (5 − 2)² ÷ 9 = 3 + 4 × 3² ÷ 9 = 3 + 4 × 9 ÷ 9 = 3 + 4 = 7
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100.
分数化小数，用分子除以分母。小数化百分数，乘以100。
To convert a percentage to a fraction, write it over 100 and simplify. A percentage increase or decrease can be found using a multiplier: increase by r% means multiply by (1 + r/100).
百分数化分数，先写成分母为100的分数再化简。求百分数增减可以用乘数：增加 r% 即乘以 (1 + r/100)。
| Common Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33.3% |
| 2/5 | 0.4 | 40% |
To add or subtract fractions, find a common denominator first. Multiply fractions by multiplying numerators and denominators separately; divide by multiplying by the reciprocal.
分数加减要先通分。分数乘法：分子乘分子，分母乘分母；分数除法：乘以倒数。
3. Algebraic Expressions | 代数表达式
Like terms can be collected together: 5x + 3y − 2x + y simplifies to 3x + 4y.
同类项可以合并：5x + 3y − 2x + y 化简为 3x + 4y。
When multiplying terms, multiply the numbers and add the powers of the same letter: 3x² × 4x³ = 12x⁵.
单项式相乘时，系数相乘，相同字母的指数相加：3x² × 4x³ = 12x⁵。
Expanding a single bracket uses the distributive law: a(b + c) = ab + ac. To factorise an expression, look for the highest common factor and write it outside the bracket.
单项括号展开用分配律：a(b + c) = ab + ac。因式分解是提取最大公因数并写在括号外面。
Expand: −2(3x − 4) = −6x + 8
Factorise: 6x² + 9x = 3x(2x + 3)
4. Linear Equations | 一元一次方程
A linear equation can be solved by performing the same operation on both sides to keep the equation balanced. Aim to isolate the unknown on one side.
解一元一次方程需要对方程两边进行相同的运算以维持平衡，目的是将未知数单独移到一边。
Steps: remove brackets, collect like terms on each side, move variable terms to one side and numbers to the other, then divide by the coefficient of the unknown.
步骤：去括号，左右分别合并同类项，将含未知数的项移到一边、常数项移到另一边，最后除以未知数的系数。
Example: 2(x + 3) = 4x − 2 → 2x + 6 = 4x − 2 → 6 + 2 = 4x − 2x → 8 = 2x → x = 4
Always check your answer by substituting it back into the original equation.
求出解后一定要代回原方程检验。
5. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference, d.
数列是按一定规则排列的一串数。等差数列中相邻两项的差是常数，这个差称为公差 d。
To find the nth term of an arithmetic sequence with first term a and common difference d, use the formula:
对于首项为 a、公差为 d 的等差数列，第 n 项公式为：
Tₙ = a + (n − 1)d
For the sequence 5, 9, 13, 17, …, a = 5, d = 4, so Tₙ = 5 + (n − 1)×4 = 4n + 1.
对于数列 5, 9, 13, 17, …，a = 5, d = 4，所以 Tₙ = 5 + (n − 1)×4 = 4n + 1。
If the rule is given as a position-to-term expression, you can generate the sequence by substituting n = 1, 2, 3, …
如果给出位置与项的关系式，代入 n = 1, 2, 3, … 即可生成数列。
6. Angles and Lines | 角与直线
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角为180°；绕一点一周的角度之和为360°；对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and interior (co‑interior) angles sum to 180°.
当两条平行线被一条截线所截时，同位角相等，内错角相等，同旁内角之和为180°。
In any triangle, the sum of the three interior angles is 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
任意三角形的三个内角之和为180°。三角形的一个外角等于与它不相邻的两个内角之和。
Triangle interior angles: ∠A + ∠B + ∠C = 180°
Exterior angle: ∠ABD = ∠A + ∠C
7. Triangles and Pythagoras’ Theorem | 三角形与勾股定理
A triangle can be classified by its sides (equilateral, isosceles, scalene) or by its angles (acute, right, obtuse). In a right‑angled triangle, the side opposite the right angle is called the hypotenuse; it is always the longest side.
三角形可按边分为等边、等腰和不等边三角形，也可按角分为锐角、直角和钝角三角形。直角三角形中，直角所对的边称为斜边，总是最长边。
Pythagoras’ theorem states that in a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
勾股定理指出，直角三角形斜边的平方等于两条直角边的平方和。
a² + b² = c²
where c is the hypotenuse and a and b are the legs. The theorem can be used to find a missing side length and to test whether a triangle is right‑angled.
其中 c 为斜边，a 和 b 为直角边。该定理可用来求未知边长，也可检验一个三角形是否为直角三角形。
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around the outside of a 2‑D shape. Area is the amount of space inside the shape, measured in square units. Volume measures the space inside a 3‑D object, in cubic units.
周长是平面图形一周的长度。面积是图形内部的大小，用平方单位计量。体积是立体图形所占的空间，用立方单位计量。
| Shape | Area Formula |
|---|---|
| Rectangle | length × width (l × w) |
| Triangle | (base × height) / 2 (½ b h) |
| Parallelogram | base × perpendicular height (b × h) |
| Trapezium | (a + b) × h / 2 (average of parallel sides × height) |
| Circle | π × r² (π ≈ 3.14) |
The circumference of a circle is C = 2πr or C = πd.
圆的周长公式为 C = 2πr 或 C = πd。
Volume of a cuboid = length × width × height. Volume of a prism = area of cross‑section × length.
长方体体积 = 长 × 宽 × 高。棱柱体积 = 横截面积 × 长度。
9. Coordinates and Straight‑Line Graphs | 坐标与直线图
A point is located on a coordinate grid by an ordered pair (x, y). The x‑axis is horizontal; the y‑axis is vertical. The origin is (0, 0).
坐标系中，点的位置用序数对 (x, y) 表示。x 轴为水平轴，y 轴为垂直轴，原点为 (0, 0)。
A straight line can be written in the form y = mx + c, where m is the gradient and c is the y‑intercept. The gradient is the change in y divided by the change in x between any two points on the line.
直线方程可写成 y = mx + c 的形式，其中 m 为斜率，c 为 y 轴截距。斜率等于直线上任意两点间 y 的变化量除以 x 的变化量。
m = (y₂ − y₁) / (x₂ − x₁)
To plot a graph from an equation, build a table of values, plot the points, and draw a straight line through them. Parallel lines have the same gradient.
根据方程画图时，先列出数值表，描点，然后用直尺连线。平行线的斜率相等。
10. Statistics and Probability | 统计与概率
Three common averages: the mean is the sum of all values divided by how many there are; the median is the middle value when data are ordered; the mode is the value that appears most often. The range is the difference between the largest and smallest values.
三种常见的平均数：平均数（均值）是所有数值之和除以数据个数；中位数是排序后处于中间位置的数；众数是出现次数最多的数。极差是最大值与最小值之差。
Mean = Σx / n
Data can be displayed in bar charts, pie charts, line graphs and frequency tables. When grouping data, make sure all intervals are equal where possible.
数据可用条形图、饼图、折线图和频数表来展示。分组时尽可能使组距相等。
Probability measures the chance of an event happening. It is always a number between 0 and 1, or a percentage between 0% and 100%.
概率用来衡量事件发生的可能性，取值总是在 0 到 1 之间，或者 0% 到 100% 之间。
Probability of an event = Number of favourable outcomes / Total number of possible outcomes
If an experiment is repeated many times, the relative frequency can be used to estimate probability. The sum of probabilities of all possible outcomes is 1.
若实验重复多次，可用相对频数来估计概率。所有可能结果的概率之和为 1。
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