📚 Year 9 CCEA Maths Formula & Theorem Quick Reference Handbook | CCEA Year 9 数学公式定理速查手册
This handbook provides a rapid reference to the essential formulas, theorems and key facts needed for Year 9 CCEA Mathematics. It is designed to support revision, homework and classroom learning by presenting each topic in a clear paired English–Chinese format.
本手册为 CCEA Year 9 数学必备的公式、定理和关键知识点提供快速查阅。通过清晰的中英对照呈现每个主题，帮助学生复习、完成作业和课堂学习。
1. Number Properties and Operations | 数的性质与运算
Prime numbers are integers greater than 1 that have exactly two factors: 1 and themselves. Examples: 2, 3, 5, 7, 11.
质数是大于 1 且只有两个因数（1 和自身）的整数。例如：2, 3, 5, 7, 11。
Factors of a number divide it exactly without leaving a remainder. Multiples are obtained by multiplying the number by an integer.
因数（因子）是能整除一个数的数。倍数是将该数乘以某个整数得到的积。
HCF (Highest Common Factor) is the largest factor shared by two or more numbers. LCM (Lowest Common Multiple) is the smallest multiple they share.
HCF（最大公因数）是两个或多个数共有的最大因数。LCM（最小公倍数）是它们共有的最小倍数。
Integer operations follow BIDMAS/BODMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
整数运算遵循 BIDMAS/BODMAS 规则：括号、指数、乘除（从左到右）、加减（从左到右）。
e numbers and cube numbers: n² = n × n, n³ = n × n × n. Square root √ returns to the base.
平方数与立方数：n² = n × n, n³ = n × n × n。平方根 √ 用于求原数。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent fractions represent the same part of a whole. Simplify by dividing numerator and denominator by their HCF.
等值分数表示相同的部分与整体关系。通过分子分母同除以它们的最大公因数进行约分。
To add or subtract fractions, write them with a common denominator, then combine numerators.
加减分数时，先通分使分母相同，再合并分子。
Multiplying fractions: multiply numerators together and denominators together. Dividing fractions: multiply by the reciprocal.
分数乘法：分子乘分子，分母乘分母。分数除法：乘以其倒数。
Convert fraction to decimal: divide numerator by denominator. Decimal to percentage: multiply by 100%.
分数转小数：分子除以分母。小数转百分数：乘以 100%。
Percentage increase/decrease = (change ÷ original) × 100%. Finding a percentage of an amount: (percentage ÷ 100) × amount.
百分比增减 = (变化量 ÷ 原值) × 100%。求一个数的百分之几：(百分比 ÷ 100) × 数值。
3. Ratio and Proportion | 比率与比例
A ratio compares part to part. The ratio a : b can be simplified by dividing both sides by their HCF.
比率用于比较部分与部分。比 a : b 可通过同时除以最大公因数进行化简。
Divide a quantity in a given ratio: 1) add the parts to find the total number of shares; 2) divide the quantity by total shares; 3) multiply by each part.
按给定比率分配数量：1) 把各部分相加得到总份数；2) 将总量除以总份数；3) 乘以每一部分的份数。
Direct proportion means two quantities increase or decrease at the same rate. Unitary method finds the value of one unit first.
正比例意味着两个量以相同速率增减。归一法先求出每一单位对应的值。
a ratio to represent real lengths. scale = drawing length : actual length (in same units).
比例尺绘图使用比率表示实际长度。比例尺 = 图上长度 : 实际长度（相同单位）。
4. Algebraic Expressions and Equations | 代数表达式与方程
Like terms contain the same variable(s) raised to the same power. Simplify expressions by collecting like terms.
同类项指含有相同变量且指数相同的项。化简表达式时合并同类项。
xpanding brackets: a(b + c) = ab + ac. Double brackets: (x + a)(x + b) = x² + (a+b)x + ab.
展开括号：a(b + c) = ab + ac。双括号：(x + a)(x + b) = x² + (a+b)x + ab。
ctorising reverses expansion: take out the highest common factor, e.g. 6x + 9 = 3(2x + 3).
因式分解是展开的逆过程：提取最大公因数，如 6x + 9 = 3(2x + 3)。
Solving linear equations: carry out inverse operations on both sides to isolate the variable.
解一元一次方程：在方程两边进行逆运算以分离变量。
Substituting into a formula: replace letters with given values and calculate using BIDMAS.
公式代入：用给定数值替换字母，并按 BIDMAS 规则计算。
5. Sequences | 数列
An arithmetic sequence has a common difference d between consecutive terms. Term-to-term rule: add d each time.
等差数列相邻两项之差为常数 d。逐项规律：每次加 d。
The nth term of an arithmetic sequence: aₙ = a₁ + (n − 1)d, where a₁ is the first term.
等差数列的第 n 项公式：aₙ = a₁ + (n − 1)d，其中 a₁ 为首项。
= dn + c. Find d from the difference between terms, then solve for c using the first term.
线性数列可写作 T(n) = dn + c。从相邻项差求出 d，再用首项求出 c。
Other sequences include square numbers (1, 4, 9, 16…), triangular numbers (1, 3, 6, 10…) and Fibonacci-type sequences.
其他数列包括平方数（1, 4, 9, 16……）、三角形数（1, 3, 6, 10……）以及类斐波那契数列。
6. Straight Line Graphs and Coordinates | 直线图与坐标
Coordinates (x, y) locate a point on the grid. The midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).
坐标 (x, y) 用于在网格上定位点。两点 (x₁, y₁) 与 (x₂, y₂) 的中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The equation of a horizontal line is y = c; a vertical line is x = k.
水平线的方程为 y = c；竖直线的方程为 x = k。
c has gradient m and y-intercept c. Gradient = rise / run = (change in y) / (change in x).
直线方程 y = mx + c 中，m 代表斜率，c 代表 y 轴截距。斜率 = 纵差 / 横差 = (y 的变化量) / (x 的变化量)。
Plot the graph by drawing a table of values, calculating y for chosen x, and joining the points with a straight line.
通过列表取值并计算对应 y 值，描点后用直线连接，即可画出图像。
7. Angles and Polygons | 角度与多边形
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的邻角之和为 180°。绕一点一周的角之和为 360°。对顶角相等。
Parallel line angles: alternate angles are equal, corresponding angles are equal, co-interior (allied) angles sum to 180°.
平行线角度：内错角相等，同位角相等，同旁内角之和为 180°。
180°, where n is the number of sides. Each interior angle of a regular polygon = sum ÷ n.
多边形内角和 = (n − 2) × 180°，其中 n 为边数。正多边形每个内角 = 内角和 ÷ n。
s of any convex polygon sum to 360°. In a regular polygon, each exterior angle = 360° ÷ n.
任意凸多边形的外角和恒为 360°。正多边形每个外角 = 360° ÷ n。
8. Perimeter, Area and Volume | 周长、面积与体积
Key formulas for 2D shapes are summarised below. All lengths must be in the same units.
以下是平面图形的关键公式。所有长度必须使用相同单位。
| Shape / 图形 | Perimeter / 周长 | Area / 面积 |
|---|---|---|
| Rectangle / 矩形 | 2(l + w) | l × w |
| Triangle / 三角形 | a + b + c | ½ × base × height |
| Parallelogram / 平行四边形 | 2(a + b) | base × perpendicular height |
| Trapezium / 梯形 | Sum of all sides | ½ (a + b) × h |
| Circle / 圆 | Circumference = 2πr or πd | πr² |
Volume of a prism = area of cross-section × length. For a cuboid: V = l × w × h.
棱柱体积 = 横截面积 × 长。长方体体积：V = l × w × h。
Surface area of a cuboid = 2(lw + lh + wh). Cylinder volume = πr²h.
长方体表面积 = 2(lw + lh + wh)。圆柱体积 = πr²h。
9. Pythagoras' Theorem | 毕达哥拉斯定理
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
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
where c is the hypotenuse, and a, b are the shorter sides. It is used to find a missing length in right-angled triangles.
其中 c 为斜边长，a 与 b 为直角边长。该定理用于求直角三角形中的未知边。
To find the hypotenuse: c = √a² + b². To find a shorter side: a = √c² − b².
求斜边：c = √a² + b²。求直角边：a = √c² − b²。
The converse: if a² + b² = c² in a triangle, then the triangle is right‑angled.
逆定理：若三角形中 a² + b² = c²，则该三角形是直角三角形。
10. Statistics and Probability | 统计与概率
middle value when data are ordered. Mode = most frequent value. Range = maximum − minimum.
平均数 = 所有数据之和 ÷ 数据个数。中位数 = 数据按序排列后的中间值。众数 = 出现频率最高的值。极差 = 最大值 − 最小值。
rom a frequency table: mean = Σ(f × x) ÷ Σf, where x is the data value and f is frequency.
频数表求平均数：平均数 = Σ(f × x) ÷ Σf，其中 x 为数据值，f 为频数。
, pie charts for proportions (angle = (frequency ÷ total) × 360°), line graphs for trends.
统计图：条形图用于离散数据，饼图用于显示比例（扇区角度 = (频数 ÷ 总数) × 360°），折线图用于趋势。
vourable outcomes ÷ total number of possible outcomes. Scale from 0 (impossible) to 1 (certain).
事件的概率 = 有利结果数 ÷ 所有可能结果数。概率值介于 0（不可能）到 1（必然）之间。
y exclusive events: P(A or B) = P(A) + P(B). The sum of probabilities of all outcomes = 1.
对于互斥事件：P(A 或 B) = P(A) + P(B)。所有可能结果的概率之和为 1。
ility × number of trials. Sample space diagrams list all possible outcomes systematically.
期望频数 = 概率 × 试验次数。样本空间图系统地列出所有可能结果。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图