📚 Year 9 Maths Formula & Theorem Quick Reference Handbook | Year 9 数学公式定理速查手册
This quick-reference guide covers the essential formulae and theorems you'll meet in the CAIE Year 9 Mathematics course. Use it to check key facts, revise for assessments, and build confidence before moving into IGCSE. Each section gives the rule in English, immediately followed by its Chinese explanation, so you can study bilingually.
本速查手册涵盖CAIE Year 9数学课程中的核心公式和定理。你可以用它核对关键知识点、复习备考，并为进入IGCSE打牢基础。每一节都先用英文给出规则，紧接着提供中文解释，方便你进行双语学习。
1. Number Systems and Operations | 数系与运算
The set of integers includes all positive and negative whole numbers and zero, denoted by ℤ.
整数集包括所有正整数、负整数和零，记作 ℤ。
The order of operations is BIDMAS: Brackets, Indices, Division/Multiplication (from left to right), Addition/Subtraction (from left to right).
运算顺序遵循BIDMAS法则：括号、指数、乘除（从左到右）、加减（从左到右）。
A prime number has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13...
质数只有两个不同的因数：1 和它本身。前几个质数是 2, 3, 5, 7, 11, 13……
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) can be found using prime factorisation: list the prime factors for each number, then for HCF take the product of the smallest powers of common primes; for LCM take the product of the largest powers of all primes.
最大公因数（HCF）和最小公倍数（LCM）可以通过质因数分解求得：列出各数的质因数，然后HCF取各公共质因数的最低次幂的乘积；LCM取所有质因数的最高次幂的乘积。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
d or subtract fractions, rewrite them with a common denominator: a/b ± c/d = (ad ± bc)/bd.
加减分数时，先通分：a/b ± c/d = (ad ± bc)/bd。
a/b × c/d = ac/bd. Divide by multiplying by the reciprocal: a/b ÷ c/d = a/b × d/c = ad/bc.
分数相乘：分子相乘，分母相乘：a/b × c/d = ac/bd。分数相除：乘以其倒数：a/b ÷ c/d = a/b × d/c = ad/bc。
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100 and add the % symbol.
分数化小数：用分子除以分母。小数化百分数：乘以100并加上%符号。
A percentage change is calculated as: (change ÷ original value) × 100%.
百分比变化计算公式：(变化量 ÷ 原值) × 100%。
3. Ratio and Proportion | 比与比例
A ratio compares two or more quantities, written as a : b or a : b : c. To simplify, divide all parts by their HCF.
比用于比较两个或以上的量，写作 a : b 或 a : b : c。化简时将所有部分除以它们的最大公因数。
re in proportion, we write a : b = c : d, and the cross-products are equal: a × d = b × c.
若两个比成比例，记作 a : b = c : d，则交叉乘积相等：a × d = b × c。
Direct proportion means y = kx for some constant k. Inverse proportion means xy = k.
正比例关系：y = kx，其中k为常数。反比例关系：xy = k。
io a : b, the first share is (a/(a+b)) × total, and the second share is (b/(a+b)) × total.
将一个量按给定比 a : b 分配时，第一份为 (a/(a+b)) × 总量，第二份为 (b/(a+b)) × 总量。
4. Basic Algebra | 代数基础
ers stand for unknown numbers or variables. An expression collects terms using +, −, ×, ÷.
代数中，字母代表未知数或变量。表达式通过 +、−、×、÷ 将项组合起来。
3a + 2a = 5a; unlike terms cannot be combined. Always simplify expressions where possible.
同类项可以合并：3a + 2a = 5a；不同类项不能合并。应尽可能化简表达式。
= ab + ac. This is used to expand brackets or to factorise by taking out a common factor.
分配律：a(b + c) = ab + ac。它用于去括号展开，或提取公因式进行因式分解。
ules (same base): aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
同底数指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。
5. Equations and Inequalities | 方程与不等式
To solve a linear equation, perform the same operation on both sides to isolate the unknown. Always check your answer by substitution.
解一元一次方程时，对方程两边进行相同的运算以求出未知数。最后务必代入检验答案。
For inequalities, the rules are similar to equations, but if you multiply or divide both sides by a negative number, you must reverse the inequality sign: < becomes >, etc.
解不等式的规则与方程类似，但当两边同乘或除以一个负数时，必须将不等号方向反转，例如 < 变为 >。
(less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
不等号符号：< (小于)，> (大于)，≤ (小于或等于)，≥ (大于或等于)。
ion can be shown on a number line: open circle for < or >, closed circle for ≤ or ≥.
解可以用数轴表示：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
6. Sequences and Patterns | 序列与规律
consecutive terms. The nth term is given by: aₙ = a₁ + (n−1)d, where a₁ is the first term.
等差数列相邻两项的差d为常数。第n项公式为：aₙ = a₁ + (n−1)d，其中a₁为首项。
equence has a constant ratio r between consecutive terms. The nth term is: aₙ = a₁ × rⁿ⁻¹.
等比数列相邻两项的比值r为常数。第n项公式为：aₙ = a₁ × rⁿ⁻¹。
To find a term-to-term rule, look for the pattern: 'add 3 each time', or 'multiply by 2 each time'. The position-to-term rule gives the nth term directly.
寻找递推规则时，观察规律，例如'每项加3'或'每项乘2'。通项公式则直接给出第n项的值。
, 25, ... (nth term = n²). Triangular numbers: 1, 3, 6, 10, 15, ... (nth term = n(n+1)/2).
平方数：1, 4, 9, 16, 25, ...（第n项 = n²）。三角形数：1, 3, 6, 10, 15, ...（第n项 = n(n+1)/2）。
7. Coordinate Geometry and Graphs | 坐标几何与图形
m is the gradient and c is the y-intercept (the point where the line crosses the y-axis).
直线的方程通常写作 y = mx + c，其中 m 为斜率，c 为 y轴截距（直线与y轴交点的纵坐标）。
− y₁)/(x₂ − x₁). A positive gradient slopes upwards, a negative gradient slopes downwards.
斜率 m = (y的变化量) ÷ (x的变化量) = (y₂ − y₁)/(x₂ − x₁)。正斜率直线向上倾斜，负斜率直线向下倾斜。
Parallel lines have the same gradient. For two lines to be parallel, m₁ = m₂.
平行线具有相同的斜率，即 m₁ = m₂。
To plot a linear graph, create a table of at least three x-values, calculate the corresponding y-values, plot the points, and draw a straight line through them.
画一次函数图像时，列出至少三个x值，计算对应的y值，描点，然后用直尺穿过这些点画出直线。
8. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
直线上的角之和为180°。同一顶点绕一周的角之和为360°。
Vertically opposite angles are equal. Corresponding and alternate angles formed by parallel lines are equal; interior angles on the same side of the transversal are supplementary (sum to 180°).
对顶角相等。平行线被一条截线所截时，同位角相等，内错角相等；同旁内角互补（和为180°）。
In any triangle, the interior angles sum to 180°. An exterior angle equals the sum of the two opposite interior angles.
任意三角形的内角和为180°。三角形的一个外角等于与它不相邻的两个内角之和。
rior angles is (n−2)×180°. The sum of exterior angles (one at each vertex) is always 360°.
n边形的内角和为 (n−2)×180°。任意凸多边形的外角和恒为360°（每个顶点取一个外角）。
es and all angles equal. Each interior angle = (n−2)×180°/n, each exterior angle = 360°/n.
正多边形的各边相等、各角相等。每个内角 = (n−2)×180°/n，每个外角 = 360°/n。
9. Perimeter, Area and Volume | 周长、面积与体积
Square: Perimeter = 4s, Area = s², where s is the side length.
正方形：周长 = 4s，面积 = s²，s为边长。
Rectangle: Perimeter = 2(l + w), Area = l × w, where l is length and w is width.
长方形：周长 = 2(l + w)，面积 = l × w，l为长，w为宽。
Triangle: Area = ½ × base × height. The perimeter is the sum of the three sides.
三角形：面积 = ½ × 底 × 高。周长为三边之和。
+ b)h, where a and b are the lengths of the parallel sides, h is the perpendicular height.
平行四边形：面积 = 底 × 高。梯形：面积 = ½ (a + b)h，其中a和b是平行边的长，h为高。
e: Circumference = 2πr or πd, Area = πr². Use π ≈ 3.14 or the π button on your calculator.
圆：周长 = 2πr 或 πd，面积 = πr²。计算时 π 取3.14或使用计算器的 π 键。
Volume of a prism: Volume = area of cross-section × length. For a cuboid, V = l × w × h.
棱柱的体积：体积 = 横截面积 × 长度。长方体体积：V = l × w × h。
10. 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: c² = a² + b², where c is the hypotenuse (the longest side, opposite the right angle).
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，其中c为斜边（最长边，直角所对的边）。
To find the length of the hypotenuse: c = √a² + b². To find one of the shorter sides: a = √c² − b² or b = √c² − a².
求斜边长：c = √a² + b²。求直角边长：a = √c² − b² 或 b = √c² − a²。
three whole numbers that satisfy a² + b² = c², e.g., (3, 4, 5), (5, 12, 13), (8, 15, 17).
勾股数是满足 a² + b² = c² 的正整数组，例如 (3, 4, 5)，(5, 12, 13)，(8, 15, 17)。
You can use the theorem to check whether a triangle is right-angled: if the longest side squared equals the sum of the squares of the other two sides, the triangle is right-angled.
可以用勾股定理判断三角形是否为直角三角形：若最长边的平方等于另两边的平方和，则该三角形是直角三角形。
11. Transformations and Symmetry | 变换与对称
The four basic transformations are translation (slide), reflection (flip), rotation (turn), and enlargement (resize). Congruent shapes have the same size and shape; similar shapes have the same shape but may differ in size.
四种基本变换是平移（滑动）、反射（翻转）、旋转（转动）和放大（缩放）。全等图形形状和大小完全相同；相似图形形状相同但大小可能不同。
Reflection: an object is mirrored in a given line (the mirror line). Each point and its image are the same perpendicular distance from the mirror line.
反射：图形沿给定直线（镜面线）进行镜像翻转。每一点与其像点到镜面线的垂直距离相等。
Rotation requires a centre of rotation, an angle, and a direction (clockwise or anticlockwise). The distance from the centre to each point stays the same.
旋转需要明确旋转中心、旋转角度和方向（顺时针或逆时针）。旋转中心到各点的距离保持不变。
Enlargement by scale factor k from a centre: all lengths are multiplied by k. If k > 1, the image is larger; if 0 < k < 1, the image is smaller.
以某点为中心、缩放因子k进行放大：所有边长乘以k。若 k > 1，图形放大；若 0 < k < 1，图形缩小。
A shape has line symmetry if it can be folded along a line so that both halves match exactly. It has rotational symmetry of order n if it looks the same after a rotation of 360°/n about its centre.
若一个图形沿某条直线对折后两边完全重合，则该图形具有线对称。若绕中心旋转 360°/n 后能与自身重合，则它具有 n 阶旋转对称。
12. Statistics and Probability | 统计与概率
The mean (average) of a set of numbers is the sum divided by the number of values. The median is the middle value when the data are ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.
一组数据的平均数（均值）等于总和除以数据个数。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。极差是最大值与最小值的差。
Σf, where f is the frequency and x is the data value (or class midpoint for grouped data).
在频数表中，均值 = Σ(fx) ÷ Σf，其中f为频数，x为数据值（分组数据时用组中点）。
ourable outcomes) ÷ (total number of possible outcomes), provided all outcomes are equally likely.
概率用分数、小数或百分数表示。事件发生的概率 = (有利结果的个数) ÷ (所有可能结果的个数)，前提是每个结果等可能。
P(A) + P(B). The sum of probabilities of all possible outcomes is 1. P(not A) = 1 − P(A).
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。P(非 A) = 1 − P(A)。
A sample space diagram or a tree diagram can be used to list all possible outcomes for combined events. Relative frequency can be used to estimate probability from experimental data.
样本空间图或树状图可用于列出复合事件的所有可能结果。相对频率可用于根据实验数据估计概率。
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