📚 Year 8 CAIE Mathematics: Formulas & Theorems Quick Reference | Year 8 CAIE 数学：公式定理速查手册
Welcome to your handy quick reference guide for Year 8 CAIE Mathematics. This handbook compiles all the essential formulas, definitions, and theorems you need to master the Stage 8 curriculum. Use it to revise systematically and boost your confidence in solving problems.
欢迎使用 Year 8 CAIE 数学速查手册。本手册汇集了 Stage 8 课程中所有关键的公式、定义和定理，帮助你系统复习，提升解题信心。
1. Number Essentials: Factors, Multiples, Primes and Roots | 数论基础：因数、倍数、质数与方根
A prime number has exactly two distinct factors: 1 and itself. Composite numbers have more than two factors.
质数恰好有两个不同的因数：1 和它本身。合数有两个以上的因数。
The highest common factor (HCF) is the largest whole number that divides two or more numbers exactly. The lowest common multiple (LCM) is the smallest multiple shared by the numbers.
最大公因数 (HCF) 是能同时整除两个或多个数的最大正整数。最小公倍数 (LCM) 是这些数共有的最小倍数。
ultiplying an integer by itself, written as n². Cube numbers are n × n × n, written as n³.
平方数是整数自乘的结果，记作 n²。立方数是 n×n×n，记作 n³。
√a × √a = a, ∛a × ∛a × ∛a = a
- Prime factorisation: every composite number can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic).
- 质因数分解：每个合数都能唯一地表示为质数的乘积（算术基本定理）。
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 convert a decimal to a percentage, multiply by 100%.
分数化为小数：分子除以分母。小数化为百分数：乘以 100%。
Fraction → Decimal: ¾ = 3 ÷ 4 = 0.75
Decimal → Percentage: 0.75 × 100% = 75%
Adding or subtracting fractions requires a common denominator. Multiply fractions by multiplying numerators and denominators separately.
分数加减需要先通分。分数相乘时，分子乘分子，分母乘分母。
a/b + c/d = (ad + bc)/bd
a/b × c/d = ac/bd
Percentage change is calculated as (change ÷ original value) × 100%.
百分比变化 = (变化量 ÷ 原始值) × 100%。
- To find a percentage of an amount, multiply by the percentage and divide by 100.
- 求一个数的百分之几，乘以百分数再除以 100。
3. Ratios and Proportions | 比与比例
A ratio compares two or more quantities in the same units. It can be simplified by dividing all parts by their highest common factor.
比用于比较两个或多个同单位的量。可通过同时除以它们的最大公因数来化简比。
When sharing an amount in a given ratio, first find the total number of parts, then work out the value of one part.
按比例分配时，先求出总份数，再算出一份的量。
Divide £50 in ratio 2:3 → total parts = 5 → one part = £10 → amounts = £20 and £30.
Equivalent ratios have the same simplest form. They can be scaled up or down by multiplying or dividing both sides by the same number.
相等的比具有相同的最简形式。可以通过在两边同时乘以或除以相同的数来扩大或缩小比。
4. Algebraic Expressions and Equations | 代数表达式与方程
Expanding brackets uses the distributive law: multiply each term inside the bracket by the term outside.
展开括号使用分配律：用括号外的项乘以括号内的每一项。
a(b + c) = ab + ac
Like terms have exactly the same variable parts and can be collected by adding or subtracting their coefficients.
同类项具有完全相同的变量部分，可以通过加减它们的系数来合并。
To solve a linear equation, use inverse operations to isolate the variable. Always keep the equation balanced by doing the same to both sides.
解一元一次方程时，用逆运算分离变量。始终保持方程平衡，两边同时进行相同操作。
Solve 2x + 3 = 11 → subtract 3: 2x = 8 → divide by 2: x = 4
For inequalities, the solution is a range. Remember: multiplying or dividing by a negative number reverses the inequality sign.
不等式的解是一个范围。记住：乘以或除以负数时，不等号方向要改变。
5. Sequences and the nth Term | 数列与第n项
An arithmetic sequence has a constant difference between consecutive terms. The nth term formula gives any term's value directly.
等差数列中相邻两项的差为常数。通项公式（第n项公式）可直接求出任意一项的值。
nth term = a + (n – 1)d or dn + (a – d)
where a is the first term and d is the common difference.
其中 a 为首项，d 为公差。
Example: sequence 5, 8, 11, 14, ... has d = 3, a = 5, so nth term = 3n + 2.
示例：数列 5, 8, 11, 14, ... 的公差 d = 3，首项 a = 5，所以第 n 项 = 3n + 2。
Other sequences include square numbers (1, 4, 9, 16, ...) and triangular numbers (1, 3, 6, 10, ...).
其他数列还包括平方数 (1, 4, 9, 16, ...) 和三角形数 (1, 3, 6, 10, ...)。
6. Linear Graphs and Coordinates | 线性图像与坐标
Coordinates are written as ordered pairs (x, y), giving horizontal then vertical position on the Cartesian plane.
坐标写作有序数对 (x, y)，表示在笛卡尔平面上先水平后竖直的位置。
t line is usually written as y = mx + c, where m is the gradient and c is the y-intercept.
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
m = rise / run = (y₂ – y₁) / (x₂ – x₁)
The midpoint of two points (x₁, y₁) and (x₂, y₂) is found by averaging the coordinates:
两点 (x₁, y₁) 和 (x₂, y₂) 的中点可通过坐标取平均值得到：
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
To plot a linear graph, create a table of values, plot at least three points, and draw a straight line through them.
绘制一次函数图像时，先列出数值表，描出至少三个点，然后过点画一条直线。
7. Angles and Shapes | 角与图形
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角（直线上的角）和为 180°。周角（绕一点的角度）和为 360°。对顶角相等。
The interior angles of a triangle always total 180°. An isosceles triangle has two equal sides and two equal base angles.
三角形内角和恒为 180°。等腰三角形有两条等边和两个相等的底角。
The sum of interior angles in a quadrilateral is 360°. A parallelogram has opposite sides parallel and equal.
四边形内角和为 360°。平行四边形对边平行且相等。
When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles sum to 180°.
当一条截线截过平行线时：同位角相等，内错角相等，同旁内角互补（和为 180°）。
| Theorem / 定理 | Diagram clue / 图形提示 | Angle relation / 角度关系 |
|---|---|---|
| Corresponding / 同位角 | F-shape | Equal / 相等 |
| Alternate / 内错角 | Z-shape | Equal / 相等 |
| Co‑interior / 同旁内角 | C-shape | Sum to 180° / 和为 180° |
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around a 2D shape. Area measures the surface enclosed, and volume measures the space a 3D object occupies.
周长是围绕二维图形一周的总长度。面积衡量封闭曲面的大小，体积衡量三维物体所占空间。
Rectangle: P = 2(l + w), A = l × w
Triangle: A = ½ b × h
Parallelogram: A = b × h
Trapezium: A = ½ (a + b) h
Circle: C = πd = 2πr, A = πr²
Pythagoras' theorem applies only to right‑angled triangles: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形：斜边的平方等于两直角边的平方和。
c² = a² + b² (c is the hypotenuse / c 为斜边)
d = length × width × height. Surface area = sum of the areas of all six rectangular faces.
长方体体积 = 长 × 宽 × 高。表面积 = 所有六个矩形面的面积之和。
V = lwh, SA = 2(lw + lh + wh)
Volume of a prism = area of cross‑section × length. For a cube of side s, V = s³, SA = 6s².
棱柱体积 = 横截面积 × 长度。边长为 s 的立方体，V = s³，SA = 6s²。
9. Transformations | 变换
Reflection flips a shape over a mirror line. Every point and its image are the same perpendicular distance from the line.
反射将图形沿对称轴翻转。每个点与其像点到对称轴的距离相等且垂直于对称轴。
Rotation turns a shape around a centre by a given angle and direction. The shape stays congruent.
旋转将图形绕一个中心按给定角度和方向转动。形状保持全等。
Translation slides a shape without turning or flipping. It is described by a column vector.
平移不转动、不翻转地滑动图形。它用一个列向量来描述。
Translation vector: (x, y) → (x + a, y + b)
Common reflections on the coordinate plane:
坐标平面上的常见反射：
- Reflection in x‑axis: (x, y) → (x, –y)
- Reflection in y‑axis: (x, y) → (–x, y)
- Reflection in y = x: (x, y) → (y, x)
- 关于 x 轴反射：(x, y) → (x, –y)；关于 y 轴反射：(x, y) → (–x, y)；关于直线 y = x 反射：(x, y) → (y, x)
Enlargement changes the size of a shape by a scale factor from a centre. (Often introduced in Stage 9, but some Stage 8 courses include simple enlargements with positive integer scale factor.)
缩放（放大）以某中心按比例因子改变图形大小。（常在 Stage 9 引入，但部分 Stage 8 课程包含正整数比例因子的简单放大。）
10. Statistics and Probability | 统计与概率
The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value, and the range is the difference between the largest and smallest values.
平均数是所有数值之和除以数值的个数。中位数是数据按顺序排列后的中间值。众数是出现次数最多的值，极差是最大值与最小值之差。
Mean = Σx / n, Range = max – min
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes:
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。对于等可能结果：
P(Event) = Number of favourable outcomes / Total number of outcomes
Experimental probability is based on the results of an experiment or trial. The more trials, the closer it tends to theoretical probability.
实验概率基于试验或实验的结果。试验次数越多，它往往越接近理论概率。
Probability of an event not happening = 1 – P(event). The sum of probabilities of all mutually exclusive outcomes is 1.
事件不发生的概率 = 1 – P(事件)。所有互斥结果的概率之和为 1。
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