📚 Advanced Mathematics for KS3: Formula Handbook | KS3 进阶数学：公式汇总手册
Welcome to your go-to formula handbook for KS3 Advanced Mathematics. This guide covers crucial formulas across algebra, geometry, trigonometry, statistics and more. Each section presents key relationships in a clear bilingual format, helping you master the foundations for GCSE and beyond.
欢迎使用 KS3 进阶数学公式手册，这里汇总了代数、几何、三角、统计等核心领域的必备公式。每个小节都以清晰的中英对照呈现关键关系，助你打牢基础，自信迈向 GCSE 及更高阶段。
1. Expanding and Factorising | 展开与因式分解
The distributive law and special products allow us to rewrite expressions in expanded or factorised forms, essential for solving equations.
分配律与特殊乘积让我们能将代数式写成展开形式或分解形式，这是解方程的关键技能。
The distributive law states: a(b + c) = ab + ac
a(b + c) = ab + ac
分配律为：a(b + c) = ab + ac
The square of a binomial gives a perfect square trinomial.
(a + b)² = a² + 2ab + b²
二项式的平方得到一个完全平方三项式。
Similarly, the difference of two squares factorises neatly.
a² – b² = (a + b)(a – b)
类似地，平方差可进行整齐的因式分解。
You can also factorise by grouping when four terms share a common structure.
ax + ay + bx + by = (a + b)(x + y)
当四项具有相同结构时，也可以通过分组进行因式分解。
2. Solving Equations | 解方程
To solve linear equations, isolate the variable using inverse operations. Always keep the equation balanced.
解一元一次方程时，利用逆运算分离变量，始终保持等式平衡。
For equations of the form ax + b = c, subtract b then divide by a.
x = (c – b) / a
形如 ax + b = c 的方程，先减 b 再除以 a。
When variables appear on both sides, collect like terms on one side first.
2x + 3 = x + 7 → x = 4
当变量出现在等号两边时，先将含变量的项移到一侧合并。
To solve two-step equations, undo addition/subtraction before multiplication/division.
5x – 2 = 13 → x = 3
解两步方程时，先处理加减法，再处理乘除法。
3. Laws of Indices | 指数定律
Indices (exponents) follow strict rules when multiplying, dividing or raising powers. Mastering these simplifies algebraic manipulation.
指数在用乘法、除法及乘方时遵循严格的规则，掌握它们能简化代数运算。
When multiplying powers with the same base, add the exponents.
am × an = am + n
同底数幂相乘时，指数相加。
When dividing, subtract the exponents.
am ÷ an = am – n
同底数幂相除时，指数相减。
Raising a power to another power multiplies the exponents.
(am)n = amn
幂的乘方，指数相乘。
A negative exponent means reciprocal.
a-n = 1 / an
负指数表示倒数。
Any non-zero number raised to the power of zero equals 1.
a0 = 1 (a ≠ 0)
任何非零数的零次幂等于 1。
Fractional indices correspond to roots: the denominator is the root index.
a1/n = √[n]{a}, am/n = (√[n]{a})m
分数指数表示根式：分母为根指数。
4. Sequences and Series | 数列与级数
Linear sequences follow a constant difference between terms. The nth term formula helps find any term directly.
线性数列的相邻两项之差为常数；第 n 项公式可直接求出任意项。
For an arithmetic sequence with first term a and common difference d:
nth term = a + (n – 1)d
首项为 a、公差为 d 的等差数列，第 n 项为：a + (n – 1)d
To identify a linear pattern, examine differences: 2, 5, 8, 11 → d=3, nth term = 3n – 1.
要识别线性规律，查看差值：2, 5, 8, 11 → 公差为3，第 n 项为 3n – 1。
Quadratic sequences have a constant second difference; their nth term is of the form an² + bn + c.
二次数列的二次差为常数，其第 n 项公式为 an² + bn + c。
Special sequences like triangular numbers and square numbers follow known patterns.
Triangular numbers: n(n+1)/2; Square numbers: n²
三角形数、平方数等特殊数列也有固定公式。
5. Straight Line Graphs | 直线与坐标系
The equation of a straight line can be expressed in different forms; the gradient and y-intercept reveal its behaviour.
直线方程可以有多种形式；斜率和 y 截距决定了直线的走向。
Slope (gradient) is calculated as rise over run between two points.
m = (y₂ – y₁) / (x₂ – x₁)
斜率（梯度）通过两点间的垂直变化除以水平变化计算。
The slope-intercept form makes plotting easy.
y = mx + c (m = gradient, c = y-intercept)
斜截式便于作图。
A horizontal line has gradient zero; a vertical line has undefined gradient.
Horizontal: y = k; Vertical: x = h
水平线斜率为零，垂直线斜率不存在。
Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = -1.
Parallel: m₁ = m₂; Perpendicular: m₁m₂ = -1
平行线斜率相等，垂直线斜率乘积为 -1。
6. Area, Perimeter and Volume | 面积、周长与体积
Memorising these geometric formulas is vital for solving problems involving shapes and solids.
牢记以下几何公式对于解决平面图形和立体图形问题至关重要。
Rectangles, triangles, parallelograms and trapeziums have key area formulas.
Rectangle: A = l × w
矩形面积 = 长 × 宽。
Triangle: A = ½ × base × height
三角形面积 = ½ × 底 × 高。
Parallelogram: A = base × perpendicular height
平行四边形面积 = 底 × 垂高。
Trapezium: A = ½ × (a + b) × h, where a, b are parallel sides.
梯形面积 = ½ × (上底 + 下底) × 高。
The circumference and area of a circle use π.
Circumference: C = 2πr or πd
圆周长 = 2πr 或 πd。
Area of circle: A = πr²
圆面积 = πr²。
For common 3D shapes:
常见立体图形的体积和表面积：
Cube: V = s³; Cuboid: V = l × w × h
正方体：V = 边长³；长方体：V = 长 × 宽 × 高。
Prism: V = area of cross-section × length
棱柱：V = 横截面面积 × 长度。
Cylinder: V = πr²h; Curved surface area = 2πrh
圆柱：V = πr²h；侧面积 = 2πrh。
7. Pythagoras’ Theorem | 毕达哥拉斯定理
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
在任何直角三角形中，斜边的平方等于两条直角边的平方和。
c² = a² + b² (where c is the hypotenuse)
c² = a² + b²（c 为斜边）。
To find a shorter side, rearrange: a² = c² – b².
需要求直角边时，移项得 a² = c² – b²。
The converse states that if a triangle satisfies a² + b² = c², it is right-angled.
逆定理表示，若三角形的三边满足 a² + b² = c²，则该三角形是直角三角形。
Applications include finding distances on coordinate grids, heights and diagonals.
该定理可应用于坐标系中求距离、求高度和对角线长度等。
8. Introduction to Trigonometry | 三角比初步
Trigonometric ratios relate the angles of a right triangle to ratios of its sides. SOH CAH TOA is the essential memory aid.
三角函数比将直角三角形中的角度与边长的比联系起来，SOH CAH TOA 是必须掌握的记忆法。
sin θ = opposite / hypotenuse
正弦 = 对边 / 斜边。
cos θ = adjacent / hypotenuse
余弦 = 邻边 / 斜边。
tan θ = opposite / adjacent
正切 = 对边 / 邻边。
To find a missing side, multiply the known side by the appropriate trigonometric ratio.
求未知边时，用已知边乘以相应的三角函数值。
To find an angle, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹.
求角度时使用反三角函数。
9. Ratio, Proportion and Percentages | 比例与百分比
Ratios compare quantities; proportions describe equal ratios. Percentages express a number as a fraction of 100.
比用于比较数量；比例描述相等比；百分数表示以 100 为分母的分数。
To share a quantity in a given ratio, first find the total number of parts.
For ratio a:b:c, share amount N as: a/(a+b+c) × N, etc.
按给定比例分配时，先求总份数，再计算各部分。
Percentage increase or decrease follows a simple formula.
% change = (difference ÷ original) × 100
百分比变化 = (变化量 ÷ 原值) × 100。
To calculate the new amount after a percentage change, multiply by the multiplier.
Increase by 15%: multiply by 1.15; decrease by 20%: multiply by 0.80
增加 15% 乘以 1.15，减少 20% 乘以 0.80。
Direct proportion: y = kx; Inverse proportion: y = k/x.
Direct: y ∝ x; Inverse: y ∝ 1/x
正比例：y = kx；反比例：y = k/x。
10. Statistics and Averages | 统计与平均数
Data sets can be summarised using measures of central tendency and spread. Knowing when to use each is key.
用集中趋势和离散程度的度量可以概括数据，知道何时使用哪一种很关键。
The mean is the sum of values divided by the number of values.
Mean = Σx / n
平均数 = 总和 ÷ 个数。
The median is the middle value when data are ordered; the mode is the most frequent value.
中位数是排序后中间的值；众数是出现次数最多的值。
The range gives a simple measure of spread.
Range = highest value – lowest value
极差 = 最大值 – 最小值。
Frequency tables and grouped data require estimated means using midpoints.
Estimated mean = Σ(f × midpoint) / Σf
分组数据的估算平均数 = Σ(频数 × 组中值) / 总频数。
11. Quadratic Equations | 二次方程
A quadratic equation takes the form ax² + bx + c = 0. Solutions can be found by factorising, using the quadratic formula, or completing the square.
二次方程的一般形式为 ax² + bx + c = 0，可通过因式分解、公式法或配方法求解。
When factorising, look for two numbers that multiply to ac and add to b.
If (x + p)(x + q) = 0, then x = -p or x = -q
因式分解时，寻找乘积为 ac 且和为 b 的两个数。
The quadratic formula works for all quadratics.
x = [ -b ± √(b² – 4ac) ] / 2a
求根公式适用于所有二次方程。
The discriminant (b² – 4ac) determines the nature of roots:
- Positive: two distinct real roots
- Zero: one repeated real root
- Negative: no real roots
判别式 (b² – 4ac) 决定了根的情况：正数有两个不等实根，零有一个重根，负数无实根。
12. Inequalities | 不等式
Inequalities express ranges of values rather than exact solutions. The rules mirror equations but with a crucial flip when multiplying or dividing by a negative.
不等式表示值的取值范围而非精确解，运算规则与方程相似，但乘或除以负数时不等号方向要改变。
Basic inequality symbols: <, >, ≤, ≥.
基础不等号：< (小于), > (大于), ≤ (小于等于), ≥ (大于等于)。
If -x > 3, dividing by -1 flips the sign: x < -3.
If a < b, then -a > -b
若 a < b，则 -a > -b。
Compound inequalities such as -2 < x ≤ 5 show intervals on a number line.
复合不等式如 -2 < x ≤ 5 可在数轴上表示区间。
Graphical inequalities shade regions on a coordinate plane; a dashed line indicates < or >, a solid line ≤ or ≥.
图像不等式在坐标平面上为区域着色；虚线表示严格不等号，实线表示带等号的不等号。
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