📚 KS3 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | KS3 CIE 进阶数学：公式定理速查手册
This compact handbook brings together all the essential formulas and theorems you need to master KS3 CIE Further Mathematics. Every rule is paired with clear explanations in both English and Chinese, helping you revise efficiently and apply concepts confidently in assessments. Keep it handy for quick reference whenever you tackle algebra, geometry, statistics or coordinate work.
这本简明手册汇集了掌握 KS3 CIE 进阶数学所需的所有核心公式和定理。每条法则都配有中英双语清晰解释，帮助你高效复习并在考试中自信运用。随时翻阅，解决代数、几何、统计与坐标相关问题都不再困难。
1. Exponents and Scientific Notation | 指数与科学记数法
Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ
乘积法则：同底数幂相乘，指数相加。
Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
商法则：同底数幂相除，指数相减。
Power of a power: (aᵐ)ⁿ = aᵐⁿ
幂的乘方：底数不变，指数相乘。
Power of a product: (ab)ⁿ = aⁿbⁿ
积的乘方：每个因式分别乘方。
Zero exponent: a⁰ = 1 (a ≠ 0)
零指数：任何非零数的 0 次方都等于 1。
Negative exponent: a⁻ⁿ = 1 / aⁿ
负指数：a⁻ⁿ 等于 aⁿ 的倒数。
Roots as exponents: √a = a1/2, ∛a = a1/3
方根与分数指数：平方根相当于 1/2 次方，立方根相当于 1/3 次方。
Scientific notation: N = a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
科学记数法：N = a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。
2. Algebraic Expressions and Factorisation | 代数表达式与因式分解
Expanding brackets: (a+b)(c+d) = ac+ad+bc+bd
去括号展开：逐一相乘再合并。
Perfect squares: (a+b)² = a² + 2ab + b² ; (a−b)² = a² − 2ab + b²
完全平方公式：两数和（差）的平方展开式。
Difference of two squares: a² − b² = (a+b)(a−b)
平方差公式：两项平方的差等于和与差的乘积。
Factorising quadratics: x² + bx + c = (x+p)(x+q) where p+q=b and pq=c.
二次三项式因式分解：寻找两数 p 和 q，使 p+q=b 且 pq=c，则 x²+bx+c=(x+p)(x+q)。
Highest common factor (HCF): Always take out the greatest common factor first.
提取公因式：因式分解时优先提取最大公因式。
3. Linear Equations | 线性方程
General form: ax + b = c. Solve by performing the same operation on both sides.
一般形式：ax + b = c。两边同时进行相同运算以求解。
Equations with brackets: Expand first, then simplify and isolate the variable.
含括号的方程：先去括号，化简后再移项求未知数。
Equations with variables on both sides: Collect like terms to one side using addition or subtraction.
两边均有未知数：通过加减法把含未知数的项集中到同一边。
Checking solution: Substitute the value back into the original equation.
验根：将解代回原方程检验等式是否成立。
4. Inequalities | 不等式
Solving linear inequalities: Treat like equations, but reverse the inequality sign when multiplying or dividing by a negative number.
解线性不等式：与解方程类似，但当两边同乘或除以负数时，不等号必须反向。
Representing on a number line: Use open circle for < or >, closed circle for ≤ or ≥.
数轴表示：用空心圆圈表示 < 或 >，实心圆点表示 ≤ 或 ≥。
Compound inequalities: a < x < b means x is greater than a and less than b.
复合不等式：a < x < b 表示 x 同时大于 a 且小于 b。
5. Sequences | 数列
Term-to-term rule: Describes how to get from one term to the next.
递推关系：描述如何从一项得到下一项的法则。
Position-to-term rule (nth term): Formula that gives the term directly from its position n.
通项公式（第 n 项）：直接根据位置 n 给出该项值的公式。
Arithmetic sequence: T(n) = a + (n−1)d, where a is the first term and d is the common difference.
等差数列：T(n) = a + (n−1)d，a 为首项，d 为公差。
Special sequences: Square numbers: T(n)=n²; Triangular numbers: T(n)=n(n+1)/2.
特殊数列：平方数 T(n)=n²；三角形数 T(n)=n(n+1)/2。
6. Angles and Straight Lines | 角与直线
Angles on a straight line add up to 180°.
平角：直线上相邻角之和为 180°。
Angles around a point sum to 360°.
周角：绕一点的所有角之和为 360°。
Vertically opposite angles are equal.
对顶角相等。
Parallel lines: Corresponding angles are equal; alternate angles are equal; co-interior (allied) angles sum to 180°.
平行线性质：同位角相等；内错角相等；同旁内角互补（和为 180°）。
Angles in a triangle: a + b + c = 180°
三角形内角和：a + b + c = 180°
7. Triangles and Pythagoras' Theorem | 三角形与毕达哥拉斯定理
Pythagoras' theorem: In a right-angled triangle with hypotenuse c, c² = a² + b².
毕达哥拉斯定理：直角三角形中，斜边为 c，则 c² = a² + b²。
Finding the hypotenuse: c = √a² + b².
求斜边：c = √a² + b²。
Finding a shorter side: a = √c² − b².
求直角边：a = √c² − b²。
Converse: If a² + b² = c² holds for three sides of a triangle, the triangle is right-angled.
逆定理：若三角形三边满足 a² + b² = c²，则该三角形为直角三角形。
Common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17) and their multiples.
常见勾股数组：(3,4,5), (5,12,13), (8,15,17) 及其倍数。
8. Perimeter, Area and Volume | 周长、面积与体积
| Shape / 图形 | Perimeter / 周长 | Area / 面积 |
|---|---|---|
| Rectangle / 矩形 (l, w) | 2(l + w) | l × w |
| Square / 正方形 (s) | 4s | s² |
| Triangle / 三角形 (b, h) | Sum of 3 sides | ½ × b × h |
| Parallelogram / 平行四边形 (b, h) | 2(a + b) | b × h |
| Trapezium / 梯形 (a, b, h) | Sum of 4 sides | ½ (a + b)h |
| Circle / 圆 (r) | 2πr (Circumference) | πr² |
Volume of a prism: V = area of cross-section × length (or height).
棱柱体积：V = 底面积 × 高。
Cube: V = s³; Cuboid: V = l × w × h; Cylinder: V = πr²h.
立方体：V = s³；长方体：V = lwh；圆柱：V = πr²h。
Surface area of a cylinder: 2πr² + 2πrh.
圆柱表面积：2πr² + 2πrh。
9. Ratio and Proportion | 比与比例
Simplifying ratios: Divide all parts by their highest common factor.
化简比：将所有部分同时除以最大公因数。
Sharing in a given ratio: Total parts = sum of ratio parts; one part = total ÷ sum of parts.
按比例分配：总份数 = 比中各项之和；一份的数量 = 总量 ÷ 总份数。
Direct proportion: y ∝ x means y = kx, where k is the constant of proportionality.
正比例：y ∝ x 即 y = kx，k 为比例常数。
Inverse proportion: y ∝ 1/x means y = k/x.
反比例：y ∝ 1/x 即 y = k/x。
10. Percentages | 百分比
Percentage change = (change ÷ original) × 100%
百分数变化：(变化量 ÷ 原值) × 100%
Increase by r%: New value = original × (1 + r/100).
增加 r%：新值 = 原值 × (1 + r/100)。
Decrease by r%: New value = original × (1 − r/100).
减少 r%：新值 = 原值 × (1 − r/100)。
Finding the original after a percentage change: Original = new value ÷ multiplier.
求原值：原值 = 新值 ÷ 乘数。
Reverse percentage: For example, if 120 is 80% of the original, original = 120 ÷ 0.8 = 150.
逆推百分数：例如 120 是原值的 80%，则原值 = 120 ÷ 0.8 = 150。
11. Statistics and Probability | 统计与概率
Mean: sum of all values ÷ number of values.
平均数：所有数据之和除以数据个数。
Median: middle value when data are ordered.
中位数：将数据排序后正中间的值。
Mode: the value that appears most often.
众数：出现次数最多的数据值。
Range: largest value − smallest value.
极差（范围）：最大值减最小值。
obability of an event = number of favourable outcomes / total number of possible outcomes.
事件概率：成功结果数 ÷ 所有可能结果总数。
Probability scale: from 0 (impossible) to 1 (certain).
概率标度：0 表示不可能，1 表示必然。
Expected frequency: probability × number of trials.
期望次数：概率 × 试验次数。
Tree diagrams: Multiply along branches for 'and' probabilities; add different branches for 'or' probabilities.
树状图：相乘分支概率表示“且”；相加不同分支概率表示“或”。
12. Coordinate Geometry | 坐标几何
Midpoint M of (x₁, y₁) and (x₂, y₂): M = ((x₁+x₂)/2 , (y₁+y₂)/2)
中点 M：M = ((x₁+x₂)/2 , (y₁+y₂)/2)
Gradient (slope) m: m = (y₂ − y₁) / (x₂ − x₁)
斜率 m：m = (y₂ − y₁) / (x₂ − x₁)
Equation of a straight line: y = mx + c, where m is gradient and c is the y-intercept.
直线方程：y = mx + c，m 为斜率，c 为 y 轴截距。
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Parallel lines have equal gradients: m₁ = m₂.
平行直线斜率相等：m₁ = m₂。
Perpendicular lines: product of gradients is −1: m₁ × m₂ = −1.
互相垂直直线：斜率乘积为 −1：m₁ × m₂ = −1。
Published by TutorHao | Further 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