Year 8 Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | Year 8 Edexcel 进阶数学：公式定理速查手册

📚 Year 8 Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | Year 8 Edexcel 进阶数学：公式定理速查手册

Welcome to your Year 8 Edexcel Further Maths quick reference guide. This handbook compiles the essential formulas, theorems and key facts you need for exam success. Each section is designed for rapid revision, with clear paired explanations in English and Chinese.

欢迎使用 Year 8 Edexcel 进阶数学速查手册。本手册汇集了考试所需的核心公式、定理及关键知识点，每节均以中英双语对照编排，便于快速复习与查漏补缺。

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1. Algebraic Expressions and Simplification | 代数表达式与化简

a(b + c) = ab + ac. Expanding brackets often uses this rule, and factorising reverses it.

通过合并同类项化简表达式。例如 3x + 2y − x + 5y = 2x + 7y。分配律为 a(b + c) = ab + ac，去括号时使用该规则，因式分解则为其逆运算。

+ 2ax + a²; (x − a)² = x² − 2ax + a². Difference of two squares: x² − a² = (x + a)(x − a).

两项式相乘时使用 FOIL 法则：(x + a)(x + b) = x² + (a+b)x + ab。完全平方：(x + a)² = x² + 2ax + a²；(x − a)² = x² − 2ax + a²。平方差公式：x² − a² = (x + a)(x − a)。

(2)² + 3(2) − 1 = 8 + 6 − 1 = 13. Always follow the order of operations (BIDMAS / PEMDAS).

代入法指用数值替换字母。若 y = 2x² + 3x − 1，当 x = 2 时，y = 2(2)² + 3×2 − 1 = 8 + 6 − 1 = 13。务必遵循运算顺序（括号、指数、乘除、加减）。

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2. Solving Linear Equations | 解一元一次方程

: x = 7. For 3x = 18, divide by 3: x = 6. Always check your solution by substituting back.

解方程时，等式两边执行相同运算。x + 5 = 12，两边减5得 x = 7。3x = 18，两边除以3得 x = 6。始终将解代回原方程进行检验。

3 → x = 4. With brackets: expand first, e.g. 2(x − 1) = 8 → 2x − 2 = 8 → 2x = 10 → x = 5.

含变量的方程在两边时：2x + 3 = x + 7 → 2x − x = 7 − 3 → x = 4。有括号时先去括号，如 2(x − 1) = 8 → 2x − 2 = 8 → 2x = 10 → x = 5。

→ multiply by 3: x + 6 = 15 → x = 9. For 2/(x+1) = 4, cross-multiply: 2 = 4(x+1) → x = −½.

含分数方程：各项乘以最简公分母。例 x/3 + 2 = 5，乘3得 x + 6 = 15 → x = 9。对于 2/(x+1) = 4，交叉相乘得 2 = 4(x+1) → x = −½。

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3. Sequences and the nth Term | 数列与第n项

a + (n − 1)d, where a is the first term. Alternatively, find the rule in the form an + b.

等差数列具有公差 d。线性数列的第 n 项公式为 T(n) = a + (n−1)d，其中 a 为首项。也可写为 an + b 形式。

y the difference coefficient of n and adjust for the zero term. For example, 5, 8, 11, 14, … has difference 3, so nth term = 3n + 2. Check: n=1 gives 5, n=2 gives 8.

求第 n 项时，先找出 n 的系数（公差），再确定常数项。如 5, 8, 11, 14, … 公差为3，第 n 项为 3n + 2。验证：n=1 得5，n=2 得8。

Special sequences: square numbers 1, 4, 9, 16, … (n²); cube numbers 1, 8, 27, 64, … (n³); triangular numbers 1, 3, 6, 10, … (½n(n+1)). Fibonacci sequence: 1, 1, 2, 3, 5, 8, … each term is sum of two preceding terms.

特殊数列：平方数 1, 4, 9, 16, … (n²)；立方数 1, 8, 27, 64, … (n³)；三角形数 1, 3, 6, 10, … (½n(n+1))。斐波那契数列：1, 1, 2, 3, 5, 8, … 每一项是前两项之和。

Term-to-term rules: 'start at 3, then add 5 each time' generates 3, 8, 13, 18, … Use position-to-term rules to jump directly to any term without listing all previous ones.

递推规则：如“从3开始，每次加5”生成 3, 8, 13, 18, …。位置规则可直接计算任意项，无需列出前面各项。

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4. Fractions, Decimals and Percentages | 分数、小数与百分数

denominators: ⅔ × ¾ = 6/12 = ½. Divide: multiply by the reciprocal: ⅔ ÷ ¾ = ⅔ × 4/3 = 8/9.

分数加减需通分：½ + ⅓ = 3/6 + 2/6 = 5/6。分数相乘：分子乘分子，分母乘分母：⅔ × ¾ = 6/12 = ½。除法：乘以倒数：⅔ ÷ ¾ = ⅔ × 4/3 = 8/9。

ion: ⅜ = 3 ÷ 8 = 0.375. Convert decimals to percentages by multiplying by 100: 0.375 × 100% = 37.5%. Percentage of amount: 20% of 45 = 0.2 × 45 = 9.

分数化小数用除法：⅜ = 3 ÷ 8 = 0.375。小数化百分数乘100：0.375 × 100% = 37.5%。求一个数的百分比：45的20%为 0.2 × 45 = 9。

Percentage increase: new value = original × (1 + r). e.g. 15% increase on £80 = 80 × 1.15 = £ riginal × (1 − r). Repeated percentage change: multiply by the multiplier for each period.

百分比增加：新值 = 原值 × (1 + r)。如 £80 增加 15% 得 80 × 1.15 = £92。百分比减少：新值 = 原值 × (1 − r)。连续百分比变化每次乘以对应的乘数。

Express one quantity as a percentage of another: (part/whole) × 100%. Example: 18 out of 60 is (18/60)×100% = 30%. Reverse percentages: if 70% is £35, the original is 35 ÷ 0.7 = £50.

一个量占另一个的百分比：(部分/整体) × 100%。如 60 中的 18 为 (18/60)×100% = 30%。逆向百分数：若 70% 为 £35，原值为 35 ÷ 0.7 = £50。

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5. Ratio, Proportion and Rates | 比、比例与速率

ts. Simplify by dividing by HCF. Example: 12:16 = 3:4. Divide a quantity in a given ratio: £50 in 2:3 gives 2 parts + 3 parts = 5 parts, one part £10, so £20 and £30.

比用于比较相同单位的量，除以最大公因数化简。12:16 = 3:4。按比例分配：将 £50 按 2:3 分配，总份数5份，每份 £10，得 £20 和 £30。

bles, y halves. Best buy problems use unit pricing: cost per unit = total cost / quantity.

正比例：y = kx。若 x=3 时 y=12，则 k=4，故 y=4x。反比例：y = k/x，x 加倍则 y 减半。最佳购买问题使用单价：单价 = 总价 ÷ 数量。

, kg/m³). Pressure = force ÷ area (N/m²). Ensure units are consistent before substituting.

速率：速度 = 距离 ÷ 时间，常用单位 m/s 或 km/h。密度 = 质量 ÷ 体积（g/cm³, kg/m³）。压强 = 力 ÷ 面积（N/m²）。代入前确保单位统一。

= 2.54x cm. Map scales: 1:25000 means 1 cm on map represents 25000 cm (250 m) in reality.

换算图与比例因子：使用归一法。如 1 英寸 = 2.54 厘米，则 x 英寸 = 2.54x 厘米。地图比例尺 1:25000 表示图上 1 cm 代表实际 25000 cm (250 m)。

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6. Pythagoras’ Theorem | 勾股定理

Pythagoras' ls the sum of the squares of the other two sides: c² = a² + b², where c is the hypotenuse.

勾股定理适用于直角三角形。斜边（最长边）的平方等于两直角边的平方和：c² = a² + b²，其中 c 为斜边。

To find the hypotenuse: c = √a² + b². To find a shorter side: a = √c² − b². Always identify the hypotenuse first and label sides accurately.

求斜边：c = √a² + b²。求一直角边：a = √c² − b²。解题时先正确标注斜边及各边。

Pythagorean triples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17) etc. Any multiple, e.g. (6, 8, 10), also works. Use these to spot right angles quickly.

勾股数：(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17) 等。其倍数如 (6, 8, 10) 同样成立，可用于快速判断直角。

gle, or distance between two points on a coordinate grid using d = √((x₂−x₁)² + (y₂−y₁)²).

应用：求矩形对角线、等腰三角形的高，或坐标网格上两点距离 d = √((x₂−x₁)² + (y₂−y₁)²)。

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7. Area and Perimeter of 2D Shapes | 平面图形面积与周长

Shape	Perimeter	Area
Rectangle	2(l + w)	l × w
Triangle	a + b + c	½ × base × height
Parallelogram	2(a + b)	base × perpendicular height
Trapezium	Sum of sides	½(a + b) × h
Circle	2πr (circumference)	πr²

记忆公式：矩形面积 = 长 × 宽，三角形面积 = ½ × 底 × 高。平行四边形面积 = 底 × 垂直高。梯形面积 = ½(上底+下底) × 高。圆的周长 C = 2πr，面积 A = πr²。

Units: perimeter in linear units (cm, m), area in square units (cm², m²). Always use the perpendicular height, not slant height. For compound shapes, split into simpler shapes, calculate individually and sum.

单位：周长用长度单位 (cm, m)，面积用平方单位 (cm², m²)。务必使用垂直高而非斜高。组合图形可拆分为简单图形，分别计算后求和。

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8. Volume and Surface Area of 3D Shapes | 立体图形体积与表面积

Solid	Volume	Surface Area
Cuboid (l×w×h)	lwh	2(lw + lh + wh)
Prism	Area of cross-section × length	Sum of all face areas
Cylinder (r, h)	πr²h	2πrh + 2πr²
Sphere (r)	4/3 πr³	4πr²
Cone (r, h, slant l)	⅓ πr²h	πrl + πr²

体积单位 cm³, m³；表面积单位 cm², m²。长方体体积 V = lwh，表面积 = 2(lw + lh + wh)。棱柱体积 = 底面积 × 高。圆柱体积 V = πr²h，曲面面积 = 2πrh，含两底总面积 = 2πrh + 2πr²。

lume = 4/3 πr³, surface area = 4πr². Learn these formulas thoroughly as they appear often.

棱锥体积 = ⅓ × 底面积 × 高。圆锥体积 = ⅓ πr²h。球体积 = 4/3 πr³，表面积 = 4πr²。这些公式需牢记，考试中频繁出现。

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9. Indices and Standard Form | 指数与标准型

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 (a≠0); a⁻ⁿ = 1/aⁿ; a^1/n = ⁿ√a; a^m/n = (ⁿ√a)ᵐ.

指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1 (a≠0)；a⁻ⁿ = 1/aⁿ；a^1/n = ⁿ√a；a^m/n = (ⁿ√a)ᵐ。

ers have positive n: 3400 = 3.4 × 10³. Small numbers have negative n: 0.0047 = 4.7 × 10⁻³.

标准型为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。大数 n 为正：3400 = 3.4 × 10³。小数 n 为负：0.0047 = 4.7 × 10⁻³。

= 6 × 10⁷. To divide: divide A and subtract exponents. Adjust A to stay between 1 and 10.

标准型乘法：A 相乘，指数相加。(3 × 10⁴) × (2 × 10³) = 6 × 10⁷。除法：A 相除，指数相减。注意调整 A 使其保持在 1 到 10 之间。

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10. Introduction to Trigonometry | 三角学入门

TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

在直角三角形中，边比定义了正弦、余弦和正切。SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

= 0.5, so opposite = 10 × 0.5 = 5 cm. Use inverse trig to find angles: θ = sin⁻¹(opp/hyp).

已知一角一边，求其他边。例：斜边 10 cm，角 30°，sin30° = 0.5，对边 = 10 × 0.5 = 5 cm。用反三角函数求角：θ = sin⁻¹(对边/斜边)。

Exact trig values for key angles: sin 0°=0, 30°=½, 45°=√2/2, 60°=√3/2, 90°=1; cos 0°=1, 30°=√3/2, 45°=√2/2, 60°=½, 90°=0; tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90° undefined.

特殊角的精确值：sin 0°=0, 30°=½, 45°=√2/2, 60°=√3/2, 90°=1；cos 0°=1, 30°=√3/2, 45°=√2/2, 60°=½, 90°=0；tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90° 无定义。

Right-angled triangle trigonometry applies to problems involving heights, distances, bearings and angles of elevation/depression. Always draw a diagram and label opposite, adjacent, hypotenuse.

直角三角形三角学用于解决高度、距离、方位角及仰角/俯角问题。始终画出示意图并标注对边、邻边和斜边。

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11. Statistical Measures and Charts | 统计度量与图表

dian is mean of two middle values. Mode is most frequent value. Range = maximum − minimum.

平均数 = 数据总和 ÷ 数据个数。中位数是排序后的中间值；偶数个时取中间两数的平均。众数是出现频率最高的值。极差 = 最大值 − 最小值。

/ Σf. Median interval found from cumulative frequency. Modal class has highest frequency.

分组数据，用组中值估计平均数。频数表：平均数 ≈ Σ(频数 × 组中值) / 总频数。中位数区间由累计频数确定。众数区间频数最高。

egories. Histograms for continuous data with equal class widths: frequency proportional to area (height).

饼图：扇区角度 = (频数/总数) × 360°。条形图用于离散类别。直方图用于等宽连续数据：频数与面积（高度）成正比。

Scatter graphs show relationship (correlation). Line of best fit can be used to make predictions. Positive correlation: as one variable increases, the other tends to increase. Negative: one increases, the other decreases.

散点图显示相关关系。最佳拟合线可用于预测。正相关：一变量增大，另一变量也趋于增大。负相关：一变量增大，另一变量减小。

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12. Probability Fundamentals | 概率基础

l number of equally likely outcomes. 0 ≤ P ≤ 1. Impossible event: P=0; certain event: P=1.

事件概率 = 有利结果数 / 等可能结果总数。0 ≤ P ≤ 1。不可能事件 P=0；必然事件 P=1。

. For non-mutually exclusive, subtract intersection: P(A or B) = P(A) + P(B) − P(A and B).

互补规则：P(非 A) = 1 − P(A)。互斥事件不能同时发生，P(A 或 B) = P(A) + P(B)。非互斥时需减去交集：P(A 或 B) = P(A) + P(B) − P(A 且 B)。

babilities along branches of a tree diagram. Independent events: P(A and B) = P(A) × P(B).

样本空间图列出所有结果。树状图的分支适用于两事件，概率相乘。独立事件：P(A 且 B) = P(A) × P(B)。

curs / total trials; it stabilises as trials increase, providing experimental probability.

期望频数 = 概率 × 试验次数。相对频率 = 事件发生次数 / 总试验次数；随着试验增加趋于稳定，可作为实验概率。

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