Year 8 AQA Further Mathematics: Formula and Theorem Quick Reference Handbook | Year 8 AQA 进阶数学：公式定理速查手册

📚 Year 8 AQA Further Mathematics: Formula and Theorem Quick Reference Handbook | Year 8 AQA 进阶数学：公式定理速查手册

Welcome to your Year 8 AQA Further Mathematics quick reference handbook. This guide collects all the essential formulas, theorems, and rules you need for revision. Keep it handy to boost your confidence and speed in tests.

欢迎使用 Year 8 AQA 进阶数学速查手册。本手册汇集了所有你需要复习的关键公式、定理和规则。把它放在手边，帮助你在测试中提升信心和解题速度。

1. Algebraic Expressions and Equations | 代数表达式与方程

Expanding a single bracket: multiply the term outside by every term inside. a(b + c) = ab + ac

单项式乘括号：将括号外的项乘以括号内的每一项。 a(b + c) = ab + ac

Expanding double brackets: use FOIL (First, Outer, Inner, Last). (x + a)(x + b) = x² + (a + b)x + ab

乘开双括号：使用 FOIL 法则（首项、外项、内项、末项）。 (x + a)(x + b) = x² + (a + b)x + ab

Factorising a quadratic: reverse of expanding. Look for two numbers that multiply to the constant and add to the coefficient of x. x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

二次三项式因式分解：将展开过程逆转。寻找两个数，它们的乘积等于常数项，和等于 x 的系数。 x² + bx + c = (x + p)(x + q)，其中 p + q = b，pq = c。

Solving linear equations: isolate the unknown by performing inverse operations on both sides. ax + b = c → x = (c − b) / a

解一元一次方程：通过在等式两边进行逆运算来隔离未知数。 ax + b = c → x = (c − b) / a

Laws of indices: a^m × aⁿ = a^{m + n}
a^m / aⁿ = a^{m − n}
(a^m)ⁿ = a^{m × n}
a⁰ = 1 (for a ≠ 0)

指数律： a^m × aⁿ = a^{m + n}
a^m / aⁿ = a^{m − n}
(a^m)ⁿ = a^{m × n}
a⁰ = 1 （a ≠ 0）

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2. Linear Inequalities | 线性不等式

Solving inequalities follows the same principles as equations, but reverse the inequality sign when multiplying or dividing by a negative number.

解不等式的方法与解方程相同，但请注意：当不等式两边同时乘以或除以一个负数时，不等号方向要改变。

Example: −3x ≤ 12 → x ≥ −4

例如： −3x ≤ 12 → x ≥ −4

Representing on a number line: use an open circle for < or > and a closed circle for ≤ or ≥.

在数轴上表示：用空心圆表示 < 或 >，用实心圆表示 ≤ 或 ≥。

Compound inequalities: a < x ≤ b means x is greater than a and less than or equal to b.

复合不等式： a < x ≤ b 表示 x 大于 a 且小于或等于 b。

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

An arithmetic sequence has a constant difference d between consecutive terms. The nth term formula is aₙ = a₁ + (n − 1)d where a₁ is the first term.

等差数列中，相邻两项的差 d 相等。第 n 项公式为 aₙ = a₁ + (n − 1)d，其中 a₁ 为首项。

Term-to-term rule: each term is formed by adding or multiplying the previous term. e.g. uₙ₊₁ = uₙ + 3

递推关系：每一项由前一项加上或乘以某个数得到。例如： uₙ₊₁ = uₙ + 3

Geometric sequences (basic): each term is multiplied by a constant r. The nth term is aₙ = a × rⁿ⁻¹.

等比数列（基础）：每一项乘以一个常数 r。第 n 项公式为 aₙ = a × rⁿ⁻¹。

Using the nth term: substitute n to find any term, or find n when given a term value.

使用第 n 项公式：代入 n 可求任意项；已知某项的值，可求出对应的 n。

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4. Pythagoras' Theorem | 毕达哥拉斯定理

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

在任何直角三角形中，斜边的平方等于两条直角边的平方之和。

c² = a² + b²

c² = a² + b²

To find the hypotenuse: c = √a² + b²

求斜边： c = √a² + b²

To find a shorter side: a = √c² − b²

求直角边： a = √c² − b²

Converse of Pythagoras: if a triangle satisfies a² + b² = c², then it is right-angled.

毕达哥拉斯定理的逆定理：若三角形满足 a² + b² = c²，则该三角形是直角三角形。

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5. Perimeter and Area | 周长与面积

Rectangle: Area = l × w, Perimeter = 2(l + w)

矩形： 面积 = 长 × 宽， 周长 = 2(长 + 宽)

Triangle: Area = ½ × b × h, Perimeter = sum of all sides

三角形： 面积 = ½ × 底 × 高， 周长 = 三边之和

Parallelogram: Area = b × h

平行四边形： 面积 = 底 × 高

Trapezium: Area = ½ (a + b)h, where a and b are the parallel sides.

梯形： 面积 = ½ (a + b)h，其中 a、b 为两平行边。

Circle: Circumference = 2πr = πd, Area = πr²

圆： 周长 = 2πr = πd， 面积 = πr²

Compound shapes: split into simpler shapes, calculate each area and sum them.

组合图形：分割成简单图形，分别计算面积后再求和。

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6. Volume and Surface Area | 体积与表面积

Cube: Volume = side³, Surface area = 6 × side²

正方体： 体积 = 边长³， 表面积 = 6 × 边长²

Cuboid (rectangular prism): Volume = l × w × h, Surface area = 2(lw + wh + hl)

长方体： 体积 = 长 × 宽 × 高， 表面积 = 2(长×宽 + 宽×高 + 高×长)

Prism: Volume = area of cross-section × length

棱柱（角柱）： 体积 = 横截面积 × 长度

Cylinder: Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πr(r + h)

圆柱： 体积 = πr²h， 侧面积 = 2πrh， 总表面积 = 2πr(r + h)

Units: 1 litre = 1000 cm³, 1 m³ = 1,000,000 cm³.

单位换算：1 升 = 1000 cm³，1 m³ = 1 000 000 cm³。

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7. Ratio and Proportion | 比与比例

Simplifying a ratio: divide all parts by their highest common factor. e.g. 8:12 = 2:3

化简比：用各项的最大公因数去除每一项。例如 8:12 = 2:3

Dividing a quantity in a given ratio: if sharing £60 in the ratio 2:3, total parts = 5, so one part = £12, amounts = £24 and £36.

按给定比分配量：若按 2:3 的比例分 £60，总份数 = 5，每份为 £12，因此分别为 £24 和 £36。

Direct proportion: y = kx, where k is the constant of proportionality. As x doubles, y doubles.

正比例： y = kx，k 为比例常数。当 x 翻倍，y 也翻倍。

Inverse proportion (introduction): y = k / x; as x increases, y decreases.

反比例（入门）： y = k / x；x 增大时 y 减小。

Scale factors and maps: actual distance = map distance × scale.

比例尺与地图：实际距离 = 图上距离 × 比例尺。

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8. Percentages | 百分比

Converting between fractions, decimals and percentages: percent = fraction × 100, decimal = percent / 100.

分数、小数与百分比的互化： 百分数 = 分数 × 100， 小数 = 百分数 / 100。

Finding a percentage of an amount: p% of A = (p/100) × A

求某数的百分之几： A 的 p% = (p/100) × A

Percentage increase/decrease: new value = original × (1 ± p/100)

百分数增加或减少： 新值 = 原值 × (1 ± p/100)

Finding original value after a change: original = new / (1 ± p/100)

已知变化后的值求原值： 原值 = 新值 / (1 ± p/100)

Percentage change: % change = (change / original) × 100%

百分比变化： 变化百分比 = (变化量 / 原值) × 100%

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9. Probability | 概率

Probability of an event = number of favourable outcomes / total number of outcomes. It is always between 0 and 1.

事件的概率 = 有利结果数 / 所有可能结果数。概率值总是在 0 与 1 之间。

Complementary events: P(not A) = 1 − P(A)

互补事件： P(非 A) = 1 − P(A)

Sum of probabilities: for mutually exclusive events, P(A or B) = P(A) + P(B).

互斥事件概率相加： P(A 或 B) = P(A) + P(B)。

Expected frequency: expected = probability × number of trials

期望频数： 期望次数 = 概率 × 试验次数

Sample space diagrams and two-way tables help list all outcomes. Probability trees (basic) multiply along branches.

样本空间图与双向表能帮助列出所有结果。概率树图（基础）沿树枝相乘概率。

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10. Statistics: Averages and Range | 统计：平均数与范围

Mean: sum of values / number of values

平均数（均值）： 数值总和 / 数据个数

Median: middle value when data is ordered. If there are two middle values, find their mean.

中位数：将数据排序后位于中间的值。若有中间两个数，则取它们的平均值。

Mode: the most frequent value(s).

众数：出现次数最多的数值。

Range: largest value − smallest value

范围（极差）： 最大值 − 最小值

From a frequency table: mean = Σ(fx) / Σf; median class can be found by cumulative frequency.

由频数表求平均数：平均值 = Σ(频数 × 组中值) / 总频数；可通过累计频数找出中位数所在的组。

Choosing the best average: median is useful when data contains outliers; mode shows the most common category.

选择最佳平均数：当数据含有异常值时中位数更有用；众数反映最常见的类别。

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