📚 Year 8 AQA Further Mathematics: Formula & Theorem Quick Reference Guide | Year 8 AQA 进阶数学：公式定理速查手册
This quick-reference handbook compiles the essential formulas, rules, and theorems you need for Year 8 AQA Further Mathematics. Use it to speed up revision, check key facts, and build confidence before assessments.
本速查手册汇集了 Year 8 AQA 进阶数学所需的核心公式、运算法则与定理。你可以用它快速回顾、查漏补缺，并在测评前增强信心。
1. Numbers and Operations | 数与运算
Fractions, decimals, percentages, indices and standard form are the building blocks of number work. Master these operations to handle more complex problems with ease.
分数、小数、百分数、指数和标准形式是数字运算的基础。掌握这些运算能让你轻松应对更复杂的问题。
a/b × c/d = ac / bd
Multiply fractions by multiplying the numerators and the denominators separately.
分数相乘时，分子乘分子，分母乘分母。
a/b ÷ c/d = a/b × d/c = ad / bc
To divide by a fraction, multiply by its reciprocal (invert the second fraction).
除以一个分数，等于乘这个分数的倒数（把第二个分数上下颠倒）。
Percentage = (Part / Whole) × 100%
Convert between fractions, decimals and percentages by remembering that percent means ‘out of 100’.
分数、小数与百分数转换时，记住百分数表示“每一百中的几”。
Percentage change = (Change / Original) × 100%
Calculate a percentage increase or decrease by finding the difference relative to the original amount.
计算增减百分比时，先求差值，再除以原量。
aᵐ × aⁿ = aᵐ⁺ⁿ
When multiplying powers with the same base, add the indices.
同底数幂相乘，指数相加。
(aᵐ)ⁿ = aᵐⁿ
When raising a power to another power, multiply the indices.
幂的乘方，指数相乘。
a⁰ = 1 (a ≠ 0)
Any non-zero number raised to the power zero equals one.
任何非零数的零次幂都等于 1。
Standard form: A × 10ⁿ, with 1 ≤ A < 10
Express very large or very small numbers using a number between 1 and 10 multiplied by a power of 10.
把非常大或非常小的数字写成 1 到 10 之间的一个数与 10 的幂相乘的形式。
2. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratio and proportion allow us to compare quantities and scale them up or down. Direct proportion and rates of change describe how one quantity varies with another.
比和比例用来比较数量并按比例缩放。正比例和变化率则描述一个量如何随另一个量变化。
Ratio a : b means for every a parts there are b parts.
比 a : b 表示每 a 份对应 b 份。总份数为 a + b。
Dividing in a ratio: share £P in the ratio a:b → one part = P/(a+b), then a×part and b×part.
按比例分配：将总量 P 按 a:b 分成两部分，一份等于 P/(a+b)，再分别乘 a 和 b。
Direct proportion: y ∝ x ⇒ y = kx
Quantities in direct proportion increase at the same rate; their graph is a straight line through the origin.
正比例的量以相同速率增长，图像为过原点的直线。
Rate of change = Change in y / Change in x
The gradient of a straight line gives the constant rate of change.
直线的斜率表示恒定的变化率。
Best buy: unit cost = total cost ÷ quantity
Compare value by working out the cost per unit (e.g. price per gram).
通过计算单位成本（如每克的价格）来比较哪个更划算。
3. Algebra – Expressions and Formulas | 代数 – 表达式与公式
Algebra uses letters to stand for unknown numbers. You need to be able to simplify, expand and factorise expressions, and to substitute into formulas.
代数用字母表示未知数。你需要掌握化简、展开、因式分解以及代入公式。
Like terms: 3x + 5x = 8x; 2a²b + 4a²b = 6a²b
Only combinations of the same variable(s) raised to the same power can be collected.
只有所含字母及其指数都相同的项才能合并。
Expanding brackets: a(b + c) = ab + ac
Multiply each term inside the bracket by the term outside. Watch the signs when the multiplier is negative.
用括号外的每一项乘括号内的每一项。当乘数为负时，注意符号变化。
Double brackets: (x + a)(x + b) = x² + (a+b)x + ab
Use the FOIL method (First, Outer, Inner, Last) to expand two binomials.
用“首外内尾”法则 (FOIL) 展开两个二项式相乘。
Factorising: ab + ac = a(b + c)
Find the highest common factor (HCF) of the terms and put it outside the bracket.
找出各项的最高公因式 (HCF) 提到括号外。
Substitution: If T = 2a + 3b and a = 4, b = 1, then T = 2(4)+3(1)=11
Replace variables with given values and follow the order of operations (BIDMAS/BODMAS).
把变量替换成已知数值，并遵循运算顺序（先乘除后加减，有括号先算）。
4. Linear Equations and Inequalities | 线性方程与不等式
Solving linear equations means finding the value of the unknown that balances both sides. Inequalities show a range of possible solutions.
解线性方程就是找出使等式两边成立的未知数值。不等式则给出一系列可能解。
Solving: 3x + 5 = 20 → 3x = 15 → x = 5
Perform inverse operations, doing the same to both sides, until the variable is isolated.
进行逆运算，等式两边同时操作，直至求出未知数。
With brackets: 2(x – 3) = 10 → 2x – 6 = 10 → 2x = 16 → x = 8
Expand brackets first, then solve as usual.
先展开括号，再按一般步骤求解。
Inequalities: x + 4 ≤ 10 → x ≤ 6
Treat inequalities like equations, but reverse the sign when multiplying or dividing by a negative number.
解不等式的方法与解方程类似，但当两边同乘或除以一个负数时，不等号要改变方向。
Number-line representation: open circle for < or >; closed circle for ≤ or ≥.
在数轴上表示时，“<”或“>”用空心圆，“≤”或“≥”用实心圆。
5. Sequences | 数列
A sequence is an ordered list of numbers that follows a rule. The nth term rule lets you find any term without listing all the earlier ones.
数列是按一定规律排列的一列数。通项公式能让你直接求出第 n 项，无需逐项列举。
Arithmetic sequence: nth term = a + (n – 1)d
Where a is the first term and d is the common difference (the amount added or subtracted each time).
其中 a 为首项，d 为公差（每次增加或减少的固定量）。
Finding d: subtract any term from the next term.
公差 d = 后一项 − 前一项。
Special sequences: square numbers 1, 4, 9, 16, … (n²); cube numbers 1, 8, 27, … (n³)
Identify patterns such as squares, cubes, triangular numbers or Fibonacci-like rules.
能识别平方数、立方数、三角数以及类似斐波那契数列的规律。
Using the nth term: for 5, 8, 11, 14, …, nth term = 3n + 2; 20th term = 3×20+2=62
Substitute n into the rule to find any term directly.
将 n 代入通项公式直接求出该项。
6. Graphs of Linear Functions | 线性函数图像
Linear graphs produce straight lines. Understanding the equation y = mx + c helps you draw and interpret these lines quickly.
线性函数的图像是一条直线。掌握方程 y = mx + c 能帮助你快速绘制和解读直线。
y = mx + c
m is the gradient (steepness), c is the y‑intercept (where the line crosses the y‑axis).
m 是斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴的交点）。
Gradient m = (change in y) / (change in x) = (y₂ – y₁) / (x₂ – x₁)
Choose two points on the line; positive m means uphill, negative m means downhill.
选取直线上两点计算斜率；m 为正时直线上升，m 为负时直线下降。
Horizontal line: y = k (m = 0); Vertical line: x = h (gradient undefined)
Horizontal lines are flat; vertical lines do not have a y = mx + c form.
水平线斜率为零，方程为 y = 常数；垂直线斜率不存在，方程为 x = 常数。
Plotting: make a table of values for x and y, plot points, then draw the straight line.
先列出 x 与 y 的对应值表，描点后用直尺连接成直线。
7. Geometry – Angles | 几何 – 角
Angle facts let you find missing angles in diagrams without measuring. You need to know rules for points, lines, triangles and parallel lines.
角度定理能让你不通过测量就求出图形中的未知角。你需要掌握点、线、三角形及平行线中的角度规则。
Angles on a straight line add up to 180°.
平角为 180°。
Angles around a point add up to 360°.
周角为 360°。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle sum to 180°.
三角形内角和等于 180°。
Angles in a quadrilateral sum to 360°.
四边形内角和等于 360°。
Parallel lines: alternate angles are equal; corresponding angles are equal; interior (co‑interior) angles sum to 180°.
平行线：内错角相等；同位角相等；同旁内角互补（和为 180°）。
8. Geometry – Perimeter, Area and Volume | 几何 – 周长、面积与体积
These formulas let you work out the size of shapes and solids. Always include the correct units (cm, m², cm³, etc.).
这些公式用于计算平面图形和立体的大小。务必带上正确的单位。
Rectangle: Perimeter = 2(L + W), Area = L × W
长方形周长 = 2×(长+宽)，面积 = 长×宽。
Triangle: Area = ½ × base × height
三角形面积 = ½ × 底 × 高。
Parallelogram: Area = base × perpendicular height
平行四边形面积 = 底 × 垂直高。
Trapezium: Area = ½ × (a + b) × h (a and b are the parallel sides)
梯形面积 = ½ × (上底+下底) × 高。
Circle: Circumference = πd or 2πr; Area = πr² (r = radius, d = 2r)
圆的周长 = πd 或 2πr；面积 = πr²。π 通常取 3.14 或保留符号。
Prism: Volume = area of cross‑section × length
棱柱体积 = 横截面积 × 长度。
Cube & cuboid: Volume = L × W × H; Surface area = 2(LW + LH + WH)
立方体和长方体体积 = 长×宽×高；表面积 = 2(长×宽 + 长×高 + 宽×高)。
9. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem connects the sides of a right‑angled triangle. It is used to find missing lengths and to check if a triangle has a right angle.
勾股定理建立了直角三角形三条边之间的关系，用来求未知边长或判断一个三角形是否为直角三角形。
a² + b² = c²
Where c is the hypotenuse (the longest side, opposite the right angle); a and b are the shorter legs.
c 为斜边（最长边，正对直角），a 和 b 为两直角边。
Find the hypotenuse: c = √(a² + b²)
Square both legs, add them, then take the square root.
计算两直角边的平方和，再开平方就得到斜边长。
Find a shorter side: a = √(c² – b²)
Subtract the square of the known leg from the square of the hypotenuse, then square root.
用斜边的平方减去已知直角边的平方，再开平方求出未知直角边。
Converse: If a² + b² = c² in a triangle, then the triangle is right‑angled.
如果一个三角形的三边满足 a² + b² = c²，那么这个三角形是直角三角形。
10. Transformations | 图形变换
Transformations move or change a shape. You must be able to describe and carry out translations, reflections, rotations and enlargements.
图形变换会移动或改变一个图形。你需要能描述和进行平移、反射、旋转和放大。
Translation: (x, y) → (x + a, y + b)
The vector ⟨a, b⟩ slides the shape a units horizontally and b units vertically.
向量 ⟨a, b⟩ 表示图形向右 a 单位、向上 b 单位平移（向左或向下为负）。
Reflection: mirror line acts as a line of symmetry.
A point and its image are the same perpendicular distance from the mirror line.
反射中，物点与像点到对称轴的垂直距离相等。
Rotation: you need centre, angle and direction (clockwise/anticlockwise).
旋转需要指定旋转中心、角度和方向。绕原点 90° 旋转时坐标发生变化。
Enlargement: (x, y) → (kx, ky) when centre is the origin.
放大时，相似图形边长变为原来的 k 倍，面积变为 k² 倍。尺度因子 k > 1 放大，0 < k < 1 缩小。
11. Probability | 概率
Probability measures how likely an event is. Probabilities can be written as fractions, decimals or percentages, and always lie between 0 and 1.
概率衡量事件发生的可能性大小。概率可用分数、小数或百分数表示，且总是在 0 到 1 之间。
P(Event) = number of favourable outcomes / total number of equally likely outcomes
The probability of an event equals the number of ways it can happen divided by the total possible outcomes.
事件发生的概率 = 该事件可能出现的结果数 ÷ 所有等可能结果总数。
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