📚 Year 7 AQA Further Maths Formula & Theorem Quick Reference Handbook | AQA 7年级进阶数学公式定理速查手册
This quick reference handbook brings together all the essential formulae, rules and theorems you need for Year 7 AQA Further Maths. Use it to revise key facts, practise applying them, and build a solid foundation for more advanced topics. Each section is presented with clear explanations in English and Chinese, and important results are highlighted for easy memorisation.
这本速查手册汇集了 Year 7 AQA 进阶数学所需的关键公式、定理和规则。你可以用它复习重点知识，练习应用这些公式，为后续学习打下坚实基础。每个小节都配有清晰的中英文解释，重要结论突出显示，便于记忆。
1. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
When an expression contains more than one operation, we must carry them out in the correct order to get the right answer. The word BIDMAS helps you remember: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Division and multiplication have equal priority – work from left to right. The same applies to addition and subtraction.
当一个算式包含多种运算时，必须按照正确的顺序进行计算。BIDMAS 能帮你记住优先级：括号、指数、除法、乘法、加法、减法。除法和乘法优先级相同，从左向右计算；加法和减法同理。
B → Brackets first
I → Indices (powers, e.g. 2²)
D → Division, M → Multiplication (left to right)
A → Addition, S → Subtraction (left to right)
B → 先算括号
I → 指数（如 2²）
D → 除法，M → 乘法（从左到右）
A → 加法，S → 减法（从左到右）
2. Directed Numbers (Negative Numbers) | 正负数运算
Rules for adding and subtracting negative numbers can be summarised with 'two signs together' shortcuts. When multiplying or dividing, two identical signs give a positive answer, and two different signs give a negative answer.
正负数的加减法可以通过“同号合并”的规律简化。乘除运算时，同号得正，异号得负。
a + (–b) = a – b
a – (–b) = a + b
(+a) × (+b) = +ab (–a) × (–b) = +ab
(+a) × (–b) = –ab (–a) × (+b) = –ab
a + (–b) = a – b
a – (–b) = a + b
正 × 正 = 正 负 × 负 = 正
正 × 负 = 负 负 × 正 = 负
3. Fractions, Decimals and Percentages | 分数、小数与百分数
These three forms are interchangeable. To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. A percentage can be turned into a fraction by writing it over 100 and simplifying.
这三种形式可以相互转换。分数转小数：分子除以分母。小数转百分数：乘以 100。百分数转分数：写成 100 分之几，再约分。
Fraction → Decimal: numerator ÷ denominator
Decimal → %: multiply by 100
% → Fraction: write over 100, then simplify
分数转小数：分子 ÷ 分母
小数转百分数：× 100
百分数转分数：写成分母为 100，再约分
When adding or subtracting fractions, find a common denominator first. To multiply fractions, multiply the numerators and multiply the denominators. To divide by a fraction, flip it and multiply (keep-change-flip).
分数加减法先通分；分数乘法，分子乘分子，分母乘分母；除以一个分数等于乘它的倒数。
a/b + c/d = (ad + bc)/bd a/b × c/d = ac/bd a/b ÷ c/d = a/b × d/c
a/b + c/d = (ad + bc)/bd a/b × c/d = ac/bd a/b ÷ c/d = a/b × d/c
4. Prime Factorisation, HCF and LCM | 质因数分解、最大公因数与最小公倍数
Every whole number greater than 1 can be written as a product of prime numbers. This is called prime factorisation. Use a factor tree to break the number down until only primes remain.
每个大于 1 的整数都可以写成质数的乘积，即质因数分解。可以通过画因数树，直到所有因数都是质数。
To find the Highest Common Factor (HCF), multiply the common prime factors using the smallest powers. For the Lowest Common Multiple (LCM), multiply each prime factor with the highest power that appears.
求最大公因数 (HCF)：取公有的质因数，用最小指数相乘。求最小公倍数 (LCM)：取所有质因数，用出现的最大指数相乘。
Example: 24 = 2³ × 3, 36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72
例：24 = 2³ × 3，36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72
5. Algebraic Expressions and Brackets | 代数表达式与去括号
Algebra uses letters to stand for unknown numbers. We simplify expressions by collecting like terms (terms with exactly the same letters and powers).
代数用字母表示未知数。通过合并同类项（字母及指数完全相同的项）可以化简表达式。
3a + 5b – a + 2b = 2a + 7b
3a + 5b – a + 2b = 2a + 7b
When multiplying a single term by a bracket, multiply the term by everything inside. To expand two brackets, multiply each term in the first bracket by every term in the second – often remembered as FOIL (First, Outer, Inner, Last).
单项式乘括号时，用单项式去乘括号里的每一项。展开两个括号时，把前一个括号中的每一项分别乘以后一个括号中的每一项，可用 FOIL 法则（先乘首项，再乘外项、内项、末项）。
a(b + c) = ab + ac
(x + a)(x + b) = x² + (a+b)x + ab
a(b + c) = ab + ac
(x + a)(x + b) = x² + (a+b)x + ab
6. Solving Linear Equations | 解一元一次方程
To solve an equation, keep it balanced by doing exactly the same operation to both sides. Aim to isolate the unknown on one side. Always check your solution by substituting it back into the original equation.
解方程时，在等式两边同时进行相同的运算，以保持平衡。目标是将未知数单独留在等式一边。求出解后务必代回原方程检验。
If 2x + 3 = 11
Subtract 3: 2x = 8
Divide by 2: x = 4
如 2x + 3 = 11
两边减 3：2x = 8
两边除以 2：x = 4
7. Sequences: The nth Term | 数列：第 n 项
A number pattern that follows a rule is a sequence. For an arithmetic (linear) sequence, the difference between consecutive terms is constant. The nth term formula lets you find any term directly.
按照一定规律排列的一列数叫做数列。在等差数列中，相邻两项的差恒定。第 n 项公式可直接求出任意项。
nth term = first term + (n – 1) × common difference
or nth term = dn + (a – d)
第 n 项 = 首项 + (n – 1) × 公差
也可写成 第 n 项 = dn + (a – d)
Example: 5, 9, 13, 17, ... has common difference 4 and first term 5. The nth term is 4n + 1.
例如：5, 9, 13, 17, … 公差为 4，首项为 5。第 n 项为 4n + 1。
8. Perimeter and Area of 2D Shapes | 二维图形的周长与面积
Perimeter is the total distance around a shape. Area is the amount of space inside it, measured in square units. Memorise the common area formulae.
周长是图形一周的总长度。面积是图形内部占用的空间，单位为平方单位。请熟记常见面积公式。
Rectangle: Perimeter = 2(l + w), Area = l × w
矩形：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 × 垂直高度
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 = 2πr, Area = πr² (π ≈ 3.14 or 22/7)
圆：周长 = 2πr，面积 = πr²（π 通常取 3.14 或 22/7）
9. Volume of Cuboids | 长方体体积
Volume measures the space a 3D object occupies, in cubic units. For a cuboid (rectangular prism), volume is found by multiplying length, width and height.
体积是三维物体所占空间的大小，单位为立方单位。长方体的体积等于长、宽、高相乘。
Volume of cuboid = length × width × height = l w h
长方体体积 = 长 × 宽 × 高 = l w h
is the side length. Surface area can also be found by summing the areas of all six faces.
正方体是特殊的长方体，所有棱长相等：体积 = a³。表面积则是六个面的面积之和。
10. Angle Facts | 角度基本定理
Knowing angle properties helps you find missing angles without measuring. The key rules cover points, straight lines, triangles and parallel lines.
掌握角度性质可以在不测量的情况下求未知角。下列关键规则涵盖点、直线、三角形和平行线。
Angles at a point sum to 360°.
一点周角为 360°。
Angles on a straight line sum to 180°.
平角之和为 180°。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle sum to 180°.
三角形内角和为 180°。
In parallel lines: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.
平行线中：内错角相等，同位角相等，同旁内角互补（和为 180°）。
11. Pythagoras' Theorem | 勾股定理
In any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
在任何直角三角形中，斜边（直角对边，最长边）的平方等于两直角边的平方和。
a² + b² = c² where c is the hypotenuse
a² + b² = c² 其中 c 为斜边
To find the hypotenuse, add the squares and square root. To find a shorter side, subtract the known square from the hypotenuse squared, then square root.
求斜边：取两直角边平方和再开平方。求直角边：用斜边平方减去已知直角边平方，再开平方。
c = √a² + b² a = √c² – b²
c = √a² + b² a = √c² – b²
12. Statistics: Averages and Range | 统计：平均数与范围
Three averages describe the centre of a data set: the mean (add all values, divide by the number of values), the median (middle value when ordered), and the mode (most frequent value). The range measures spread: largest value – smallest value.
描述数据中心趋势的三个量：平均数（所有数值之和除以个数）、中位数（排序后位于中间的值）和众数（出现次数最多的值）。极差表示数据的离散程度，即最大值减最小值。
Mean = sum of data ÷ number of data values
Range = maximum – minimum
平均数 = 数据总和 ÷ 数据个数
极差 = 最大值 – 最小值
When the number of data values is even, the median is the mean of the two middle numbers. A data set may have one mode, more than one mode, or no mode at all.
当数据个数为偶数时，中位数是中间两个数的平均数。一组数据可能有一个众数、多个众数，或没有众数。
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