📚 Year 7 Cambridge Advanced Mathematics: Formula & Theorem Quick Reference Handbook | 剑桥7年级进阶数学：公式定理速查手册
This handbook provides a concise and clear collection of the essential formulas, rules, and theorems that every Year 7 Cambridge Advanced Mathematics student needs to know. It is designed as a quick revision tool to help you master key concepts across number, algebra, geometry, statistics, and probability. Keep this guide handy while practising past paper questions or completing homework tasks.
本手册简明扼要地整理了每一位剑桥7年级进阶数学学生必须掌握的核心公式、法则与定理。它作为一本快速复习工具，帮助你掌握数论、代数、几何、统计与概率等领域的关键概念。在做真题练习或完成作业时，随时查阅这本手册将事半功倍。
1. Integers and Order of Operations | 整数与运算顺序
When performing calculations with integers, the order of operations must be followed strictly: Brackets, Indices (powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This is often remembered as BIDMAS or BODMAS.
进行整数运算时，必须严格遵守运算顺序：括号、指数（乘方和开方）、除法和乘法（从左至右）、加法和减法（从左至右）。这通常被记忆为 BIDMAS 或 BODMAS 法则。
Integer rules for addition and subtraction: adding a positive integer moves right on the number line; adding a negative integer moves left. Subtracting a negative integer is equivalent to adding its positive counterpart.
整数的加减法则：加上一个正整数在数轴上向右移动；加上一个负整数则向左移动。减去一个负数等价于加上它的相反正数。
= a positive; a positive × a negative = a negative. The same sign rules apply to division.
乘除法则：正 × 正 = 正；负 × 负 = 正；正 × 负 = 负。相同的符号规则适用于除法。
| Operation | Example | Result |
|---|---|---|
| (-5) + (-3) | Add two negatives | -8 |
| (-4) × 6 | Negative × positive | -24 |
| (-9) ÷ (-3) | Negative ÷ negative | 3 |
2. Factors, Multiples and Primes | 因数、倍数与质数
A factor of a whole number is any integer that divides it exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is obtained by multiplying the number by any integer; the multiples of 5 are 5, 10, 15, 20, …
一个整数的因数是指能够整除该数且没有余数的任何整数。例如，12 的因数有 1、2、3、4、6、12。倍数则是将该数乘以任意整数得到的积；5 的倍数有 5、10、15、20……
ctorisation of a number expresses it as a product of prime numbers; for example, 60 = 2² × 3 × 5.
质数是大于 1 且只有两个因数（1 和它本身）的整数。前几个质数为 2、3、5、7、11、13、17、19。一个数的质因数分解将其表示为质数的乘积；例如 60 = 2² × 3 × 5。
The highest common factor (HCF) of two numbers is the largest factor they share. The lowest common multiple (LCM) is the smallest multiple they share. Use prime factorisation to find HCF and LCM efficiently.
两个数的最大公因数 (HCF) 是它们共有的最大因数。最小公倍数 (LCM) 是它们共有的最小倍数。使用质因数分解可以高效求得 HCF 和 LCM。
HCF(24, 36) = 12, LCM(24, 36) = 72
3. Powers and Roots | 乘方与开方
Here, 3 is the base and 4 is the exponent or index. The square of a number is the number raised to the power 2; the cube is raised to the power 3.
乘方（或指数）表示底数自乘的次数。例如，3⁴ 表示 3 × 3 × 3 × 3 = 81。这里 3 是底数，4 是指数。一个数的平方是该数的 2 次幂；立方是 3 次幂。
A square root reverses squaring: √a asks 'what number multiplied by itself gives a?'. The cube root ³√a works similarly. The symbol √ denotes the principal (non‑negative) square root.
平方根是平方的逆运算：√a 问“哪个数自乘得到 a？”。立方根 ³√a 同理。符号 √ 表示主（非负）平方根。
Key index laws for positive integer exponents:
- Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ – 同底数幂相乘，指数相加。
- Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n) – 同底数幂相除，指数相减。
- Power of a power: (aᵐ)ⁿ = aᵐⁿ – 幂的乘方，指数相乘。
Any number raised to the power 0 equals 1: a⁰ = 1 (a ≠ 0).
任何非零数的 0 次幂等于 1：a⁰ = 1 (a ≠ 0)。
| Expression | Simplified |
|---|---|
| 2³ × 2⁴ | 2⁷ = 128 |
| 5⁶ ÷ 5² | 5⁴ = 625 |
| (4²)³ | 4⁶ = 4096 |
4. Fractions, Decimals and Percentages | 分数、小数与百分数
A fraction represents a part of a whole: numerator/denominator. Equivalent fractions are obtained by multiplying or dividing both numerator and denominator by the same non‑zero number. To simplify a fraction, divide both by their HCF.
分数表示整体的一部分：分子/分母。将分子和分母同时乘以或除以同一个非零数，可以得到等值分数。化简分数时，将分子和分母同时除以它们的最大公因数。
To add or subtract fractions, first find a common denominator. To multiply fractions, multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal.
加减分数时，先通分找到公分母。乘法直接分子乘分子、分母乘分母。除以一个分数等于乘以它的倒数。
Converting between forms:
- Fraction to decimal: divide numerator by denominator. – 分数化小数：分子除以分母。
- Decimal to percentage: multiply by 100%. – 小数化百分数：乘以 100%。
- Percentage to fraction: write over 100 and simplify. – 百分数化分数：写成 /100 再化简。
A percentage increase or decrease can be calculated using a multiplier. For an increase of r%, multiply by (1 + r/100); for a decrease, multiply by (1 − r/100).
计算百分数增减可以使用乘数。增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 − r/100)。
25% of 200 = 0.25 × 200 = 50
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities of the same kind, written as a : b. Ratios can be simplified by dividing all parts by a common factor. The order of a ratio matters.
比用于比较两个或多个同类量，写作 a : b。通过将所有部分除以公因数可以化简比。比的顺序很重要。
ven ratio, the total number of parts is the sum of the ratio terms. For example, to divide £60 in the ratio 2 : 3, total parts = 5, so each part is £12; the amounts are £24 and £36.
按给定比例分配一个量时，总份数等于比项之和。例如，将 £60 按 2 : 3 分配，总份数 = 5，每份 £12，所得为 £24 和 £36。
s directly proportional to x, we write y = kx, where k is the constant of proportionality.
正比例：两个量以相同速率增减。若 y 与 x 成正比，记作 y = kx，其中 k 为比例常数。
increases, the other decreases at a rate such that their product remains constant: xy = k.
反比例：一个量增加时，另一个量以乘积保持不变的方式减少：xy = k。
6. Algebraic Expressions and Linear Equations | 代数表达式与线性方程
Algebraic form is built from variables (letters), constants, and operations. Terms are separated by + or − signs. Like terms have the same variable raised to the same power and can be combined by adding or subtracting their coefficients.
代数式由变量（字母）、常数和运算符号构成。项由加号或减号分隔。同类项含有相同的变量且指数相同，可通过系数加减合并。
de: a(b + c) = ab + ac. To factorise, reverse this process by identifying a common factor.
去括号时，将外面的项乘以括号内的每一项：a(b + c) = ab + ac。因式分解则通过找出公因式反向操作。
A linear equation can be solved by performing the same operation on both sides to isolate the unknown. The goal is to get x on one side of the equation. Always check your solution by substitution.
解一元一次方程时，对等式两边进行相同运算以分离未知数。目标是将 x 单独留在等式一边。务必用代入法检验解。
Example: 3x + 2 = 14 → 3x = 12 → x = 4
Forming equations from word problems involves identifying the unknown quantity, representing it with a variable, and translating the given relationships into an equation.
根据文字题列方程时，先明确未知量，用变量表示，再将给定的关系转化为方程。
7. Sequences and Patterns | 序列与模式
A number sequence is an ordered list of numbers governed by a rule. Each number in the sequence is called a term. The nth term rule allows us to find any term in the sequence without listing all preceding terms.
数列是一组按规则排列的有序数字。数列中的每个数称为项。第 n 项公式使我们无需列出前面的所有项即可求得任意项。
rm formula is: T(n) = a + (n − 1)d, where a is the first term, d is the common difference.
等差数列中，连续两项的差为常数。第 n 项公式为：T(n) = a + (n − 1)d，其中 a 为首项，d 为公差。
To find whether a number is in a sequence, set T(n) equal to that number and solve for n. If n is a positive integer, the number belongs to the sequence.
判断一个数是否在数列中，令 T(n) 等于该数并解出 n。若 n 为正整数，则该数属于数列。
Non‑linear sequences such as square numbers (1, 4, 9, 16, …) or triangular numbers (1, 3, 6, 10, …) follow their own patterns. Sometimes differences between terms form a linear pattern, allowing us to generate further terms.
非线性数列，如平方数 (1, 4, 9, 16, …) 或三角形数 (1, 3, 6, 10, …)，各有其模式。有时项之间的差形成线性模式，可据此生成更多项。
8. Angles and Lines | 角与线
Angles are measured in degrees (°). Basic angle facts: a right angle is 90°, an acute angle is less than 90°, an obtuse angle is between 90° and 180°, a straight angle is 180°, and a reflex angle is greater than 180° but less than 360°.
角以度 (°) 测量。基本角事实：直角 90°，锐角小于 90°，钝角介于 90° 和 180° 之间，平角 180°，优角大于 180° 且小于 360°。
Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal when two lines intersect.
直线上的角之和为 180°。围绕一点的角之和为 360°。两条直线相交时，对顶角相等。
When a transversal crosses parallel lines:
- Corresponding angles are equal. – 同位角相等。
- Alternate interior (or exterior) angles are equal. – 内错角（或外错角）相等。
- Interior angles on the same side of the transversal are supplementary (sum to 180°). – 同旁内角互补（和为 180°）。
| Angle relationship | Diagram clue | Value |
|---|---|---|
| Corresponding | Same position on parallel lines | Equal |
| Alternate | Inside between lines, opposite sides of transversal | Equal |
| Co‑interior | Inside between lines, same side of transversal | Sum = 180° |
9. Properties of Triangles and Quadrilaterals | 三角形与四边形的性质
The sum of the interior angles of any triangle is 180°. An equilateral triangle has three equal sides and three equal angles of 60° each. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has all sides and angles different.
任意三角形的内角和为 180°。等边三角形三条边相等，三个角均为 60°。等腰三角形有两条边相等，两个底角相等。不等边三角形的所有边和角均不相等。
A right‑angled triangle contains one 90° angle. The side opposite the right angle is called the hypotenuse; at this stage, we are not using Pythagoras, but we recognise the right‑angle notation.
直角三角形含有一个 90° 角。直角对边称为斜边；现阶段我们虽然不涉及勾股定理，但需识别直角标记。
Quadrilaterals have interior angles summing to 360°. Special quadrilaterals:
- Square: four equal sides, four right angles. – 正方形：四条边相等，四个直角。
- Rectangle: opposite sides equal, four right angles. – 矩形：对边相等，四个直角。
- Parallelogram: opposite sides parallel and equal, opposite angles equal. – 平行四边形：对边平行且相等，对角相等。
- Rhombus: four equal sides, opposite angles equal. – 菱形：四条边相等，对角相等。
- Trapezium (UK): one pair of parallel sides. – 梯形：至少有一对平行边。
The diagonals of a quadrilateral often have special properties, such as bisecting each other in a parallelogram.
四边形的对角线常有特殊性质，例如在平行四边形中互相平分。
10. Perimeter, Area and Volume | 周长、面积与体积
otal length of its sides. For a rectangle: P = 2(l + w), where l is length and w is width.
多边形的周长是其所有边长的总和。矩形：P = 2(l + w)，其中 l 为长，w 为宽。
Area is the amount of space inside a 2D shape, measured in square units (cm², m², etc.). Key formulas:
- Rectangle: A = l × w – 矩形面积 = 长 × 宽。
- Triangle: A = ½ × base × height – 三角形面积 = ½ × 底 × 高。
- Parallelogram: A = base × perpendicular height – 平行四边形面积 = 底 × 垂直高。
- Trapezium: A = ½(a + b) × h, where a and b are the parallel sides – 梯形面积 = ½ (a + b) × 高，其中 a、b 为平行边。
- : circumference C = 2πr or πd, area A = πr². Use π ≈ 3.14 or the π button on a calculator. – 圆：周长 C = 2πr 或 πd，面积 A = πr²。π 取 3.14 或使用计算器上的 π 键。
s the space inside a 3D solid. For a cuboid: V = l × w × h. For a prism: V = area of cross‑section × length. Volume is measured in cubic units (cm³, m³).
体积衡量三维立体内部空间。长方体：V = l × w × h。棱柱：V = 横截面积 × 长。体积单位为立方单位（cm³, m³）。
| Shape | Formula |
|---|---|
| Rectangle area | A = lw |
| Circle area | A = πr² |
11. Coordinate Geometry and Straight‑Line Graphs | 坐标几何与直线图像
The Cartesian plane consists of two perpendicular number lines: the x‑axis (horizontal) and the y‑axis (vertical). A point is given by an ordered pair (x, y). The origin is (0, 0).
笛卡尔平面由两条互相垂直的数轴构成：x 轴（水平）和 y 轴（垂直）。点用有序对 (x, y) 表示。原点为 (0, 0)。
The midpoint of two points (x₁, y₁) and (x₂, y₂) is found by averaging the x‑coordinates and the y‑coordinates separately: ((x₁+x₂)/2, (y₁+y₂)/2).
由两点 (x₁, y₁) 和 (x₂, y₂) 构成的线段中点，计算方法为分别对 x 坐标和 y 坐标取平均值：((x₁+x₂)/2, (y₁+y₂)/2)。
ht line is often written as y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis). Gradient = (change in y)/(change in x) = (y₂ − y₁)/(x₂ − x₁).
直线方程常写作 y = mx + c，其中 m 为斜率（坡度），c 为 y 轴截距（直线与 y 轴的交点）。斜率 = (y 的变化量)/(x 的变化量) = (y₂ − y₁)/(x₂ − x₁)。
To plot a linear graph, create a table of values, substitute x‑values into the equation to find corresponding y‑values, plot the points, and join them with a straight line.
绘制线性图像时，建立数值表，将 x 值代入方程求得对应的 y 值，描点后用直线连接。
12. Statistics and Probability | 统计与概率
Mean, median, mode, and range are measures of central tendency and spread. The mean is the sum of all values divided by the number of values. The median is the middle value when data is ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.
平均数、中位数、众数和极差是集中趋势与离散程度的度量。平均数是所有数值之和除以数据个数。中位数是排序后位于中间的数值。众数是出现频次最高的数值。极差是最大值与最小值的差。
hart shows proportions: the angle for each category is (frequency/total frequency) × 360°.
呈现数据时常用条形图、饼图、折线图和频数表。饼图显示比例：每个类别的扇区角度 = (频数/总数) × 360°。
lity of an event = (number of favourable outcomes) / (total number of equally likely outcomes).
概率衡量事件发生的可能性，可用 0（不可能）到 1（必然）之间的分数、小数或百分数表示。事件的概率 = (有利结果数) / (所有等可能结果总数)。
robabilities of all possible outcomes is 1. The probability of an event not happening is 1 − P(event).
两个独立事件同时发生的概率是各自概率的乘积：P(A 且 B) = P(A) × P(B)。所有可能结果的概率之和为 1。事件不发生的概率为 1 − P(事件发生)。
13. Units of Measurement and Conversions | 计量单位与换算
Common metric units: length – mm, cm, m, km; mass – g, kg; capacity – on powers of ten: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m; 1 kg = 1000 g; 1 l = 1000 ml.
常用公制单位：长度 – 毫米 (mm)、厘米 (cm)、米 (m)、千米 (km)；质量 – 克 (g)、千克 (kg)；容量 – 毫升 (ml)、升 (l)。公制内部换算基于 10 的幂：1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m；1 kg = 1000 g；1 l = 1000 ml。
ic conversions; however, knowing that 1 inch ≈ 2.54 cm and 1 mile ≈ 1.6 km can be helpful.
时间单位：60 秒 = 1 分钟，60 分钟 = 1 小时，24 小时 = 1 天。题目可能给出英制单位，但剑桥大纲侧重公制换算；了解 1 英寸 ≈ 2.54 厘米，1 英里 ≈ 1.6 千米会有帮助。
and cube it for volume. For example, 1 m² = 10000 cm² (since 1 m = 100 cm, 100² = 10000).
换算面积或体积单位时，面积需将长度换算因子平方，体积则立方。例如，1 m² = 10000 cm²（因为 1 m = 100 cm，100² = 10000）。
h, m/s). To find distance: distance = speed × time. To find time: time = distance ÷ speed.
速度、距离和时间的关系公式为：速度 = 距离 ÷ 时间。单位必须保持一致（如 km/h、m/s）。求距离：距离 = 速度 × 时间。求时间：时间 = 距离 ÷ 速度。
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