📚 Year 7 Cambridge Advanced Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 7 Cambridge 进阶数学：公式定理速查手册
This handbook is designed for Year 7 students following the Cambridge Advanced Mathematics curriculum. It summarises the essential formulas, properties and theorems you need to recall quickly for homework, class tests and end-of-term examinations. Keep it beside your exercise book, use it to check a rule before you start a question, and read the worked statements again whenever a topic feels shaky.
本手册专为学习 Cambridge 进阶数学课程的 Year 7 学生编写，总结了作业、随堂测验和期末考试中需要快速回忆的重要公式、性质与定理。请把它放在练习册旁边，做题前先核对规则，遇到不熟悉的知识点就重读一遍对应条目。
1. Number Operations and Properties | 数的运算与性质
The place value system with decimals: each digit in a number has a value depending on its position. For example, in 5 724.639, the digit 7 is worth 7 hundreds (700), and the digit 3 is worth 3 hundredths (0.03). Multiplying by 10 moves every digit one place to the left, and dividing by 10 moves every digit one place to the right.
带小数的位值系统：一个数中每一位数字根据其位置有不同的值。例如在 5 724.639 中，数字 7 代表 7 个百 (700)，数字 3 代表 3 个百分之一 (0.03)。乘以 10 使每一位数字向左移动一位，除以 10 则向右移动一位。
The three basic laws of arithmetic let you rearrange calculations safely. They hold for all numbers, including decimals, fractions and negative numbers.
算术的三大基本定律让你可以安全地重新排列计算顺序。它们对所有数都成立，包括小数、分数和负数。
Commutative law: a + b = b + a and a × b = b × a
Associative law: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
Distributive law: a × (b + c) = a × b + a × c
Order of operations (BIDMAS): Brackets, Indices, Division and Multiplication (left to right), then Addition and Subtraction (left to right). Every inverse operation undoes its partner: addition undoes subtraction, multiplication undoes division, and squaring is undone by taking the square root.
运算顺序 (BIDMAS)：括号、指数、除法与乘法（从左到右），然后加法与减法（从左到右）。每个逆运算都能抵消其对应的运算：加法抵消减法，乘法抵消除法，平方由开平方抵消。
2. Factors, Multiples, Primes and Indices | 因数、倍数、质数与指数
F) is the largest factor shared by two numbers, and the lowest common multiple (LCM) is the smallest multiple they share. For 24 and 36: HCF = 12 and LCM = 72, and HCF × LCM = 24 × 36 for any pair of positive integers.
因数能整除一个数而没有余数；倍数是一个数乘以整数所得的结果。最大公因数 (HCF) 是两个数共有的最大因数，最小公倍数 (LCM) 是它们共有的最小倍数。对于 24 和 36：HCF = 12，LCM = 72，且对任意正整数对都有 HCF × LCM = 两数之积。
ten as a unique product of primes, which is called its prime factorisation: 360 = 2³ × 3² × 5.
质数恰好有两个因数，即 1 和它本身。30 以内的质数是 2、3、5、7、11、13、17、19、23 和 29。每个大于 1 的整数都能唯一地写成质数的乘积，称为质因数分解：360 = 2³ × 3² × 5。
The index laws describe how powers combine. They are the backbone of algebra in later years, so learn them as words as well as symbols.
指数定律描述了幂如何合并。它们是以后代数学习的核心，因此不仅要记符号，还要会用语言表述。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 a⁻ⁿ = 1 ÷ aⁿ a1/2 = √a a1/3 = ∛a
Squares and cubes worth memorising: 1² to 15² and 1³ to 5³, together with √1 to √144. These values appear constantly in Pythagoras and area questions.
值得背熟的平方与立方：1² 到 15² 以及 1³ 到 5³，还有 √1 到 √144。这些数值在勾股定理和面积题中反复出现。
3. Fractions, Decimals and Percentages | 分数、小数与百分数
To add or subtract fractions, first write them with a common denominator. To multiply, multiply numerators together and denominators together. To divide, multiply by the reciprocal of the second fraction.
分数加减时，先通分到公分母。乘法时分子乘分子、分母乘分母。除法时乘以第二个分数的倒数。
+ c/d = (ad + bc) ÷ bd a/b × c/d = ac ÷ bd a/b ÷ c/d = a/b × d/c
ten: 0.375 = 375/1000 = 3/8. A percentage is simply a fraction with denominator 100, so 45% t a fraction to a percentage, divide the numerator by the denominator and multiply by 100.
小数化分数时，把它写在 10 的幂之上：0.375 = 375/1000 = 3/8。百分数就是分母为 100 的分数，所以 45% = 45/100 = 0.45 = 9/20。分数化百分数时，用分子除以分母再乘以 100。
Percentage change compares the change with the original amount, and multipliers make repeated change quick to handle.
百分比变化是把变化量与原来的量相比，而用乘数处理连续变化会非常快捷。
Percentage change = (change ÷ original) × 100%
Increase by p%: × (1 + p/100) Decrease by p%: × (1 − p/100)
4. Ratio, Proportion and Rates | 比、比例与速率
A ratio compares quantities of the same kind and is simplified by dividing all parts by their HCF: 45 : 60 : 75 simplifies to 3 : 4 : 5. To share an amount in a given ratio, add the parts to find the total number of shares, divide the amount by the total, then multiply by each part.
比用来比较同类量，用各部分的 HCF 同除即可化简：45 : 60 : 75 化简为 3 : 4 : 5。按给定比分配一个量时，先把各部分相加得到总份数，用总量除以总份数，再分别乘以每一份。
A proportion states that two ratios are equal, and problems are solved by finding the multiplier or by cross-multiplication.
比例表示两个比相等，解题时可找出倍数关系或使用交叉相乘。
If a/b = c/d then a × d = b × c
Rates compare quantities of different kinds. The three compound measures you meet most often are speed, density and pressure, each built from the same pattern: the first quantity divided by the second.
速率比较不同类型的量。最常见的三个复合量度是速度、密度和压强，它们的结构相同：第一个量除以第二个量。
distance ÷ time Density = mass ÷ volume Pressure = force ÷ area
5. Algebraic Expressions, Expansion and Factorisation | 代数式、展开与因式分解
Algebra uses letters to stand for numbers, so every law of arithmetic applies unchanged. A term is a single product such as 5x²y, a coefficient is the number multiplying a letter, and like terms have exactly the same letters with the same powers. Only like terms may be collected: 3x + 5y − x + 2y simplifies to 2x + 7y.
代数用字母代表数，因此所有算术定律原样适用。项是像 5x²y 这样的单个乘积，系数是乘在字母前面的数，同类项则字母及其指数完全相同。只有同类项才能合并：3x + 5y − x + 2y 化简为 2x + 7y。
To expand a bracket, multiply every term inside by the term outside. To expand a pair of brackets, multiply every term in the first by every term in the second, then collect like terms. Factorising reverses expansion by taking out the HCF of the terms.
展开括号时，用括号外的项乘括号内每一项。展开两个括号时，用第一个括号的每一项乘第二个括号的每一项，然后合并同类项。因式分解是展开的逆过程，方法是提取各项的 HCF。
(a + b)(c + d) = ac + ad + bc + bd
(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²
Difference of two squares: (a + b)(a − b) = a² − b²
tions. If x = −3, then x² = 9 but −x² = −9: the bracket, not the letter, decides the sign.
代入求值时，把每个字母换成给定数值，再按运算顺序计算。若 x = −3，则 x² = 9，而 −x² = −9：决定正负号的是括号，而不是字母本身。
6. Solving Equations and Inequalities | 解方程与不等式
An equation is balanced, so whatever you do to one side you must do to the other. The aim is to collect the unknown terms on one side and the numbers on the other, then divide by the coefficient of the unknown. Always check your answer by substituting it back into the original equation.
方程是平衡的，所以对一边做了什么运算，另一边也要做同样的运算。目标是先把含未知数的项集中到一边、常数集中到另一边，然后除以未知数的系数。求出答案后一定要代回原方程检验。
If 5x − 3 = 2x + 12 then 3x = 15 and x = 5
Inequalities are solved in exactly the same way, with one critical extra rule: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
不等式的解法完全相同，但有一条关键附加规则：当两边同时乘以或除以一个负数时，不等号方向要反转。
−2x < 8 ⇒ x > −4 (sign flips because both sides are divided by −2)
Simultaneous equations in two unknowns are solved by elimination or substitution. Elimination adds or subtracts the equations to remove one letter; substitution expresses one unknown in terms of the other and puts it into the second equation.
含两个未知数的联立方程可用消元法或代入法求解。消元法通过相加或相减消去一个字母；代入法则用另一个未知数表示其中一个未知数，再代入第二个方程。
7. Sequences, Patterns and the nth Term | 数列、规律与第 n 项
A sequence is an ordered list of numbers called terms, written t₁, t₂, t₃, … A term-to-term rule describes how to move from one term to the next, while a position-to-term rule, or nth term formula, lets you jump straight to any term.
数列是按顺序排列的一列数，称为项，记作 t₁、t₂、t₃……递推规则描述如何从一项走到下一项，而通项（第 n 项）公式让你直接跳到任意一项。
For a linear sequence the difference between consecutive terms is constant, and the nth term always takes the form dn + c, where d is the common difference.
线性数列中相邻两项之差恒定，其第 n 项总具有 dn + c 的形式，其中 d 是公差。
nth term of a linear sequence = dn + c where d = t₂ − t₁
Some special sequences are worth recognising on sight: square numbers 1, 4, 9, 16, 25, … (n²); triangular numbers 1, 3, 6, 10, 15, … (n(n + 1) ÷ 2); cube numbers 1, 8, 27, 64, … (n³); and the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, … where each term is the sum of the previous two.
有些特殊数列值得一眼认出：平方数 1、4、9、16、25……（n²）；三角形数 1、3、6、10、15……（n(n + 1) ÷ 2）；立方数 1、8、27、64……（n³）；以及斐波那契数列 1、1、2、3、5、8、13……，其中每一项是前两项之和。
8. Coordinates, Graphs and Straight Lines | 坐标、图象与直线
Coordinates are written as ordered pairs (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The axes divide the plane into four quadrants, and points on the axes belong to no quadrant.
坐标写成有序数对 (x, y)，其中 x 是距原点的水平距离，y 是垂直距离。坐标轴把平面分为四个象限，落在坐标轴上的点不属于任何象限。
gradient and c is the y-intercept. The gradient measures steepness and can be found from any two points on the line.
每条直线图象都能写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。斜率衡量倾斜程度，可由直线上任意两点求出。
Gradient m = (y₂ − y₁) ÷ (x₂ − x₁)
Midpoint of AB = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
t has a positive gradient, a falling line has a negative gradient, a horizontal line has gradient 0, and a vertical line has an undefined gradient. Graphs are also used to read off solutions: the x-value where a line crosses the x-axis is the solution of the related equation.
平行线斜率相同 (m₁ = m₂)。从左向右上升的直线斜率为正，下降的直线斜率为负，水平线斜率为 0，竖直线斜率不存在。图象也可用来读数求解：直线与 x 轴交点的横坐标就是相应方程的解。
9. Angle Facts and Geometric Reasoning | 角的事实与几何推理
Angles are measured in degrees, and a full turn is 360°. Angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal. These three facts alone solve most Year 7 angle puzzles.
角以度为单位，一整圈是 360°。同一直线上的角之和为 180°，围绕一点的角之和为 360°，对顶角相等。仅凭这三条事实就能解决 Year 7 的大部分角度问题。
When two parallel lines are cut by a transversal, three pairs of equal or supplementary angles appear: corresponding angles (F-shape) are equal, alternate angles (Z-shape) are equal, and co-interior angles (C-shape) add to 180°.
当两条平行线被一条截线所截时，会出现三组相等或互补的角：同位角（F 形）相等，内错角（Z 形）相等，同旁内角（C 形）之和为 180°。
| Shape | 图形 | Angle sum | 内角和 | Example | 示例 |
| Triangle 三角形 | 180° | 60° + 70° + 50° = 180° |
| Quadrilateral 四边形 | 360° | rectangle 矩形 |
| Pentagon 五边形 | 540° | (5 − 2) × 180° |
| n-sided polygon n 边形 | (n − 2) × 180° | sum of interior angles |
The exterior angle of a triangle equals the sum of the two opposite interior angles, and the exterior angles of any polygon always add to 360°, whatever the number of sides. An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three equal sides and three angles of 60°.
三角形的外角等于两个不相邻内角之和，而任何多边形的外角和恒为 360°，与边数无关。等腰三角形有两条相等的边和两个相等的底角；等边三角形三条边相等，三个角都是 60°。
10. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around a shape, area is the amount of surface it covers and is measured in square units, and volume is the space a solid occupies and is measured in cubic units. Always write the unit with your answer.
周长是图形一周的总长度，面积是它覆盖的表面大小，以平方单位计量，体积是立体占据的空间，以立方单位计量。答案一定要带上单位。
| Shape 图形 | Perimeter 周长 | Area 面积 |
| Rectangle 长方形 | 2(l + w) | l × w |
| Triangle 三角形 | a + b + c | ½ × base × height |
| Parallelogram 平行四边形 | 2(a + b) | base × perpendicular height |
| Trapezium 梯形 | sum of four sides | ½ × (a + b) × h |
| Circle 圆 | πd = 2πr | πr² |
For solids, volume is found by multiplying the area of cross-section by the length or height. Surface area is the total area of every face, so draw the net of the solid before you start adding.
对于立体图形，体积等于横截面积乘以长度或高度。表面积是所有面的面积之和，动笔前先把展开图画出来。
Cuboid: V = l × w × h Prism: V = area of cross-section × length
Cylinder: V = πr²h Surface area of a cube = 6 × side²
11. Transformations, Symmetry and Similarity | 变换、对称与相似
A transformation changes the position or size of a shape. Translation slides a shape without turning it, reflection flips it across a mirror line, rotation turns it about a centre by a given angle and direction, and enlargement changes its size by a scale factor about a centre of enlargement.
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