📚 Year 7 Edexcel Further Maths Quick Reference Handbook | Year 7 Edexcel 进阶数学公式定理速查手册
This quick reference handbook covers all the essential formulae, theorems and key concepts from the Year 7 Edexcel Further Mathematics syllabus. Use it to revise number properties, algebraic manipulation, geometric reasoning and data handling. Each rule is presented in clear English, immediately followed by its Chinese equivalent, ensuring bilingual understanding and exam readiness.
这本速查手册涵盖了 Year 7 Edexcel 进阶数学课程中的所有重要公式、定理和关键概念。无论是数的性质、代数运算、几何推理还是数据处理，每条规则都以清晰的英文呈现，并紧跟中文对应表述，帮助你在双语环境中巩固知识、备战考试。
1. Number Properties and Operations | 数的性质与运算
Whole numbers can be classified as even (divisible by 2) or odd (not divisible by 2). A prime number has exactly two distinct factors: 1 and itself.
整数可分为偶数（能被 2 整除）和奇数（不能被 2 整除）。质数恰好有两个不同的因数：1 和它本身。
A factor is a number that divides another exactly, while a multiple is the product of a number and any integer. The highest common factor (HCF) and lowest common multiple (LCM) are found by prime factorisation.
因数是可以整除另一个数的数，倍数是一个数与任意整数的乘积。最大公因数 (HCF) 和最小公倍数 (LCM) 可通过质因数分解求得。
Square number: a number multiplied by itself, e.g. n² = n × n. Cube number: n³ = n × n × n.
平方数：一个数自乘的积，如 n² = n × n。立方数： n³ = n × n × n。
The order of operations is remembered by BIDMAS: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right).
运算顺序用 BIDMAS 记忆：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To add or subtract fractions, rewrite them with a common denominator. To multiply fractions, multiply the numerators and multiply the denominators.
分数加减：先化为同分母再计算。分数乘法：分子乘分子，分母乘分母。
a/b × c/d = (a × c) / (b × d)
a/b × c/d = (a × c) / (b × d)
To divide by a fraction, multiply by its reciprocal (keep, change, flip).
除以一个分数等于乘以它的倒数（保留、改号、翻转）。
A terminating decimal has a finite number of digits; a recurring decimal repeats a pattern forever. To convert a fraction to a decimal, divide the numerator by the denominator.
有限小数位数有限；循环小数则无限重复某个数字串。分数化小数：用分子除以分母。
To change a percentage to a decimal, divide by 100. To change a decimal to a percentage, multiply by 100.
百分数化小数：除以 100。小数化百分数：乘以 100。
Percentage of an amount: (percentage ÷ 100) × amount. Percentage change: (difference ÷ original) × 100%.
求一个数的百分之几：(百分数 ÷ 100) × 该数。百分比增减：(变化量 ÷ 原量) × 100%。
3. Algebraic Expressions and Formulae | 代数表达式与公式
Algebra uses letters (variables) to stand for unknown numbers. A term is either a single number, a variable, or numbers and variables multiplied together, e.g. 5x²y.
代数使用字母（变量）表示未知数。项可以是一个数、一个字母或数与字母的乘积，如 5x²y。
Like terms contain the same variable raised to the same power and can be combined by adding or subtracting their coefficients.
同类项所含字母及指数完全相同，可以合并系数进行加减。
Multiplying terms: multiply the coefficients and add the indices of identical letters (xⁿ × xᵐ = xⁿ⁺ᵐ). Dividing terms: divide the coefficients and subtract the indices (xⁿ ÷ xᵐ = xⁿ⁻ᵐ).
项相乘：系数相乘，同底数幂的指数相加 (xⁿ × xᵐ = xⁿ⁺ᵐ)。项相除：系数相除，同底数幂的指数相减 (xⁿ ÷ xᵐ = xⁿ⁻ᵐ)。
When substituting values into a formula, replace each variable with the given number and follow the order of operations.
将数值代入公式时，用给定的数替换每个变量，并遵循运算顺序。
4. Linear Equations | 一次方程
A linear equation contains a variable raised only to the power of 1. To solve, perform the same operation on both sides to isolate the variable.
一次方程中的变量指数仅为 1。解法：对方程两边同时进行相同的运算，以求出变量。
Example: 3x + 5 = 20 → subtract 5: 3x = 15 → divide by 3: x = 5.
例如： 3x + 5 = 20 → 两边减 5： 3x = 15 → 两边除以 3： x = 5。
Always check your solution by substituting it back into the original equation.
务必将解代入原方程检验。
Equations may contain brackets: expand first, then simplify, then solve.
方程中若含有括号，先展开再化简，然后求解。
5. Sequences and Patterns | 数列与规律
A sequence is an ordered list of numbers governed by a rule. The term-to-term rule tells you how to get from one term to the next.
数列是按照某种规则排列的一列数。项间规则说明了如何从一项得到下一项。
An arithmetic sequence has a constant difference between consecutive terms, called the common difference (d). The nth term is given by: a + (n − 1)d, where a is the first term.
等差数列相邻两项的差恒定，这个差称为公差 (d)。第 n 项公式为： a + (n − 1)d，其中 a 为首项。
Examples of special sequences: square numbers 1, 4, 9, 16, … (n²); triangle numbers 1, 3, 6, 10, … [½ n(n+1)].
特殊数列举例：平方数 1, 4, 9, 16, … (n²)；三角形数 1, 3, 6, 10, … [½ n(n+1)]。
A position-to-term rule (nth term) lets you find any term directly from its position.
位置–项规则（第 n 项公式）可直接由位置求出该项。
6. Geometry: Angles and Polygons | 几何：角与多边形
Angles are measured in degrees (°). An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°.
角度以度 (°) 为单位。锐角小于 90°，直角等于 90°，钝角在 90° 到 180° 之间，优角大于 180°。
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
平角（一直线上邻角之和）为 180°。周角（绕一点的角之和）为 360°。
When two lines intersect, vertically opposite angles are equal.
两直线相交，对顶角相等。
In a triangle, the sum of interior angles is 180°. An isosceles triangle has two equal sides and two equal base angles.
三角形内角和为 180°。等腰三角形有两条边相等，两个底角也相等。
For any polygon, the sum of interior angles = (n − 2) × 180°, where n is the number of sides.
任意多边形内角和 = (n − 2) × 180°，其中 n 为边数。
7. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around a 2D shape. For a rectangle, P = 2(l + w). For a square, P = 4 × side.
周长是二维图形一周的长度。矩形周长： P = 2(长 + 宽)。正方形周长： P = 4 × 边长。
Area of rectangle = length × width
矩形面积 = 长 × 宽
Area of triangle = ½ × base × height. Area of parallelogram = base × perpendicular height.
三角形面积 = ½ × 底 × 高。平行四边形面积 = 底 × 垂直高。
Area of a trapezium = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.
梯形面积 = ½ × (a + b) × h，其中 a 与 b 为平行的两边，h 为它们之间的垂直距离。
Volume of a cuboid = length × width × height. Surface area is the total area of all faces.
长方体体积 = 长 × 宽 × 高。表面积是所有面的面积之和。
Units: area in cm², m²; volume in cm³, m³. Remember to convert units consistently.
单位：面积用 cm²、m²，体积用 cm³、m³。计算时注意统一单位。
8. Coordinates and Graphs | 坐标与图像
Coordinates are written as (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The origin is (0, 0).
坐标表示为 (x, y)，其中 x 是距原点的水平距离，y 是垂直距离。原点为 (0, 0)。
The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).
连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y‑intercept.
直线方程可写为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Gradient = rise / run = (change in y) / (change in x). A positive gradient slopes upward; a negative gradient slopes downward.
斜率 = 垂直距离 / 水平距离 = (y 的变化量) / (x 的变化量)。正斜率向上倾斜，负斜率向下倾斜。
To plot a linear graph, find a set of (x, y) pairs that satisfy the equation and join them with a straight line.
绘制一次函数图像：找出若干组满足方程的 (x, y) 值，描点并用直线连接。
9. Ratio and Proportion | 比与比例
A ratio compares amounts of the same kind. Simplify a ratio by dividing all parts by their HCF. The order of a ratio must match the order stated.
比用于比较同类的量。化简比时，所有项同除以它们的最大公因数。比的顺序必须与所给顺序一致。
To share a quantity in a given ratio, find the total number of parts, then divide the quantity by that total and multiply by each part in turn.
按给定比例分配：先求总份数，用总量除以总份数，再依次乘以各比例的份数。
Direct proportion means that as one quantity increases, the other increases at the same rate. y ∝ x can be written as y = kx, where k is the constant of proportionality.
正比例意味着一个量增加时，另一个量以相同速率增加。y ∝ x 可写作 y = kx，其中 k 是比例常数。
Inverse proportion: as one quantity increases, the other decreases. y ∝ 1/x is written as y = k/x.
反比例：一个量增加时，另一个量减少。y ∝ 1/x 可写作 y = k/x。
Best buys can be found by calculating the unit price (cost per gram, per litre, etc.) for each option.
通过计算单位价格（每克、每升等的费用）可判断最优购买方案。
10. Statistics and Probability | 统计与概率
The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.
平均数（均值）是所有数据之和除以数据个数。中位数是排序后位于中间的数。众数是出现次数最多的数。极差是最大值与最小值的差。
A frequency table can be used to organise data. To find the mean from a frequency table, use: mean = Σ(fx) / Σf.
频率表用于整理数据。从频率表求均值：均值 = Σ(fx) / Σf。
Probability is a measure of likelihood, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes, probability = (number of favourable outcomes) / (total number of outcomes).
概率衡量可能性大小，范围从 0（不可能）到 1（必然发生）。对于等可能结果，概率 = 有利结果数 / 可能结果总数。
The probability of an event not happening = 1 − probability of the event happening.
事件不发生的概率 = 1 − 事件发生的概率。
In a sample space diagram, all possible outcomes are listed to find probabilities of combined events.
在样本空间图中，列出所有可能的结果，以计算复合事件的概率。
11. Units and Measures | 单位与度量
Common metric conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm; 1 tonne = 1000 kg, 1 kg = 1000 g; 1 litre = 1000 ml; 1 hour = 60 minutes, 1 minute = 60 seconds.
常见公制单位换算： 1 千米 = 1000 米， 1 米 = 100 厘米， 1 厘米 = 10 毫米； 1 吨 = 1000 千克， 1 千克 = 1000 克； 1 升 = 1000 毫升； 1 小时 = 60 分钟， 1 分钟 = 60 秒。
When converting between area units, square the linear conversion factor (e.g. 1 m² = 10000 cm², because 1 m = 100 cm, and 100² = 10000). For volume, cube the factor (1 m³ = 1 000 000 cm³).
面积单位换算时，需平方长度换算系数（如 1 m² = 10000 cm²，因为 1 m = 100 cm， 100² = 10000）。体积单位换算时，需立方换算系数（1 m³ = 1 000 000 cm³）。
Speed = distance ÷ time. Compound measures may require consistent units (e.g. km/h, m/s).
速度 = 距离 ÷ 时间。复合单位需保持一致（如 km/h、m/s）。
12. Problem-Solving Strategies | 解题策略
Read the problem carefully, identify what is given and what needs to be found. Break multi-step problems into smaller, logical steps.
仔细读题，明确已知条件和求解目标。将多步问题分解为若干个逻辑清晰的小步骤。
Draw a diagram, make a table, or look for a pattern to represent the information visually.
通过画图、列表或找规律，将信息可视化。
Work backwards when the final result is known but initial steps are missing. Check your answer against the original problem to ensure it makes sense.
当已知最终结果而缺少起始步骤时，可采用逆向推理。最后将答案代入原题检验其合理性。
Estimate before calculating to gauge the size of the answer. Round numbers sensibly depending on the context.
计算前先估算答案的大小。根据实际情况合理取整。
Write down each step clearly — this helps avoid mistakes and makes it easier to check your work.
清晰写出每一步，这有助于减少错误，也使检查更加方便。
Published by TutorHao | Further Maths Revision Series | aleveler.com
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