📚 Year 7 CIE Further Mathematics: Quick Reference Formulas & Theorems | Year 7 CIE 进阶数学：公式定理速查手册
This handy reference sheet compiles all the essential formulas, rules, and theorems you need for Year 7 CIE Further Mathematics. Use it for quick revision and to reinforce your understanding of key concepts.
这份实用的速查表汇编了 Year 7 CIE 进阶数学所需的所有关键公式、规则和定理。你可以用它快速复习，巩固对核心概念的理解。
1. Order of Operations (BIDMAS/PEMDAS) | 运算顺序 (BIDMAS/PEMDAS)
The acronym BIDMAS tells us the correct order: Brackets, Indices (powers), Division & Multiplication (left to right), Addition & Subtraction (left to right).
助记口诀 BIDMAS 告诉我们正确的运算顺序：括号、指数（幂）、除法和乘法（从左到右）、加法和减法（从左到右）。
For division and multiplication, work from left to right as they have equal priority. The same rule applies to addition and subtraction.
除法和乘法具有相同的优先级，计算时从左到右进行。加法和减法同理。
Example: 8 + 2 × (3² − 1) = 8 + 2 × (9 − 1) = 8 + 2 × 8 = 8 + 16 = 24.
示例：8 + 2 × (3² − 1) = 8 + 2 × (9 − 1) = 8 + 2 × 8 = 8 + 16 = 24。
2. Integers and Directed Numbers | 整数与有向数
dding a negative number is equivalent to subtracting its positive value: a + (−b) = a − b.
加上一个负数等于减去它的相反数：a + (−b) = a − b。
ubtracting a negative number is equivalent to adding its positive value: a − (−b) = a + b.
减去一个负数等于加上它的相反数：a − (−b) = a + b。
Multiplying or dividing two numbers with the same sign gives a positive result: (+) × (+) = (+), (−) × (−) = (+).
同号两数相乘或相除，结果为正：(+) × (+) = (+)，(−) × (−) = (+)。
Multiplying or dividing two numbers with different signs gives a negative result: (+) × (−) = (−), (−) × (+) = (−).
异号两数相乘或相除，结果为负：(+) × (−) = (−)，(−) × (+) = (−)。
3. Fractions, Decimals, and Percentages | 分数、小数与百分数
a fraction to a decimal, divide the numerator by the denominator. e.g., ⅜ = 3 ÷ 8 = 0.375.
分数转小数：用分子除以分母。例如 ⅜ = 3 ÷ 8 = 0.375。
To convert a decimal to a percentage, multiply by 100%. e.g., 0.65 = 0.65 × 100% = 65%.
小数转百分数：乘以 100%。例如 0.65 = 0.65 × 100% = 65%。
To find a percentage of a quantity: Percentage of amount = (percentage ÷ 100) × amount.
求一个数量的百分之几：数量的百分数 = (百分数 ÷ 100) × 数量。
subtracting fractions: find a common denominator first. e.g., ⅓ + ¼ = ⁴/₁₂ + ³/₁₂ = ⁷/₁₂.
分数加减：先通分。例如 ⅓ + ¼ = ⁴/₁₂ + ³/₁₂ = ⁷/₁₂。
Multiplying fractions: multiply numerators and multiply denominators. e.g., ⅔ × ⅖ = (2×2)/(3×5) = ⁴/₁₅.
分数乘法：分子乘分子，分母乘分母。例如 ⅔ × ⅖ = (2×2)/(3×5) = ⁴/₁₅。
Dividing fractions: multiply by the reciprocal. e.g., ⅔ ÷ ⅘ = ⅔ × ⁵/₄ = ¹⁰/₁₂ = ⅚.
分数除法：乘以除数的倒数。例如 ⅔ ÷ ⅘ = ⅔ × ⁵/₄ = ¹⁰/₁₂ = ⅚。
4. Algebraic Expressions and Simplification | 代数表达式与化简
Like terms contain exactly the same variable(s) raised to the same power(s). Only like terms can be combined.
同类项包含完全相同的变量且变量的指数相同。只有同类项才能合并。
Simplify by collecting like terms: 3a + 5b − a + 2b = (3a − a) + (5b + 2b) = 2a + 7b.
通过合并同类项化简：3a + 5b − a + 2b = (3a − a) + (5b + 2b) = 2a + 7b。
Expanding a bracket: multiply the term outside by every term inside. a(b + c) = ab + ac.
去括号：将括号外的项与括号内的每一项相乘。a(b + c) = ab + ac。
find the highest common factor and write it outside the bracket. e.g., 6x + 9 = 3(2x + 3).
因式分解是去括号的逆运算：找出最大公因数并写在括号外。例如 6x + 9 = 3(2x + 3)。
eplace letters with numbers, then use BIDMAS. If x = 2, 3x² − 5 = 3(2)² − 5 = 3×4 − 5 = 7.
代入：将字母替换为数字，然后按照 BIDMAS 计算。若 x = 2，3x² − 5 = 3(2)² − 5 = 3×4 − 5 = 7。
5. Linear Equations | 线性方程
A linear equation has the unknown raised to the power of 1. The goal is to isolate the unknown by performing the same operation on both sides.
线性方程中未知数的次数为 1。目标是通过在等号两边进行相同运算，将未知数单独解出。
Inverse operations: addition ↔ subtraction, multiplication ↔ division.
逆运算：加法与减法互逆，乘法与除法互逆。
Example: Solve 2x + 3 = 11. Subtract 3: 2x = 8. Divide by 2: x = 4.
示例：解方程 2x + 3 = 11。两边减 3：2x = 8。两边除以 2：x = 4。
Always check your solution by substituting it back into the original equation.
始终将解代入原方程进行检验。
6. Ratio and Proportion | 比与比例
can be simplified by dividing all parts by their highest common factor. e.g., 12:8 = 3:2.
比用于比较两个或多个量。可将比的各项除以它们的最大公因数进行化简。例如 12:8 = 3:2。
o, find the total number of parts, then divide. For ratio 3:2, total parts = 5. A share of £50: 3 parts = (3/5)×50 = £30, 2 parts = £20.
按比例分配：求出总份数，再分配。例如按 3:2 分配 £50，总份数 = 5，3 份 = (3/5)×50 = £30，2 份 = £20。
t proportion: y = kx, where k is the constant of proportionality. As x doubles, y doubles.
正比例：y = kx，k 是比例常数。x 翻倍时，y 也翻倍。
Using the unitary method: find the value of one unit first, then multiply to find the required amount.
单位法：先求出单一量，再乘以所需份数。
7. Geometry: Angles and Lines | 几何：角与线
Angles on a straight line sum to 180°.
平角（直线上的角）之和为 180°。
Angles around a point sum to 360°.
绕一点的周角之和为 360°。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle sum to 180°.
三角形内角和为 180°。
Angles in a quadrilateral sum to 360°.
四边形内角和为 360°。
For parallel lines: alternate angles are equal; corresponding angles are equal; co-interior (allied) angles sum to 180°.
平行线性质：内错角相等；同位角相等；同旁内角之和为 180°。
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°.
锐角小于 90°，直角为 90°，钝角介于 90° 到 180° 之间，优角大于 180°。
8. Perimeter, Area, and Volume | 周长、面积与体积
Perimeter of a rectangle: P = 2(l + w) or P = 2l + 2w.
矩形的周长：P = 2(l + w) 或 P = 2l + 2w。
Area of a rectangle: A = l × w.
矩形的面积：A = l × w。
Area of a triangle: A = ½ × b × h, where b is base and h is perpendicular height.
三角形的面积：A = ½ × b × h，其中 b 为底，h 为垂直高。
Area of a parallelogram: A = b × h.
平行四边形的面积：A = b × h。
Area of a trapezium: A = ½ × (a + b) × h, where a and b are the parallel sides.
梯形的面积：A = ½ × (a + b) × h，其中 a 和 b 为平行边。
Circumference of a circle: C = πd or C = 2πr (π ≈ 3.14 or ²²/₇).
圆的周长：C = πd 或 C = 2πr（π ≈ 3.14 或 ²²/₇）。
Area of a circle: A = πr².
圆的面积：A = πr²。
Volume of a cuboid (rectangular prism): V = l × w × h.
长方体（矩形棱柱）的体积：V = l × w × h。
Volume of a cube: V = s³, where s is the side length.
立方体的体积：V = s³，其中 s 为棱长。
9. Coordinates and Graphs | 坐标与图表
A point is written as (x, y). The x-coordinate gives the horizontal position, the y-coordinate gives the vertical position.
点的坐标记作 (x, y)。x 坐标表示水平位置，y 坐标表示垂直位置。
The midpoint of two points (x₁, y₁) and (x₂, y₂) is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 和 (x₂, y₂) 的中点坐标为 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
graphs are often in the form y = mx + c, where m is the gradient and c is the y-intercept.
直线图像通常形如 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Gradient between two points: m = (y₂ − y₁) / (x₂ − x₁).
两点间的斜率（梯度）：m = (y₂ − y₁) / (x₂ − x₁)。
Horizontal lines have equation y = constant, vertical lines have equation x = constant.
水平线的方程为 y = 常数，竖直线的方程为 x = 常数。
10. Statistics: Averages and Charts | 统计：平均数与图表
all values and dividing by the number of values: Mean = (sum of data) ÷ (number of data).
平均数（均值）是将所有数据值相加再除以数据个数：均值 = (数据总和) ÷ (数据个数)。
The median is the middle value when the data are arranged in order. If there are two middle values, take their mean.
中位数是将数据排序后中间的那个值。如果有两个中间值，则取它们的均值。
The mode is the value that appears most frequently. A set may have one mode, more than one mode, or no mode.
众数是出现次数最多的值。一组数据可能有一个众数、多个众数或无众数。
The range is the difference between the largest and smallest values: Range = max − min.
极差是最大值与最小值的差：极差 = 最大值 − 最小值。
For frequency tables, mean = Σ(fx) / Σf, where x is the data value and f is its frequency.
对于频数表，均值 = Σ(fx) / Σf，其中 x 为数据值，f 为对应的频数。
ions (each sector angle = (frequency/total) × 360°), and line graphs for trends over time.
常用图表：条形图用于分类数据，饼图显示比例（每个扇形角 = (频数/总数) × 360°），折线图展示随时间变化的趋势。
11. Probability | 概率
Probability measures how likely an event is to happen. It is always between 0 (impossible) and 1 (certain).
概率衡量事件发生的可能性，总是在 0（不可能）到 1（必然）之间。
ility of an event = (number of favourable outcomes) / (total number of possible outcomes).
事件的概率 = (有利结果的数量) / (所有可能结果的总数)。
d up to 1. If the probability of rain is 0.3, the probability of no rain is 1 − 0.3 = 0.7.
所有可能结果的概率之和为 1。如果下雨的概率是 0.3，则不下雨的概率为 1 − 0.3 = 0.7。
The sample space lists all possible outcomes. For rolling a fair six-sided die, the probability of rolling a 4 is ⅙.
样本空间列出所有可能的结果。抛掷一枚公平的六面骰子，掷出 4 的概率为 ⅙。
of times event occurs) / (total number of trials). It becomes more accurate with more trials.
实验概率 = (事件发生的次数) / (试验总次数)。试验次数越多，结果越准确。
12. Sequences | 数列
A sequence is an ordered list of numbers. Each number is called a term.
数列是按顺序排列的一列数。每一个数称为一项。
difference between consecutive terms. e.g., 3, 7, 11, 15, ... has common difference d = 4.
等差数列中相邻两项的差为常数。例如：3, 7, 11, 15, ... 的公差 d = 4。
nth term of an arithmetic sequence can be written as: nth term = first term + (n − 1) × d.
等差数列的第 n 项可表示为：第 n 项 = 首项 + (n − 1) × d。
For the sequence 5, 8, 11, 14, ... the nth term is 5 + (n − 1)×3 = 3n + 2.
对于数列 5, 8, 11, 14, ... 第 n 项为 5 + (n − 1)×3 = 3n + 2。
Term-to-term rule describes how to get from one term to the next (e.g., 'add 3'). The position-to-term rule (nth term) links the term directly to its position.
邻项规则描述了如何从一项得到下一项（例如“加 3”）。位置规则（第 n 项公式）则将项直接与其位置关联。
a common ratio between consecutive terms. e.g., 2, 6, 18, 54, ... has common ratio r = 3.
等比数列中相邻两项之比为常数。例如 2, 6, 18, 54, ... 的公比 r = 3。
9, 16, 25, ... nth term = n². Triangle numbers: 1, 3, 6, 10, 15, ... nth term = n(n+1)/2.
平方数数列：1, 4, 9, 16, 25, ... 第 n 项 = n²。三角形数数列：1, 3, 6, 10, 15, ... 第 n 项 = n(n+1)/2。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图