📚 Year 8 CIE Mathematics Formula & Theorem Quick Reference Handbook | 剑桥Year 8数学公式定理速查手册
This quick-reference guide compiles all essential formulas, theorems, and key facts needed for Year 8 CIE Mathematics. It is designed to help students revise efficiently before tests and examinations. Each section presents points in pairs of English and Chinese, making it suitable for bilingual learners. Keep this handbook handy for daily practice and last-minute checks.
本速查手册汇集了Year 8 CIE数学所需的所有核心公式、定理和关键知识点，旨在帮助学生考前高效复习。每个部分都以英中双语成对展示要点，适合双语学习者。请将本手册放在手边，用于日常练习和考前快速回顾。
1. Number Properties and Operations | 数的性质与运算
Place value: In a whole number, each digit has a value ten times greater than the digit to its right. For example, in 3472, the digit 4 represents 400.
数位：在一个整数中，每个数位的值是其右边数位的十倍。例如，在 3472 中，数字 4 表示 400。
Order of operations (BIDMAS/BODMAS): Brackets, Indices (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right).
运算顺序（BIDMAS/BODMAS）：括号、指数（幂）、除法和乘法（从左到右）、加法和减法（从左到右）。
Factors and multiples: A factor divides exactly into a number. A multiple is the product of a number and an integer. The highest common factor (HCF) and lowest common multiple (LCM) are found by listing factors or prime factorisation.
因数与倍数：因数是能整除某数的数，倍数是某数与整数的乘积。最大公因数 (HCF) 和最小公倍数 (LCM) 可以通过列举因数或质因数分解求得。
Prime numbers: A prime number has exactly two distinct factors: 1 and itself. (2, 3, 5, 7, 11, …). 1 is not prime.
质数：质数恰好有两个不同的因数：1 和它本身（如 2, 3, 5, 7, 11 …）。1 不是质数。
Integers and directed numbers: Adding a negative is the same as subtracting its opposite. The product of two negative numbers is positive.
整数与有向数：加上一个负数等同于减去它的相反数。两个负数相乘得正数。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Equivalent fractions: Multiply or divide both numerator and denominator by the same non‑zero number to get an equivalent fraction.
等值分数：将分子和分母同时乘以或除以同一个非零数，即可得到等值分数。
Adding/subtracting fractions: Rewrite fractions with a common denominator, then add or subtract the numerators. Simplify if possible.
分数加减：先通分使分母相同，再将分子相加或相减，最后化简。
Multiplying fractions: Multiply numerators together and denominators together. Cancel common factors first to simplify.
分数乘法：分子乘分子，分母乘分母。先约分能简化计算。
Dividing fractions: Multiply by the reciprocal (flip the second fraction).
分数除法：乘以除数的倒数（将第二个分数的分子分母颠倒后相乘）。
Converting between fractions, decimals and percentages: To change a decimal to a percentage, multiply by 100. Percentage → decimal: divide by 100.
分数、小数与百分数互化：分数化小数用分子除以分母；小数化百分数乘以100；百分数化小数除以100。
Finding a percentage of a quantity: Multiply the quantity by the percentage as a decimal. (e.g., 15% of 80 = 0.15 × 80).
求一个量的百分之几：将该量乘以百分数对应的小数（例如，80的15% = 0.15 × 80）。
3. Ratios and Proportions | 比和比例
Ratios compare quantities: The ratio a : b means for every a units of the first quantity there are b units of the second. Simplify ratios by dividing by common factors.
比用来比较数量：a : b 表示第一量每 a 个单位对应第二量 b 个单位。通过除以公因数化简比。
Dividing a quantity in a given ratio: Add the parts of the ratio to find the total number of shares. Divide the quantity by the total shares and multiply by each part.
按给定比分配：将比的各项相加得出总份数，用总量除以总份数，再乘以各项份数得到各部分。
Direct proportion: Two quantities are directly proportional if their ratio remains constant. If y ∝ x, then y = kx for a constant k.
正比例：若两个量的比值保持不变，则它们成正比例关系。若 y ∝ x，则 y = kx（k 为常数）。
Scale drawings and maps: The scale is a ratio comparing the length on the drawing to the real length. For example, 1 : 100 means 1 cm on the map represents 100 cm in reality.
比例尺与地图：比例尺是比较图上长度与实际长度的比。例如 1 : 100 表示图上 1 cm 代表实际 100 cm。
4. Algebraic Expressions and Simplification | 代数表达式与化简
Using letters for numbers: A variable represents an unknown or changing quantity. Expressions like 3x + 2y contain terms (numbers and variables combined by +, −).
用字母表示数：变量表示未知或变化的量。表达式如 3x + 2y 包含项（数字和变量通过 +、− 组合）。
Collecting like terms: arts can be combined. For example, 5a + 3a = 8a, but 5a + 3b cannot be simplified further.
合并同类项：只有变量部分完全相同的项才能合并。例如 5a + 3a = 8a，但 5a + 3b 不能进一步化简。
Expanding brackets: : Multiply each term inside the bracket by the term outside. Example: 3(x + 4) = 3x + 12; 2(3y − 5) = 6y − 10.
展开括号：将括号外的项乘以括号内的每一项。例：3(x + 4) = 3x + 12; 2(3y − 5) = 6y − 10。
Factorising simple expressions: est common factor of all terms and write it outside a bracket. Example: 4x + 8 = 4(x + 2).
简单因式分解：找出所有项的最高公因式并写在括号外。例：4x + 8 = 4(x + 2)。
5. Linear Equations | 线性方程
Solving equations: The goal is to isolate the variable on one side. Perform the same operation on both sides to maintain balance (add, subtract, multiply, divide).
解方程：目标是将变量孤立在等号一侧。在等式两边同时进行相同的运算（加、减、乘、除）以保持平衡。
One-step equations: Example: x + 7 = 12 → subtract 7 from both sides → x = 5.
一步方程：例：x + 7 = 12 → 两边同减 7 → x = 5。
Two-step equations: Example: 2x − 3 = 9 → add 3 to both sides → 2x = 12 → divide by 2 → x = 6.
两步方程：例：2x − 3 = 9 → 两边同加 3 → 2x = 12 → 两边同除以 2 → x = 6。
Equations with brackets: Expand brackets first, then solve. Example: 2(x + 5) = 18 → 2x + 10 = 18 → 2x = 8 → x = 4.
含括号的方程：先展开括号，再求解。例：2(x + 5) = 18 → 2x + 10 = 18 → 2x = 8 → x = 4。
6. Geometry: Angles and Lines | 几何：角与线
Types of angles: , Right (= 90°), Obtuse (> 90° and < 180°), Straight (= 180°), Reflex (> 180° but < 360°).
角的类型：锐角（< 90°），直角（= 90°），钝角（> 90° 且 < 180°），平角（= 180°），优角（> 180° 且 < 360°）。
Angles on a straight line add up to 180°. Angles around a point add up to 360°.
直线上的角之和为 180°。一点周围的角之和为 360°。
Vertically opposite angles are equal when two lines intersect.
对顶角相等（两直线相交时）。
Angles in parallel lines: Alternate angles are equal. Corresponding angles are equal. Co‑interior (allied) angles sum to 180°.
平行线中的角：内错角相等，同位角相等，同旁内角互补（和为 180°）。
Angle sum of a triangle 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和 = 180°。三角形的外角等于与它不相邻的两个内角之和。
7. Geometry: 2D Shapes | 几何：二维图形
Triangles by sides: Equilateral (3 equal sides, 3 equal 60° angles), Isosceles (2 equal sides, 2 equal base angles), Scalene (no equal sides).
按边分类的三角形：等边三角形（三边相等，三个 60° 角），等腰三角形（两腰相等，两底角相等），不等边三角形（各边不等）。
Quadrilaterals: Square (4 equal sides, 4 right angles), Rectangle (opposite sides equal, 4 right angles), Parallelogram (opposite sides parallel and equal, opposite angles equal), Rhombus (4 equal sides, opposite angles equal), Trapezium (one pair of parallel sides), Kite (two pairs of adjacent equal sides).
四边形：正方形（四边相等，四直角），长方形（对边相等，四直角），平行四边形（对边平行且相等，对角相等），菱形（四边相等，对角相等），梯形（一组对边平行），风筝形（两对邻边相等）。
Perimeter: The total length around the shape. For a rectangle: P = 2(l + w). For a square: P = 4s.
周长：图形一圈的总长度。长方形周长：P = 2(长 + 宽)。正方形周长：P = 4 × 边长。
Area formulas: : A = ½ × base × height. Parallelogram: A = base × height. Trapezium: A = ½ × (a + b) × h.
面积公式：长方形：A = 长 × 宽。正方形：A = 边长²。三角形：A = ½ × 底 × 高。平行四边形：A = 底 × 高。梯形：A = ½ × (上底 + 下底) × 高。
Circle parts and formulas: iameter d = 2r. Circumference C = πd = 2πr. Area A = πr². (Approximate π as 3.14 or 22/7.)
圆的组成部分与公式：半径 r，直径 d = 2r。周长 C = πd = 2πr。面积 A = πr²。(π 近似取 3.14 或 22/7。)
8. Geometry: 3D Shapes and Volume | 几何：三维图形与体积
Common 3D shapes: Cube, cuboid, prism, cylinder, pyramid, cone, sphere.
常见三维图形：立方体、长方体、棱柱、圆柱、棱锥、圆锥、球体。
Volume of a cuboid: V = length × width × height. For a cube: V = s³.
长方体体积：V = 长 × 宽 × 高。立方体体积：V = 边长³。
Volume of a prism: V = area of cross‑section × length. This includes triangular prisms and cylinders (cylinder: V = πr²h).
棱柱体积：V = 横截面积 × 长度。适用于三棱柱和圆柱（圆柱：V = πr²h)。
Surface area: areas of all faces. For a cuboid: SA = 2(lw + lh + wh). For a cylinder: SA = 2πr² + 2πrh.
表面积：所有面的面积之和。长方体表面积：SA = 2(lw + lh + wh)。圆柱表面积：SA = 2πr² + 2πrh。
9. Coordinates and Graphs | 坐标与图像
Coordinate plane: The x‑axis is horizontal, the y‑axis is vertical. A point is written as (x, y). The origin is (0, 0).
坐标平面：x 轴为水平轴，y 轴为垂直轴。点的记法为 (x, y)。原点为 (0, 0)。
Plotting linear graphs: For an equation like y = mx + c, choose several x‑values, calculate the matching y‑values, and plot the ordered pairs. Connect them with a straight line.
绘制线性图像：对于 y = mx + c 这样的方程，选择若干个 x 值，计算对应的 y 值，描点后用直线连接。
Gradient (slope): change in x). A positive gradient slopes up to the right; a negative gradient slopes down to the right.
斜率：斜率 = 垂直变化 / 水平变化 = (y 的变化量) / (x 的变化量)。正斜率向右上方倾斜；负斜率向右下方倾斜。
Midpoint between two points (x₁, y₁) and (x₂, y₂): x-coordinate = (x₁ + x₂)/2, y-coordinate = (y₁ + y₂)/2.
两点间的中点：(x₁, y₁) 和 (x₂, y₂) 的中点 x 坐标 = (x₁ + x₂)/2，y 坐标 = (y₁ + y₂)/2。
10. Statistics | 统计
Mean (average): Sum of all data values divided by the number of values.
平均数：所有数据值的总和除以数据的个数。
Median: The middle value when the data set is ordered. If there is an even number of values, take the mean of the two middle values.
中位数：数据按大小顺序排列后位于中间的数值。若数据个数为偶数，则取中间两个数的平均数。
Mode: The value that appears most frequently. A set may have no mode, one mode, or more than one.
众数：出现次数最多的数据值。一组数据可能没有众数、有一个或多个众数。
Range: Largest value minus smallest value. It measures the spread of the data.
极差：最大值减最小值。用于衡量数据的离散程度。
Frequency tables: Mean from a frequency table: multiply each data value by its frequency, sum, then divide by total frequency.
频率表：由频率表求平均数：将每个数据值乘以其频数，求和后除以总频数。
11. Probability | 概率
Probability scale: From 0 (impossible) to 1 (certain). Probability can be expressed as a fraction, decimal, or percentage.
概率尺度：从 0（不可能）到 1（必然）。概率可以用分数、小数或百分数表示。
Basic probability: vourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
基本概率计算：事件发生的概率 = 有利结果的数量 / 所有可能结果的总数，前提是所有结果是等可能的。
Complementary events: The probability of an event not happening = 1 − probability that it does happen.
互补事件：事件不发生的概率 = 1 − 事件发生的概率。
Sample space diagrams list all possible outcomes. They help calculate probabilities for combined events.
样本空间图列出所有可能的结果，有助于计算组合事件的概率。
12. Pythagoras Theorem | 勾股定理
Right‑angled triangle: The side opposite the right angle is the hypotenuse, the longest side.
直角三角形：直角所对的边是斜边，为最长边。
Pythagoras theorem: In a right‑ , where c is the length of the hypotenuse and a, b are the lengths of the other two sides.
勾股定理：在直角三角形中，a² + b² = c²，其中 c 为斜边长，a 和 b 为另外两条直角边长。
Finding the hypotenuse: c = √a² + b². For example, if a = 3, b = 4, then c = √3² + 4² = √25 = 5.
求斜边：c = √a² + b²。例：若 a = 3，b = 4，则 c = √3² + 4² = √25 = 5。
Finding a shorter side: a = √c² − b². Example: If c = 13, b = 5, then a = √13² − 5² = √169 − 25 = √144 = 12.
求直角边：a = √c² − b²。例：若 c = 13，b = 5，则 a = √13² − 5² = √169 − 25 = √144 = 12。
Pythagorean triples: Sets of integers that satisfy a² + b² = c², like (3, 4, 5), (5, 12, 13), (6, 8, 10).
勾股数：满足 a² + b² = c² 的整数组，如 (3, 4, 5)、(5, 12, 13)、(6, 8, 10)。
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