📚 Year 9 CIE Maths: Formula & Theorem Quick Reference Handbook | CIE 数学公式定理速查手册
This comprehensive quick reference handbook covers all the essential formulas and theorems for the Year 9 CIE Lower Secondary Mathematics curriculum. Use it as a revision aid to master key concepts in numbers, algebra, geometry, measurement, statistics, and probability. Each formula is presented with clear explanations and practical examples.
这本全面的速查手册涵盖了 Year 9 CIE 初中数学课程中的所有重要公式与定理。你可以将其作为复习工具，轻松掌握数、代数、几何、测量、统计与概率等关键概念。每一个公式都配有清晰的解释和实用示例。
1. Number Facts | 数与算术
, 23 and 29. Prime factorisation expresses a number as a product of prime factors, e.g. 60 = 2² × 3 × 5.
质数是大于 1 的自然数，除了 1 和它本身以外不再有其他因数。最初的几个质数是 2、3、5、7、11、13、17、19、23 和 29。质因数分解是将一个数写成质因数乘积的形式，例如 60 = 2² × 3 × 5。
LCM) is the smallest number that is a multiple of each. For 12 and 18, HCF = 6 and LCM = 36.
两个或多个数的最大公因数 (HCF) 是能够整除这些数的最大数。最小公倍数 (LCM) 是这些数的倍数中最小的一个。对于 12 和 18，HCF = 6，LCM = 36。
and 0.0032 = 3.2 × 10⁻³. This is particularly useful for very large or very small numbers.
标准形式（科学记数法）将一个数写成 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。例如 4 500 = 4.5 × 10³，而 0.0032 = 3.2 × 10⁻³。这种形式在处理极大或极小数时非常有用。
When rounding numbers, we look at the digit immediately to the right of the place value we are rounding to. If it is 5 or more, we round up; otherwise we round down. Significant figures count all non‑zero digits, zeros between non‑zero digits and trailing zeros after a decimal point.
进行四舍五入时，我们要看所要取舍数位右边的那一位数字。如果是 5 或更大，则进位；否则舍去。有效数字包括所有非零数字、非零数字之间的零以及小数点后的末尾零。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
gits; a recurring decimal has one or more digits that repeat indefinitely, such as 1/3 = 0.333...
将分数转化为小数，用分子除以分母。例如 3/8 = 0.375。有限小数有有限位数；循环小数则有一位或几位无限重复，如 1/3 = 0.333…。
To change a decimal to a percentage, multiply by 100 and add the % sign. Conversely, to change a percentage to a decimal, divide by 100. Hence 0.85 = 85% and 7% = 0.07.
将小数转化为百分比，只需乘以 100 并加上百分号。反之，将百分比化为小数，则除以 100。因此 0.85 = 85%，7% = 0.07。
centage of an amount, multiply the amount by the percentage as a decimal. For instance, 15% × 0.15 = 30. Percentage increase or decrease is calculated using (change ÷ original) × 100%.
求一个数量的百分比时，把该数量乘以转化为小数的百分比。例如，200 的 15% 是 200 × 0.15 = 30。百分比增减使用（变化量 ÷ 原值）× 100% 计算。
3. Algebraic Expressions | 代数表达式
For double brackets use the FOIL method: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6.
展开括号时，用括号外的项乘以括号内的每一项。例如 3(2x + 5) = 6x + 15。对于两个括号相乘，使用 FOIL 方法：(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6。
Key identities for squaring binomials are essential:
(a + b)² = a² + 2ab + b²
二次展开的关键恒等式：
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a – b)² = a² – 2ab + b²
The difference of two squares factors as:
a² – b² = (a + b)(a – b)
平方差公式因式分解为：
a² – b² = (a + b)(a – b)
s with four terms can often be factorised by grouping: ax + ay + bx + by = (a + b)(x + y).
因式分解是展开的逆过程。始终先提取公因数。例如 4x + 12 = 4(x + 3)。含有四项的表达式常可通过分组分解：ax + ay + bx + by = (a + b)(x + y)。
4. Equations and Inequalities | 方程与不等式
8 → x = 4. Always check your solution by substituting it back into the original equation.
解一元一次方程时，对等式两边进行相同的运算以分离未知数。例如 2x + 5 = 13 → 2x = 8 → x = 4。始终将解代入原方程验算。
appears on both sides, collect all terms containing the variable to one side. Example: 5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4.
当未知数同时出现在等式两边时，将含变量的项全部移到一边。例：5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4。
Inequalities are solved like equations, but remember: multiplying or dividing by a negative number reverses the inequality sign. The solution can be shown on a number line with open or closed circles.
不等式的解法与方程类似，但切记：乘以或除以负数时，不等号方向要反转。解集可在数轴上用空心或实心圆圈表示。
5. Sequences | 数列
An arithmetic sequence has a constant difference, d, between consecutive terms. The nth term of an arithmetic sequence is given by:
aₙ = a₁ + (n – 1)d
等差数列相邻两项的差为常数 d。其第 n 项公式为：
aₙ = a₁ + (n – 1)d
r the sequence 5, 8, 11, 14, ..., the first term is 5 and d = 3, so the nth term is 5 + (n – 1)×3 = 3n + 2.
其中 a₁ 为首项，n 为项数的位置。对于数列 5, 8, 11, 14, ...，首项为 5，公差 d = 3，因此第 n 项为 5 + (n – 1)×3 = 3n + 2。
Recognising patterns in sequences of squares, cubes or triangles also helps. The sequence of square numbers is 1, 4, 9, 16, ... and its nth term is n². Triangular numbers follow the rule n(n+1)/2.
识别平方数、立方数或三角形数等数列的规律同样重要。平方数数列为 1, 4, 9, 16, ...，其第 n 项是 n²。三角形数遵循公式 n(n+1)/2。
6. Ratio, Proportion and Rates | 比、比例与速率
products are equal: ad = bc. This property is used to find missing values in proportions.
比用于比较两个或多个量。如果 a : b = c : d，则交叉乘积相等：ad = bc。此性质可用于求比例中的未知值。
nship is y = kx, where k is the constant of proportionality. For example, if 5 apples cost $2, then 20 apples cost $8.
正比例意味着一个量增加时，另一个量以相同的比率增加。关系式为 y = kx，其中 k 是比例常数。例如，5 个苹果 2 美元，那么 20 个苹果 8 美元。
Speed, distance and time are linked by the rate formula:
Speed = Distance ÷ Time
速度、距离和时间通过速率公式联系：
速度 = 距离 ÷ 时间
nd pressure are other common compound rates: Density = Mass/Volume; Pressure = Force/Area.
密度和压强是另外两个常见复合速率：密度 = 质量/体积；压强 = 力/面积。
7. Geometry: Angles and Lines | 几何：角与线
Vertically opposite angles are equal. Angles on a straight line sum to 180°, and angles around a point sum to 360°.
对顶角相等。平角（直线上的角）之和为 180°，绕一点一周的角之和为 360°。
When parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles sum to 180°. These angle facts are fundamental for solving geometry problems.
当平行线被一条截线所截时，同位角相等，内错角相等，同旁内角互补（和为 180°）。这些角的关系是解决几何问题的基础。
quadrilateral, the sum is 360°. The general formula for the sum of interior angles of an n‑sided polygon is (n – 2) × 180°.
任意三角形的内角和为 180°。任意四边形的内角和为 360°。n 边形内角和的一般公式是 (n – 2) × 180°。
8. Pythagoras' Theorem | 勾股定理
In a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
c² = a² + b²
在直角三角形中，斜边的平方等于两条直角边的平方和：
c² = a² + b²
the longest side, opposite the right angle). To find the length of the hypotenuse, use c = √a² + b². To find a shorter side, rearrange: a = √c² – b².
其中 c 是斜边（最长的边，直角所对的边）。求斜边长度时，使用 c = √a² + b²。求直角边时，移项得 a = √c² – b²。
Pythagoras' theorem is widely used to check if a triangle is right‑angled: if a² + b² = c² holds, then the triangle is right‑angled. It also applies to finding distances in coordinate geometry and real‑life problems.
勾股定理常用于检验三角形是否为直角三角形：若 a² + b² = c² 成立，则该三角形为直角三角形。它还可用于坐标几何中的距离计算以及现实生活问题。
9. Perimeter, Area and Volume | 周长、面积与体积
e: P = 2(l + w). Area of a rectangle: A = l × w. For a triangle: Area = ½ × base × height.
矩形周长：P = 2(长 + 宽)。矩形面积：A = 长 × 宽。三角形面积 = ½ × 底 × 高。
)h, where a and b are the parallel sides and h is the perpendicular distance between them.
平行四边形面积：A = 底 × 垂直高。梯形面积：A = ½ (a + b)h，其中 a 与 b 为两平行边，h 为它们之间的垂直距离。
2πr or C = πd. Area of a circle: A = πr². Here r is the radius and d = 2r is the diameter.
圆的周长 C = 2πr 或 C = πd。圆面积 A = πr²。其中 r 为半径，d = 2r 为直径。
Volume of a prism (including cylinder): V = area of cross‑ = ⅓ × base area × height. Spheres are introduced at higher tiers, with volume V = ⁴⁄₃ πr³.
棱柱（含圆柱）体积：V = 横截面积 × 长度。圆柱体积 V = πr²h。棱锥或圆锥体积：V = ⅓ × 底面积 × 高。球体在更高层级中引入，体积 V = ⁴⁄₃ πr³。
10. Transformations | 变换
A translation moves a shape by a vector (x y) without rotating, reflecting or resizing. The shape remains congruent to the original.
平移通过向量 (x y) 移动图形，不进行旋转、反射或缩放。图形与原图全等。
A reflection flips the shape across a mirror line. Each point and its image are the same perpendicular distance from the mirror line. On a coordinate grid, reflecting in the x‑axis changes the sign of the y‑coordinate; reflecting in the y‑axis changes the sign of the x‑coordinate.
反射将图形沿镜线翻转。每个点与其像到镜线的垂直距离相等。在坐标网格上，关于 x 轴反射会改变 y 坐标的符号；关于 y 轴反射会改变 x 坐标的符号。
A rotation turns the shape about a fixed centre. Rotation is described by the angle (90°, 180°, 270°) and direction (clockwise or anticlockwise). An enlargement changes the size of a shape by a scale factor from a centre of enlargement. If the scale factor is greater than 1, the shape becomes larger; if between 0 and 1, it becomes smaller.
旋转将图形绕一个固定中心转动。旋转由角度（90°、180°、270°）和方向（顺时针或逆时针）描述。放大从放大中心按比例因子改变图形大小。比例因子大于 1 时图形变大；介于 0 和 1 之间时图形
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