📚 Year 7 CAIE Maths: Formula & Theorem Quick Reference | Year 7 CAIE 数学：公式定理速查手册
This quick-reference handbook summarises the essential formulae and theorems covered in the Year 7 CAIE Mathematics curriculum. It is designed for rapid revision and to help students recall key facts confidently.
本速查手册总结了 Year 7 CAIE 数学课程中的核心公式和定理，旨在帮助同学们快速复习，自信回忆关键知识点。
1. Number Types and Operations | 数的种类与运算
Natural numbers are the counting numbers: 1, 2, 3, … They are positive whole numbers.
自然数是计数的数字：1, 2, 3, …，它们是正整数。
Integers include all whole numbers, both positive and negative, and zero: …, -2, -1, 0, 1, 2, …
整数包括所有正整数、负整数和零：…, -2, -1, 0, 1, 2, …
Addition and multiplication follow the commutative law: a + b = b + a and a × b = b × a.
加法和乘法遵循交换律：a + b = b + a，以及 a × b = b × a。
The associative law holds: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
结合律成立：(a + b) + c = a + (b + c)，且 (a × b) × c = a × (b × c)。
The distributive law connects multiplication and addition: a × (b + c) = a × b + a × c.
分配律将乘法与加法联系起来：a × (b + c) = a × b + a × c。
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. (For example, 2³ × 2² = 2⁵.)
同底数的幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。如 2³ × 2² = 2⁵。
The order of operations is BIDMAS: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right).
运算顺序遵循 BIDMAS：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
Zero as an additive identity: a + 0 = a. One as a multiplicative identity: a × 1 = a.
零是加法恒等元：a + 0 = a。1 是乘法恒等元：a × 1 = a。
2. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a whole number that divides another number exactly without a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, 12.
因数是可以整除另一个数的整数。例如，12 的因数有 1, 2, 3, 4, 6, 12。
A multiple is the product of a number and any integer. The first few multiples of 5 are 5, 10, 15, 20, …
倍数是一个数与任意整数的乘积。5 的前几个倍数是 5, 10, 15, 20, …
A prime number has exactly two distinct positive factors: 1 and itself. The first five primes are 2, 3, 5, 7, 11. Note that 1 is not a prime.
质数恰好有两个不同的正因数：1 和它本身。前五个质数是 2, 3, 5, 7, 11。注意 1 不是质数。
A composite number has more than two factors. For example, 10 is composite (factors 1, 2, 5, 10).
合数的因数多于两个。例如 10 是合数（因数：1, 2, 5, 10）。
Every composite number can be written as a product of prime factors. This is called prime factorisation.
每个合数都可以写成质因数的乘积，这叫做质因数分解。
The Highest Common Factor (HCF) of two numbers is the largest factor they share. The Lowest Common Multiple (LCM) is the smallest multiple they share.
两个数的最大公因数 (HCF) 是它们共有的最大因数。最小公倍数 (LCM) 是它们共有的最小倍数。
3. Fractions, Decimals and Percentages | 分数、小数与百分数
To convert a fraction to a decimal, divide the numerator by the denominator: ⅜ = 3 ÷ 8 = 0.375.
分数化小数：用分子除以分母。⅜ = 3 ÷ 8 = 0.375。
To convert a decimal to a percentage, multiply by 100: 0.45 = 0.45 × 100% = 45%.
小数化百分数：乘以 100。0.45 = 0.45 × 100% = 45%。
To add or subtract fractions, find a common denominator first. a/b + c/d = (ad + bc)/bd.
分数加减时，先通分。a/b + c/d = (ad + bc)/bd。
Multiplying fractions: a/b × c/d = (a × c) / (b × d). Simplify before multiplying if possible.
分数乘法：a/b × c/d = (a × c) / (b × d)。能约分的先约分再乘。
Dividing by a fraction is the same as multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.
除以一个分数等于乘它的倒数：a/b ÷ c/d = a/b × d/c。
To find a percentage of a quantity, write the percentage as a fraction over 100 and multiply. For example, 20% of 80 = (20/100) × 80 = 16.
求一个数的百分之几：将百分数写成分母为 100 的分数再乘。如 80 的 20% = (20/100) × 80 = 16。
Percentage increase = (increased amount / original amount) × 100%. Percentage decrease works similarly.
增长百分比 = (增长量 / 原量) × 100%。减少百分比同理计算。
4. Algebraic Expressions | 代数表达式
An algebraic expression uses letters (variables) and numbers combined with operations. Example: 3x + 2y – 4.
代数表达式使用字母（变量）和数字，通过运算组合而成。例如 3x + 2y – 4。
A term is a single number, variable, or product of both. In 5x² + 3x – 7, the constant term is -7 and the coefficient of x is 3.
项是一个单独的数、变量或两者的乘积。在 5x² + 3x – 7 中，常数项是 -7，x 的系数是 3。
Like terms contain exactly the same variables raised to the same powers. Combine them by adding or subtracting the coefficients: 2a + 5a = 7a.
同类项含有完全相同的变量且次数相同。合并时系数相加减：2a + 5a = 7a。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. Also, (x + 2)(x + 3) = x² + 5x + 6 (using FOIL method).
去括号使用分配律：a(b + c) = ab + ac。同样，(x + 2)(x + 3) = x² + 5x + 6（使用 FOIL 法则）。
Factorising is the reverse of expanding. For example, 6x + 9 = 3(2x + 3). Always take out the highest common factor.
因式分解是展开的逆运算。例如 6x + 9 = 3(2x + 3)。一定要提取最大公因数。
5. Equations and Formulas | 方程与公式
An equation is a mathematical statement that two expressions are equal, using an equals sign. E.g., 2x + 3 = 11.
方程是使用等号表明两个表达式相等的数学陈述，例如 2x + 3 = 11。
To solve an equation, perform inverse operations to both sides to isolate the variable. x + 5 = 9 → x = 9 – 5 → x = 4.
解方程时，两边同时进行逆运算以分离变量。x + 5 = 9 → x = 9 – 5 → x = 4。
If an equation has division, multiply both sides by the denominator. x/4 = 3 → x = 12.
若方程中有除法，两边同乘分母。x/4 = 3 → x = 12。
A formula shows a relationship between quantities. For example, the area of a rectangle is given by A = l × w.
公式表示量之间的关系。例如矩形的面积由 A = l × w 给出。
Substitute known values into a formula to find an unknown. If l = 6 and w = 4, then A = 6 × 4 = 24 square units.
将已知值代入公式以求得未知量。若 l = 6, w = 4，则 A = 6 × 4 = 24 平方单位。
6. Ratios and Proportions | 比率与比例
A ratio compares two or more quantities. It can be written as a:b or a to b. All quantities must be in the same units.
比率用来比较两个或多个量。可写成 a:b 或 a 比 b。所有量的单位必须相同。
Simplify a ratio by dividing all parts by their HCF. For example, 12:8 simplifies to 3:2 (divide by 4).
化简比率：将各项除以它们的最大公因数。如 12:8 化简为 3:2（除以 4）。
To divide a quantity in a given ratio, first find the total number of parts. Split £50 in the ratio 2:3 → total parts = 5, so one part = £10; shares are £20 and £30.
按给定比率分配数量时，先求出总份数。将 £50 按 2:3 分配 → 总份数 = 5，每份 £10；得到的分别是 £20 和 £30。
Two quantities are in direct proportion if they increase or decrease together at the same rate. If y ∝ x, then y = kx for a constant k.
若两个量以相同的比率同时增大或减小，则它们成正比。若 y ∝ x，则 y = kx，k 为常数。
In a recipe for 4 people we need 2 eggs. For 10 people we can use proportion: (10/4) × 2 = 5 eggs.
一份供 4 人食用的食谱需要 2 个鸡蛋。供 10 人食用可按比例计算：(10/4) × 2 = 5 个鸡蛋。
7. Measurement and Conversions | 度量与换算
Metric length units: 1 cm = 10 mm; 1 m = 100 cm; 1 km = 1000 m.
公制长度单位：1 cm = 10 mm；1 m = 100 cm；1 km = 1000 m。
Metric mass units: 1 kg = 1000 g; 1 tonne = 1000 kg.
公制质量单位：1 kg = 1000 g；1 吨 = 1000 kg。
Metric capacity: 1 L = 1000 mL; 1 cm³ = 1 mL.
公制容量：1 L = 1000 mL；1 cm³ = 1 mL。
Time conversions: 1 minute = 60 seconds; 1 hour = 60 minutes; 1 day = 24 hours.
时间换算：1 分钟 = 60 秒；1 小时 = 60 分钟；1 天 = 24 小时。
To convert a larger unit to a smaller one, multiply. (1.2 m to cm: 1.2 × 100 = 120 cm). To convert smaller to larger, divide.
较大单位换算为较小单位需乘以进率。（如 1.2 m 换为 cm：1.2 × 100 = 120 cm）。较小单位换较大单位则除以进率。
Area and volume conversions: 1 m² = 10,000 cm² (100 × 100); 1 m³ = 1,000,000 cm³ (100 × 100 × 100).
面积和体积换算：1 m² = 10,000 cm² (100×100)；1 m³ = 1,000,000 cm³ (100×100×100)。
8. Angles and Lines | 角与线
An acute angle measures between 0° and 90°. A right angle is exactly 90°. An obtuse angle is between 90° and 180°.
锐角在 0° 到 90° 之间。直角恰好是 90°。钝角在 90° 到 180° 之间。
A straight angle is 180° and a full turn (complete angle) is 360°.
平角为 180°，周角（整圈）为 360°。
Vertically opposite angles are formed when two lines intersect; they are equal. If ∠A = 70°, the opposite angle is also 70°.
对顶角在两条直线相交时产生，它们相等。若 ∠A = 70°，对顶角也是 70°。
Angles on a straight line add up to 180°.
直线上角度之和为 180°。
Angles around a point add up to 360°.
绕一点一周的角度之和为 360°。
When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, interior (co-interior) angles sum to 180°.
当一条截线与平行线相交时：同位角相等，内错角相等，同旁内角之和为 180°。
The sum of interior angles of a triangle is always 180°.
三角形的内角和总是 180°。
The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形的一个外角等于与它不相邻的两个内角之和。
9. 2D Shapes: Perimeter and Area | 二维图形：周长与面积
Perimeter is the total length around a shape. It is measured in units such as cm or m.
周长是图形一周的总长度，单位为 cm、m 等。
| Shape 图形 | Perimeter 周长 | Area 面积 |
|---|---|---|
| Square (side s) | P = 4s | A = s² |
| Rectangle (length l, width w) | P = 2(l + w) | A = lw |
| Triangle (base b, height h) | Sum of three sides | A = ½ × b × h |
| Parallelogram (base b, height h) | Sum of all four sides | A = b × h |
| Trapezium (parallel sides a, b; height h) | Sum of four sides | A = ½(a + b)h |
| Circle (radius r) | C = 2πr or C = πd | A = πr² |
Use π ≈ 3.14 or the π key on a calculator unless told otherwise. Always include the correct unit: cm² for area, cm for perimeter.
除非另有说明，π 取 3.14 或用计算器上的 π 键。务必带对单位：面积用 cm²，周长用 cm。
10. 3D Shapes: Volume and Surface Area | 立体图形：体积与表面积
Volume measures the space inside a 3D object. It is given in cubic units: cm³, m³, etc.
体积测量三维物体内部的空间，单位为立方单位如 cm³、m³。
Surface area is the total area of all faces of a solid.
表面积是立体所有面的总面积。
Cube of edge length a: Volume V = a³. Surface area SA = 6a².
棱长为 a 的立方体：体积 V = a³，表面积 SA = 6a²。
Cuboid (rectangular prism) with length l, width w, height h: V = l × w × h. SA = 2(lw + lh + wh).
长 l、宽 w、高 h 的长方体：V = l × w × h。SA = 2(lw + lh + wh)。
Volume of any prism = area of cross-section × length.
所有棱柱的体积 = 横截面积 × 长度。
For a cylinder with radius r and height h: V = πr²h. (This may be introduced in Later Year 7/early Year 8, but the concept is helpful.)
半径为 r、高为 h 的圆柱体：V = πr²h。（这可能在高年级引入，但概念有助于理解。）
11. Data Handling: Averages and Range | 数据处理：平均数与极差
The mean (average) is found by adding all values and dividing by the number of values.
平均数（均值）的计算：将所有数值相加再除以数值的个数。
Mean = (sum of data values) / (number of values)
平均数 = (数据值之和) / (数据个数)
The median is the middle value when the data are arranged in order. For an even number of values, the median is the mean of the two middle numbers.
中位数是将数据排序后位于中间的值。若数据个数为偶数，中位数是中间两个数的平均值。
The mode is the value that appears most often. A data set can have one mode, more than one mode, or no mode at all.
众数是出现次数最多的值。一组数据可以有一个众数、多个众数，或没有众数。
The range is the difference between the largest and smallest values: Range = maximum – minimum.
极差是最大值与最小值之差：极差 = 最大值 – 最小值。
Example: The data 3, 7, 7, 2, 9 have mean = (3+7+7+2+9)/5 = 5.6, median = 7, mode = 7, range = 9 – 2 = 7.
例子：数据 3, 7, 7, 2, 9 的平均数 = (3+7+7+2+9)/5 = 5.6，中位数 = 7，众数 = 7，极差 = 9 – 2 = 7。
12. Coordinates and Graphs | 坐标与图像
A point in the coordinate plane is written as (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
坐标平面上的点记作 (x, y)，x 是到原点的水平距离，y 是垂直距离。
The origin is the point (0, 0). The x-axis is horizontal; the y-axis is vertical.
原点是 (0, 0)。x 轴为水平方向，y 轴为垂直方向。
Plot points by moving along the x-axis first, then up or down parallel to the y-axis.
描点时，先沿 x 轴移动，再平行于 y 轴上下移动。
A straight-line graph shows a constant rate of change. The general equation is y = mx + c, where m is the slope and c is the y-intercept. (This formal form is usually developed later, but the idea of plotting points from a rule is covered.)
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