📚 Year 7 CAIE Advanced Mathematics Formula & Theorem Quick Reference Manual | Year 7 CAIE 进阶数学公式定理速查手册
This concise quick-reference manual summarises the most important formulas, theorems and definitions you will encounter in the Year 7 CAIE Advanced Mathematics course. Use it alongside your class notes and textbook to reinforce key concepts and build confidence when tackling problems. Each section is presented in paired English and Chinese paragraphs so that you can study effectively in both languages.
这份简明速查手册汇总了你在 Year 7 CAIE 进阶数学课程中将遇到的最重要的公式、定理和定义。配合课堂笔记与课本使用，可以巩固核心概念，并在解题时建立信心。每一个要点均以英文与中文配对段落呈现，方便你进行双语学习。
1. Number Properties: Prime, HCF and LCM | 数的性质：质数、最大公约数、最小公倍数
A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and the number itself. The number 1 is not prime. Composite numbers have more than two factors.
质数是大于 1 且恰好只有两个不同因数的整数：1 和它本身。数字 1 不是质数。合数拥有两个以上的因数。
Fundamental Theorem of Arithmetic: Every integer greater than 1 can be written uniquely as a product of prime numbers (up to the order of the factors). This is called the prime factorisation.
算术基本定理：每一个大于 1 的整数都可以唯一地表示为质数的乘积（因数顺序除外）。这称为质因数分解。
To find the Highest Common Factor (HCF) of two or more numbers, list the prime factors of each number and multiply the smallest powers of common prime factors. To find the Lowest Common Multiple (LCM), multiply the highest powers of all prime factors that appear in any of the numbers.
求两个或多个数的最大公约数（HCF）时，列出每个数的质因数，再乘以各个公共质因数中指数最小的那个。求最小公倍数（LCM）时，则将所有出现过的质因数的最高次幂相乘。
The relationship HCF(a, b) × LCM(a, b) = a × b holds for any two positive integers a and b.
对于任意两个正整数 a 和 b，满足关系 HCF(a, b) × LCM(a, b) = a × b。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
A fraction represents a part of a whole. The numerator (top) indicates how many parts are taken; the denominator (bottom) shows the total number of equal parts. Equivalent fractions represent the same quantity and can be found by multiplying or dividing both terms by the same non-zero number.
分数表示整体的一部分。分子（上方）表示取了多少份；分母（下方）表示整体被等分成多少份。等值分数表示相同的量，可以通过将分子分母同乘或同除一个非零数而得到。
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with a power of ten as denominator and simplify.
将分数转换为小数时，用分子除以分母。将小数转换为分数时，将小数写成分母为 10 的幂次的分数，再化简。
To switch between percentages and decimals, multiply or divide by 100. To find a percentage of a quantity, multiply the quantity by the percentage expressed as a decimal or fraction.
在百分数和小数之间转换时，乘以或除以 100。要计算某个量的百分比，将该量乘以用小数或分数表示的百分比。
Fraction → Decimal: divide numerator by denominator. Decimal → Percentage: × 100
分数 → 小数：分子除以分母。小数 → 百分数：× 100
3. Algebraic Simplification | 代数化简
In algebra, letters (variables) represent unknown numbers. Terms are separated by + or − signs. Like terms have the same variable part and can be combined by adding or subtracting their coefficients.
在代数中，字母（变量）代表未知数。项与项之间由 + 或 − 号分隔。同类项含有相同的变量部分，可以通过加减其系数来合并。
Laws of indices (for integer exponents):
指数定律（适用于整数指数）：
- am × an = am+n — 同底数幂相乘，指数相加。
- am ÷ an = am−n — 同底数幂相除，指数相减。
- (am)n = am×n — 幂的乘方，指数相乘。
- a0 = 1 (a ≠ 0) — 任何非零数的 0 次方等于 1。
The distributive law states that a(b + c) = ab + ac. Expanding brackets and factorising (writing an expression as a product) rely on this law.
分配律指出 a(b + c) = ab + ac。展开括号和因式分解（将表达式写成乘积形式）都依据这一规律。
Always simplify expressions by collecting like terms and using the index laws before substituting values.
在代入数值之前，应始终通过合并同类项并运用指数定律来化简表达式。
4. Linear Equations | 一元一次方程
A linear equation in one variable takes the form ax + b = c, where a, b, c are constants and a ≠ 0. The goal is to isolate the variable on one side of the equation.
一元一次方程的形式为 ax + b = c，其中 a、b、c 为常数且 a ≠ 0。解题目标是将变量单独置于等式的一侧。
Golden rules for solving equations: Do the same operation to both sides of the equation to maintain balance. You can add, subtract, multiply or divide by the same non-zero number.
解方程的黄金法则：在方程两边进行相同的运算以保持平衡。你可以在两边同时加、减、乘或除以同一个非零数。
Check your solution by substituting it back into the original equation. If both sides are equal, the solution is correct.
通过将解代回原方程来检验。如果两边相等，则解为正确。
Eg: 3x + 7 = 19 → 3x = 12 → x = 4
例：3x + 7 = 19 → 3x = 12 → x = 4
5. Sequences and Patterns | 数列与规律
uence is called a term. The position of a term is usually denoted by n (n = 1, 2, 3, ...).
数列是按一定规则排列的有序数字列表。数列中的每个数称为项。项的位置通常用 n 表示（n = 1, 2, 3, ...）。
An arithmetic sequence has a constant difference between consecutive terms. The nth term of an arithmetic sequence is given by an = a + (n − 1)d, where a is the first term and d is the common difference.
等差数列的相邻两项之差恒定。其第 n 项（通项）公式为 an = a + (n − 1)d，其中 a 为首项，d 为公差。
Other patterns may follow geometric progressions or involve squares and cubes. Recognising the rule allows you to predict later terms and state a formula for the nth term.
其他规律可能遵循等比数列或包含平方、立方模式。识别规则后，即可预测后续项，并写出第 n 项的公式。
6. Ratio and Proportion | 比和比例
A ratio compares two or more quantities of the same kind. It is written as a : b and has no units. Ratios can be simplified by dividing all parts by their HCF.
比用于比较两个或多个同类的量，写成 a : b 的形式，不带单位。比可通过将各部分同时除以它们的最大公约数来化简。
When a quantity is divided in a given ratio, work out the total number of parts and then find the value of one part. Multiply that value by the required number of parts.
当按给定比例分配某一数量时，先计算总份数，再求出一份的值，最后将该值乘以所需的份数即可。
Direct proportion means two quantities increase or decrease at the same rate. Their ratio remains constant. Inverse proportion means one quantity increases as the other decreases, so their product is constant.
正比例指两个量以相同的速率增加或减少，它们的比值保持不变。反比例指一个量增加时另一个量减少，两者的乘积保持恒定。
Direct: y = kx. Inverse: y = k/x (k constant)
正比例：y = kx。反比例：y = k/x（k 为常数）
7. Angles: Types, Rules and Parallel Lines | 角：类型、规则与平行线
Angles are measured in degrees (°). Acute: 0° < θ < 90°. Right angle: exactly 90°. Obtuse: 90° < θ < 180°. Straight line: 180°. Reflex: 180° < θ < 360°.
角度以度（°）为单位。锐角：0° < θ < 90°。直角：严格等于 90°。钝角：90° < θ < 180°。平角：180°。优角：180° < θ < 360°。
Angles at a point add up to 360°. Angles on a straight line add up to 180°. Vertically opposite angles are equal. Complementary angles sum to 90°, supplementary angles sum to 180°.
绕一点一周的角之和为 360°。直线上的邻角之和为 180°。对顶角相等。互余角之和为 90°，互补角之和为 180°。
When a transversal cuts two parallel lines (∥), alternate interior angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.
当一条截线与两条平行线（∥）相交时，内错角相等，同位角相等，同旁内角之和为 180°。
Parallel line rules: ∠ alternate = ∠ corresponding; co-interior sum = 180°
平行线规则：内错角 = 同位角；同旁内角之和 = 180°
8. 2D Shapes: Perimeter and Area | 平面图形：周长与面积
The perimeter of a polygon is the total length of its sides. For a rectangle with length l and width w: P = 2(l + w). For a square of side s: P = 4s.
多边形的周长是其所有边长之和。对于长为 l、宽为 w 的矩形：P = 2(l + w)。对于边长为 s 的正方形：P = 4s。
Area measures the surface inside a closed shape. Key area formulas:
面积用于衡量封闭图形内部区域的大小。关键面积公式：
| Shape | Formula | 中文 |
|---|---|---|
| Rectangle | A = l × w | 矩形面积 = 长 × 宽 |
| Square | A = s² | 正方形面积 = 边长² |
| Triangle | A = ½ × base × height | 三角形面积 = ½ × 底 × 高 |
| Parallelogram | A = base × perpendicular height | 平行四边形面积 = 底 × 垂直高度 |
| Trapezium | A = ½ × (a + b) × h | 梯形面积 = ½ × (上底 + 下底) × 高 |
The height in triangle and parallelogram formulas must be the perpendicular distance between the base and the opposite side.
三角形与平行四边形公式中的高必须是底边与对边之间的垂直距离。
9. 3D Shapes: Volume and Surface Area | 立体图形：体积与表面积
Volume measures the space an object occupies, while surface area is the total area of all its faces. Common prism-shaped objects are cuboids and cubes.
体积用于衡量物体占据的空间大小，表面积指所有面的总面积之和。常见的柱体形状有长方体和立方体。
For a cuboid with length l, width w, height h: Volume = l × w × h. Surface area = 2(lw + lh + wh). For a cube of side s: V = s³, SA = 6s².
对于长为 l、宽为 w、高为 h 的长方体：体积 = l × w × h。表面积 = 2(lw + lh + wh)。对于边长为 s 的立方体：V = s³，SA = 6s².
The volume of any prism is found by multiplying the area of its constant cross-section by the length. Units: cm³ for volume, cm² for surface area.
任意直棱柱的体积可通过底面积（恒定横截面积）乘以长度来计算。单位：体积用 cm³，表面积用 cm²。
10. Coordinate Geometry Basics | 坐标几何基础
The coordinate plane consists of two perpendicular number lines: the x-axis (horizontal) and y-axis (vertical), intersecting at the origin (0,0). Points are written as ordered pairs (x, y).
坐标平面由两条相互垂直的数轴构成：x 轴（水平）和 y 轴（垂直），它们相交于原点 (0,0)。点用有序对 (x, y) 表示。
The midpoint between two points (x₁, y₁) and (x₂, y₂) is given by:
两点 (x₁, y₁) 和 (x₂, y₂) 之间的中点坐标为：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The straight-line distance (length) between the same points, derived from Pythagoras' theorem, is:
基于勾股定理，同两点之间的直线距离（长度）公式为：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
dient m and y-intercept c. Horizontal lines have equation y = k; vertical lines are x = h.
绘制形如 y = mx + c 的线性方程会得到一条斜率为 m、y 轴截距为 c 的直线。水平线方程为 y = k，垂直线方程为 x = h。
11. Symmetry and Transformations | 对称与变换
A shape has line symmetry if it can be folded along a line so that the two halves match exactly. The number of such lines is the order of line symmetry.
如果一个图形可以沿某条线对折，使两部分完全重合，则该图形具有线对称。这类线的条数即线对称的阶数。
Rotational symmetry occurs when a shape can be rotated about its centre and still look the same in less than a full turn. The number of distinct positions is the order of rotational symmetry.
旋转对称指一个图形绕其中心旋转小于一周的某个角度后，仍与原有图形重合。不同重合位置的次数即为旋转对称的阶数。
The three basic transformations are translation (sliding without turning or flipping), reflection (flipping over a mirror line) and rotation (turning about a centre). A shape and its image are congruent after any of these.
三种基本变换为：平移（滑动，不转向不翻转）、反射（以一条对称线翻折）和旋转（绕一个中心转动）。图形经过这些变换后，它与像全等。
sends (x, y) → (−x, y). Rotation by 90° clockwise about the origin sends (x, y) → (y, −x).
关于 x 轴的反射：(x, y) → (x, −y)；关于 y 轴的反射：(x, y) → (−x, y)。关于原点顺时针旋转 90°：(x, y) → (y, −x)。
12. Statistics and Probability | 统计与概率
Statistics involves collecting, organising and interpreting data. Key measures of central tendency are the mean, median and mode. The range gives a measure of spread.
统计涉及数据的收集、整理和解读。重要的集中趋势量数包括平均数、中位数和众数。极差用于衡量数据的分散程度。
re ordered (or the mean of the two middle values). Mode = most frequently occurring value.
平均数 = 所有数据之和 ÷ 数据个数。中位数 = 排序后处于中间位置的数值（若有偶数个，则取中间两数的平均数）。众数 = 出现次数最多的数值。
Range = largest value − smallest value.
极差 = 最大值 − 最小值。
Probability measures how likely an event is. It is a number between 0 and 1, often expressed as a fraction, decimal or percentage.
概率用于度量事件发生的可能性，是一个介于 0 到 1 之间的数，常用分数、小数或百分比表示。
Probability of an event = (number of favourable outcomes) ÷ (total number of outcomes)
事件概率 = （有利结果数）÷ （可能结果总数）
Probabilities of all possible outcomes sum to 1. If the probability of an event is p, the probability of it not occurring is 1 − p.
所有可能结果的概率之和为 1。如果某事件发生的概率为 p，则其不发生的概率为 1 − p。
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