📚 Year 7 SQA Advanced Mathematics: Formula & Theorem Quick Reference Handbook | 苏格兰资格认证局七年级进阶数学：公式定理速查手册
This quick reference handbook compiles essential formulas and theorems for the Year 7 SQA Advanced Mathematics course. It is designed to help students revise key concepts and apply them accurately in assessments.
本速查手册汇集了苏格兰资格认证局（SQA）七年级进阶数学课程的核心公式与定理，旨在帮助学生梳理重点概念，并能准确应用于测试中。
1. Number Types and Properties | 数字类型与性质
Natural numbers are positive counting numbers: 1, 2, 3, …
自然数是正整数：1, 2, 3, …
Whole numbers include all natural numbers and zero.
整数包括所有自然数和零。
Integers are whole numbers that can be positive, negative, or zero.
整数系包含正整数、负整数和零。
Rational numbers can be expressed as a fraction a/b, where a and b are integers and b is not zero.
有理数可以表示为分数 a/b，其中 a 和 b 是整数且 b 不为零。
Irrational numbers cannot be written as a simple fraction; their decimal expansions are non-terminating and non-repeating. Examples: √2, π.
无理数不能写成简单的分数，其小数形式无限不循环，例如 √2、π。
A prime number has exactly two distinct factors, 1 and itself. A composite number has more than two factors.
质数有且仅有两个不同的因数（1 和它本身）。合数则有多于两个因数。
The highest common factor (HCF) and lowest common multiple (LCM) can be found by listing factors or prime factorisation. For two numbers a and b: Product of numbers = HCF × LCM.
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。对于两个数 a 和 b：两数的乘积 = HCF × LCM。
2. 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: 3/4 = 3 ÷ 4 = 0.75.
分数化小数：用分子除以分母，如 3/4 = 3 ÷ 4 = 0.75。
To convert a decimal to a percentage, multiply by 100%: 0.75 = 0.75 × 100% = 75%.
小数化百分数：乘以 100%，如 0.75 = 0.75 × 100% = 75%。
To convert a percentage to a fraction, write the percentage over 100 and simplify: 75% = 75/100 = 3/4.
百分数化分数：将百分数写成分母为 100 的分数，再化简，如 75% = 75/100 = 3/4。
Adding or subtracting fractions requires a common denominator. Use equivalent fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
分数加减法需要先通分，利用等值分数：1/2 + 1/3 = 3/6 + 2/6 = 5/6。
To multiply fractions, multiply the numerators and denominators: 2/3 × 4/5 = 8/15.
分数乘法：分子相乘、分母相乘，如 2/3 × 4/5 = 8/15。
To divide by a fraction, multiply by its reciprocal: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
分数除法：除以一个分数等于乘以它的倒数，2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。
3. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. It can be simplified just like a fraction: 10:15 = 2:3.
比用来比较两个或多个量，可以像分数一样化简：10:15 = 2:3。
To divide a quantity in a given ratio, add the parts together, find the value of one part, then multiply. For example, share £40 in the ratio 3:5: total parts = 8, one part = £5, so amounts are £15 and £25.
按比例分配：将总份数相加，求出每份的值再相乘。例如按 3:5 分配 £40，总份数 8，每份 £5，所以两部分分别为 £15 和 £25。
Direct proportion: two quantities increase or decrease at the same rate. Equation: y = kx, where k is the constant of proportionality.
正比例：两个量以相同速率增加或减少。公式：y = kx，其中 k 为比例常数。
Rates like speed, density, and unit price are examples of compound units. Speed = distance ÷ time.
速度、密度和单价等属于复合单位。速度 = 距离 ÷ 时间。
4. Algebraic Expressions | 代数表达式
In an algebraic term like 5x², 5 is the coefficient, x is the variable, and ² is the exponent.
在代数项 5x² 中，5 是系数，x 是变量，² 是指数。
Like terms have the same variable raised to the same power. They can be combined: 3x + 2y – 5x = -2x + 2y.
同类项含相同变量的相同次幂，可以合并：3x + 2y – 5x = -2x + 2y。
When expanding brackets, use the distributive law: a(b + c) = ab + ac. For example, 3(x + 4) = 3x + 12.
去括号时使用分配律：a(b + c) = ab + ac，例如 3(x + 4) = 3x + 12。
Factorising is the reverse process: 6x + 9 = 3(2x + 3). Look for the highest common factor.
因式分解是展开的逆运算：6x + 9 = 3(2x + 3)，找出最大公因数。
5. Solving Linear Equations | 解线性方程
The equation is a statement that two expressions are equal. The goal is to find the value of the unknown.
方程表示两个表达式相等，目的是找到未知数的值。
Use inverse operations to isolate the variable: x + 5 = 11 → x = 11 – 5 → x = 6.
使用逆运算分离变量：x + 5 = 11 → x = 11 – 5 → x = 6。
For multiplication: 4x = 20 → x = 20 ÷ 4 → x = 5.
乘法方程：4x = 20 → x = 20 ÷ 4 → x = 5。
Two-step equations: 2x + 3 = 11 → subtract 3 → 2x = 8 → x = 4.
两步方程：2x + 3 = 11 → 减 3 → 2x = 8 → x = 4。
Equations with variables on both sides: 3x + 2 = x + 10 → 2x = 8 → x = 4.
变量在方程两边：3x + 2 = x + 10 → 2x = 8 → x = 4。
6. Inequalities | 不等式
Inequalities compare two expressions using <, >, ≤ or ≥. The solution is a range of values.
不等式用 <、>、≤ 或 ≥ 比较两个表达式，解是一个取值范围。
Solve inequalities like equations, but if you multiply or divide by a negative number, reverse the inequality sign: -2x < 6 → x > -3.
解不等式与解方程类似，但当乘或除以负数时，不等号要改变方向：-2x < 6 → x > -3。
Represent inequalities on a number line using open or closed circles.
可以在数轴上用空心或实心圆点表示不等式。
7. Sequences and Patterns | 数列与规律
A number sequence follows a rule. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference (d).
数列按规则排列。在等差数列中，相邻两项的差恒定，称为公差（d）。
The nth term of an arithmetic sequence is given by: T(n) = a + (n – 1)d, where a is the first term, n is the term number, and d is the common difference.
T(n) = a + (n – 1)d
等差数列的第 n 项公式：T(n) = a + (n – 1)d，其中 a 为首项，n 为项数，d 为公差。
Example: For the sequence 4, 7, 10, 13, … a = 4, d = 3, so T(n) = 4 + (n – 1) × 3 = 3n + 1.
例如数列 4, 7, 10, 13, …，a = 4，d = 3，所以第 n 项 T(n) = 4 + (n – 1) × 3 = 3n + 1。
8. Geometry: Angles and Lines | 几何：角与线
Angles on a straight line add up to 180°. Angles around a point sum to 360°.
平角（一直线上的角）之和为 180°。绕一点的周角之和为 360°。
Vertically opposite angles are equal when two lines intersect.
两直线相交时对顶角相等。
In parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.
平行线中，同位角相等，内错角相等，同旁内角互补（和为 180°）。
The sum of interior angles in a triangle is 180°. In a quadrilateral, the sum is 360°.
三角形内角和为 180°。四边形内角和为 360°。
Types of angles: acute (<90°), right (=90°), obtuse (>90° and <180°), straight (=180°), reflex (>180°).
角分类：锐角（小于 90°），直角（等于 90°），钝角（大于 90° 且小于 180°），平角（等于 180°），优角（大于 180°）。
9. Geometry: Perimeter, Area and Volume | 周长、面积与体积
Perimeter of a rectangle: P = 2(l + w), where l is length and w is width.
矩形周长：P = 2(l + w)，l 为长，w 为宽。
Area of a rectangle: A = l × w.
矩形面积：A = l × w。
Area of a parallelogram: A = b × h, where b is base and h is perpendicular height.
平行四边形面积：A = b × h，b 为底，h 为垂直高。
Area of a triangle: A = ½ × b × h.
三角形面积：A = ½ × b × h。
Area of a trapezium: A = ½ × (a + b) × h, where a and b are parallel sides and h is height.
梯形面积：A = ½ × (a + b) × h，a 和 b 为平行边，h 为高。
Circumference of a circle: C = π × d or C = 2πr, where r is radius and d is diameter.
圆周长：C = π × d 或 C = 2πr，r 为半径，d 为直径。
Area of a circle: A = πr².
圆面积：A = πr²。
Volume of a cuboid: V = l × w × h. Volume of a cube: V = a³, where a is edge length.
长方体体积：V = l × w × h。正方体体积：V = a³，a 为棱长。
Volume of a prism: V = area of cross-section × length. For a cylinder: V = πr²h.
棱柱体积：V = 底面积 × 高。圆柱体积：V = πr²h。
10. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides: a² + b² = c².
a² + b² = c²
在直角三角形中，斜边（c）的平方等于两条直角边的平方和：a² + b² = c²。
Use this theorem to find a missing side: if a = 3 and b = 4, then c = √(3² + 4²) = 5.
可用该定理求未知边长：若 a = 3，b = 4，则 c = √(3² + 4²) = 5。
To check if a triangle is right-angled, verify if the sides satisfy a² + b² = c².
验证一个三角形是否为直角三角形，可检查其边长是否满足 a² + b² = c²。
11. Statistics: Averages and Spread | 统计量：平均数与离散度
Mean = sum of values ÷ number of values. For data {2, 3, 5, 10}, mean = (2+3+5+10) ÷ 4 = 5.
平均数 = 数据总和 ÷ 数据个数。数据 {2, 3, 5, 10} 的平均数为 (2+3+5+10) ÷ 4 = 5。
Median is the middle value when data is ordered. For an even number of values, it is the mean of the two middle numbers.
中位数是排列后位于中间的数据点。若数据个数为偶数，则取中间两个数的平均数。
Mode is the value that appears most often.
众数是出现次数最多的数据值。
Range = maximum – minimum. It measures the spread of data.
极差（全距）= 最大值 – 最小值，用来描述数据的离散程度。
12. Index Laws and Standard Form | 指数定律与科学记数法
Multiplying powers with the same base: aᵐ × aⁿ = aᵐ⁺ⁿ. Example: 2³ × 2² = 2⁵.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。例如 2³ × 2² = 2⁵。
Dividing powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0). Example: 5⁴ ÷ 5² = 5².
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)。例如 5⁴ ÷ 5² = 5²。
Power of a power: (aᵐ)ⁿ = aᵐⁿ. Example: (3²)³ = 3⁶.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。例如 (3²)³ = 3⁶。
Any non-zero number raised to the power of zero equals 1: a⁰ = 1.
任何非零数的零次方等于 1：a⁰ = 1。
A negative index indicates reciprocal: a⁻ⁿ = 1/aⁿ. Example: 10⁻³ = 1/1000.
负指数表示倒数：a⁻ⁿ = 1/aⁿ。例如 10⁻³ = 1/1000。
Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Example: 4500 = 4.5 × 10³.
科学记数法将数字写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。例如 4500 = 4.5 × 10³。
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