📚 Year 7 OCR Further Maths: Formula & Theorem Quick Reference | 七年级OCR进阶数学：公式定理速查手册
This handbook compiles the essential formulas and theorems for Year 7 OCR Further Mathematics. It serves as a quick revision guide covering number theory, algebra, geometry, measurement, statistics, and probability. Mastering these fundamentals will build a strong foundation for advanced mathematical reasoning.
本手册汇编了七年级OCR进阶数学的基本公式和定理。它是涵盖数论、代数、几何、测量、统计和概率的快速复习指南。掌握这些基础知识将为高级数学推理打下坚实的基础。
1. Number Properties and Prime Factorization | 数的性质与质因数分解
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples: 2, 3, 5, 7, 11.
质数是大于1的自然数，且只有两个不同的正因数：1和它本身。例如：2, 3, 5, 7, 11。
A composite number is a positive integer greater than 1 that has more than two divisors.
合数是大于1且有多于两个因数的正整数。
Prime factorization expresses a number as a product of prime numbers. Example: 60 = 2² × 3 × 5.
质因数分解将一个数表示为质数的乘积。例如：60 = 2² × 3 × 5。
The Greatest Common Divisor (GCD) of two numbers is the largest positive integer that divides both numbers without remainder. Using prime factors: pick common prime factors with the lowest exponents.
两个数的最大公因数（GCD）是能同时整除这两个数的最大正整数。用质因数法：选取共同的质因数，指数取最小的。
The Least Common Multiple (LCM) is the smallest positive integer divisible by both numbers. Relation: LCM(a, b) × GCD(a, b) = a × b.
最小公倍数（LCM）是能被两个数整除的最小正整数。关系：LCM(a, b) × GCD(a, b) = a × b。
2. Fractions, Decimals, and Percentages | 分数、小数和百分比
Adding fractions with different denominators: find a common denominator, then add the numerators. a/b + c/d = (ad + bc) / bd.
异分母分数相加：求公分母，然后分子相加。法则：a/b + c/d = (ad + bc) / bd。
Multiplying fractions: multiply numerators and multiply denominators. (a/b) × (c/d) = ac / bd. Dividing fractions: multiply by the reciprocal. (a/b) ÷ (c/d) = a/b × d/c = ad / bc.
分数乘法：分子乘分子，分母乘分母。法则：(a/b) × (c/d) = ac / bd。分数除法：乘以倒数。(a/b) ÷ (c/d) = a/b × d/c = ad / bc。
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, use place value and simplify.
将分数转换为小数：分子除以分母。将小数转换为分数：根据数位写出分母，然后约分。
Percent means ‘per hundred’. To change a fraction to a percent, multiply by 100%. Example: 3/4 = 0.75 = 75%.
百分比表示“每一百”。将分数转换为百分数，乘以100%。例如：3/4 = 0.75 = 75%。
3. Powers and Roots | 幂与根
A power aⁿ means a multiplied by itself n times (n is a positive integer). Example: 2³ = 2 × 2 × 2 = 8.
幂 aⁿ 表示 a 自乘 n 次（n 为正整数）。例如：2³ = 2 × 2 × 2 = 8。
The square root √a is the number that when squared gives a. √9 = 3 because 3² = 9. The cube root ∛a satisfies (∛a)³ = a. ∛8 = 2.
平方根 √a 是平方后得到 a 的数。√9 = 3 因为 3² = 9。立方根 ∛a 满足 (∛a)³ = a。∛8 = 2。
Laws of exponents: aᵐ × aⁿ = a^(m+n), (aᵐ)ⁿ = a^(mn), aᵐ ÷ aⁿ = a^(m-n) for m > n.
指数法则：aᵐ × aⁿ = a^(m+n)，(aᵐ)ⁿ = a^(mn)，aᵐ ÷ aⁿ = a^(m-n)（当 m > n 时）。
4. Algebraic Expressions | 代数表达式
Like terms contain exactly the same variables raised to the same powers. Combine like terms by adding or subtracting their coefficients.
同类项包含完全相同的变量和指数。通过加减系数合并同类项。
Example: 3x + 2y – x + 5y = (3x – x) + (2y + 5y) = 2x + 7y.
例如：3x + 2y – x + 5y = (3x – x) + (2y + 5y) = 2x + 7y。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. Double brackets: (a + b)(c + d) = ac + ad + bc + bd.
展开括号使用分配律：a(b + c) = ab + ac。双重括号：(a + b)(c + d) = ac + ad + bc + bd。
Factorising extracts a common factor: ab + ac = a(b + c).
因式分解提取公因式：ab + ac = a(b + c)。
5. Solving Linear Equations | 解线性方程
To solve an equation, perform the same operation on both sides to isolate the variable.
解方程时，对方程两边执行相同的运算以隔离变量。
One-step addition: x + a = b → x = b – a. Subtraction: x – a = b → x = b + a. Multiplication: ax = b → x = b/a (a ≠ 0).
一步加法方程：x + a = b → x = b – a。减法：x – a = b → x = b + a。乘法：ax = b → x = b/a（a ≠ 0）。
Two-step equation example: 2x + 3 = 9 → 2x = 6 → x = 3.
两步方程示例：2x + 3 = 9 → 2x = 6 → x = 3。
Always check your solution by substituting it back into the original equation.
始终将解代回原方程进行检验。
6. Sequences and Patterns | 序列与规律
A number sequence follows a rule. The term-to-term rule explains how to move from one term to the next.
数列按规律排列。逐项规则解释如何从一项得到下一项。
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is given by aₙ = a₁ + (n – 1)d, or simplified as aₙ = dn + c.
等差数列的相邻两项之差为常数 d。第 n 项公式为 aₙ = a₁ + (n – 1)d，或简化为 aₙ = dn + c。
Example: 3, 7, 11, 15,… difference d = 4, so nth term = 4n – 1.
例如：3, 7, 11, 15,… 差 d = 4，所以第 n 项 = 4n – 1。
7. Angles and Polygons | 角与多边形
Angle types: acute < 90°, right = 90°, obtuse > 90° and < 180°, straight = 180°, reflex > 180°.
角的类型：锐角 < 90°，直角 = 90°，钝角 > 90° 且 < 180°，平角 = 180°，优角 > 180°。
Complementary angles sum to 90°; supplementary angles sum to 180°.
互余角之和为 90°；互补角之和为 180°。
Vertically opposite angles are equal. Angles on a straight line add to 180°; angles around a point add to 360°.
对顶角相等。直线上邻角之和为 180°；绕点一周的角之和为 360°。
Parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles sum to 180°.
平行线：同位角相等，内错角相等，同旁内角互补（和为 180°
Published by TutorHao | Year 7 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导