📚 Year 7 OCR Further Mathematics: Formula & Theorem Quick Reference | 7年级OCR进阶数学公式定理速查手册
This article serves as a handy quick-reference guide for Year 7 students studying OCR Further Mathematics. It compiles essential formulas, theorems and key concepts, with each point provided in both English and Chinese to support bilingual learning and revision. The content aligns with the OCR Year 7 Further Mathematics syllabus and is designed for rapid recall and exam preparation.
本文为学习OCR进阶数学的7年级学生提供便捷的速查手册，汇集了重要公式、定理与核心概念，每个知识点均提供中英双语对照，便于复习巩固。内容紧扣OCR Year 7进阶数学大纲，适合快速回顾与备考。
1. Algebraic Expressions and Identities | 代数表达式与恒等式
rm inside the bracket by the term outside. The distributive law states a(b + c) = ab + ac.
单项括号展开：将括号外的项乘以括号内的每一项。分配律为 a(b + c) = ab + ac。
brackets: (x + a)(x + b) = x² + (a + b)x + ab. This is a special product of two binomials.
双括号展开：(x + a)(x + b) = x² + (a + b)x + ab。这是两个一次二项式的特殊乘积。
common factor: ab + ac = a(b + c). Always look for the highest common factor in each term.
提公因式分解因式：ab + ac = a(b + c)。始终提取各项的最大公因数。
uares: a² - b² = (a + b)(a - b). This identity factorises a difference of perfect squares.
平方差公式：a² - b² = (a + b)(a - b)。该恒等式用于分解两个完全平方数的差。
- b)² = a² - 2ab + b². These are useful for expanding or factorising squares of binomials.
完全平方恒等式：(a + b)² = a² + 2ab + b² 且 (a - b)² = a² - 2ab + b²。它们用于展开或分解二项式的平方。
2. Linear Equations | 线性方程
+ b = c, subtract b from both sides to get ax = c - b, then divide by a: x = (c - b) / a.
解一元一次方程需通过逆运算隔离变量。对于 ax + b = c，两边同时减去 b 得到 ax = c - b，再除以 a 得 x = (c - b) / a。
If the equation contains brackets, expand them first. Always perform the same operation on both sides to maintain balance.
若方程含有括号，先展开括号。始终在等式两边执行相同运算以保持平衡。
and constants on the other. For example, 3x + 5 = x + 9 → 3x - x = 9 - 5 → 2x = 4 → x = 2.
两边均含变量的方程：将变量项移到一边，常数项移到另一边。例如 3x + 5 = x + 9 → 3x - x = 9 - 5 → 2x = 4 → x = 2。
Check your solution by substituting it back into the original equation. Both sides must be equal.
通过将解代回原方程检验解的正确性，两边必须相等。
3. Sequences and Patterns | 数列与规律
rm formula is uₙ = a + (n - 1)d, where a is the first term and d is the common difference.
等差数列相邻项的差为常数。第 n 项公式为 uₙ = a + (n - 1)d，其中 a 是首项，d 是公差。
A sequence of square numbers is given by n²: 1, 4, 9, 16, 25, ... The nth term is simply n².
平方数数列由 n² 给出：1, 4, 9, 16, 25, … 第 n 项即为 n²。
Triangular numbers: 1, 3, 6, 10, 15, ... The nth triangular number is Tₙ = n(n + 1)/2.
三角形数：1, 3, 6, 10, 15, … 第 n 个三角形数公式为 Tₙ = n(n + 1)/2。
To find a term in a geometric sequence, multiply the previous term by a constant ratio r. The nth term is arn-1. However, in Year 7 Further Mathematics, most sequences are arithmetic or follow a simple pattern described by position-to-term rules.
在等比数列中，每一项乘以常数比 r 得到后一项。第 n 项为 arn-1。但在7年级进阶数学中，多数数列为等差数列或遵循用位置描述项的简单规律。
4. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. To simplify a ratio, divide all parts by their greatest common factor. For example, 8:12 simplifies to 2:3.
比用于比较各部分。化简比时，将所有部分除以它们的最大公因数。例如 8:12 可化简为 2:3。
To divide a quantity in a given ratio, find the total number of parts, then calculate the value of one part and scale up. For a ratio a:b, the share for each part is a/(a+b) and b/(a+b) of the total.
按给定比例分配数量时，先求总份数，计算一份的量，再按比例缩放。对于比 a:b，两部分分别占总量的 a/(a+b) 和 b/(a+b)。
here k is the constant of proportionality. As x increases, y increases at a constant rate.
正比例：若 y = kx，则两量成正比，k 为比例常数。x 增大时，y 以恒定速率增大。
/x. As x increases, y decreases. This concept may be introduced at the higher end of Year 7 Further Mathematics.
反比例（拓展）：y = k/x。x 增大时 y 减小。该概念可能在7年级进阶数学的高阶部分引入。
5. Fractions, Decimals and Percentages | 分数、小数与百分比
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%.
分数化小数：分子除以分母。小数化百分数：乘以100%。
Adding fractions: a/b + c/d = (ad + bc)/bd. Find a common denominator to combine.
分数加法：a/b + c/d = (ad + bc)/bd。先通分再相加。
fractions: (a/b) × (c/d) = ac/bd. Multiply numerators together and denominators together.
分数乘法：(a/b) × (c/d) = ac/bd。分子相乘，分母相乘。
g fractions: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc. Flip the second fraction and multiply.
分数除法：(a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc。将第二个分数取倒数后相乘。
× (1 + percentage/100). Percentage decrease: new value = original × (1 - percentage/100).
百分数增加：新值 = 原值 × (1 + 百分数/100)。百分数减少：新值 = 原值 × (1 - 百分数/100)。
Expressing one quantity as a percentage of another: (part/whole) × 100%.
一个量占另一个量的百分比：(部分/整体) × 100%。
6. Geometry: Angles and Lines | 几何：角与直线
Vertically opposite angles are equal when two lines intersect. Angles on a straight line add up to 180°.
两直线相交时，对顶角相等。平角（一条直线上的角）之和为180°。
Angles around a point sum to 360°. Complementary angles add to 90°, supplementary angles add to 180°.
绕一点的角之和为360°。互余角之和为90°，互补角之和为180°。
When a transversal crosses parallel lines: corresponding angles are equal, alternate interior angles are equal, and co-interior (allied) angles sum to 180°.
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°。
Angle sum of a triangle is 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为180°。三角形的一个外角等于与它不相邻的两个内角之和。
sided polygon: (n - 2) × 180°. Sum of exterior angles (one at each vertex) is always 360°.
n 边形内角和：(n - 2) × 180°。外角和（每个顶点取一个外角）恒为360°。
7. Triangles and Pythagoras' Theorem | 三角形与勾股定理
Pythagoras' t-angled triangles. If c is the hypotenuse, then a² + b² = c², where a and b are the legs.
勾股定理适用于直角三角形。若 c 为斜边，则 a² + b² = c²，其中 a 和 b 为两直角边。
gled, verify whether the side lengths satisfy a² + b² = c², with c being the longest side.
判断三角形是否为直角三角形：验证三边长度是否满足 a² + b² = c²，其中 c 为最长边。
Pythagorean triples are sets of integers that satisfy the theorem, such as (3, 4, 5), (5, 12, 13) and (8, 15, 17). Multiples of these also work.
勾股数是指满足该定理的整数三元组，如 (3, 4, 5)、(5, 12, 13) 和 (8, 15, 17)。这些数的倍数同样成立。
Triangle inequality: the sum of any two sides of a triangle must be greater than the third side.
三角形不等式：三角形任意两边之和必大于第三边。
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter of a rectangle: P = 2(l + w). Area of a rectangle: A = l × w.
矩形周长：P = 2(l + w)。矩形面积：A = l × w。
Area of a triangle: A = ½ × base × height. The height must be perpendicular to the base.
三角形面积：A = ½ × 底 × 高。高必须与底边垂直。
ram: A = b × h. Area of a trapezium: A = ½ (a + b)h, where a and b are the parallel sides.
平行四边形面积：A = b × h。梯形面积：A = ½ (a + b)h，其中 a 和 b 为两平行边。
le: C = πd = 2πr. Area of a circle: A = πr². Use π ≈ 3.14 or the π button on a calculator.
圆的周长：C = πd = 2πr。圆的面积：A = πr²。使用 π ≈ 3.14 或计算器上的 π 键。
Volume of a cuboid: V = l × w × h. Volume of a prism: V = area of cross-section × length.
长方体体积：V = l × w × h。棱柱体积：V = 底面积 × 长（或高）。
πr²h. Surface area of a cylinder: the total of two circles and the curved rectangle is 2πr² + 2πrh.
圆柱体积（拓展）：V = πr²h。圆柱表面积：两个圆面积加侧面矩形面积为 2πr² + 2πrh。
9. Data Handling and Averages | 数据处理与平均数
Mean is the sum of all data values divided by the number of values: mean = Σx / n.
平均数（均值）等于所有数据值之和除以数据的个数：平均数 = Σx / n。
Median is the middle value when the data is ordered. If there are two middle values, the median is their mean.
中位数是将数据排序后位于中间的值。若中间有两个值，中位数取两者的平均数。
t frequently. Range is the difference between the largest and smallest values: range = max – min.
众数是出现频率最高的值。极差（范围）为最大值与最小值之差：极差 = 最大值 – 最小值。
ated using (Σ f × x) / Σ f, where x is the data value or the midpoint of a class interval.
在频数表中，平均数估算为 (Σ f × x) / Σ f，其中 x 为数据值或组中值。
To draw a pie chart, calculate sector angle = (frequency / total) × 360°.
绘制饼图时，计算扇形角度 = (频数/总数) × 360°。
10. Probability Basics | 概率基础
vourable outcomes / total number of possible outcomes. It is expressed as a fraction, decimal or percentage between 0 and 1.
某事件的概率 = 有利结果数 / 所有可能结果总数。概率用分数、小数或百分数表示，范围在0到1之间。
lity of an event not occurring is 1 minus the probability that it does occur: P(not A) = 1 – P(A).
某事件不发生的概率等于1减去它发生的概率：P(非 A) = 1 – P(A)。
ded die, P(rolling a 6) = 1/6. All outcomes must be equally likely for these calculations.
抛一枚均匀硬币，P(正面) = ½。掷一个均匀六面骰子，P(掷出6) = 1/6。所有结果必须等可能。
, if the probability of rain is 0.3, over 100 days you would expect rain on about 30 days.
在 n 次试验中的期望频数 = 概率 × n。例如，若下雨概率为0.3，则100天内大约会有30天下雨。
11. Number Theory: Factors, Multiples and Primes | 数论：因数、倍数与质数
A prime number has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, ... Note that 1 is not a prime.
质数有且仅有两个不同的因数：1和它本身。前几个质数为2, 3, 5, 7, 11, 13, … 注意1不是质数。
: write the number as a product of prime numbers using index notation, e.g., 72 = 2³ × 3².
合数可表示为质因数的乘积。质因数分解：用指数记法将数写成质数的乘积，如 72 = 2³ × 3²。
Greatest common divisor (GCD) of two numbers is the largest number that divides both without remainder. Lowest common multiple (LCM) is the smallest number that is a multiple of both.
两数的最大公因数（GCD）是能同时整除它们的最大数。最小公倍数（LCM）是能同时被它们整除的最小数。
To find the LCM using prime factors: multiply the highest power of each prime that appears in the factorisations. For GCD, multiply the lowest power of each common prime.
用质因数求 LCM：将每个出现质数的最高次幂相乘。求 GCD：将每个共有质数的最低次幂相乘。
12. Indices and Standard Form | 指数与标准形式
= aᵐ⁻ⁿ (where m > n, otherwise results in a negative exponent), (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ.
正整数指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ（当 m>n，否则产生负指数），(aᵐ)ⁿ = aᵐⁿ，(ab)ⁿ = aⁿbⁿ。
sed to the power 0 is 1: a⁰ = 1. A negative exponent indicates the reciprocal: a⁻ⁿ = 1/aⁿ.
任何非零数的0次幂为1：a⁰ = 1。负指数表示倒数：a⁻ⁿ = 1/aⁿ。
integer. Large numbers have positive n; small numbers have negative n. Example: 4500 = 4.5
Published by TutorHao | Year 7 进阶数学 Revision Series | aleveler.com
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