📚 Year 7 OCR Maths: Formula & Theorem Quick Reference | Year 7 OCR 数学：公式定理速查手册
Welcome to the Year 7 OCR Maths Formula & Theorem Quick Reference. This guide brings together the essential rules, properties and formula you will meet during your first year of secondary mathematics. Use it to check key facts, support your homework and build confidence for class tests.
欢迎使用 Year 7 OCR 数学公式定理速查手册。本指南汇集了你在初中第一年将接触到的基础法则、性质和公式。你可以用它来核对要点、辅助作业，并为课堂测验树立信心。
1. Basic Integer Operations | 整数基本运算
Addition is commutative: a + b = b + a
加法满足交换律：a + b = b + a
Multiplication is commutative: a × b = b × a
乘法满足交换律：a × b = b × a
The order of operations is remembered with BIDMAS: Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
运算顺序遵循 BIDMAS 法则：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
Example: 5 + 3 × 2² = 5 + 3 × 4 = 5 + 12 = 17
例子：5 + 3 × 2² = 5 + 3 × 4 = 5 + 12 = 17
Adding a negative number is the same as subtracting its positive: a + (−b) = a − b
加上一个负数等于减去它的正数：a + (−b) = a − b
Subtracting a negative number means adding the positive: a − (−b) = a + b
减去一个负数等于加上它的正数：a − (−b) = a + b
2. Fractions, Decimals & Percentages | 分数、小数与百分数
To convert a fraction to a decimal, divide the numerator by the denominator: e.g., 3/4 = 3 ÷ 4 = 0.75
分数化小数：分子除以分母。例如：3/4 = 3 ÷ 4 = 0.75
To convert a decimal to a percentage, multiply by 100: e.g., 0.45 = 45%
小数化百分数：乘以 100。例如：0.45 = 45%
To find a percentage of an amount, write the percentage as a fraction of 100 and multiply: e.g., 20% of 60 = (20/100) × 60 = 12
求一个数量的百分之几：将百分数写成分母为 100 的分数再相乘。例如：60 的 20% = (20/100) × 60 = 12
When adding or subtracting fractions, find a common denominator first: e.g., 1/3 + 1/4 = 4/12 + 3/12 = 7/12
分数加减时先通分：例如 1/3 + 1/4 = 4/12 + 3/12 = 7/12
Multiplying fractions: multiply numerators together and denominators together: a/b × c/d = (a×c) / (b×d)
分数乘法：分子相乘作分子，分母相乘作分母：a/b × c/d = (a×c)/(b×d)
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以它的倒数：a/b ÷ c/d = a/b × d/c
3. Algebraic Expressions & Equations | 代数表达式与方程
A term is a number, a variable or a product of numbers and variables, e.g., 3x, −2y², 7
项是一个数字、一个变量或者数字与变量的乘积，如 3x, −2y², 7。
Like terms can be combined: 4a + 3a = 7a, but 4a + 3b cannot be simplified further.
同类项可以合并：4a + 3a = 7a，但 4a + 3b 无法进一步化简。
The distributive law: a(b + c) = ab + ac
分配律：a(b + c) = ab + ac
To solve a one-step equation, perform the inverse operation: x + 5 = 12 → x = 12 − 5 = 7
解一步方程时使用逆运算：x + 5 = 12 → x = 12 − 5 = 7
For two-step equations, undo addition or subtraction first, then multiplication or division: 2x + 3 = 11 → 2x = 8 → x = 4
解两步方程时先消去加减，再消去乘除：2x + 3 = 11 → 2x = 8 → x = 4
4. Sequences & Patterns | 数列与模式
An arithmetic sequence has a common difference between consecutive terms, e.g., 4, 7, 10, 13 … (difference = 3)
等差数列中相邻项的差相等，如 4, 7, 10, 13 …（公差 = 3）。
The term-to-term rule tells you how to get from one term to the next, often by adding or multiplying a constant.
项间法则告诉你如何从前一项得到后一项，通常是加上或乘以一个固定数。
The nth term of a linear sequence can be written as an + b, where a is the common difference. Example: 5, 8, 11, 14 … nth term = 3n + 2
线性数列的第 n 项可写作 an + b，其中 a 为公差。例如：5, 8, 11, 14 … 第 n 项 = 3n + 2。
Check the nth term by substituting n = 1, 2, 3 …
用 n = 1, 2, 3 … 代入验证第 n 项公式。
5. Angles & Lines | 角与线
Angles are measured in degrees (°). Acute angles: less than 90°; right angle: exactly 90°; obtuse: between 90° and 180°; reflex: between 180° and 360°.
角度用度（°）衡量。锐角：小于 90°；直角：恰好 90°；钝角：90° 到 180° 之间；优角：180° 到 360° 之间。
Angles on a straight line add up to 180°.
直线上的角之和为 180°。
Angles around a point add up to 360°.
绕一点一周的角之和为 360°。
Vertically opposite angles are equal.
对顶角相等。
On parallel lines: corresponding angles are equal, alternate angles are equal, interior (co-interior) angles add to 180°.
平行线性质：同位角相等，内错角相等，同旁内角之和为 180°。
In a triangle, the three interior angles sum to 180°.
三角形内角和为 180°。
6. Perimeter & Area of 2D Shapes | 二维图形的周长与面积
Perimeter of a rectangle: P = 2(l + w), where l = length, w = 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 triangle: A = ½ × base × height (height must be perpendicular to the base).
三角形面积：A = ½ × 底 × 高（高必须与底垂直）。
Area of a parallelogram: A = base × perpendicular height
平行四边形面积：A = 底 × 垂直高度
Area of a trapezium: A = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular height.
梯形面积：A = ½ × (a + b) × h，a 和 b 为平行边，h 为垂直高度。
For a circle: circumference C = π × d = 2πr, area A = πr² (π ≈ 3.14 or 22/7).
圆的周长：C = π × d = 2πr；面积：A = πr²（π 约等于 3.14 或 22/7）。
7. Surface Area & Volume of 3D Shapes | 三维图形的表面积与体积
Volume of a cube: V = side³ = a³
立方体体积：V = 边长³ = a³
Volume of a cuboid: V = length × width × height = l × w × h
长方体体积：V = 长 × 宽 × 高 = l × w × h
Surface area of a cuboid: SA = 2(lw + lh + wh)
长方体表面积：SA = 2(lw + lh + wh)
The volume of a prism is found by multiplying the area of the cross-section by the length: V = area of cross-section × length.
棱柱体积 = 截面面积 × 长度。
Remember that capacity is often measured in litres: 1 cm³ = 1 ml, 1000 cm³ = 1 litre.
容量常用升表示：1 cm³ = 1 ml，1000 cm³ = 1 升。
8. Unit Conversions | 单位换算
Length: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
长度：1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m
Mass: 1 kg = 1000 g, 1 tonne = 1000 kg
质量：1 kg = 1000 g，1 吨 = 1000 kg
Capacity: 1 litre = 1000 ml = 1000 cm³
容量：1 升 = 1000 毫升 = 1000 cm³
Time: 1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours
时间：1 分钟 = 60 秒，1 小时 = 60 分钟，1 天 = 24 小时
When converting between units, multiply by the conversion factor for larger to smaller units, and divide for smaller to larger.
单位换算时，从大单位化为小单位乘以进率，从小单位化为大单位除以进率。
9. Statistics: Mean, Median, Mode & Range | 统计：平均数、中位数、众数与极差
The mean is found by adding all the values and dividing by the number of values: Mean = sum of values ÷ number of values.
平均数：将所有数值相加再除以数值的个数。平均数 = 总和 ÷ 个数。
The median is the middle value when the data is arranged in order. If there are two middle numbers, take their average.
中位数：将数据按顺序排列后位于中间的数。如果中间有两个数，则取其平均数。
The mode is the value that appears most often. A set of data can have more than one mode or no mode at all.
众数是出现次数最多的数值。一组数据可以有一个以上的众数，也可以没有众数。
The range is the difference between the largest and the smallest value: Range = maximum − minimum.
极差是最大值与最小值的差：极差 = 最大值 − 最小值。
Bar charts, pictograms and line graphs are used to display data clearly. Always label axes and include a title.
条形图、象形图和折线图用来清晰展示数据。务必给坐标轴添加标签并写上标题。
10. Ratio & Proportion | 比与比例
A ratio compares two or more quantities in the same unit, e.g., 2 : 3.
比用来比较同一单位下的两个或多个量，例如 2 : 3。
Ratios can be simplified like fractions by dividing all parts by a common factor: 10 : 15 = 2 : 3 (divide by 5).
比可以像分数一样化简，各部分同时除以公因数：10 : 15 = 2 : 3（除以 5）。
To share an amount in a given ratio, first find the total number of parts, then divide the amount by the total, and multiply by each part.
按给定比例分配：先算出总份数，将总量除以总份数，再乘以各自的份数。
Direct proportion means that as one quantity doubles, the other doubles. The ratio between them remains constant.
正比例意味着一个量翻倍时另一个量也翻倍，它们之间的比值保持不变。
Use the unitary method: find the value of 1 unit first, then scale up or down.
使用归一法：先求出 1 个单位的值，再按需要放大或缩小。
11. Coordinates & Graphs | 坐标与图表
Coordinates are written as (x, y) where x is the horizontal distance from the origin and y is the vertical distance.
坐标写作 (x, y)，其中 x 是横轴方向上离原点的距离，y 是纵轴方向上的距离。
The origin is the point (0, 0).
原点是 (0, 0)。
Move right for positive x, left for negative x; move up for positive y, down for negative y.
x 为正数时向右移动，为负数时向左移动；y 为正数时向上移动，为负数时向下移动。
A straight-line graph has an equation of the form y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
直线图像具有形式为 y = mx + c 的方程，其中 m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴交点的 y 坐标）。
To plot a graph, create a table of x values, calculate the corresponding y values, and plot the points.
绘制图像时，先列一张 x 值的表格，计算对应的 y 值，然后描点连线。
12. Introduction to Probability | 概率入门
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
概率衡量事件发生的可能性大小，其值介于 0（不可能）到 1（必然）之间。
Probability of an event = number of favourable outcomes ÷ total number of possible outcomes, provided all outcomes are equally likely.
事件的概率 = 有利结果数 ÷ 所有可能结果的总数，前提是每种结果发生的可能性相等。
The sum of probabilities of all possible outcomes is 1.
所有可能结果的概率之和为 1。
Expected frequency = probability × number of trials. For example, if you roll a fair die 60 times, the expected number of sixes is (1/6) × 60 = 10.
期望频数 = 概率 × 试验次数。例如，掷一枚均匀骰子 60 次，出现 6 的期望次数是 (1/6) × 60 = 10。
A probability closer to 0 means the event is unlikely; closer to 1 means it is likely.
概率越接近 0，事件越不可能发生；越接近 1，事件越可能发生。
Probabilities can be written as fractions, decimals or percentages.
概率可以用分数、小数或百分数表示。
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