📚 Year 7 OCR Maths: Formula & Theorems Quick Reference Handbook | 七年级 OCR 数学：公式定理速查手册
This quick reference handbook provides a compact summary of all the essential formulas, rules and theorems you need to master in Year 7 OCR Mathematics. Use it to review key topics before tests, complete homework with confidence, or refresh your memory whenever you get stuck.
本速查手册为您总结七年级 OCR 数学中所有必须掌握的公式、法则与定理。可在考前复习、自信完成作业或随时需要唤起记忆时使用，是您的随身数学工具包。
1. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS tells you the correct sequence to follow when a calculation involves more than one operation. The letters stand for Brackets, Indices (powers and roots), Division, Multiplication, Addition and Subtraction. Division and multiplication have equal priority – work them from left to right. The same rule applies to addition and subtraction.
BIDMAS 指明了当一道算式中含有多种运算时的正确计算顺序。字母分别代表：括号、指数（乘方与开方）、除法、乘法、加法和减法。除法和乘法优先级相同，按从左到右的顺序计算；加法和减法也是如此。
Example: 10 – 2 × 3 = 10 – 6 = 4 (not 8 × 3)
Always deal with what is inside brackets first, then evaluate any powers, next perform divisions and multiplications as they appear from left to right, and finally handle additions and subtractions from left to right. A common mistake is to always do addition before subtraction – remember they are equal, so just move left to right.
始终先算括号内的部分，然后计算指数，接着从左到右进行除法和乘法，最后从左到右处理加法和减法。一个常见错误是总以为加法先于减法——请记住它们优先级相同，只需从左往右计算即可。
2. Negative Numbers | 负数
Adding a negative number is the same as subtracting its positive value. Subtracting a negative number means you add. For multiplication and division, two like signs give a positive answer; two unlike signs give a negative answer.
加上一个负数等于减去它的相反正数。减去一个负数则变成加法。对于乘法和除法，同号得正，异号得负。
e.g., 5 + (-3) = 5 – 3 = 2; (-4) – (-2) = -4 + 2 = -2; (-3) × (-4) = 12; 15 ÷ (-3) = -5
Visualise a number line: moving right for adding a positive, left for adding a negative. When subtracting a negative, you reverse direction, effectively moving right. These rules are essential for algebra and solving equations later.
想象一条数轴：加正数向右移动，加负数向左移动；减负数时方向反转，等同于向右移动。这些规则对后续的代数和方程求解至关重要。
3. Fractions, Decimals & Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of representing parts of a whole. To convert a fraction to a decimal, divide the numerator by the denominator. To turn a decimal into a percentage, multiply by 100. To go from a percentage to a decimal, divide by 100.
分数、小数和百分数是表示整体部分的三种方式。将分数化为小数，用分子除以分母；将小数化为百分数，乘以100；由百分数化为小数，则除以100。
e.g., 3/4 = 0.75 = 75%; 40% = 0.4 = 2/5
To add or subtract fractions, find a common denominator first, then add or subtract the numerators and keep the denominator the same. For multiplication, simply multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal (flip the second fraction).
分数加减时，先找到公分母，再将分子相加或相减，分母保持不变。乘法直接将分子相乘、分母相乘。除以一个分数等于乘以它的倒数（将第二个分数的分子分母颠倒）。
Example: 2/3 × 3/4 = 6/12 = 1/2; 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 1 1/4
When finding a percentage of an amount, convert the percentage to a decimal and multiply. To increase by a percentage, multiply by (1 + p/100); to decrease, multiply by (1 – p/100).
求一个数的百分之几，先将百分数转为小数再相乘。增加一个百分数，乘以 (1 + p/100)；减少一个百分数，乘以 (1 – p/100)。
4. Perimeter | 周长
The perimeter of a shape is the total distance around its outer edges. For any polygon, simply add the lengths of all sides. A rectangle has two pairs of equal sides, so its perimeter can be found quickly.
周长是图形外边界的总长度。对于任何多边形，只需将所有边长相加。长方形有两组相等的对边，因而可以快速计算周长。
Perimeter of a rectangle = 2 × (length + width) or P = 2(l + w)
For a square, all four sides are equal: P = 4 × side (P = 4s). The perimeter of any triangle is simply the sum of its three side lengths: P = a + b + c.
正方形的四条边都相等：P = 4 × 边长 (P = 4s)。任何三角形的周长就是三条边长之和：P = a + b + c。
When dealing with compound shapes made of rectangles, break the shape into known lengths and add up only the outer edges – do not include internal shared edges.
处理由矩形组成的复合图形时，将图形拆分为已知边长，只将外围边相加，内部共享边不计入周长。
5. Area | 面积
Area measures the space inside a flat shape. It is expressed in square units such as cm² or m². Each shape has its own formula; you must learn these by heart.
面积衡量平面图形内部空间的大小，单位为平方单位，如 cm² 或 m²。每种图形都有自己的公式，需要熟记。
Rectangle: Area = length × width, A = lw
Square: Area = side × side, A = s²
Triangle: Area = ½ × base × perpendicular height, A = ½bh
Parallelogram: Area = base × perpendicular height, A = bh
Trapezium: Area = ½ × (a + b) × perpendicular height, A = ½(a + b)h
For a rectangle and square, multiply the two perpendicular dimensions. The triangle formula can be remembered as ‘half the base times the height’ – the height must be at right angles to the chosen base. A parallelogram follows the same rule as a rectangle, but you must use the vertical height, not the slanted side length. The trapezium formula adds the two parallel sides (a and b), multiplies by the height and then halves.
长方形和正方形将两个垂直尺寸相乘。三角形公式可记为“底乘高的一半”——高必须与所选底边垂直。平行四边形与长方形规则类似，但需使用垂直高，而非斜边长度。梯形公式将两条平行边（a 和 b）相加，乘以高再除以2。
6. Volume | 体积
Volume is the amount of space a solid object occupies, measured in cubic units such as cm³ or m³. In Year 7, you mainly work with cuboids (rectangular prisms) and cubes.
体积是立体物所占空间的大小，以立方单位如 cm³ 或 m³ 计量。七年级主要学习长方体（矩形棱柱）和立方体。
Cuboid: Volume = length × width × height, V = l × w × h
Cube: Volume = side³, V = s³
To find the volume of a cuboid, multiply the three dimensions together – the order does not matter. A cube is a special cuboid where all edges are equal, so you raise the side length to the power of three. Remember that volume is not the same as surface area; volume is how much it can hold inside.
计算长方体体积，将三条边长相乘，顺序不限。立方体是长宽高均相等的特殊长方体，因此将边长三次方即可。要记住体积与表面积不同；体积表示内部能容纳多少东西。
7. Angles | 角度
Angles are measured in degrees (°). A right angle is exactly 90°, a straight line is 180°, and a full turn is 360°. When two lines intersect, vertically opposite angles are equal.
角度以度 (°) 为单位。直角是90°，平角是180°，周角是360°。两条直线相交时，对顶角相等。
Angles on a straight line add up to 180°.
Angles around a point add up to 360°.
Vertically opposite angles are equal.
Complementary angles add up to 90°, and supplementary angles add up to 180°. Inside a triangle, the three interior angles always sum to 180°. For any quadrilateral, the sum is 360°. In a regular polygon, all sides and angles are equal, and the sum of interior angles is (n – 2) × 180°, where n is the number of sides.
互余角之和为90°，互补角之和为180°。三角形三个内角之和恒为180°，任意四边形内角和为360°。在正多边形中，所有边和内角均相等，内角和公式为 (n – 2) × 180°，其中 n 为边数。
8. Algebra: Expressions | 代数：表达式
In algebra, letters (variables) are used to stand for unknown numbers. A term is a combination of numbers and variables, like 3x or -5y. The number in front is called the coefficient. When terms have exactly the same variable part, they are called ‘like terms’ and can be combined.
在代数中，字母（变量）用来表示未知数。项是数字和变量的组合，如 3x 或 -5y。字母前的数字称为系数。当若干项的变量部分完全相同时，称为“同类项”，可以合并。
Simplify: 4a + 2b + 3a – b = 7a + b
To expand brackets, multiply every term inside the bracket by the term outside. For example, 3(x + 2) = 3x + 6. This is the distributive law. Factorising is the reverse process – write a common factor outside a bracket.
展开括号时，用括号外的每一项乘以括号内的每一项。例如 3(x + 2) = 3x + 6，这是分配律。因式分解是其逆过程——把公因式提到括号外。
Expand: -2(4y – 3) = -8y + 6
Substituting a number for a variable means replacing the letter with a given value and then calculating using BIDMAS. Always write the substitution clearly to avoid mistakes.
代入数值指的是将字母替换为给定的数字，再按照 BIDMAS 规则计算。务必清晰书写代入过程，避免出错。
9. Solving Equations | 解方程
An equation shows that two expressions are equal, like a balanced scale. To solve it, you perform the same operation on both sides to keep the balance and find the value of the unknown.
方程表示两个表达式相等，就像一架平衡的天平。解方程时，两边同时进行相同运算以保持平衡，从而求出未知数的值。
One-step equations: x + 7 = 12 → x = 5; 4x = 20 → x = 5; x / 3 = 6 → x = 18
For two-step equations, undo the operations in reverse BIDMAS order – deal with addition/subtraction first, then multiplication/division. For example, 2x + 5 = 17: subtract 5 from both sides to get 2x = 12, then divide by 2, giving x = 6.
对于两步方程，按逆 BIDMAS 顺序消去运算——先处理加减，再处理乘除。例如 2x + 5 = 17：两边同时减去5得 2x = 12，再除以2，得到 x = 6。
Always check your answer by substituting it back into the original equation. If both sides give the same value, your solution is correct. Equations with unknowns on both sides will be covered later, but the principle of balance remains the same.
始终要通过代回原方程检验答案。若两边结果相等，解就正确。未知数出现在两边的方程将在后续学习，但平衡原理不变。
10. Statistics: Averages & Range | 统计：平均数与极差
Statistics is about collecting, organising and interpreting data. The mean, median, mode and range are used to summarise a set of numbers.
统计学关注数据的收集、整理与解读。平均数、中位数、众数和极差用于概括一组数据。
Mean = (sum of all values) ÷ (number of values)
Median = middle value when data is ordered
Mode = value that appears most often
Range = largest value – smallest value
The mean gives the ‘fair share’ average but can be affected by extreme values. The median is the central number; if there is an even number of data items, the median is the mean of the two middle numbers. The mode is useful for finding the most common category. The range shows how spread out the data is.
平均数给出“平均分配”的值，但易受极端值影响。中位数是中间的数；若数据个数为偶数，中位数取中间两个数的平均数。众数有助于找出最常见的类别。极差反映数据的分散程度。
When data is presented in frequency tables, the mode is the value with the highest frequency. The range is still max – min. The mean is calculated by multiplying each value by its frequency, summing these products and dividing by the total frequency.
在频数表中，众数是频数最高的值；极差依然是最大值减最小值。计算平均数时，先将每个值乘以频数，再将这些乘积相加，最后除以总频数。
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