📚 Year 8 OCR Maths: Formula & Theorem Quick Reference | Year 8 OCR 数学：公式定理速查手册
This quick-reference handbook brings together the essential formulas, theorems and key facts needed for the Year 8 OCR Mathematics curriculum. Use it to strengthen your understanding, check a rule quickly, or revise before an assessment. Every topic is presented in clear English–Chinese pairs so you can learn the terminology in both languages.
这本速查手册汇集了 Year 8 OCR 数学课程所需的核心公式、定理和关键知识点。你可以用它加深理解、快速查询规则，或在评估前进行复习。每个主题都以清晰的英文–中文对照呈现，帮助你掌握双语术语。
1. Number and Place Value | 数字与位值
Place value tells us the value of each digit in a number. For whole numbers, the columns are units, tens, hundreds, thousands and so on.
位值表示数字中每一位数字的价值。对于整数，数位从右向左依次是个位、十位、百位、千位等。
A positive number is greater than zero; a negative number is less than zero. The symbols < (less than) and > (greater than) are used to compare numbers.
正数大于零；负数小于零。比较数字时用符号 < (小于) 和 > (大于)。
Rounding to the nearest 10, 100 or 1000: look at the digit immediately to the right – if it is 5 or more, round up; otherwise, round down.
四舍五入到最近的10、100或1000：看右边第一位数字，如果大于等于5则进一，否则舍去。
When multiplying by powers of 10, digits move to the left; when dividing, they move to the right.
乘以10的幂时，数字向左移动；除以10的幂时，数字向右移动。
d multiplication: 3² = 3 × 3 = 9, 2³ = 2 × 2 × 2 = 8. The square root √ reverses squaring: √9 = 3.
指数表示重复相乘：3² = 3 × 3 = 9，2³ = 2 × 2 × 2 = 8。平方根 √ 是平方的逆运算：√9 = 3。
2. Fractions, Decimals & Percentages | 分数、小数与百分数
ent fractions have the same value but different numerators and denominators, e.g. ½ = ²⁄₄.
分数表示整体的一部分。等值分数数值相同但分子分母不同，如 ½ = ²⁄₄。
To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100 and add the % sign.
分数化小数：分子除以分母。小数化百分数：乘以100并加上百分号。
Adding or subtracting fractions requires a common denominator. Multiply numerator and denominator to make the denominators the same.
分数加减法需要通分。将分子和分母同乘以一个数，使分母相同。
t, multiply the amount by the percentage written as a decimal or fraction. For example, 15% of 80 = 0.15 × 80 = 12.
求一个量的百分数，用这个量乘以百分数表示的小数或分数。例如，80的15% = 0.15 × 80 = 12。
entage increase or decrease: new amount = original × (1 ± percentage change as a decimal).
百分比增减：新量 = 原量 × (1 ± 变化率的小数形式)。
3. Ratio and Proportion | 比和比例
A ratio compares two or more quantities. It can be written as a : b or a/b, and should be simplified by dividing both parts by their highest common factor.
比用来比较两个或多个数量，可写成 a : b 或 a/b，并应通过除以最大公因数来化简。
Sharing in a given ratio: add the parts of the ratio to find the total number of shares, then divide the total amount by this number and multiply by each part.
按比例分配：将比的各项相加得到总份数，再用总量除以总份数，然后乘以每一项。
ease at the same rate. Their relationship can be written as y = kx, where k is a constant.
如果两个量以相同速率增减，则成正比例。关系可表示为 y = kx，其中 k 是常数。
The unitary method finds the value of one unit first, then scales up to the required amount – very useful for solving proportion problems.
归一法先求出一个单位的量，再扩展到所需数量——这对解决比例问题非常有用。
4. Expressions, Equations & Inequalities | 表达式、方程与不等式
and operation symbols. Like terms (same variable and power) can be combined: 3a + 2a = 5a.
代数表达式包含数字、字母和运算符号。同类项（变量和指数相同）可以合并：3a + 2a = 5a。
the distributive law: a(b + c) = ab + ac. Watch the signs when the bracket is subtracted.
展开括号使用分配律：a(b + c) = ab + ac。括号前有减号时要注意变号。
Factorising is the reverse of expanding – take out the highest common factor: 4x + 6 = 2(2x + 3).
因式分解是展开的逆运算，提取最大公因子：4x + 6 = 2(2x + 3)。
An equation is a statement that two expressions are equal. Solve by performing the same operation on both sides, keeping the balance.
方程说明两个表达式相等。解方程时，两边同时进行相同运算，保持平衡。
olved like equations, but if you multiply or divide by a negative number, you must flip the inequality sign.
不等式用 <、>、≤、≥ 表示。解法与方程类似，但如果两边同乘以或除以一个负数，必须反转不等号。
Substitution means replacing letters with given numbers and working out the value. Always follow BIDMAS (order of operations).
代入求值是指用给定数字替换字母并计算。始终遵循运算顺序（先乘除后加减，有括号先算括号）。
5. Linear Sequences | 线性序列
A linear sequence has a constant difference between consecutive terms. The nth term formula gives any term directly.
线性序列相邻两项的差相等。第 n 项公式可直接求出任意一项。
T(n) = a + (n – 1)d
where a = first term, d = common difference, and n = term position.
其中 a = 首项，d = 公差，n = 项数。
To find the nth term from a pattern: work out the term-to-term rule (difference) and then adjust the multiple of n to fit the starting number.
从规律中寻找第 n 项：先找出相邻项之差，然后调整 n 的倍数使其符合首项。
Generating the sequence from the nth term is simply substituting n = 1, 2, 3… into the formula.
根据第 n 项公式生成序列，只需将 n = 1、2、3...代入公式即可。
6. Coordinates & Straight-Line Graphs | 坐标与直线图
Coordinates are written as (x, y) and plotted on a grid with four quadrants. The x-coordinate (abscissa) tells you how far to move horizontally from the origin (0,0); the y-coordinate (ordinate) tells you how far to move vertically.
坐标写为 (x, y)，绘制在具有四个象限的网格上。x 坐标（横坐标）表示从原点 (0,0) 水平移动的距离，y 坐标（纵坐标）表示垂直移动的距离。
The midpoint between two points (x₁, y₁) and (x₂, y₂) is found by averaging the x- and y-coordinates:
两点 (x₁, y₁) 和 (x₂, y₂) 的中点等于坐标的平均值：
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
re m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
直线图方程通常写为 y = mx + c，其中 m 是斜率（倾斜程度），c 是 y 截距（直线与 y 轴交点的纵坐标）。
To draw a straight-line graph, create a table of values, work out the y-values for at least three x-values, plot the points and join them with a ruler.
绘制直线图时，先列出自变量取值表，计算出至少三个 x 对应的 y 值，描点后用直尺连成直线。
7. Angles & Parallel Lines | 角与平行线
Angles on a straight line add up to 180°. Angles around a point add up to 360°.
平角等于180°。周角等于360°。
Vertically opposite angles (formed when two lines cross) are equal.
对顶角（两直线相交形成）相等。
When a transversal cuts a pair of parallel lines:
当一条截线与一对平行线相交时：
Corresponding angles are equal. Alternate angles are equal. Co-interior (allied) angles sum to 180°.
同位角相等。内错角相等。同旁内角互补（和为180°）。
An angle bisector divides an angle into two equal parts. Perpendicular lines meet at a right angle (90°).
角平分线将一个角平分为两个相等的角。垂直线相交成直角（90°）。
8. Properties of Triangles & Quadrilaterals | 三角形与四边形的性质
The interior angles of any triangle always add up to 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
任意三角形内角和恒为180°。三角形的一个外角等于与它不相邻的两个内角之和。
Triangles are classified by sides: equilateral (three equal sides, three 60° angles), isosceles (two equal sides, two equal base angles) and scalene (no equal sides). By angles: acute, right and obtuse.
三角形按边分类：等边三角形（三边相等，三个角均为60°），等腰三角形（两腰相等，两底角相等）和不等边三角形（三边均不等）。按角分类：锐角、直角和钝角三角形。
The sum of interior angles of a quadrilateral is 360°. Special quadrilaterals include the square (four equal sides, four right angles), rectangle (opposite sides equal, four right angles), rhombus (four equal sides, opposite angles equal), parallelogram (opposite sides parallel and equal), trapezium (at least one pair of parallel sides) and kite (two pairs of adjacent equal sides, one pair of equal opposite angles).
四边形内角和为360°。特殊四边形包括：正方形（四边相等，四个直角），长方形（对边相等，四个直角），菱形（四边相等，对角相等），平行四边形（对边平行且相等），梯形（至少一对平行边）和风筝形（两对邻边相等，一对对角相等）。
For any polygon with n sides, the sum of interior angles = (n – 2) × 180°.
任意 n 边形的内角和 = (n – 2) × 180°。
9. Perimeter, Area & Volume | 周长、面积与体积
Perimeter is the distance around the outside of a shape. Area is the amount of surface a shape covers, measured in square units. Volume measures the space a solid occupies, in cubic units.
周长是图形外部边界的总长度。面积是图形所占表面的大小，以平方单位计量。体积测量立体所占空间，以立方单位计量。
| Shape / Formula (English) | 形状 / 公式（中文） |
|---|---|
| Rectangle: A = l × w | 长方形：面积 = 长 × 宽 |
| Square: A = a² | 正方形：面积 = 边长² |
| Triangle: A = ½ × b × h | 三角形：面积 = ½ × 底 × 高 |
| Parallelogram: A = b × h | 平行四边形：面积 = 底 × 高 |
| Trapezium: A = ½ (a + b) h | 梯形：面积 = ½ (上底 + 下底) × 高 |
| Circle: C = 2πr = πd | 圆：周长 = 2πr = πd |
| Circle: A = πr² | 圆：面积 = π × 半径² |
Volume of a cuboid (rectangular prism): V = l × w × h. For a cube: V = a³.
长方体体积：V = 长 × 宽 × 高。正方体体积：V = 边长³。
rism, the volume is given by V = area of cross-section × length. For a cylinder, V = πr²h.
任何棱柱的体积 = 横截面积 × 长度。圆柱体体积：V = πr²h。
Surface area of a solid is the total area of all its faces. For a cube: SA = 6a².
立体表面积是指所有面的总面积。正方体表面积：SA = 6 × 边长²。
10. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem applies only to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它表明斜边的平方等于两直角边的平方和。
a² + b² = c²
(where c is the hypotenuse, the longest side opposite the right angle)
（其中 c 为斜边，直角所对的最长边）
To find the length of the hypotenuse, add the squares of the legs and then take the square root. To find a shorter side, subtract the square of the known leg from the square of the hypotenuse and then take the square root.
求斜边长：将两直角边的平方相加，再开平方根。求一直角边：用斜边平方减去另一直角边平方，再开平方根。
eorem: if the sides of a triangle satisfy a² + b² = c², then the triangle is right-angled.
也可以使用逆定理：若三角形三边满足 a² + b² = c²，则该三角形是直角三角形。
11. Transformations | 变换
A translation moves a shape without turning or flipping it. It is described by a column vector that tells you how many units to move left/right and up/down.
平移移动图形时不旋转也不翻转。用列向量描述，指明向右/左和向上/下移动的单位数。
A reflection produces a mirror image of the shape. The mirror line is a straight line; points on the original shape and its image are the same perpendicular distance from this line.
反射生成图形的镜像。对称轴是一条直线；原图形的每一点与其像到该直线的垂直距离相等。
A rotation turns a shape about a fixed point, the centre of rotation. You must state the centre, the angle of rotation and the direction (clockwise or anticlockwise).
旋转使图形绕一个固定点（旋转中心）转动。需说明旋转中心、旋转角度和方向（顺时针或逆时针）。
An enlargement changes the size of a shape by a scale factor about a centre of enlargement. If the scale factor is greater than 1, the image is larger; if between 0 and 1, the image is smaller. A negative scale factor also changes the orientation.
放大是以放大中心为基点，按比例因子改变图形大小。比例因子大于1，像变大；介于0和1之间，像缩小。负
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