📚 Year 8 OCR Maths: Quick Reference Handbook of Formulas and Theorems | Year 8 OCR 数学：公式定理速查手册
This concise handbook collects the key formulas, theorems and rules you will meet in Year 8 OCR Mathematics. Use it for rapid revision, homework support and end‑of‑topic checks. Every entry is explained in clear English and paired with its Chinese translation so that you can build a deeper understanding in both languages.
这本速查手册汇集了 Year 8 OCR 数学课程中的核心公式、定理和规则。你可以用它快速复习、辅助作业和进行单元检测。每个条目都用清晰的英文解释并配以中文翻译，帮助你在两种语言中建立起深刻的理解。
1. Number Properties and Operations | 数的性质与运算
Multiplication and division with powers of the same base follow three index laws. These are essential for simplifying calculations and algebraic expressions.
同底数幂的乘法和除法遵循三条指数律。它们是简化计算和代数表达式的基础。
am × an = am+n | am ÷ an = am−n | (am)n = am×n
For any non‑zero base a, a0 = 1. Negative indices give reciprocals: a−n = 1 / an. These rules hold for all rational bases.
任何非零底数 a 都有 a0 = 1。负指数表示倒数：a−n = 1 / an。这些规则对所有有理底数都成立。
The order of operations is remembered as BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction). Always work from left to right when operations have equal priority.
运算顺序可记为 BIDMAS（括号、指数、除法/乘法、加法/减法）。当运算具有相同优先级时，始终从左向右计算。
A fraction a / b is equivalent to (a × k) / (b × k) for any non‑zero k. To add or subtract fractions we write them with a common denominator using the LCM of the denominators.
对于任意非零 k，分数 a / b 等价于 (a × k) / (b × k)。加减分数时，我们利用分母的最小公倍数将它们化为同分母分数。
fraction, multiply by its reciprocal: (a / b) ÷ (c / d) = (a / b) × (d / c).
分数乘法：(a / b) × (c / d) = (a × c) / (b × d)。分数除法：除以一个分数等于乘以它的倒数：(a / b) ÷ (c / d) = (a / b) × (d / c)。
Converting between fractions, decimals and percentages: a percentage p% means p / 100. To change a decimal to a percentage, multiply by 100; to change a fraction to a decimal, divide the numerator by the denominator.
分数、小数和百分比的转换：百分数 p% 表示 p / 100。将小数化为百分数，乘以 100；将分数化为小数，用分子除以分母。
al × (1 − r/100). Reverse percentages are found by dividing by the multiplier.
百分比增加：新值 = 原值 × (1 + r/100)。百分比减少：新值 = 原值 × (1 − r/100)。逆向百分比通过除以乘数来计算。
2. Algebra: Simplifying and Expanding | 代数：化简与展开
Like terms contain the same variable raised to the same power. We combine them by adding or subtracting their coefficients. Unlike terms cannot be combined.
同类项含有相同的变量和相同的指数。我们通过加减它们的系数来合并同类项。不同类项不能合并。
ing a product of two binomials we multiply each term in the first bracket by each term in the second.
分配律用于展开括号：a(b + c) = ab + ac。展开两个二项式的乘积时，要把第一个括号中的每一项乘以第二个括号中的每一项。
s. For example, 6x + 9 = 3(2x + 3). Always check your factorisation by expanding mentally.
因式分解是展开的逆运算。我们寻找所有项的最高公因式 (HCF)。例如，6x + 9 = 3(2x + 3)。总是通过心算展开来检验因式分解的结果。
e careful to reverse the sign of every term inside: −(2x − 5) = −2x + 5.
当括号前带有负号时，化简时要小心地将括号内每一项变号：−(2x − 5) = −2x + 5。
= 3 and y = −2, the value of 4x − y is 4(3) − (−2) = 12 + 2 = 14.
代入法是指用给定的数字替换字母。例如，若 x = 3、y = −2，则 4x − y 的值为 4(3) − (−2) = 12 + 2 = 14。
3. Solving Linear Equations | 解一元一次方程
An equation states that two expressions are equal. To solve it we isolate the unknown on one side by using inverse operations, keeping the equation balanced.
方程表示两个表达式相等。求解时我们通过逆运算将未知数孤立在等式的一边，并始终保持等式平衡。
side and the constants on the other. Example: 5x + 3 = 2x + 9 ⇒ 3x = 6 ⇒ x = 2.
当未知数出现在等式两边时，先将含变量的项移到一边，常数项移到另一边。例如：5x + 3 = 2x + 9 ⇒ 3x = 6 ⇒ x = 2。
Equations with brackets: expand first, then solve. With fractions, multiply every term by the lowest common denominator to clear the fractions before solving.
带括号的方程：先展开，再求解。含有分数时，先乘以分母的最小公倍数消去分母，然后再求解。
An inequality uses one of <, >, ≤ or ≥. Solving inequalities follows the same steps as equations except that multiplying or dividing by a negative number reverses the inequality sign.
不等式使用 <、>、≤ 或 ≥。解不等式的步骤与方程相同，但当乘以或除以一个负数时，不等号的方向要反转。
Represent inequalities on a number line: an open circle for < or >, a filled circle for ≤ or ≥. A solid line continues to the direction of the solution.
在数轴上表示不等式：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。实线向解集的方向延伸。
4. Sequences and Patterns | 数列与规律
A sequence is an ordered list of numbers that follows a rule. A term‑to‑term rule tells you how to get the next term from the current one; a position‑to‑term rule gives the nth term directly.
数列是按照一定规则排列的一列数。项间规则告诉你如何从当前项得到下一项；位置规则直接给出第 n 项的表达式。
In an arithmetic sequence the difference between consecutive terms is constant, called the common difference d. The nth term is a + (n − 1)d, where a is the first term.
在等差数列中，相邻两项的差是常数，称为公差 d。第 n 项公式为 a + (n − 1)d，其中 a 为首项。
If the pattern of differences is not constant, check whether the sequence is geometric (constant ratio r) or follows a quadratic or other rule. Fibonacci sequences give each term as the sum of the two previous ones.
如果差值不固定，检查数列是否为等比数列（固定公比 r），或者遵循二次或其他规则。斐波那契数列中每一项都是前两项之和。
To find a missing term inside a sequence, use the term‑to‑term rule forwards or backwards. Always test your rule on several given terms to confirm it is correct.
要找数列中缺失的项，向前或向后运用项间规则。务必用多个已知项检验你的规则是否正确。
5. Ratio, Proportion and Percentages | 比，比例与百分比
A ratio compares parts of a whole. The ratio a : b can be simplified by dividing both parts by their highest common factor. For every 3 parts in a 3 : 5 ratio, there are 8 equal parts altogether.
比表示整体中各部分的关系。比 a : b 可以通过同时除以最大公因数来化简。例如在 3 : 5 的比中，每包含 3 份就有总共 8 份等份。
To share a quantity in a given ratio: find the total number of parts, work out the value of one part, then multiply by each ratio number.
按给定比例分配一个量：求出总份数，计算每一份的值，然后分别乘以比的每个数字。
io is constant, y = kx. Graphs of direct proportion are straight lines through the origin.
正比例：如果两个量的比值恒定，则它们成正比例，y = kx。正比例的图像是通过原点的直线。
Percentages can express a part as a fraction of 100. A percentage change is calculated as (change / original) × 100%. Using multipliers avoids repeated additions: a 15% increase multiplies by 1.15.
百分数可以表示以 100 为分母的分数。百分比变化计算公式为 (变化量 / 原量) × 100%。使用乘数可以避免重复相加：增加 15% 相当于乘以 1.15。
t: I = P × r × t, where P is principal, r the annual rate and t time in years.
单利仅根据本金计算：I = P × r × t，其中 P 为本金，r 为年利率，t 为以年为单位的时长。
6. Geometry: Angles in Parallel Lines and Polygons | 几何：平行线与多边形的角
Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°；围绕一个点的周角之和为 360°。对顶角相等。
When a transversal intersects parallel lines: corresponding angles are equal, alternate interior angles are equal, and co‑interior angles sum to 180°.
当一条截线与平行线相交时：同位角相等，内错角相等，同旁内角互补（和为 180°）。
g;. For any polygon with n sides, sum of interior angles = (n − 2) × 180°.
三角形的内角和总是 180°。四边形的内角和为 360°。对于任意 n 边形，内角和 = (n − 2) × 180°。
Each interior angle of a regular n‑gon is (n − 2) × 180° / n. The exterior angle at any vertex is 180° − interior angle, and the sum of exterior angles of any polygon is 360°.
正 n 边形的每个内角为 (n − 2) × 180° / n。任何一个顶点的外角 = 180° − 内角，且任何多边形的外角和都是 360°。
7. Perimeter, Area and Volume | 周长、面积与体积
= 2l + 2w. For a circle, the circumference C = πd or C = 2πr, where r is the radius.
周长是二维图形边界的总长度。长为 l、宽为 w 的矩形周长 P = 2l + 2w。圆的周长 C = πd 或 C = 2πr，其中 r 为半径。
Area measures the surface inside a shape. Key formulas are summarised in the table below.
面积度量图形内部的表面大小。关键公式总结在下表中。
| Shape | Area Formula |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × base × height |
| Parallelogram | A = base × perpendicular height |
| Trapezium | A = ½ (a + b) × h, where a and b are parallel sides |
| Circle | A = πr2 |
Area of compound shapes can be found by splitting the shape into known figures, calculating each area and adding or subtracting as needed.
组合图形的面积可以通过将图形分割成已知的图形，分别计算面积，然后相加或相减来求得。
Volume of a prism = area of cross‑ gth l, width w and height h, V = lwh. For a cylinder with radius r and height h, V = πr2h.
棱柱体积 = 横截面积 × 长度。长为 l、宽为 w、高为 h 的长方体体积 V = lwh。底面半径为 r、高为 h 的圆柱体积 V = πr2h。
a of a cuboid = 2(lw + lh + wh). Units for volume are cubic units, e.g. cm³, m³.
长方体的表面积 = 2(lw + lh + wh)。体积的单位是立方单位，如 cm³、m³。
8. Pythagoras' Theorem | 毕达哥拉斯定理
In a right‑angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
在直角三角形中，斜边（直角对边）的平方等于其他两条边的平方和。
a2 + b2 = c2
Where a and b are the legs and c is the hypotenuse. This theorem only works if the triangle is right‑angled.
其中 a 和 b 是直角边，c 是斜边。该定理仅在直角三角形中成立。
We can use Pythagoras' theorem to find a missing side: to find the hypotenuse, c = √(a2 + b2); to find a leg, a = √(c2 − b2).
我们可以用毕达哥拉斯定理求未知边长：求斜边，c = √(a2 + b2)；求直角边，a = √(c2 − b2)。
Pythagorean triples are sets of three integers that satisfy a2 + b2 = c2, such as (3, 4, 5) and (5, 12, 13). Recognising them can speed up problem solving.
勾股数是指满足 a2 + b2 = c2 的整数三元组，例如 (3, 4, 5) 和 (5, 12, 13)。识别它们能加快解题速度。
The theorem can be applied in real life to find distances, heights and diagonals, such as the length of a ladder leaning against a wall or the diagonal of a rectangular screen.
该定理可以实际应用于求距离、高度和对角线，例如靠墙的梯子长度或长方形屏幕的对角线。
9. Transformations and Coordinates | 变换与坐标
A translation moves every point of a shape by the same vector. It is described by a column vector (x, y) where x gives the horizontal movement and y the vertical movement.
平移将图形的每一个点按照相同的向量移动。平移可用列向量 (x, y) 描述，其中 x 表示水平移动，y 表示垂直移动。
on flips a shape over a mirror line. The shape is congruent to the original and each point' to the mirror line is preserved. Common mirror lines are the axes, y = x and y = −x.
反射将图形沿一条镜像线翻转。图形与原图形全等，且每个点到镜像线的距离保持不变。常见的镜像线有坐标轴、y = x 和 y = −x。
Rotation turns a shape about a centre through a given angle. Three pieces of information are needed: centre of rotation, angle (90°, 180° etc) and direction (clockwise or anticlockwise).
旋转使图形绕一个中心转动给定的角度。需要三个要素：旋转中心、旋转角度（如 90°、180°）和方向（顺时针或逆时针）。
An enlargement changes the size of a shape by a scale factor k about a centre. If k > 1 the shape gets larger; if 0 < k < 1 it gets smaller. Distances from the centre are multiplied by k.
放大以缩放因子 k 绕中心改变图形的大小。若 k > 1，图形变大；若 0 < k < 1，图形变小。各点到中心的距离乘以 k。
Coordinates in all four quadrants use negative and positive values. After a transformation, we can write the new coordinates of each vertex by applying the mapping rule.
四个象限中的坐标使用正值和负值。变换后，我们可以通过应用映射规则写出每个顶点的新坐标。
10. Statistics: Averages and Charts | 统计：平均数与图表
The mean is the arithmetic average, found by sum of all values divided by the number of values. The median is the middle value when data are ordered; the mode is the most frequent value.
平均数是算术平均值，等于所有数值之和除以数据个数。中位数是数据排序后位于中间的值；众数是出现次数最多的值。
rgest value − smallest value. Outliers are extreme values that can affect the mean significantly but have less effect on the median.
极差衡量数据的分散程度：极差 = 最大值 − 最小值。异常值是可能显著影响均值的极端数值，但对中位数影响较小。
A bar chart displays categorical data with bars of equal width but varying height. The height represents frequency. Gap between bars shows the categories are separate.
条形图用于分类数据，条形宽度相同而高度不同。高度代表频数。条形之间的间隔表示类别是独立的。
Pie charts show proportions of a whole. The angle for a sector is calculated as (frequency / total frequency) × 360°. Always check that the angles sum to 360°.
饼图表示整体中各部分的比例。扇形的角度计算公式为 (频数 / 总频数) × 360°。务必检查各角度之和是否为 360°。
A scatter graph shows the relationship between two variables. If the points lie close to a straight line we can say there is a correlation (positive, negative or none). A line of best fit can be used to estimate missing values.
散点图显示两个变量之间的关系。如果点大致分布在一条直线附近，我们就可以说有相关关系（正相关、负相关或无相关）。最佳拟合线可以用来估计缺失的数值。
11. Probability: Single Events and Sample Spaces | 概率：单事件与样本空间
al number of equally likely outcomes. Probabilities range from 0 (impossible) to 1 (certain).
某个事件的概率 P(事件) = 有利结果的数量 / 所有等可能结果的总数。概率的取值范围从 0（不可能）到 1（必然）。
P(A). This is useful when it is easier to find the probability of the complementary event.
所有互斥结果的概率之和为 1。P(非 A) = 1 − P(A)。当计算对立事件的概率更容易时，这个公式很有用。
A sample space lists all possible outcomes. For two events, a two‑way table or a sample space diagram helps to count outcomes systematically. For example, in rolling two dice there are 36 possible pairs.
样本空间列出所有可能的结果。对于两个事件，双向表或样本空间图有助于系统地计算结果数量。例如，掷两个骰子共有 36 种可能的结果对。
imes; number of trials. If a fair coin is tossed 150 times, we expect heads about 75 times, though the actual result may vary.
期望频数 = 概率 × 试验次数。若抛掷一枚公平硬币 150 次，我们期望正面朝上
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