📚 Year 9 OCR Maths: Quick Reference Handbook of Formulas and Theorems | Year 9 OCR 数学：公式定理速查手册
This handbook brings together the essential formulas, rules and theorems you will meet in Year 9 OCR mathematics. Use it to revise key facts from number work, algebra, geometry, measures and statistics – each section is written in both English and Chinese so you can check your understanding in either language.
这本手册汇总了 Year 9 OCR 数学中最重要的公式、法则和定理。你可以用它复习数与运算、代数、几何、测量和统计等核心知识。每个部分都采用中英双语，方便你随时用任意一种语言核对理解。
1. Number and Operations | 数与运算
The order of operations is captured by BIDMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction. Always work from left to right when only division and multiplication (or addition and subtraction) remain.
运算顺序遵循 BIDMAS 法则：括号、指数、除/乘、加/减。当只剩下乘除或加减运算时，从左向右依次计算。
Brackets → Indices → ÷/× → +/−
A number written in standard form is expressed as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Large numbers give a positive power; small decimals give a negative power.
标准形式将一个数写成 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。较大的数指数为正，较小的纯小数为负指数。
Standard form: a × 10ⁿ
When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Raising a power to another power multiplies the indices: (aᵐ)ⁿ = aᵐⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。相除则指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方指数相乘：(aᵐ)ⁿ = aᵐⁿ。
2. Fractions, Decimals, Percentages | 分数、小数、百分数
To add or subtract fractions, first find a common denominator. For multiplication, multiply the numerators and multiply the denominators. For division, multiply by the reciprocal of the second fraction.
分数加减先通分，找公分母。乘法：分子相乘、分母相乘。除法：乘以第二个分数的倒数。
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c
A percentage is a fraction out of 100. To convert a decimal to a percentage, multiply by 100. To find a percentage of an amount, multiply the amount by the percentage written as a decimal.
百分数表示以 100 为分母的分数。小数化为百分数乘 100。求一个数量的百分之几，用该数量乘百分数的小数形式。
Percentage increase is calculated as: new value = original × (1 + percentage increase/100). Percentage decrease uses (1 − percentage decrease/100).
百分比增加：新值 = 原值 × (1 + 增幅%/100)。百分比减少：新值 = 原值 × (1 − 减幅%/100)。
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. To split a quantity in a given ratio, add the parts, then divide the total by that sum to find one share; multiply each part of the ratio by the value of one share.
比用于比较整体的各个部分。按比例分配时，先将比的各部分相加，用总数除以总份数得到一份的值，再用各部分的份数乘一份的值。
Direct proportion means two quantities increase or decrease at the same rate, so y = kx. Inverse proportion means y = k/x. The constant k is found from a known pair of values.
正比例表示两个量以相同速率变化，关系为 y = kx。反比例则是 y = k/x。常数 k 可以由一对已知值求出。
Direct: y = kx
Inverse: y = k/x
4. Algebraic Expressions | 代数表达式
Like terms can be combined by adding or subtracting their coefficients. For example, 3x + 2x = 5x.
同类项可以通过加减系数合并。例如 3x + 2x = 5x。
When expanding brackets, multiply every term inside by the factor outside. With double brackets, use FOIL (First, Outer, Inner, Last) or the distributive law: (a + b)(c + d) = ac + ad + bc + bd.
展开括号时，用括号外的因数乘括号内的每一项。双括号展开可以用 FOIL 法（先、外、内、后）或分配律：(a + b)(c + d) = ac + ad + bc + bd。
(a + b)(c + d) = ac + ad + bc + bd
Factorising is the reverse of expanding. Look for a common factor in all terms and put it outside a bracket. For quadratic expressions like x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆运算。先找出各项的公因数放在括号外。对于二次式 x² + bx + c，要找到两个相乘得 c、相加得 b 的数。
x² + (p+q)x + pq = (x + p)(x + q)
5. Equations and Inequalities | 方程与不等式
To solve a linear equation, perform the same operation on both sides to isolate the unknown. For example, 2x + 3 = 7 → subtract 3 → 2x = 4 → divide by 2 → x = 2.
解一元一次方程时，在等式两边执行相同运算以隔离未知数。如 2x + 3 = 7 → 减 3 → 2x = 4 → 除以 2 → x = 2。
Inequalities behave like equations but reverse the sign when multiplying or dividing by a negative number. Represent solutions on a number line with open circles for < or >, closed circles for ≤ or ≥.
不等式的解法与方程类似，但乘或除以负数时要反转不等号。解集可用数轴表示：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
If −a < −b, then a > b (sign reverses)
6. Sequences and Graphs | 数列与图像
Arithmetic sequences have a constant difference. The nth term is given by: nth term = a + (n − 1)d, where a is the first term and d is the common difference. Often written as Un = dn + (a − d).
等差数列有固定的公差。第 n 项公式为：第 n 项 = a + (n − 1)d，其中 a 是首项，d 是公差。常写作 Un = dn + (a − d)。
Un = dn + (a − d)
The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Gradient m = (change in y)/(change in x). Parallel lines have equal gradients.
直线方程是 y = mx + c，m 为斜率，c 为 y 轴截距。斜率 m = Δy/Δx。平行线斜率相等。
y = mx + c
m = (y₂ − y₁)/(x₂ − x₁)
To plot a linear graph, make a table of at least three x-values, calculate the corresponding y-values, and draw a straight line through the points. The midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).
绘制一次函数图像时，至少选取三个 x 值，计算对应的 y 值，用直线连接这些点。两点 (x₁, y₁) 和 (x₂, y₂) 的中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
7. Geometry: Angles and Lines | 几何：角与直线
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角相加为 180°。绕一点的周角为 360°。对顶角相等。
When a transversal crosses parallel lines, alternate angles are equal, corresponding angles are equal, and interior (co-interior) angles sum to 180°.
一条截线与平行线相交时，内错角相等，同位角相等，同旁内角互补（和为 180°）。
In any triangle, the interior angles sum to 180°. An exterior angle equals the sum of the two opposite interior angles. In a quadrilateral, the interior angles sum to 360°.
任意三角形内角和为 180°。外角等于与它不相邻的两个内角之和。四边形内角和为 360°。
8. Perimeter, Area, Volume | 周长、面积、体积
Perimeter is the distance around a shape. For a rectangle, P = 2(l + w). Area of a rectangle: A = lw. Area of a triangle: A = ½ b h. Area of a parallelogram: A = b h. Area of a trapezium: A = ½ (a + b) h.
周长是围绕图形一周的长度。矩形周长 P = 2(l + w)。面积：矩形 A = lw；三角形 A = ½ b h；平行四边形 A = b h；梯形 A = ½ (a + b) h。
Triangle area: A = ½ × base × height
Trapezium area: A = ½ (a + b)h
The area of a circle is πr²; the circumference is 2πr or πd. Use π ≈ 3.14 or the π button on a calculator. For compound shapes, split the shape into simpler parts and sum their areas.
圆的面积是 πr²，周长是 2πr 或 πd。π 通常取 3.14 或使用计算器上的 π 键。组合图形的面积可以先分割为基本图形再相加。
Circle: A = πr², C = 2πr = πd
Volume of a cuboid: V = l × w × h. Volume of a triangular prism: V = area of cross-section × length. Surface area is the total area of all faces.
长方体体积 V = l × w × h。三棱柱体积 V = 横截面积 × 长。表面积是所有面的总面积。
9. Pythagoras Theorem | 毕达哥拉斯定理
In a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Use it to find a missing side: c² = a² + b², where c is the hypotenuse.
在直角三角形中，斜边的平方等于两直角边的平方和。用来求未知边长：c² = a² + b²，其中 c 为斜边。
c² = a² + b²
c = √(a² + b²)
To check if a triangle is right‑angled, verify that the three sides satisfy a² + b² = c² with the longest side as c. The theorem is reversible: if a² + b² = c², the triangle is right‑angled.
判断三角形是否为直角三角形，可检验三边是否满足 a² + b² = c²（最长边为 c）。定理可逆：若 a² + b² = c²，则三角形是直角三角形。
10. Statistics and Probability | 统计与概率
The mean is found by summing all values and dividing by the number of values. The median is the middle value when data are ordered; for an even number, it is the mean of the two middle values. The mode is the most frequent value. The range is the difference between the largest and smallest values.
平均数（均值）= 总和 ÷ 数据个数。中位数是排序后居中的数；若数据个数为偶数，取中间两数的均值。众数是出现次数最多的值。极差 = 最大值 − 最小值。
Mean = Σx/n
Range = max − min
Probability is a number between 0 and 1, written as P(event) = (number of favourable outcomes)/(total number of possible outcomes). For equally likely outcomes, all single outcomes have the same probability.
概率是 0 到 1 之间的数，P(事件) = 有利结果数 / 所有可能结果的总数。对于等可能的结果，每个单一结果的概率相同。
For mutually exclusive events A and B, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B). The sum of probabilities of all possible outcomes is 1.
互斥事件 A 或 B 的概率：P(A 或 B) = P(A) + P(B)。独立事件同时发生的概率：P(A 且 B) = P(A) × P(B)。所有可能结果的概率之和为 1。
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