📚 Year 9 OCR Maths: Quick-Reference Formula and Theorem Handbook | Year 9 OCR 数学：公式定理速查手册
Welcome to your essential quick-reference handbook for Year 9 OCR Mathematics. This guide gathers the most important formulas, theorems and rules you need to master across number, algebra, geometry, statistics and more. Each entry is presented with clear explanations in both English and Chinese, so you can revise efficiently and tackle exam questions with confidence.
欢迎使用 Year 9 OCR 数学速查手册。本手册汇集了数、代数、几何、统计等模块中你必须掌握的核心公式、定理和法则。每个条目都配有清晰的中英双语解释，帮助你高效复习，自信应对考试题目。
1. Basic Arithmetic and Number Properties | 算术与运算性质
Integer addition and subtraction: When adding a positive and a negative integer, find the difference between their absolute values and give the answer the sign of the number with the larger absolute value. For subtraction, remember that subtracting a number is the same as adding its opposite: a – b = a + (-b).
整数加法和减法：正数与负数相加时，求出两者绝对值之差，并取绝对值较大的数的符号。做减法时，记住减去一个数等于加上它的相反数：a – b = a + (-b)。
a – (-b) = a + b
Multiplication and division with negatives: When you multiply or divide two numbers with the same sign, the result is positive. When the signs are different, the result is negative.
负数的乘除：同号两数相乘或相除结果为正；异号两数相乘或相除结果为负。
(+a) × (+b) = +ab (-a) × (-b) = +ab (+a) × (-b) = -ab
Prime numbers: A prime number has exactly two distinct positive factors: 1 and itself. 1 is not a prime number. The first few primes are 2, 3, 5, 7, 11, 13, …
质数：质数有且只有两个不同的正因数：1 和它本身。1 不是质数。开头几个质数是 2, 3, 5, 7, 11, 13 ……
Highest Common Factor (HCF) and Lowest Common Multiple (LCM): The HCF is the largest number that divides into two or more numbers without a remainder. The LCM is the smallest number that is a multiple of each of the given numbers. Use prime factorisation to find them efficiently.
最大公因数 (HCF) 和最小公倍数 (LCM)：HCF 是能同时整除两个或多个数的最大数。LCM 是能够被这些数同时整除的最小数。使用质因数分解可以高效地求出它们。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Converting between fractions, decimals and percentages: To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify.
分数、小数与百分数的互化：分数化小数：用分子除以分母。小数化百分数：乘以 100。百分数化分数：将百分数写成分母为 100 的分数并化简。
Fraction → Decimal: 3/4 = 3 ÷ 4 = 0.75 Decimal → Percentage: 0.75 × 100 = 75%
Adding and subtracting fractions: Find a common denominator first. Convert each fraction to an equivalent fraction with that denominator, then add or subtract the numerators. Keep the denominator the same.
分数加减法：先找到公分母。将每个分数转化成以该公分母为分母的等值分数，然后对分子进行加减，分母保持不变。
Multiplying and dividing fractions: To multiply fractions, multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal (turn the second fraction upside down).
分数乘除法：分数相乘：分子乘分子，分母乘分母。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）。
a/b × c/d = (a × c)/(b × d) a/b ÷ c/d = a/b × d/c
Percentage increase and decrease: Find the percentage of the original amount and add or subtract it. Alternatively, use a multiplier: for a 15% increase multiply by 1.15; for a 15% decrease multiply by 0.85.
百分数增减：求出原数量的百分数，然后加上或减去该数量。也可以使用乘数法：增加 15% 就乘以 1.15；减少 15% 就乘以 0.85。
3. Indices and Standard Form | 指数与标准形式
Powers and index notation: The expression aⁿ means a multiplied by itself n times. a is the base and n is the index (or exponent). For example, 2³ = 2 × 2 × 2 = 8.
幂与指数记号：aⁿ 表示 a 自乘 n 次。a 是底数，n 是指数。例如 2³ = 2 × 2 × 2 = 8。
Laws of indices: When multiplying with the same base, add the indices. When dividing, subtract the indices. When raising a power to another power, multiply the indices.
指数律：同底数幂相乘，指数相加。同底数幂相除，指数相减。幂的乘方，指数相乘。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐ×ⁿ
Negative and zero indices: Any non-zero number raised to the power 0 equals 1. A negative index represents the reciprocal: a⁻ⁿ = 1 / aⁿ.
负指数与零指数：任何非零数的 0 次幂等于 1。负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
a⁰ = 1 (a ≠ 0) a⁻ⁿ = 1 / aⁿ
Standard form (scientific notation): A number is in standard form when written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. This helps to write very large or very small numbers compactly.
标准形式（科学记数法）：当一个数写成 A × 10ⁿ 的形式，且 1 ≤ A < 10，n 为整数时，它就是标准形式。这用于简洁地表示极大或极小的数。
4800 = 4.8 × 10³ 0.00065 = 6.5 × 10⁻⁴
4. Algebraic Expressions and Simplification | 代数表达式与化简
Collecting like terms: Terms that contain exactly the same variables to the same powers can be combined by adding or subtracting their coefficients. For example, 3x + 5x = 8x, but 3x and 3x² are not like terms.
合并同类项：含有完全相同字母且指数相同的项可以合并，只需将其系数相加或相减。例如 3x + 5x = 8x，但 3x 和 3x² 不是同类项。
Expanding brackets: Multiply the term outside the bracket by each term inside the bracket. If there is a negative sign in front, it changes the signs of all terms inside.
去括号展开：将括号外的项乘以括号内的每一项。若括号前为负号，则括号内各项均要变号。
a(b + c) = ab + ac -2(3x – 4) = -6x + 8
Factorising expressions: This is the reverse of expanding. Look for the highest common factor of the terms and place it outside the bracket. Always check your answer by expanding.
因式分解：这是展开的逆过程。找出各项的最大公因数并将其提到括号外面。最后记得用展开检验你的答案。
6x + 9 = 3(2x + 3) x² + 5x = x(x + 5)
Substitution into formulas: Replace each variable with its given numerical value, then work out the calculation following the BIDMAS/BODMAS order. Clearly show each step to avoid mistakes.
代入公式求值：将每个变量替换为给定的数值，然后按照运算顺序（括号、指数、乘除、加减）进行计算。清晰地写出每一步以避免错误。
5. Linear Equations and Inequalities | 线性方程与不等式
Solving one-step equations: Perform the inverse operation on both sides of the equation. For instance, if x + 5 = 12, subtract 5 from both sides to get x = 7.
解一步方程：对等式两边同时进行逆运算。例如，若 x + 5 = 12，两边同时减去 5 即得 x = 7。
Solving multi-step linear equations: Start by removing brackets, then collect like terms on each side, and finally isolate the variable by using inverse operations. Always keep the equation balanced.
解多步线性方程：先去括号，然后合并每一边的同类项，最后通过逆运算将变量单独分离出来。始终要保持等式的平衡。
2(x – 3) = x + 4 ⇒ 2x – 6 = x + 4 ⇒ x = 10
Solving inequalities: Inequalities are solved in the same way as equations, but remember: if you multiply or divide by a negative number, you must reverse the inequality sign.
解不等式：不等式的解法与方程类似，但要特别注意：如果两边同时乘或除以一个负数，不等号的方向必须改变。
-2x < 6 ⇒ x > -3
Representing inequalities on a number line: Use a hollow circle (◦) for strict inequalities (< or >) and a filled circle (•) for inclusive inequalities (≤ or ≥). Draw an arrow in the direction of the solution set.
在数轴上表示不等式：严格不等式（< 或 >）用空心圆圈，包含等号的不等式（≤ 或 ≥）用实心圆点。然后向解集方向画箭头。
6. Sequences | 序列
Finding the nth term of a linear sequence: For a sequence with a constant difference d between terms, the nth term is given by an + b. Find d by subtracting any term from the next. Then find b by checking the term when n = 1 and adjusting the formula. Always test your rule.
求线性序列的第 n 项：若序列相邻两项的差为常数 d，则第 n 项公式为 an + b。用后一项减去前一项求出 d，再将 n = 1 代入调整得出 b。务必检验你的通项公式。
Example: 4, 7, 10, 13,… ⇒ nth term = 3n + 1
Term-to-term rules: A term-to-term rule tells you how to get from one term to the next, for example, ‘add 3 each time’ or ‘multiply by 2 then subtract 1’. This is different from the position-to-term (nth term) rule.
邻项关系法则：邻项关系法则说明如何从一项得到下一项，比如“每次加 3”或“乘以 2 再减 1”。这与用位置求项 (第 n 项) 的法则不同。
Other sequences: Recognize square numbers (1, 4, 9, 16,…), cube numbers (1, 8, 27, 64,…) and Fibonacci-type sequences where each term is the sum of the two previous terms.
其他序列：能识别平方数序列 (1, 4, 9, 16,…)、立方数序列 (1, 8, 27, 64,…) 以及斐波那契型序列，即每一项都是前两项之和。
7. Ratio and Proportion | 比率与比例
Simplifying ratios: Divide all parts of the ratio by their highest common factor to write it in its simplest form. A ratio should be expressed using whole numbers where possible.
化简比：将比的所有项都除以它们的最大公因数，写成最简形式。比应尽可能使用整数来表示。
18 : 24 simplifies to 3 : 4 (divide both by 6)
Sharing in a ratio: Add the parts of the ratio to find the total number of parts. Divide the total quantity by the total number of parts to find the value of one part, then multiply by the number of parts for each share.
按比例分配：将比的各部分相加得出总份数。用总量除以总份数求出一份的量，然后分别乘以每个份额的份数。
Direct proportion: Two quantities are in direct proportion if their ratio stays constant. This means y ∝ x, which gives the equation y = kx. The graph of y against x is a straight line through the origin.
正比例：若两个量的比值保持不变，它们就成正比例。即 y ∝ x，从而得出方程 y = kx。y 关于 x 的图像是一条过原点的直线。
y = kx
Best buys and value for money: Calculate the unit price (price per gram, per litre, etc.) by dividing the total cost by the quantity. Compare unit prices between different offers to identify the best value.
最佳购买与性价比：用总价除以数量求出单位价格（每克、每升等的价格）。比较不同优惠产品的单位价格，从而找出最划算的选择。
8. Measurement and Geometry Formulas | 测量与几何公式
Perimeter: The perimeter of a shape is the total distance around its boundary. For a rectangle: P = 2l + 2w, where l is length and w is width. For a triangle, simply add the three side lengths.
周长：周长是一个图形边界的总长度。矩形周长：P = 2l + 2w，其中 l 为长，w 为宽。三角形周长为三边长之和。
Rectangle P = 2(l + w)
Area of common shapes: Learn the key area formulas. Area is measured in square units (cm², m², etc.).
常见图形面积：熟记关键面积公式。面积以平方单位（cm², m² 等）计量。
Square: A = s² Rectangle: A = l × w Triangle: A = ½ × base × height
Parallelogram: A = base × height Trapezium: A = ½(a + b)h
Area and circumference of a circle: For any circle with radius r, use the formulas involving π (pi, approximately 3.14).
圆的面积和周长：对于半径为 r 的圆，使用包含 π（约 3.14）的公式。
Circumference C = 2πr Area A = πr²
Volume of prisms: The volume of any prism is found by multiplying the area of the cross-section by its length. For a cuboid, V = l × w × h.
棱柱的体积：任何棱柱的体积都等于底面积乘以高（或长）。长方体的体积：V = l × w × h。
Volume of a prism = area of cross-section × length
9. Angle Rules and Properties of Polygons | 角度规则与多边形性质
Angles on a straight line and around a point: Angles on a straight line add up to 180°. Angles around a point add up to 360°.
平角与周角：直线上的角之和为 180°。绕一点的角之和为 360°。
Angles in triangles: The sum of the interior angles of any triangle is always 180°. In an equilateral triangle, each angle is 60°; in an isosceles triangle, the base angles are equal.
三角形内角：任何三角形的内角和恒为 180°。等边三角形每个角为 60°；等腰三角形两底角相等。
a + b + c = 180°
Parallel lines and angles: When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles add up to 180°.
平行线与角：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°）。
Interior angles of polygons: For an n-sided polygon, the sum of interior angles is (n – 2) × 180°. For a regular polygon, each interior angle = (n – 2) × 180° / n.
多边形内角和：n 边形内角和为 (n – 2) × 180°。正多边形每个内角等于 (n – 2) × 180° / n。
Sum of interior angles = (n – 2) × 180°
Exterior angles of polygons: The sum of the exterior angles of any convex polygon is always 360°. In a regular polygon, each exterior angle = 360° / n.
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