📚 Year 9 Cambridge Maths: Formula & Theorem Quick Reference | Year 9 剑桥数学公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems and shortcuts covered in the Year 9 Cambridge Mathematics curriculum. Use it to check key results while solving problems, completing homework or preparing for assessments. Each section is presented in clear English and Chinese to support bilingual learners.
这本速查手册汇集了 Year 9 剑桥数学课程中的重要公式、定理与快捷方法。你可以在解题、完成作业或准备考试时快速查阅。每个部分均以清晰的中英双语呈现，方便双语学习者使用。
1. Algebraic Expressions and Simplification | 代数表达式与化简
Algebraic expressions contain numbers, variables and operation symbols, such as 4x + 3y − 7.
代数表达式由数字、变量和运算符号构成，例如 4x + 3y − 7。
To simplify, collect like terms: 5a + 2b − 3a + b = (5a − 3a) + (2b + b) = 2a + 3b.
化简时需合并同类项：5a + 2b − 3a + b = (5a − 3a) + (2b + b) = 2a + 3b。
When expanding a bracket, multiply each term inside by the factor outside: 3(2x − 4) = 6x − 12.
去括号时，将括号外的因数乘以括号内的每一项：3(2x − 4) = 6x − 12。
Factorising reverses the process: 8y + 12 = 4(2y + 3), taking out the highest common factor.
因式分解是去括号的逆运算：8y + 12 = 4(2y + 3)，提取最大公因数。
The product (a + b)² expands to a² + 2ab + b², and (a − b)² = a² − 2ab + b².
完全平方公式：(a + b)² = a² + 2ab + b²，(a − b)² = a² − 2ab + b²。
2. Solving Linear Equations | 解线性方程
A linear equation can be written in the form ax + b = c. The goal is to isolate the variable.
线性方程可写成 ax + b = c 的形式，目标是解出未知数。
Perform the same operation on both sides: subtract b, then divide by a. For example, 3x + 5 = 20 → 3x = 15 → x = 5.
方程两边同时进行相同运算：先减 b，再除以 a。例如 3x + 5 = 20 → 3x = 15 → x = 5。
If the variable appears on both sides, collect like terms first: 4x − 3 = 2x + 7 → 2x = 10 → x = 5.
若未知数出现在等式两边，先合并同类项：4x − 3 = 2x + 7 → 2x = 10 → x = 5。
Equations with fractions are cleared by multiplying every term by the least common denominator.
含有分数的方程，可每项乘以最小公分母以消去分母。
3. Inequalities | 不等式
Inequalities use the symbols > (greater than), < (less than), ≥ (greater than or equal to) and ≤ (less than or equal to).
不等号包括 >（大于）、<（小于）、≥（大于或等于）和 ≤（小于或等于）。
Solving inequalities follows the same steps as equations, except: when multiplying or dividing by a negative number, reverse the inequality sign.
解不等式与解方程步骤相同，唯一区别是：当乘以或除以负数时，不等号方向要改变。
Example: −2x ≤ 8 → x ≥ −4.
例如：−2x ≤ 8 → x ≥ −4。
Inequalities can be shown on a number line: an open circle for < or >, a closed circle for ≤ or ≥.
不等式可用数轴表示：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
4. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. The ratio a : b is equivalent to a/b.
比用来比较两个或两个以上的量。比 a : b 等价于 a/b。
To simplify a ratio, divide all parts by their greatest common divisor. 15 : 10 simplifies to 3 : 2.
化简比时，将每项除以它们的最大公约数。15 : 10 化简为 3 : 2。
When two quantities are in direct proportion, y = kx, where k is the constant of proportionality.
两个量成正比例时，y = kx，k 是比例常数。
Inverse proportion means y = k/x; as x increases, y decreases.
反比例关系为 y = k/x；x 增大时 y 减小。
Unitary method: find the value of one unit first, then scale up or down.
归一法：先求出一个单位对应的值，再进行放大或缩小。
5. Sequences and the nth Term | 数列与第 n 项
A linear sequence has a constant difference between consecutive terms. If the first term is a and the common difference is d, the nth term is given by Uₙ = a + (n − 1)d.
线性数列相邻两项的差为常数。若首项为 a，公差为 d，则第 n 项公式为 Uₙ = a + (n − 1)d。
For example, the sequence 5, 8, 11, 14, … has a = 5, d = 3 → nth term = 5 + 3(n − 1) = 3n + 2.
例如数列 5, 8, 11, 14, …，首项 5，公差 3，第 n 项 = 5 + 3(n − 1) = 3n + 2。
Quadratic sequences have a second difference that is constant. Their nth term contains an n² term.
二次数列的二次差为常数，第 n 项表达式包含 n² 项。
To find the next terms, use the difference pattern; to generate terms from the nth term, substitute n = 1, 2, 3,…
求后续项时利用差值规律；根据第 n 项公式生成各项时，分别代入 n = 1, 2, 3…
6. Straight Line Graphs | 直线图像
The equation of a straight line is usually written as y = mx + c, where m is the gradient and c is the y-intercept.
y = m x + c
直线的方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Gradient m = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁).
斜率 m = (y 的变化量) / (x 的变化量) = (y₂ − y₁) / (x₂ − x₁)。
A horizontal line has m = 0; a vertical line has equation x = k, with undefined gradient.
水平线的斜率 m = 0；垂直线的方程为 x = k，其斜率无定义。
To draw a line, find the y-intercept, then use the gradient to plot a second point.
画直线时，先标出 y 轴截距，再利用斜率找到第二个点。
7. Pythagoras’ Theorem | 勾股定理
In any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
a² + b² = c²
在任意直角三角形中，斜边的平方等于两直角边的平方和。
Here c is the hypotenuse (the side opposite the right angle), and a, b are the legs.
其中 c 是斜边（直角的对边），a 和 b 是直角边。
To find a missing leg, rearrange: a = √(c² − b²).
求直角边时可用变形公式：a = √(c² − b²)。
Triples like 3‑4‑5 and 5‑12‑13 satisfy the theorem and help quickly identify right angles.
像 3-4-5 和 5-12-13 这样的勾股数满足定理，有助于快速识别直角。
8. Area and Perimeter of 2D Shapes | 平面图形的面积与周长
Rectangle: Area = length × width, Perimeter = 2(l + w).
长方形：面积 = 长 × 宽，周长 = 2(长 + 宽)。
Triangle: Area = ½ × base × height. For a right triangle, the legs can serve as base and height.
三角形：面积 = ½ × 底 × 高。在直角三角形中，直角边可直接作为底和高。
Parallelogram: Area = base × vertical height.
平行四边形：面积 = 底 × 垂直高。
Trapezium: Area = ½ × (a + b) × h, where a and b are the parallel sides.
梯形：面积 = ½ × (上底 + 下底) × 高。
Circle: Circumference = 2πr or πd, Area = πr². (Use π ≈ 3.14 or the π button on a calculator.)
圆：周长 = 2πr 或 πd，面积 = πr²。（π 取 3.14 或使用计算器的 π 键。）
9. Volume and Surface Area of 3D Solids | 立体图形的体积与表面积
Cuboid (rectangular prism): Volume = length × width × height, Surface area = 2(lw + lh + wh).
长方体：体积 = 长 × 宽 × 高，表面积 = 2(长×宽 + 长×高 + 宽×高)。
Prism: Volume = area of cross-section × length. The surface area is the sum of all face areas.
棱柱：体积 = 底面积 × 高（棱长）。表面积为各个面的面积之和。
Cylinder: Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πr(r + h).
圆柱：体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)。
Pyramid: Volume = ⅓ × base area × vertical height.
棱锥：体积 = ⅓ × 底面积 × 垂直高。
Cone: Volume = ⅓ πr²h, Slant height l = √(r² + h²), Curved surface area = πrl.
圆锥：体积 = ⅓ πr²h，母线长 l = √(r² + h²)，侧面积 = πrl。
Sphere: Volume = ⁴⁄₃ πr³, Surface area = 4πr².
球：体积 = ⁴⁄₃ πr³，表面积 = 4πr²。
10. Transformations | 图形变换
Reflection: a mirror image. Reflection in the x‑axis maps (x, y) → (x, −y); in the y‑axis maps (x, y) → (−x, y).
反射：镜像。关于 x 轴的反射将 (x, y) 映射为 (x, −y)；关于 y 轴映射为 (−x, y)。
Rotation: turning around a centre. A 90° clockwise rotation about the origin maps (x, y) → (y, −x).
旋转：绕一个中心转动。绕原点顺时针旋转 90° 将 (x, y) 变为 (y, −x)。
Translation: sliding a shape without turning. Written as a column vector (x shift, y shift).
平移：不转动地移动图形。用列向量 (水平位移, 垂直位移) 表示。
Enlargement: a scale factor k from a centre. Side lengths multiply by k, area multiplies by k².
放大：以某点为中心按比例因子 k 放大。边长变为 k 倍，面积变为 k² 倍。
A negative scale factor produces an inverted image on the opposite side of the centre.
负比例因子会在中心的另一侧生成倒立图像。
11. Statistics – Averages and Charts | 统计——平均数与图表
Mean = sum of all values ÷ number of values. For data set 3, 7, 8, 8, 14: mean = (3+7+8+8+14)/5 = 8.
平均数 = 所有数据的总和 ÷ 数据个数。数据集 3, 7, 8, 8, 14：平均数 = 40/5 = 8。
Median: the middle value when data are ordered. For an even number of values, it is the mean of the two middle numbers.
中位数：排序后位于中间的值。数据个数为偶数时，取中间两数的平均数。
Mode: the value that appears most often. Range = maximum value − minimum value.
众数：出现次数最多的值。极差 = 最大值 − 最小值。
A bar chart shows frequency for categories; a pie chart represents proportions as angles (angle = proportion × 360°).
条形图表示各类别的频数；饼图用扇形的角度表示比例（角度 = 比例 × 360°）。
Scatter graphs show the relationship between two variables. A line of best fit can be drawn when correlation is clear.
散点图展示两个变量之间的关系。当相关性明显时，可绘制最佳拟合线。
12. Probability Basics | 概率基础
Probability of an event = number of favourable outcomes / total number of equally likely outcomes.
P(event) = number of favourable outcomes / total number of outcomes
事件概率 = 有利结果的数量 / 所有等可能结果的总数。
Probabilities range from 0 (impossible) to 1 (certain). The sum of probabilities of all possible outcomes is 1.
概率取值范围从 0（不可能）到 1（必然）。所有可能结果的概率之和为 1。
For mutually exclusive events, P(A or B) = P(A) + P(B).
互斥事件满足 P(A 或 B) = P(A) + P(B)。
Sample space diagrams and two‑way tables help list all outcomes for two events.
样本空间图与双向表格有助于列出两个事件的所有可能结果。
Expected frequency = probability × number of trials.
期望频数 = 概率 × 试验次数。
In a tree diagram, multiply probabilities along branches to find combined probabilities.
在树状图中，沿分支将概率相乘可得到组合概率。
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