📚 Essential Maths Book 9C Complete Revision Guide | 《Essential Maths Book 9C》完整复习指南
This revision guide covers the core topics in Essential Maths Book 9C, a widely used Year 9 mathematics course. The notes are organised by topic to help you revise efficiently and build confidence for end-of-year assessments.
本复习指南涵盖《Essential Maths Book 9C》的核心内容，这是一套广泛使用的九年级数学教材。笔记按主题编排，帮助你高效复习，为年终测评建立信心。
1. Number and Place Value | 数与位值
Always follow the order of operations: brackets first, then indices, then division and multiplication from left to right, and finally addition and subtraction from left to right. Many students lose marks by calculating left to right without considering priority.
始终遵循运算顺序：先算括号，再算指数，然后从左到右计算除法和乘法，最后从左到右计算加法和减法。很多学生因为不按优先级从左到右计算而失分。
efore addition. Similarly, (3 + 4) × 2 = 7 × 2 = 14 because the bracket changes the order.
例如，3 + 4 × 2 = 3 + 8 = 11，而不是 14，因为乘法先于加法。类似地，(3 + 4) × 2 = 7 × 2 = 14，括号改变了运算顺序。
g or dividing two negatives gives a positive result, while one negative and one positive gives a negative result.
对于负数，记住加上一个负数等同于减去一个正数：5 + (-3) = 5 - 3 = 2。两个负数相乘或相除得到正数，而一负一正相乘或相除得到负数。
0 can be written as 4.56 × 10⁴ in standard form when rounded to three significant figures.
有效数字和科学计数法有助于处理非常大或非常小的数。例如，45,600 保留三位有效数字时可以写成 4.56 × 10⁴。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
multiply fractions, multiply the numerators and denominators separately: 2/3 × 4/5 = 8/15.
分数加减时，先找公分母。例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12。分数相乘时，分子乘分子、分母乘分母：2/3 × 4/5 = 8/15。
r example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. Always simplify your final answer where possible.
除以一个分数等于乘以它的倒数。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8。最后的答案应尽可能化简。
ultiply by 100; to change a percentage to a decimal, divide by 100. For example, 0.35 = 35% and 7% = 0.07.
小数与百分比的互换是许多实际问题的基础。小数化为百分比乘以 100；百分比化为小数除以 100。例如，0.35 = 35%，7% = 0.07。
Percentage increase and decrease calculations often involve a multiplier. To increase 80 by 15%, multiply 80 by 1.15 to get 92. To find the original amount after a 20% decrease that results in 64, divide 64 by 0.80 to get 80.
百分比增减计算通常使用乘数。将 80 增加 15%，用 80 乘以 1.15 得到 92。已知减少 20% 后为 64，求原数时用 64 除以 0.80 得到 80。
3. Ratio and Proportion | 比与比例
Ratios can be simplified like fractions by dividing every part by the same number. For example, 12:18:24 simplifies to 2:3:4 after dividing each part by 6.
比可以像分数一样通过同时除以相同的数来化简。例如，12:18:24 每部分都除以 6 后化简为 2:3:4。
To share an amount in a given ratio, add the parts, divide the total by that sum, then multiply by each part. For example, sharing £60 in the ratio 2:3 gives £24 and £36.
按给定比例分配一个量时，先求比例的总份数，用总量除以总份数，再分别乘以每份的份数。例如，按 2:3 分配 60 英镑，得到 24 英镑和 36 英镑。
Direct proportion means both quantities change at the same rate. If 5 pens cost £3, then 20 pens cost £12 because the quantity is multiplied by 4. Inverse proportion means one quantity decreases as the other increases.
正比例意味着两个量以相同的比率变化。如果 5 支笔花费 3 英镑，那么 20 支笔花费 12 英镑，因为数量乘以了 4。反比例意味着一个量增加时另一个量减少。
Scale drawings use ratios to represent real distances. A map scale of 1:50,000 means 1 cm on the map represents 50,000 cm, or 0.5 km, in real life.
比例尺使用比来表示实际距离。地图比例尺 1:50,000 表示地图上的 1 厘米代表实际 50,000 厘米，即 0.5 公里。
4. Algebraic Expressions and Equations | 代数表达式与方程
ide. For example, 3(x + 4) = 3x + 12, and 2(x - 5) + 3(x + 1) = 2x - 10 + 3x + 3 = 5x - 7.
展开括号就是用括号外的项乘以括号内的每一项。例如，3(x + 4) = 3x + 12，而 2(x - 5) + 3(x + 1) = 2x - 10 + 3x + 3 = 5x - 7。
out the highest common factor. For example, 6x + 9 = 3(2x + 3), and 4x² + 8x = 4x(x + 2).
因式分解是展开的逆运算。始终提取最大公因数。例如，6x + 9 = 3(2x + 3)，4x² + 8x = 4x(x + 2)。
For 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide by 3 to get x = 5.
解线性方程时，对等式两边进行相同的运算以保持平衡。对于 3x + 5 = 20，两边同时减去 5 得到 3x = 15，再同时除以 3 得到 x = 5。
denominator. For x/2 + 3 = 7, subtract 3 to get x/2 = 4, then multiply by 2 to get x = 8.
当方程含有分数时，可以将每一项都乘以公分母。对于 x/2 + 3 = 7，先减去 3 得到 x/2 = 4，再乘以 2 得到 x = 8。
5. Linear Graphs and Sequences | 线性图像与数列
mx + c, where m is the gradient and c is the y-intercept. The gradient measures steepness: a larger positive m gives a steeper upward slope, while a negative m gives a downward slope.
直线图像的一般方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率衡量陡峭程度：m 为较大的正数时，直线向上更陡；m 为负数时，直线向下倾斜。
change in x. For points (1, 3) and (4, 9), the gradient is (9 - 3) ÷ (4 - 1) = 6 ÷ 3 = 2.
求两点之间的斜率时，用纵坐标的变化量除以横坐标的变化量。对于点 (1, 3) 和 (4, 9)，斜率为 (9 - 3) ÷ (4 - 1) = 6 ÷ 3 = 2。
d is the common difference. For 5, 9, 13, 17, ..., the nth term is 5 + 4(n - 1) = 4n + 1.
等差序列按照固定的差值递增或递减。第 n 项的公式为 a + (n - 1)d，其中 a 是首项，d 是公差。对于数列 5, 9, 13, 17, ...，第 n 项为 5 + 4(n - 1) = 4n + 1。
ercepts. For example, y = 3x + 2 and y = 3x - 5 are parallel because both have gradient 3.
平行直线具有相同的斜率但不同的 y 轴截距。例如，y = 3x + 2 和 y = 3x - 5 平行，因为两者的斜率都是 3。
6. Geometry: Angles and Polygons | 几何：角与多边形
Basic angle facts help solve many geometry problems. Angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal.
基本角度规律有助于解决许多几何问题。直线上的角之和为 180°，围绕一个点的角之和为 360°，对顶角相等。
any polygon, the sum of interior angles is (n - 2) × 180°, where n is the number of sides.
三角形的内角和为 180°，四边形的内角和为 360°。对于任何多边形，内角和为 (n - 2) × 180°，其中 n 是边的数量。
The exterior angles of any polygon add up to 360°. In a regular polygon, each exterior angle equals 360° ÷ n, and each interior angle equals 180° minus the exterior angle.
任何多边形的外角和为 360°。在正多边形中，每个外角等于 360° ÷ n，每个内角等于 180° 减去外角。
When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles add up to 180°. Recognising these relationships is key to finding missing angles.
当一条截线穿过平行线时，同位角相等，内错角相等，同旁内角之和为 180°。识别这些关系是求未知角的关键。
7. Perimeter, Area and Volume | 周长、面积与体积
ight, parallelogram = base × height, and trapezium = ½ × (sum of parallel sides) × height.
常见平面图形的面积公式要牢记：矩形 = 长 × 宽，三角形 = ½ × 底 × 高，平行四边形 = 底 × 高，梯形 = ½ × (两底之和) × 高。
r is the radius. When only the diameter is given, divide it by 2 to find the radius first.
圆的周长公式为 C = 2πr，面积公式为 A = πr²，其中 r 是半径。当只给出直径时，先将直径除以 2 求出半径。
volume = length × width × height. Remember that 1 litre = 1000 cm³ when converting units.
棱柱的体积等于横截面积乘以长度。对于长方体，体积 = 长 × 宽 × 高。换算单位时记住 1 升 = 1000 立方厘米。
For compound shapes, split the shape into simpler parts, find each area or volume separately, then add or subtract as needed. Always check that all dimensions are in the same unit before calculating.
对于组合图形，先把图形分割成简单部分，分别求出面积或体积，再根据需要相加或相减。计算前要检查所有尺寸的单位是否一致。
8. Pythagoras' Theorem | 勾股定理
Pythagoras' here c is the hypotenuse, the longest side opposite the right angle, and a and b are the other two sides.
勾股定理只适用于直角三角形。它表明 a² + b² = c²，其中 c 是斜边，即直角所对的最长边，a 和 b 是另外两条直角边。
he two shorter sides, add them, then take the square root. For sides of 6 cm and 8 cm, c = √6² + 8² = √36 + 64 = √100 = 10 cm.
求斜边时，先将两条直角边分别平方，相加后再开平方。例如直角边为 6 厘米和 8 厘米时，c = √6² + 8² = √36 + 64 = √100 = 10 厘米。
To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root. For hypotenuse 13 cm and one side 5 cm, the other side is √13² - 5² = √169 - 25 = √144 = 12 cm.
求一条直角边时，用斜边的平方减去另一条直角边的平方，再开平方。例如斜边为 13 厘米，一条直角边为 5 厘米，另一条直角边为 √13² - 5² = √169 - 25 = √144 = 12 厘米。
Real-life problems often require drawing a right-angled triangle and identifying the hypotenuse first. For a ladder leaning against a wall, the ladder length is the hypotenuse, and the ground and wall are the shorter sides.
实际生活问题通常需要先画出直角三角形并确定斜边。例如梯子斜靠在墙上时，梯子的长度是斜边，地面和墙分别是两条直角边。
9. Statistics and Probability | 统计与概率
Data can be displayed in bar charts, line graphs, pie charts, and scatter diagrams. Choose the most suitable type for the data: bar charts for categories, line graphs for trends over time, and scatter diagrams for relationships between two variables.
数据可以用柱状图、折线图、饼图和散点图展示。选择最适合数据的图表类型：柱状图适合分类数据，折线图适合时间趋势，散点图适合两个变量之间的关系。
The mean is found by adding all values and dividing by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value, and the range is the largest value minus the smallest.
平均数的求法是将所有数值相加再除以数值个数。中位数是数据按顺序排列后的中间值。众数是出现次数最多的值，极差是最大值减最小值。
outcomes ÷ total number of outcomes. The probabilities of all possible mutually exclusive events add up to 1.
概率的范围是从 0 到 1。对于等可能的结果，概率 = 有利结果的数量 ÷ 所有可能结果的总数。所有互斥事件发生的概率之和为 1。
Expected frequency can be calculated by multiplying probability by the number of trials. If the probability of rolling a six on a fair die is 1/6, then in 300
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