📚 KS3 Advanced Mathematics: End-of-Term Revision Guide | KS3 进阶数学：期末复习提纲
This comprehensive revision guide covers the essential topics for KS3 advanced mathematics, helping you consolidate key concepts, master problem-solving techniques and approach your end-of-term exam with confidence. Each section combines succinct explanations with practical examples to support independent study.
这份全面的复习提纲涵盖了 KS3 进阶数学的核心主题，帮助你巩固关键概念、掌握解题技巧，并以十足的信心迎接期末考试。每个小节都将简洁的解释与实际示例相结合，方便你自主学习。
1. Number Skills and Operations | 数字技巧与运算
A solid grasp of number operations is the foundation of all mathematics. At the advanced level, you must use the correct order of operations (BIDMAS/BODMAS) automatically: Brackets, Indices (powers and roots), Division and Multiplication (working left to right), Addition and Subtraction (left to right).
扎实掌握数字运算是所有数学的基础。在进阶阶段，你必须能熟练使用正确的运算顺序（BIDMAS/BODMAS）：先括号，再指数（乘方和开方），然后乘除（从左向右），最后加减（从左向右）。
Negative numbers often cause mistakes. Remember that adding a negative is subtraction, and subtracting a negative is addition. For multiplication and division: when the signs are the same the result is positive; when they are different the result is negative.
负数的处理常常出错。记住，加上一个负数就是减法，减去一个负数相当于加法。乘除时：同号得正，异号得负。
Estimation and approximation let you check if an answer is reasonable. Rounding to one significant figure before calculating gives a quick ‘ballpark’ answer.
估算与近似值可以帮助你检验答案是否合理。先按一位有效数字进行舍入，再计算，就能得到一个快速的“大致”结果。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Being able to move flexibly between fractions, decimals and percentages is a key skill. Convert a fraction to a decimal by dividing the numerator by the denominator; to a percentage by multiplying the decimal by 100. For example, 3/8 = 3 ÷ 8 = 0.375 = 37.5%.
能够在分数、小数和百分比之间灵活转换是一项核心技能。分数化小数用分子除以分母；化百分比再将小数乘以 100。例如，3/8 = 3 ÷ 8 = 0.375 = 37.5%。
With fractions, revise addition and subtraction by finding a common denominator, and multiplication by multiplying numerators and denominators separately. When dividing, keep the first fraction, change ÷ to ×, and flip the second fraction (the reciprocal).
对于分数，复习通过寻找公分母来加减；乘法时分子相乘、分母相乘。除法要保留第一个分数、除号变乘号，并将第二个分数上下翻转（取倒数）。
Percentage increase and decrease often appear in real-life contexts. To increase by 12%, multiply by 1.12; to decrease by 12%, multiply by 0.88. Always check whether you are finding a percentage of an amount or expressing one quantity as a percentage of another.
百分比的增加和减少在实际生活中很常见。增加 12% 可乘以 1.12；减少 12% 则乘以 0.88。要始终分清你是求某个数量的百分比，还是表示一个量占另一个量的百分比。
3. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratios compare parts of a whole. If the ratio of boys to girls is 3 : 5, the total number of parts is 8. To share £72 in this ratio, each part is worth £72 ÷ 8 = £9, giving £27 to boys and £45 to girls. Simplify ratios by dividing both sides by their highest common factor.
比是用来比较整体中的各个部分。如果男生与女生的比是 3 : 5，则总份数是 8。按此比例分配 £72，每份金额为 £72 ÷ 8 = £9，男生得 £27，女生得 £45。化简比时，两边同除以最大公因数。
Direct proportion means two quantities increase at the same rate: if 5 pens cost £3.50, then 8 pens cost (8/5) × £3.50 = £5.60. The unitary method (finding the value of one item first) always works.
正比例表示两个量同速率增加：如果 5 支笔价格是 £3.50，那么 8 支笔的价格就是 (8/5) × £3.50 = £5.60。归一法（先求一个单位的对应值）总是能解决问题。
Speed, distance and time relationships involve constant rates. Use the formula triangle: Speed = Distance ÷ Time. When the units are mixed, convert first – for example, to km/h from m/s multiply by 3.6.
速度、距离和时间的关系涉及恒定变化率。使用公式三角形：速度 = 距离 ÷ 时间。当单位不一致时需要先换算——例如，将 m/s 换算成 km/h 需乘以 3.6。
4. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra uses letters to represent numbers. You must be confident collecting like terms: 5a + 3b – 2a + 7b simplifies to 3a + 10b. Remember that a² and a are not like terms—you cannot combine them.
代数用字母表示数字。你必须熟练掌握合并同类项：5a + 3b – 2a + 7b 化简为 3a + 10b。注意 a² 和 a 不是同类项——不能合并。
Expanding brackets means multiplying each term inside the bracket by the term outside. For 3(2x – 5), expand to 6x – 15. With double brackets such as (x + 4)(x – 3), use the FOIL method: First, Outer, Inner, Last, then collect like terms to obtain x² + x – 12.
展开括号就是把括号中的每一项都与外面的项相乘。对于 3(2x – 5)，展开得到 6x – 15。双括号如 (x + 4)(x – 3)，可用 FOIL 法则——先乘首项、再外项、内部、末项——然后合并同类项得到 x² + x – 12。
Factorising is the reverse of expanding. Look for the highest common factor first: 8y + 12 factorises to 4(2y + 3). For quadratics like x² + 5x + 6, find two numbers that multiply to 6 and add to 5, giving (x + 2)(x + 3).
因式分解是展开的逆运算。先提取最大公因数：8y + 12 分解为 4(2y + 3)。对于 x² + 5x + 6 这样的二次式，找到两个乘积为 6、和为 5 的数字，得到 (x + 2)(x + 3)。
5. Solving Linear Equations | 解一元一次方程
Solving an equation means finding the value of the unknown that makes the statement true. Always maintain balance—whatever you do to one side, you must do to the other. For x + 7 = 15, subtract 7: x = 8.
解方程就是找到使等式成立的未知数的值。始终要保持方程平衡——对等式一边进行的任何运算，也要对另一边做同样的运算。对于 x + 7 = 15，两边同减 7 得到 x = 8。
When the unknown appears on both sides, collect all x terms on one side and numbers on the other. Example: 5x – 3 = 2x + 9. Subtract 2x: 3x – 3 = 9. Add 3: 3x = 12, so x = 4. Always check your answer by substituting it back into the original equation.
当未知数出现在等式两边时，把所有含 x 的项移到一边，数字移到另一边。例如：5x – 3 = 2x + 9。两边同减 2x：3x – 3 = 9。再加 3：3x = 12，因此 x = 4。务必代入原方程检验答案。
Equations involving fractions can be cleared by multiplying every term by the lowest common denominator. For x/2 + 3 = x/4, multiply through by 4: 2x + 12 = x, giving x = -12.
含有分数的方程可以通过乘以最小公分母来消去分母。对于 x/2 + 3 = x/4，两边同时乘以 4，得到 2x + 12 = x，解得 x = -12。
6. Inequalities and Number Lines | 不等式与数轴
Inequalities use symbols < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). They are solved much like equations, but with one critical rule: when multiplying or dividing by a negative number, reverse the inequality sign.
不等式使用符号 <（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）。解不等式的方法与解方程类似，但有一条关键规则：当对不等式两边同乘或同除以一个负数时，必须反转不等号方向。
Represent the solution set on a number line. An open circle shows the value is not included (< or >); a closed circle shows it is included (≤ or ≥). For -2 < x ≤ 3, draw an open circle at -2, a closed circle at 3, and shade the line between them.
在数轴上表示解集。空心圆圈表示该值不包括在内（< 或 >）；实心圆圈表示包含该值（≤ 或 ≥）。对于 -2 < x ≤ 3，在 -2 处画空心圈，3 处画实心圈，然后将两点之间的线段涂黑。
Double inequalities like 1 < 2x + 3 ≤ 9 can be solved in one go: subtract 3 from all parts to get -2 < 2x ≤ 6, then divide by 2: -1 < x ≤ 3. Always aim to isolate the x in the middle.
像 1 < 2x + 3 ≤ 9 这样的双边不等式可以一次解出：三部分同减 3 得到 -2 < 2x ≤ 6，再同除以 2 得到 -1 < x ≤ 3。目标是让中间的变量 x 单独出现。
7. Sequences and the nth Term | 数列与第n项
A sequence is a list of numbers following a rule. An arithmetic sequence increases or decreases by a constant difference. To find the nth term of 4, 7, 10, 13…, notice the common difference is 3, so the nth term is 3n + 1 (when n=1, 3×1+1=4).
数列是按照某种规则排列的一组数字。等差数列按固定的差值递增或递减。要求数列 4, 7, 10, 13… 的第 n 项，看到公差是 3，因此第 n 项为 3n + 1（当 n=1 时，3×1+1=4）。
The nth term formula allows you to find any term without listing all previous ones. For a descending sequence like 20, 17, 14, 11…, the difference is -3, and the nth term is -3n + 23 (since 23 – 3×1 = 20). Check by generating the first few terms.
有了第 n 项公式，你无需列出前面的所有项就能找到任意项。对于递减数列 20, 17, 14, 11…，公差为 -3，第 n 项为 -3n + 23（因为 23 – 3×1 = 20）。可通过生成前几项来验证。
Other sequences include geometric progressions (multiplying by a constant) and special sequences like square numbers (n²), cube numbers (n³) and triangular numbers. Recognise patterns both in numbers and diagrams.
其他数列包括等比数列（乘以一个常数）以及特殊的数列，例如平方数 (n²)、立方数 (n³) 和三角形数。要学会在数字和图形中辨识规律。
8. Graphs of Linear Functions | 一次函数图像
A linear function produces a straight-line graph. Its equation is usually written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). For y = 2x + 3, the gradient is 2 and the intercept is 3.
一次函数形成一条直线图像。它的方程通常写为 y = mx + c，其中 m 是斜率（倾斜度），c 是 y 轴截距（直线与 y 轴的交点）。对于 y = 2x + 3，斜率为 2，截距为 3。
To plot a linear graph, choose at least three x-values, calculate their y-values, plot the points and join them with a straight line. Horizontal lines have the form y = a (gradient 0); vertical lines are x = b (undefined gradient).
画一次函数图像时，至少选取三个 x 值，计算出对应的 y 值，描点后用直线连接。水平线特征为 y = a（斜率为 0）；竖直线则为 x = b（斜率无定义）。
Parallel lines have the same gradient. Perpendicular lines have gradients that are negative reciprocals of each other—for example, if one gradient is 2, the perpendicular gradient is -1/2. Understanding this helps solve coordinate geometry problems.
平行直线斜率相等。互相垂直的直线，其斜率互为负倒数——例如，若一条直线斜率为 2，则垂直线斜率为 -1/2。理解这一点有助于解决坐标几何问题。
9. Geometry: Angles and Parallel Lines | 几何：角与平行线
Angle facts must be second nature. Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In a triangle, the interior angles add up to 180°; in a quadrilateral, they add up to 360°.
角的性质必须烂熟于心。平角之和为 180°，周角为 360°，对顶角相等。三角形的内角和是 180°，四边形的内角和是 360°。
When a transversal crosses parallel lines, special angle relationships appear: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°. Sketching the Z, F and C shapes will help you spot them quickly.
当一条截线与两条平行线相交时，会出现特殊角度关系：内错角相等，同位角相等，同旁内角之和为 180°。画一画 Z 形、F 形和 C 形有助于快速辨认。
Angle problems often combine these rules with algebra. For instance, if two angles in a triangle are given as x and 2x, you can set up the equation x + 2x + third angle = 180° and solve. Always write down which angle rule you are using.
角度问题常结合代数一起考查。例如，若三角形中两个角分别表示为 x 和 2x，可列出方程 x + 2x + 第三个角 = 180° 并求解。解题时务必写出所引用的角度定理。
10. Area, Perimeter and Circles | 面积、周长与圆
Perimeter is the distance around a shape. For rectangles, P = 2(l + w); for a composite shape, add the outer side lengths carefully. The area of a rectangle is length × width; a triangle is ½ × base × height. For a parallelogram, area = base × perpendicular height, not slant height.
周长是图形一周的长度。长方形周长 P = 2(l + w)；对于组合图形，则仔细将各外边长度相加。长方形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高；平行四边形的面积 = 底 × 垂直高，而非斜高。
The area of a trapezium is calculated as the average of the parallel sides multiplied by the distance between them: Area = ½(a + b)h. Always check that the height is perpendicular to the parallel sides.
梯形面积等于上下底之和的平均数乘以两底间的距离：面积 = ½(a + b)h。务必注意高要垂直于平行边。
For circles, learn the key formulas: Circumference = π × d = 2πr and Area = πr². Use the π button on your calculator unless told otherwise. When calculating arc length or sector area, set up a fraction based on the central angle out of 360°.
对于圆，记牢关键公式：周长 = π × d = 2πr，面积 = πr²。除非另有要求，可使用计算器的 π 键。计算弧长或扇形面积时，按圆心角与 360° 的比值建立分数关系。
11. Pythagoras’ Theorem | 毕达哥拉斯定理
Pythagoras’ theorem applies only to right-angled triangles. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.
毕达哥拉斯定理只适用于直角三角形。其表述为：斜边（最长边，直角对边）的平方等于另两条直角边的平方和：a² + b² = c²，其中 c 为斜边。
To find a missing hypotenuse, square the two shorter sides, add, then take the square root. For sides of 6 cm and 8 cm: c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then square root: a = √(c² – b²).
求未知的斜边时，将两直角边分别平方后相加，再开方。若直角边分别为 6 cm 和 8 cm：c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm。求直角边时，用斜边平方减去已知直角边的平方，再开方：a = √(c² – b²)。
Always check if your answer is sensible: the hypotenuse must be the longest side. Pythagoras-style problems often involve coordinates (distance between two points) or real-life contexts like ladders leaning against walls.
始终检验答案的合理性：斜边必须是最长的边。毕达哥拉斯定理的题目常涉及坐标（两点间距离）或实际情境，如梯子靠墙问题。
12. Statistics and Probability | 统计与概率
Average is a measure of central tendency. The mode is the most frequent value, the median is the middle value when data are ordered, and the mean is calculated by summing all values and dividing by the number of values. The range shows the spread of data: largest minus smallest.
平均数是描述数据集中趋势的度量。众数是出现次数最多的数值，中位数是数据排序后位于中间的数值，平均数则通过所有数值之和除以数据个数来计算。极差（全距）表示数据的分散程度：最大值减最小值。
Charts and diagrams summarise data effectively. Bar charts compare categories; pie charts show proportions; scatter graphs reveal correlation. When drawing graphs, always label axes, use appropriate scales and give your chart a title.
统计图能够有效概括数据。条形图用于比较类别，饼图显示各部分比例，散点图揭示相关性。绘制图表时，务必标注坐标轴、使用合适的刻度并给予标题。
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, Probability = number of favourable outcomes / total number of outcomes. In an advanced KS3 context, you might list all outcomes using sample space diagrams or tree diagrams to calculate combined probabilities.
概率衡量某个事件发生可能性的大小，范围从 0（不可能）到 1（必然）。对于等可能的结果，概率 = 有利结果数 / 总结果数。在 KS3 进阶层次，你可能需要利用样本空间图或树状图列出所有可能的结果，从而计算组合概率。
Relative frequency is an estimate of probability based on experiments: Relative frequency = number of successful trials / total number of trials. The more trials you carry out, the closer the relative frequency usually gets to the theoretical probability.
相对频率是基于实验给出的概率估计值：相对频率 = 成功次数 / 总试验次数。试验次数越多，相对频率通常会越接近理论上的概率。
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