📚 GCSE CIE Maths: Last-Minute Revision Notes | GCSE CIE 数学：考前冲刺笔记
This is your final stop before the CIE GCSE Mathematics exam – a concise, high‑impact revision guide covering the core topics, essential formulas, common pitfalls, and exam‑winning strategies. Work through each section, actively test yourself, and focus on the areas where you feel least confident.
这是你在 CIE GCSE 数学考试前的最后一站——一份简洁高效的冲刺笔记，覆盖核心考点、必背公式、常见失分点和制胜考场策略。逐节复习、主动自测，把精力留给最薄弱的地方。
1. Number and Operations | 数字与运算
Master the four operations with fractions, decimals, and negative numbers. Remember to apply BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction) strictly when simplifying expressions.
熟练掌握分数、小数和负数的四则运算。化简表达式时必须严格遵循运算顺序：先括号，再乘方，然后乘除，最后加减。
When converting recurring decimals to fractions, set up an equation, multiply to shift the decimal point, and subtract. For example, 0.3̅ = 1/3, but 0.2̅7̅ = 27/99 = 3/11.
将循环小数转化成分数时，设未知数列方程、移位相减。如：0.3̅ = 1/3，而 0.2̅7̅ = 27/99 = 3/11。
Standard form is written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. To multiply, add the indices; to divide, subtract the indices.
标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数。相乘时指数相加，相除时指数相减。
Upper and lower bounds are crucial for measurements. If a length is given as 8.0 cm to the nearest millimetre, the true length lies between 7.95 cm and 8.05 cm.
上界和下界对测量值至关重要。若某长度给定为 8.0 cm（精确到毫米），真实长度介于 7.95 cm 至 8.05 cm 之间。
Surds: √a × √b = √(ab), and (√a + √b)(√a – √b) = a – b. Always rationalise denominators that contain a surd, e.g. 1/√2 → √2/2.
根式运算法则：√a × √b = √(ab)，且 (√a + √b)(√a – √b) = a – b。务必有理化含根式的分母，如 1/√2 → √2/2。
2. Algebra Basics and Equations | 代数基础与方程
Expand brackets carefully, multiplying each term inside by the term outside. Simplify like terms immediately. For quadratics, recognise the difference of two squares: a² – b² = (a + b)(a – b).
去括号要认真，外项乘以内项的每一项，随即合并同类项。遇到二次式时，要能辨认平方差公式：a² – b² = (a + b)(a – b)。
Solving linear equations: aim to isolate x using inverse operations. Always perform the same operation on both sides of the equation.
解一元一次方程：用逆运算把 x 单独留在一边，每一步必须对方程两边同时进行相同操作。
For quadratic equations ax² + bx + c = 0, either factorise, or use the quadratic formula:
x = (–b ± √(b² – 4ac)) / 2a
. The discriminant b² – 4ac tells you the number of real roots.
对于一元二次方程 ax² + bx + c = 0，或因式分解，或使用求根公式：x = (–b ± √(b² – 4ac)) / 2a。判别式 b² – 4ac 可判断实根个数。
Simultaneous equations can be solved by elimination (add or subtract equations to cancel one variable) or substitution. Always check your solution satisfies both original equations.
解联立方程组可用消元法（两式相加或相减去消去一个变量）或代入法。记得将所得解代回原方程组验证。
Rearranging formulae: treat the subject variable as the unknown and use inverse operations step by step. The same rules apply whether the formula is linear, quadratic, or contains roots.
转化公式的主项：把目标字母当作未知数，逐步使用逆运算。无论公式是线性、二次还是带根号，规则都一样。
3. Functions and Graphs | 函数与图像
A function maps each input x to exactly one output f(x). The domain is the set of all possible inputs; the range is the set of all possible outputs.
函数将每个输入 x 映射到唯一的输出 f(x)。定义域是所有可能输入值的集合，值域是所有可能输出值的集合。
Composite functions fg(x) mean apply g first, then f. Inverse function f⁻¹(x) reverses the effect; its graph is a reflection in the line y = x.
复合函数 fg(x) 表示先作用 g，再作用 f。反函数 f⁻¹(x) 逆转原函数的映射，其图像关于直线 y = x 对称。
The gradient of a straight line is rise / run. The equation of a line is
y = mx + c
where m is the gradient and c is the y‑intercept.
直线的斜率 = 纵差 / 横差。直线方程 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Quadratic graphs y = ax² + bx + c are parabolas. If a > 0, the parabola opens upward (∪); if a < 0, it opens downward (∩). The vertex is found by completing the square or using x = –b/(2a).
二次函数图像 y = ax² + bx + c 是抛物线。当 a > 0 时开口向上 (∪)，a < 0 时开口向下 (∩)。顶点可通过配方法或 x = –b/(2a) 求得。
Exponential graphs y = aˣ with a > 1 grow rapidly and never cross the x‑axis (asymptote at y = 0). Recognise shapes of y = x³, y = 1/x, and trigonometric graphs.
指数函数 y = aˣ (a > 1) 增长迅速，永不与 x 轴相交（渐近线为 y = 0）。记住 y = x³、y = 1/x 及三角函数的图像形状。
4. Ratio, Proportion and Rates of Change | 比例、比率与变化率
Simplify ratios by dividing all parts by their highest common factor. A ratio of 8:12 simplifies to 2:3. Always express ratios in their simplest integer form unless directed otherwise.
将比的所有部分同除以最大公约数化至最简。如 8:12 化简为 2:3。除非题目另有要求，否则比都应写成最简整数比。
Direct proportion: y ∝ x means y = k x, graph is a straight line through the origin. Inverse proportion: y ∝ 1/x means y = k/x, graph is a hyperbola.
正比例：y ∝ x 即 y = kx，图像为一条过原点的直线。反比例：y ∝ 1/x 即 y = k/x，图像为双曲线。
For speed, density, and pressure, use compound measures: Speed = Distance ÷ Time, Density = Mass ÷ Volume, Pressure = Force ÷ Area. Rearrange according to the unknown.
速度、密度、压强等复合量要灵活运用公式：速度 = 路程 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 压力 ÷ 面积。根据所求变量变形公式。
Percentage change: % change = (change / original) × 100. For repeated percentage changes, use a multiplier (e.g., 1.05 for a 5% increase) raised to a power.
百分比变化：变化百分比 = (变化量 / 原值) × 100。反复百分比变化时，使用乘数（如 5% 增长对应乘数 1.05）并作乘方运算。
5. Geometry – Angles and Triangles | 几何——角度与三角形
Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的邻角互余，和为 180°；绕一点的所有角之和为 360°。对顶角相等。
Parallel lines: corresponding angles are equal, alternate angles are equal, and interior angles sum to 180°. Use these to find missing angles in complex diagrams.
平行线中，同位角相等，内错角相等，同旁内角之和为 180°。在复杂图形中利用这些性质求未知角度。
Triangle angle sum is 180°. An exterior angle equals the sum of the two opposite interior angles. Base angles of an isosceles triangle are equal.
三角形内角和为 180°。外角等于两个不相邻内角之和。等腰三角形的两个底角相等。
Congruent triangles: SSS, SAS, ASA, AAS, RHS. Similar triangles have equal corresponding angles and proportional sides; use scale factor to find lengths.
全等三角形的判定条件：SSS、SAS、ASA、AAS、RHS。相似三角形对应角相等、对应边成比例，借助比例因子可求未知边长。
6. Geometry – Circles and Polygons | 圆与多边形
Circle theorems you must know: angle at centre = 2 × angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal.
必须掌握的圆定理：圆心角是圆周角的两倍；半圆上的圆周角是直角；同弧上的圆周角相等。
Opposite angles of a cyclic quadrilateral sum to 180°. A tangent is perpendicular to the radius at the point of contact.
圆内接四边形的对角互补，和为 180°。切线垂直于过切点的半径。
For polygons, sum of interior angles = (n – 2) × 180°, where n is the number of sides. Sum of exterior angles is always 360°.
多边形内角和 = (n – 2) × 180°，其中 n 为边数。外角和恒等于 360°。
Regular polygons have equal sides and equal angles. Each exterior angle = 360°/n; each interior angle = 180° – exterior angle.
正多边形的各边相等、各角相等。每个外角 = 360°/n；每个内角 = 180° – 外角。
7. Mensuration (Area, Volume, etc.) | 测量（面积、体积等）
Area formulas must be memorised: triangle = ½ × base × height; parallelogram = base × height; trapezium = ½ × (a + b) × h; circle = πr².
面积公式必须熟记：三角形 = ½ × 底 × 高；平行四边形 = 底 × 高；梯形 = ½ × (上底 + 下底) × 高；圆 = πr²。
Circumference of a circle = 2πr or πd. Arc length = (θ/360) × 2πr, sector area = (θ/360) × πr². θ is the angle at the centre.
圆的周长 = 2πr 或 πd。弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。θ 为圆心角。
Volume of a prism = area of cross‑section × length. Cylinder: V = πr²h. Pyramid and cone: V = ⅓ × base area × height. Sphere: V = ⁴⁄₃πr³.
棱柱体积 = 横截面积 × 长度。圆柱：V = πr²h。棱锥与圆锥：V = ⅓ × 底面积 × 高。球体：V = ⁴⁄₃πr³。
Surface area: sphere = 4πr²; cone (curved surface) = πrl where l is slant height. For composite solids, treat each face separately.
表面积：球体 = 4πr²；圆锥侧面积 = πrl，其中 l 为斜高。对于组合立体，分面逐一计算。
8. Trigonometry | 三角学
In a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOH CAH TOA is your friend.
直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。牢牢记住 SOH CAH TOA。
For non‑right triangles, use the sine rule: a/sin A = b/sin B = c/sin C, or the cosine rule: a² = b² + c² – 2bc cos A.
非直角三角形中，使用正弦定理 a/sin A = b/sin B = c/sin C，或余弦定理 a² = b² + c² – 2bc cos A。
The area of any triangle can be found using ½ ab sin C, where C is the included angle between sides a and b.
任意三角形的面积可用公式 ½ ab sin C 求得，其中 C 为边 a 与 b 的夹角。
Angles of elevation and depression are always measured from the horizontal. Draw a clear diagram and label the sides relative to the given angle.
仰角和俯角总是从水平线量起。画草图、明确标注各边与已知角的关系是解题关键。
Exact trigonometric values for 0°, 30°, 45°, 60°, 90° are often tested. Learn them: sin 30° = ½, cos 45° = √2/2, tan 60° = √3, etc.
0°, 30°, 45°, 60°, 90° 的特殊角三角函数值常考。务必熟记：sin 30° = ½, cos 45° = √2/2, tan 60° = √3 等。
9. Probability | 概率
Probability of an event = (number of favourable outcomes) / (total number of possible outcomes). Probabilities lie between 0 (impossible) and 1 (certain).
事件概率 = (有利结果的数目) / (所有可能结果的数目)。概率取值范围在 0 (不可能) 到 1 (必然) 之间。
If two events are independent, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
若两个事件独立，P(A 且 B) = P(A) × P(B)。若两事件互斥，P(A 或 B) = P(A) + P(B)。
Tree diagrams help organise combined events. Multiply along branches for ‘and’, and add the final outcomes for ‘or’. The sum of probabilities on branches from a point is 1.
树状图可清晰整理复合事件。沿分支相乘得“且”的概率，再将各结局概率相加得“或”的概率。从同一点出发的各分支概率和为 1。
Conditional probability refers to the probability of A given B has occurred, written P(A|B). Calculation: P(A|B) = P(A and B) / P(B).
条件概率指在事件 B 已发生的条件下事件 A 发生的概率，记作 P(A|B)。计算公式：P(A|B) = P(A 且 B) / P(B)。
10. Statistics | 统计
Mean = sum of values / number of values. Median is the middle value when data is ordered. Mode is the most frequent value. Range = maximum – minimum.
平均数 = 总和 / 数据个数。中位数是排序后位于中间的数据。众数是出现频率最高的值。极差 = 最大值 – 最小值。
For grouped data, estimate the mean using midpoints. The modal class is the class with the highest frequency. Median is found using cumulative frequency.
分组数据中，用组中值估算平均数。众数组是频率最高的组。中位数通过累积频率图确定。
Cumulative frequency graphs and box plots show the spread of data. The interquartile range (IQR) = Q₃ – Q₁, measures the middle 50% of the data.
累积频率图和箱线图可展示数据分布。四分位距 IQR = 上四分位数 (Q₃) – 下四分位数 (Q₁)，描述中间 50% 数据的离散程度。
Scatter graphs show correlation. A line of best fit can be used to make predictions, but caution: correlation does not imply causation.
散点图展示相关性。最佳拟合线可作预测，但需谨慎：相关不意味因果。
Histograms: for equal class widths, the height represents frequency. For unequal widths, the area represents frequency (frequency density = frequency / class width).
直方图：等组距时，高表示频数；不等组距时，面积表示频数（频数密度 = 频数 / 组距）。
11. Common Mistakes to Avoid | 常见错误拎重点
Forgetting that a negative number squared gives a positive result, e.g., (–3)² = 9, but writing –3² as –9 is a classic error when brackets are omitted.
忘记负数平方得正值，如 (–3)² = 9。但漏写括号将 –3² 算作 –9 是经典错误。
Confusing factorising quadratics: check your expansion always gives the original expression. Missing the common factor first often leads to an incomplete factorisation.
二次式因式分解容易混淆： 务必乘开来检验。漏提公因式常导致分解不彻底。
In geometry, mislabelling angles or not stating the theorem used can lose marks. Always give reasons (e.g., ‘alternate angles are equal’) in your working.
几何题中角度标注错误或不写出所用定理会失分。解题步骤中务必给出理由（如“内错角相等”）。
In probability, forgetting to reduce fractions or leaving answers as ‘1/2, 1/2’ when they should be ‘1/4’ after multiplication. Read the question: are events independent?
概〓题中忘记约分，或将相乘后的概率仍写作 1/2 而非 1/4。仔细审题：事件是否独立？
Rounding errors: carry exact values through your working and round only at the final answer. Premature rounding distorts results, especially in trigonometry and bounds.
四舍五入失误：计算过程中保留精确值，只在最终答案中近似。过早舍入会扭曲结果，尤其在三角学和界值问题中。
12. Exam Technique and Tips | 应试技巧与提醒
Read the question twice before starting. Highlight key information – units, degree of accuracy, what form the answer should take.
做题前把题目读两遍。圈出关键信息：单位、精确度要求、答案应呈现的形式。
Show every step of your working. Even if your final answer is wrong, method marks can still be awarded. A clear, logical layout helps both you and the examiner.
写出每一步推导过程。即使最终答案有误，也能获得方法分。清晰有条理的卷面对你和阅卷人都有好处。
Manage your time: spend no more than one minute per mark. If stuck, mark the question and move on, then return if time allows. Do not leave any multiple‑choice blank.
把握好时间：每分不过一分钟。若卡住，先做标记跳过，时间充裕时再回头。选择题决不留空。
Check your answers, especially the reasonability. Does a length of 1500 metres make sense in the context? Does your angle sum correctly? Re‑read the question to confirm you answered exactly what was asked.
检查答案，特别是其合理性。1500 米的长度在题目情境中合理吗？角度加起来对吗？重读题目，确认你回答的正是所问。
Use your calculator wisely: know how to use the fraction button, the ANS key, and store intermediate values. Switch between degrees and radians appropriately.
善用计算器：熟悉分数键、ANS 键和存储中间值功能。根据题目在角度制和弧度制之间正确切换。
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