📚 GCSE Edexcel Mathematics: End-of-Term Revision Guide | GCSE Edexcel 数学：期末复习提纲
This comprehensive revision guide covers the key topics from the GCSE Edexcel Mathematics syllabus, providing a structured recap of formulas, methods, and problem-solving strategies. Use it alongside past papers to target your weak areas and build exam confidence.
这份全面的复习提纲涵盖了 GCSE Edexcel 数学大纲的核心主题，系统地梳理了公式、方法和解题策略。搭配历年真题使用，可帮助你针对薄弱环节并建立考试信心。
1. Number: Fractions, Decimals, Percentages | 数与运算：分数、小数、百分数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 gives 3 ÷ 8 = 0.375.
要将分数转为小数，用分子除以分母。例如 3/8，计算 3 ÷ 8 = 0.375。
Converting a percentage to a decimal involves dividing by 100, so 45% = 0.45. To find a percentage of a quantity, multiply by the decimal form.
将百分数转为小数需除以 100，因此 45% = 0.45。计算一个量的百分之几时，乘以该小数形式。
Percentage increase and decrease are applied using a multiplier: new value = original × (1 ± percentage/100). For compound changes, raise the multiplier to the power of the number of time periods.
百分比增减可使用乘数：新值 = 原值 × (1 ± 百分比/100)。对于重复变化，将乘数按周期数取幂。
Working with standard form: a number is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. When multiplying or dividing, handle the powers of 10 separately.
标准形式：数写为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。乘除时，分别处理 10 的幂次。
2. Algebra: Expressions, Equations and Sequences | 代数：表达式、方程与数列
Simplify expressions by collecting like terms: 3x + 2y + 5x – y simplifies to 8x + y.
化简表达式时合并同类项：3x + 2y + 5x – y 简化为 8x + y。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. For double brackets, (x + a)(x + b) = x² + (a+b)x + ab.
展开括号运用分配律：a(b + c) = ab + ac。双括号如 (x + a)(x + b) = x² + (a+b)x + ab。
Factorising is the reverse of expanding. Always look for a common factor first, then recognise special forms such as the difference of two squares: a² – b² = (a + b)(a – b).
因式分解是展开的逆运算。先提取公因式，再识别特殊形式，如平方差：a² – b² = (a + b)(a – b)。
To solve a quadratic equation by factorising, set each factor equal to zero. If it does not factorise neatly, use the quadratic formula:
x = [ –b ± √(b² – 4ac) ] / (2a)
若无法轻松因式分解，使用二次公式：
x = [ –b ± √(b² – 4ac) ] / (2a)
For linear sequences, the nth term is given by an + b, where a is the common difference. For quadratic sequences, the second differences are constant, and the nth term takes the form an² + bn + c.
线性数列的第 n 项为 an + b，其中 a 是公差。二次数列的第二差为常数，第 n 项形式为 an² + bn + c。
3. Graphs: Linear, Quadratic and Real-Life | 图像：线性、二次与实际应用
The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines have the same gradient.
直线方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。平行线斜率相同。
To plot a quadratic graph, calculate y values for a range of x values. The graph is a parabola; its x-intercepts give the solutions to the corresponding quadratic equation.
绘制二次图像时，选择一组 x 值计算对应的 y。图像为抛物线，与 x 轴的交点即为对应二次方程的解。
Distance–time graphs: the gradient represents speed. A horizontal line indicates the object is stationary. In a velocity–time graph, the gradient gives acceleration and the area under the graph represents distance travelled.
距离–时间图中，斜率表示速度，水平线表示物体静止。速度–时间图中，斜率表示加速度，图下方面积表示行进距离。
4. Ratio, Proportion and Rates of Change | 比、比例与变化率
When sharing in a ratio, add the parts to find the total number of shares, then divide the quantity accordingly. For example, sharing £60 in the ratio 2:3 gives £24 and £36.
按比例分配时，将比值相加得到总份数，再相应分配。例如，按 2:3 分配 60 英镑，得到 24 镑和 36 镑。
Direct proportion means y = kx for some constant k; as x doubles, y doubles. Inverse proportion has the form y = k/x; as x doubles, y halves.
正比例关系为 y = kx（k 为常数），x 翻倍时 y 也翻倍。反比例形式为 y = k/x，x 翻倍时 y 减半。
Compound measures: speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area. Always check units for consistency.
复合度量：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 力 ÷ 面积。注意单位是否统一。
5. Geometry: Angles and Shapes | 几何：角与形状
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.
直线上的邻角之和为 180°，绕一点的周角为 360°，对顶角相等。
In parallel lines, alternate angles are equal, corresponding angles are equal, and interior (co-interior) angles sum to 180°.
平行线中，内错角相等，同位角相等，同旁内角互补（和为 180°）。
For polygons, the sum of interior angles is (n – 2) × 180°, and the sum of exterior angles is always 360°. A regular polygon has equal sides and equal angles.
多边形内角和为 (n – 2) × 180°，外角和恒为 360°。正多边形各边等长，各角相等。
6. Congruence and Similarity | 全等与相似
Congruent shapes are identical in size and shape. The four conditions for triangle congruence are SSS, SAS, ASA and RHS (right angle, hypotenuse, side).
全等图形大小和形状完全相同。三角形全等的四个判定条件为：SSS、SAS、ASA 和 RHS。
Similar shapes have equal angles and proportional side lengths. If the length scale factor is k, the area scale factor is k² and the volume scale factor is k³.
相似形各角相等，对应边成比例。若长度比例因子为 k，则面积比例因子为 k²，体积比例因子为 k³。
7. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
在任何直角三角形中，斜边的平方等于两直角边的平方和：
a² + b² = c²
The trigonometric ratios are defined by SOH CAH TOA:
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
三角比由 SOH CAH TOA 定义：
sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边
Exact trigonometric values must be memorised for 0°, 30°, 45°, 60° and 90°. For instance, sin 30° = ½, cos 45° = 1/√2, tan 60° = √3.
需熟记 0°、30°、45°、60° 和 90° 的精确三角值，如 sin 30° = ½，cos 45° = 1/√2，tan 60° = √3。
8. Perimeter, Area and Volume | 周长、面积与体积
Area of a rectangle: length × width. Area of a triangle: ½ × base × height. Area of a trapezium: ½ × (a + b) × h, where a and b are the parallel sides.
矩形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高；梯形面积 = ½ × (a + b) × h，其中 a、b 为平行边。
Circle formulas: circumference = 2πr (or πd) and area = πr². Use the π button on your calculator for accuracy.
圆：周长 = 2πr（或 πd），面积 = πr²。考试中使用计算器的 π 键以提高精度。
Volume of a prism = area of cross-section × length. For a cylinder, volume = πr²h. Surface area of a cone or sphere is given on the formula sheet – use it carefully.
棱柱体积 = 横截面积 × 长度；圆柱体积 = πr²h。圆锥和球的表面积需要参考公式表并正确代入。
9. Statistics: Averages, Charts and Data | 统计：平均数、图表与数据
The mean is the sum of values divided 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 difference between the largest and smallest.
平均数为总和除以个数；中位数为排序后中间的值；众数为出现频率最高的值；极差为最大值与最小值之差。
For grouped data, the modal class is the class with the highest frequency. The estimated mean uses midpoints: sum of (frequency × midpoint) ÷ total frequency.
对于分组数据，众数所在组为频率最高的组。估算平均数使用组中值：总和 (频率 × 组中值) ÷ 总频率。
In histograms, the area of each bar represents frequency – use frequency density = frequency ÷ class width. Cumulative frequency graphs help find medians and quartiles; the interquartile range describes spread.
直方图中，每个长方形面积代表频率，频率密度 = 频率 ÷ 组距。累积频率图用于求中位数和四分位数，四分位距描述离散程度。
10. Probability | 概率
Probabilities lie between 0 (impossible) and 1 (certain). The sum of probabilities of all mutually exclusive outcomes is 1.
概率介于 0（不可能）和 1（必然）之间。互斥事件所有可能结果的概率之和为 1。
Expected frequency = probability × number of trials. Use two-way tables and sample space diagrams to list possible outcomes systematically.
期望频数 = 概率 × 试验次数。使用双向表和样本空间图系统地列出所有可能的结果。
On a tree diagram, multiply along branches for ‘and’ events and add probabilities for ‘or’ events. For independent events, P(A and B) = P(A) × P(B). For conditional probability, the probability of B given A is P(B|A) = P(A and B) / P(A).
树形图中，沿分支相乘表示“且”事件，相加表示“或”事件。独立事件满足 P(A 且 B) = P(A) × P(B)。条件概率 P(B|A) = P(A 且 B) / P(A)。
11. Transformations | 变换
Translation is described by a column vector (x over y), which moves the shape without changing size or orientation.
平移用列向量 (x y) 描述，移动图形但不改变大小和方向。
Reflection requires a mirror line, such as x = 2 or y = –x. The image is the same perpendicular distance behind the mirror as the object is in front.
反射需指定对称轴，如 x = 2 或 y = –x。像与物体到对称轴的垂直距离相等。
Rotation is defined by centre, angle and direction (clockwise or anticlockwise). Tracing paper can be used to visualise the turn.
旋转由中心、角度和方向（顺时针或逆时针）确定。可使用描图纸辅助想象转动。
Enlargement changes size by a scale factor from a centre of enlargement. A fractional scale factor reduces the shape, while a negative scale factor places the image on the opposite side of the centre.
放大变换以某点为中心，按比例因子改变大小。分数比例因子使图形缩小，负比例因子将像放在中心的另一侧。
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