📚 GCSE Maths: Key Topic Comparisons | GCSE 数学：知识点对比
In GCSE Mathematics, many topics seem closely related but have distinct rules and applications. Understanding these subtle differences can prevent common mistakes and boost your confidence in the exam. This article compares twelve pairs of concepts that students often mix up, with clear side-by-side explanations.
在 GCSE 数学中，许多知识点看似相近，却有截然不同的规则和应用。理解这些细微差别能避免常见错误，并提升考试信心。本文以清晰的并列解释，对比了十二组学生常混淆的概念。
1. Mean, Median and Mode | 平均数、中位数与众数
Mean is calculated by summing all values and dividing by the number of values. It is sensitive to extreme values (outliers). Median is the middle value when data is ordered; it is not affected by outliers. Mode is the most frequent value, useful for categorical data.
平均数是将所有数值相加后除以数据个数，它受极端值（离群值）影响较大。中位数是排序后位于中间的数，不受极端值影响。众数是出现频率最高的值，适合类别数据。
Example: Data set 2, 3, 3, 5, 8 gives mean = (2+3+3+5+8)/5 = 4.2, median = 3, mode = 3. Add an outlier 50, the mean becomes 11.8 but median stays 3.
例子：数据集 2, 3, 3, 5, 8 的平均数为 (2+3+3+5+8)/5 = 4.2，中位数为 3，众数为 3。若加入极端值 50，平均数变为 11.8，而中位数依然是 3。
2. Direct vs Inverse Proportion | 正比例与反比例
Direct proportion means as one quantity increases, the other increases at the same rate: y = kx, where k is constant. The graph is a straight line through the origin. Inverse proportion means as one increases, the other decreases: y = k/x, giving a curved graph called a hyperbola.
正比例表示一个量增加，另一个量以相同速率增加：y = kx，其中 k 是常数。图像是一条过原点的直线。反比例表示一个量增加，另一个量减少：y = k/x，图像是一条曲线（双曲线）。
To identify, check if doubling x doubles y (direct) or halves y (inverse). Use the product xy = k for inverse proportion.
识别时，检查 x 翻倍是否使 y 翻倍（正比例）或减半（反比例）。反比例可利用乘积 xy = k。
3. Permutations vs Combinations | 排列与组合
Permutations count arrangements where order matters (e.g. race finishes). Formula: nPr = n!/(n − r)!. Combinations count selections where order does not matter (e.g. choosing a team). Formula: nCr = n!/(r!(n − r)!).
排列计数顺序重要的安排（如比赛名次）。公式：nPr = n!/(n − r)!。组合计数顺序无关的选择（如挑选队员）。公式：nCr = n!/(r!(n − r)!).
For example, from letters A, B, C, the permutations of 2 letters are AB, BA, AC, CA, BC, CB (6); combinations are AB, AC, BC (3). Always decide if order changes the outcome.
例如从字母 A, B, C 中，取 2 个排列有 AB, BA, AC, CA, BC, CB（6 种）；组合仅有 AB, AC, BC（3 种）。务必先判断顺序是否影响结果。
4. Linear vs Quadratic Equations | 线性方程与二次方程
A linear equation has the form ax + b = 0 and produces a straight-line graph. It has at most one solution. A quadratic equation ax² + bx + c = 0 (a ≠ 0) gives a parabola, and can have 0, 1 or 2 real solutions, found by factorising, completing the square or the quadratic formula.
线性方程形如 ax + b = 0，图像为直线，最多有一个解。二次方程ax² + bx + c = 0（a ≠ 0）图像为抛物线，可有 0、1 或 2 个实数解，通过因式分解、配方法或二次公式求解。
The quadratic formula is x = [−b ± √(b² − 4ac)] / 2a. Remember to set the equation to zero before solving.
二次公式为 x = [−b ± √(b² − 4ac)] / 2a。解方程前务必先整理成等于零的形式。
5. Perimeter vs Area | 周长与面积
Perimeter is the distance around a 2D shape, measured in linear units (cm, m). Area is the space inside, measured in square units (cm², m²). Common formulas: rectangle perimeter = 2(l + w), area = lw; circle circumference = 2πr, area = πr².
周长是二维图形一周的长度，单位是线性单位（cm, m）。面积是图形内部的大小，单位是平方单位（cm², m²）。常用公式：矩形周长 = 2(l + w)，面积 = lw；圆周长 = 2πr，面积 = πr²。
Do not confuse the two when a question asks for fencing (perimeter) or tiling (area). Always include units.
题目中问围栏材料用长要算周长，问铺瓷砖则算面积，切勿混淆。始终注明单位。
6. Surface Area vs Volume | 表面积与体积
Surface area is the total area of all faces of a 3D solid, measured in square units. Volume is the amount of space inside, measured in cubic units (cm³, m³). For a cube of side a: surface area = 6a², volume = a³.
表面积是三维物体所有面的总面积，单位为平方单位。体积是物体内部空间的大小，单位为立方单位（cm³, m³）。对边长为 a 的立方体：表面积 = 6a²，体积 = a³。
Be careful with composite solids — find volume by separating into simpler parts; surface area may require hiding shared faces.
处理组合体时要小心——体积可拆分为简单形体计算；表面积要注意隐藏的接触面。
7. Theoretical vs Experimental Probability | 理论概率与实验概率
Theoretical probability is what should happen based on equally likely outcomes: P(event) = favourable outcomes / total outcomes. Experimental probability (relative frequency) is based on actual trials: P = number of times event occurred / total trials.
理论概率基于等可能性：P(事件) = 有利结果数 / 总结果数。实验概率（相对频率）基于实际试验：P = 事件发生次数 / 总试验次数。
As the number of trials increases, experimental probability tends to get closer to theoretical probability (the Law of Large Numbers). Both can be expressed as fractions, decimals or percentages.
随着试验次数增加，实验概率会趋近理论概率（大数定律）。两者均可用分数、小数或百分数表示。
8. Discrete vs Continuous Data | 离散数据与连续数据
Discrete data can only take specific values (often integers), like shoe sizes or number of pets. It is shown on bar charts or frequency diagrams. Continuous data can take any value within a range, like height or time. It is displayed in histograms or line graphs.
离散数据只能取特定值（常为整数），如鞋码或宠物数量，用条形图或频数图表示。连续数据可在某一范围内取任意值，如身高、时间，用直方图或折线图表示。
When grouping continuous data, use inequalities: e.g. 150 ≤ h < 155. The class width affects histogram frequency density.
对连续数据分组时使用不等式，如 150 ≤ h < 155。组距影响直方图中的频率密度。
9. Sine, Cosine and Tangent | 正弦、余弦和正切
In a right‑angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use SOH CAH TOA to remember. They apply only to right‑angled triangles; for non‑right triangles use the sine or cosine rule.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用 SOH CAH TOA 记忆。这些定义仅适用于直角三角形；非直角三角形需用正弦定理或余弦定理。
sin⁻¹, cos⁻¹, tan⁻¹ are used to find angles when the ratio is known. Always label sides relative to the angle you are working with.
当已知比值时，可用 sin⁻¹, cos⁻¹, tan⁻¹ 求角度。务必根据所考虑的角来标记边。
10. Factorising vs Expanding | 因式分解与展开
Expanding removes brackets: a(b + c) = ab + ac, and (x + a)(x + b) = x² + (a+b)x + ab. Factorising is the reverse process: writing an expression as a product of its factors. e.g. x² + 5x + 6 factorises to (x+2)(x+3).
展开是去括号：a(b + c) = ab + ac，以及 (x + a)(x + b) = x² + (a+b)x + ab。因式分解是逆过程：将表达式写成因式的乘积，如 x² + 5x + 6 分解为 (x+2)(x+3)。
Always look for a common factor first. For quadratics, find two numbers that multiply to the constant and add to the coefficient of x.
分解时先找公因式。对于二次式，寻找两个数，其积等于常数项、其和等于 x 项系数。
11. Inequalities vs Equations | 不等式与方程
Equations state that two expressions are equal, using =, with a fixed solution set. Inequalities use <, >, ≤, ≥ to show a range of values. Solving is similar, but multiplying or dividing by a negative flips the inequality sign.
方程用等号 = 连接两个表达式，解为固定值。不等式使用 <, >, ≤, ≥ 表示取值范围。解法类似，但乘或除以负数时不等号方向反转。
Represent solutions on a number line: open circle for strict inequality, closed circle for ≤, ≥. Always list integer solutions if required.
在数轴上表示解集：严格不等式用空心圆，≤, ≥ 用实心圆。如题目要求，需列出所有整数解。
12. Functions vs Inverse Functions | 函数与反函数
A function f(x) maps each input x to a unique output. The inverse function f⁻¹(x) reverses this mapping, taking the output back to the original input. To find an inverse, write y = f(x), swap x and y, and solve for y.
函数 f(x) 把每个输入 x 对应到唯一输出。反函数 f⁻¹(x) 逆转这一映射，将输出还原为原始输入。求反函数时，写出 y = f(x)，交换 x 和 y，然后解出 y。
The domain of f is the range of f⁻¹ and vice versa. Graphically, the inverse is a reflection in the line y = x. Not all functions have an inverse unless they are one‑to‑one.
f 的定义域是 f⁻¹ 的值域，反之亦然。图像上，反函数关于直线 y = x 对称。不是所有函数都有反函数，除非是一一映射。
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