📚 Common Mistakes in IGCSE Edexcel Maths | IGCSE Edexcel 数学：易错题精讲
IGCSE Edexcel Mathematics often tests not only your knowledge but also your attention to detail and precision. Many students lose marks not because they don’t understand the concepts, but because they fall into the same predictable traps. This article carefully selects some of the most common pitfalls across Algebra, Geometry, Probability and more, analyses why they occur, and shows you how to avoid them. Use these worked examples to sharpen your exam technique and secure the top grades.
IGCSE Edexcel 数学考试不仅考查知识掌握，更考验你的细心与严谨。不少同学丢分并非因为不懂概念，而是反复掉进同样的陷阱。本文精选了代数、几何、概率等模块中最高频的易错点，分析犯错的原因，并讲解正确解法。通过这些精讲例题，你可以打磨解题技巧，稳稳拿下高分。
1. Misreading the Question – BIDMAS and Negative Numbers | 题目看漏细节——运算法则与负数
A classic mistake occurs when evaluating expressions such as -3². Many students interpret this incorrectly as (-3)² = 9, whereas the correct application of BIDMAS treats it as -(3²) = -9. Remember that the index applies only to the number directly attached, not to the sign in front.
计算 -3² 这类表达式时，经典错误是把它当作 (-3)² = 9。正确运用 BIDMAS 法则，应视为 -(3²) = -9。记住指数只作用于紧靠它的底数，前面的负号不参与平方。
Correction: Use brackets to avoid confusion: if you mean negative three squared, write (-3)². If you mean the negative of three squared, write -3², and evaluate 3² first, then apply the minus.
正确做法：为避免混淆，可以使用括号。如果要求负三的平方，写 (-3)²；如果要表示三的平方的相反数，写 -3²，先算平方再取负。
2. Expanding Brackets with Minus Signs | 含负号的括号展开错误
When expanding an expression like 5 – 2(x – 3), students often multiply the -2 only by the x, forgetting to multiply by the -3. This gives 5 – 2x – 6 instead of 5 – 2x + 6. The sign of every term inside the bracket must be reversed when multiplying by a negative.
展开 5 – 2(x – 3) 时，很多同学只是用 -2 乘以 x，却忘记乘以 -3，结果错成 5 – 2x – 6，正确应为 5 – 2x + 6。括号外的负号要分配给括号内的每一项，符号要全部变号。
Think of -2(x – 3) as -2 × x + (-2) × (-3). Writing the intermediate step helps reduce careless mistakes.
可以把 -2(x – 3) 看作 -2 × x + (-2) × (-3)，写出中间步骤能大幅减少粗心错误。
3. Solving Linear Equations and Losing the Denominator | 解一次方程时丢分母
A frequent error when solving equations like (x/2) + 3 = 7 is to incorrectly subtract 3 and then divide, or to multiply only some terms by 2. Students might write x + 3 = 14, forgetting to multiply the +3 by 2 as well. The correct method multiplies every term by 2 first: x + 6 = 14, so x = 8.
解方程如 (x/2) + 3 = 7 时，常见错误是先减 3 再错误操作，或只将部分项乘以 2。有的同学会写成 x + 3 = 14，忘记 +3 也要乘 2。正确做法是等式两边每一项同时乘以 2，得 x + 6 = 14，因此 x = 8。
Always apply the multiplication to the entire side of the equation. Use brackets mentally if needed: 2 × ((x/2) + 3) = 2 × 7.
记住乘法要作用于等号一侧的所有项，必要时可以在脑内加括号：2 × ((x/2) + 3) = 2 × 7。
4. Factorising Quadratics When a ≠ 1 | 二次项系数不为 1 时的因式分解误区
Factorising expressions like 2x² + 7x + 3 confuses many. A common wrong attempt is (2x + 1)(x + 3), which multiplies out to 2x² + 7x + 3 correctly in this case, but errors arise in signs and coefficient pair selection. The real mistake is not checking by expansion. Another typical blunder is mismatching inner and outer terms.
分解 2x² + 7x + 3 时，许多同学容易混乱。虽然 (2x + 1)(x + 3) 碰巧在这里正确，但在符号和系数配对选择时极易出错。最常见的问题是不进行展开验算，以及搞混交叉相乘的顺序。
Use the systematic method: multiply a by c (2 × 3 = 6), find two numbers that multiply to 6 and add to 7 (6 and 1). Then split the middle term: 2x² + 6x + x + 3, factor into pairs: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). This approach avoids guessing.
采用系统方法：a×c = 2×3 = 6，找两个数乘积为 6 且和为 7（6 与 1），分裂中间项：2x² + 6x + x + 3，分组分解：2x(x+3) + 1(x+3) = (2x+1)(x+3)。这样做可以避免盲目猜测。
5. Gradient of a Straight Line – Change in y over Change in x | 直线斜率——纵差与横差的顺序颠倒
When finding the gradient between two points (x₁, y₁) and (x₂, y₂), a common slip is reversing the formula to (x₂ – x₁) / (y₂ – y₁). Gradient is rise over run, i.e. (y₂ – y₁) / (x₂ – x₁). Mixing them up leads to the reciprocal of the correct slope.
求两点 (x₁, y₁) 和 (x₂, y₂) 间斜率时，常有同学把公式颠倒成 (x₂ – x₁) / (y₂ – y₁)。斜率是纵差除以横差，即 (y₂ – y₁) / (x₂ – x₁)。颠倒后会得到正确斜率的倒数。
Another related error is using the same point ordering in numerator and denominator inconsistently. Always ensure both y’s are taken in the same direction as the x’s.
另一个相关错误是分子分母的点序不一致，必须保证纵差和横差用同一个起点和终点。
6. Working with Bounds – Rounding and Calculations | 上下界运算——四舍五入的复合计算
When a length is given as 5.6 cm correct to 1 decimal place, students often use 5.6 directly in further calculations without considering the upper and lower bounds. The actual value lies between 5.55 cm and 5.65 cm. To find the maximum possible area of a square with that side, you must use the upper bound of the side length.
当题目给出长度 5.6 cm 保留一位小数时，许多同学在后续计算中直接用 5.6，而忽略上下界。实际值为 5.55 cm 到 5.65 cm 之间。如果求该边长正方形的最大可能面积，必须使用边长的上界。
For compound calculations, remember to combine bounds correctly: for maximum sum, use upper bounds; for maximum product, use upper bounds; for maximum quotient, use upper bound of numerator and lower bound of denominator. Carefully analyse each operation.
复合计算时，正确组合上下界：最大和用上界；最大乘积用上界；最大商则用分子的上界除以分母的下界。需要根据运算类型逐一分析。
7. Probability of Combined Events – ‘At Least One’ Problems | 组合概率事件——“至少一个”问题
Questions like “A bag contains 4 red and 6 blue counters. Two are drawn without replacement. Find the probability of at least one red.” often tempt students to list all favourable cases but miscalculate denominators or miss conditional probabilities. The error arises when they add P(RR) + P(RB) + P(BR) but use wrong second-stage probabilities.
袋子有 4 红 6 蓝，无放回取两次，求至少一个红的概率。这类题容易让人列出所有有利情况，却算错第二阶段的条件概率。常见错误是在 P(RR) + P(RB) + P(BR) 时用了错误的分母或概率。
The neatest method is 1 – P(no red) = 1 – P(BB). P(BB) = (6/10) × (5/9) = 30/90 = 1/3, so probability of at least one red = 1 – 1/3 = 2/3. This avoids the mess and reduces errors.
最简洁的方法是 1 – P(无红) = 1 – P(BB)。P(BB) = (6/10) × (5/9) = 30/90 = 1/3，因此至少一红的概率 = 1 – 1/3 = 2/3。这样计算整洁且不易出错。
8. Trigonometry – Selecting the Correct Ratio | 三角函数——选错比值
In right-angled triangle problems, students often mix up sin, cos, and tan, especially when they mislabel the sides. Remember SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. A common mistake is using tan when the hypotenuse is involved, which must use sin or cos.
在直角三角形问题中，学生经常分不清 sin、cos、tan，尤其当错误标注对边和邻边时。记住 SOH CAH TOA：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。常见错误是涉及斜边时错用了 tan，斜边必须用 sin 或 cos。
Always identify the hypotenuse first (longest side, opposite the right angle), then label the opposite and adjacent relative to the angle you are using. Double-check your ratio selection before calculating.
一定要先找出斜边（最长边，直角对边），再根据使用的角标注对边和邻边，计算前再次确认比值选择。
9. Histograms – Frequency Density and Bar Width | 直方图——频率密度与组距误导
When drawing or interpreting histograms with unequal class widths, a common error is to plot frequency directly on the vertical axis. In a histogram, it’s the area of the bar that represents frequency, so the vertical axis must show frequency density = frequency / class width. Using frequency directly distorts the distribution.
画或解读组距不等的直方图时，常见错误是在纵轴直接标频率。直方图用面积表示频率，所以纵轴应是频率密度 = 频率 ÷ 组距。如果直接标频率，会扭曲分布形态。
Another linked mistake is incorrectly calculating frequency from a bar: multiply frequency density by the class width. Many forget to do this and read the frequency density as the frequency itself.
另一个相关错误是由直条求频率：必须用频率密度乘以组距。不少同学忘记这一步，把频率密度当成了频率。
10. Vectors – Direction and Notation Errors | 向量——方向与符号错误
Students often confuse a vector with its magnitude, or overlook the negative sign when reversing direction. For example, if AB = b – a (position vector of B minus position vector of A), then BA = a – b. Writing vector AB but calculating BA is a frequent slip.
同学们经常混淆向量与其大小（模），或在反向时忽略负号。比如，AB = b – a（B 的位置向量减 A 的位置向量），那么 BA = a – b。写的是 AB 却算了 BA，是常见失误。
Also, when adding or subtracting column vectors, arithmetic errors occur when dealing with negative components. Take extra care: (3, -2) + (-1, 4) = (2, 2), not (2, -6).
同样，进行列向量加减时，涉及负分量容易算错。务必小心：(3, -2) + (-1, 4) = (2, 2)，而不是 (2, -6)。
11. Number – Prime Factorisation and HCF/LCM from Index Form | 数字——质因数分解与指数形式的最大公因数/最小公倍数
Once numbers are written in index form, finding HCF and LCM causes trouble. Mistake: for HCF, taking the higher power instead of the lower. Correct: HCF takes the smallest power of each common prime factor; LCM takes the highest power of every prime factor present in either number.
将数字写成指数形式后，求 HCF 和 LCM 时容易出错。错误做法：求 HCF 时取了较大的指数。正确：HCF 取各公共质因数的最小指数；LCM 取所有出现的质因数的最大指数。
Example: A = 2³ × 3² × 5, B = 2² × 3⁴ × 7. HCF = 2² × 3², LCM = 2³ × 3⁴ × 5 × 7. Check by calculating HCF × LCM = A × B.
例如 A = 2³ × 3² × 5, B = 2² × 3⁴ × 7，HCF = 2² × 3²，LCM = 2³ × 3⁴ × 5 × 7。可以用 HCF × LCM = A × B 验证。
12. Rounding and Significant Figures in Intermediate Steps | 计算过程中四舍五入与有效数字错误
A common source of accumulated error is rounding off intermediate answers too early. Students might round the result of a division to 3 significant figures and then use that rounded value in the next part of the calculation. The final answer then drifts away from the accurate value. Always use the full calculator display, or at least keep several extra digits, and only round the final answer.
累积误差的一大来源是中间步骤过早四舍五入。有的同学会把某步除法的结果先保留三位有效数字，再用这个约值进行后续计算，导致最终答案偏离正确值。应该使用计算器显示的完整数值，或至少多保留几位，只在最后答案处按要求四舍五入。
Examiners often check “use of correct values not rounded prematurely” in mark schemes. Even if your method is correct, a too-early rounding can lead to an answer outside the acceptable range.
评分标准中经常强调“使用未经提前四舍五入的正确数值”。方法即使正确，过早舍入也可能使最终答案超出可接受的范围。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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