📚 IB WJEC Mathematics: Common Mistakes Explained | IB WJEC 数学：易错题精讲
In both IB and WJEC Mathematics, students frequently fall into similar traps that cost them marks in algebra, calculus, statistics and beyond. This article examines ten of the most common pitfalls, pairing each with a worked example, the typical wrong approach and the correct solution. Understanding why these mistakes happen – and how to sidestep them – will sharpen your exam technique and deepen your mathematical intuition.
在 IB 和 WJEC 数学考试中，学生经常会在代数、微积分、统计等专题里掉进相似的失分陷阱。本文精选十个最常见的易错题，每道题都结合了典型错误解法和正确做法。弄清这些错误为何发生、学会避开它们，既能改善考试技巧，也能加深对数学本质的理解。
1. Misusing the Distributive Law with Negative Signs | 负号分配律的误用
A common slip occurs when expanding expressions involving subtraction. For example, simplify 4(2x – 3) – 3(x + 2).
一个常见错误发生在展开带有减号的式子时。例如，化简 4(2x – 3) – 3(x + 2)。
Typical mistake: 4(2x – 3) – 3(x + 2) = 8x – 12 – 3x + 6 = 5x – 6. The error is applying the minus sign only to the first term of the bracket, ignoring the second.
典型错误：4(2x – 3) – 3(x + 2) = 8x – 12 – 3x + 6 = 5x – 6。错误在于只把负号分配给括号里的第一项，忽略了第二项。
Correct method: Treat the subtraction as adding the negative: 4(2x – 3) + [-3(x + 2)] = 8x – 12 – 3x – 6 = 5x – 18.
正确方法：把减法视为加上负值：4(2x – 3) + [-3(x + 2)] = 8x – 12 – 3x – 6 = 5x – 18。
Always rewrite subtractive brackets as a plus-negative term before expanding, especially in IB HL algebra and WJEC Pure topics.
在展开前，始终坚持把减号括号改写为加负括号，这对 IB HL 代数和 WJEC 纯数题目都很关键。
2. Negative and Fractional Indices Confusions | 负指数与分数指数混淆
Simplify (8x⁻³)^(2/3), assuming x > 0. Many students mishandle the negative exponent or the fractional power.
化简 (8x⁻³)^(2/3)，假设 x > 0。很多学生处理不好负指数或分数次幂。
Common error: Writing (8x⁻³)^(2/3) = 8^(2/3) * x^(-3 * 2/3) = 4x⁻², then thinking the answer is 4/x² (which is actually correct in this case!) but often they forget the x is in the denominator and treat x⁻² incorrectly when combining. The more serious error is writing (x⁻³)^(2/3) = x^(-3 × 2/3) = x⁻², but then adding an extra negative sign.
常见错误：写成 (8x⁻³)^(2/3) = 8^(2/3) * x^(-3 * 2/3) = 4x⁻²，然后认为答案是 4/x²（其实这道题结果确实如此），但经常会有学生把 x⁻² 误当成 x^(2/3) 甚至写出 1/(4x²)。最大的麻烦出在对 x⁻² 的含义模糊。
Correct step-by-step: 8^(2/3) = (8^(1/3))² = 2² = 4. x^(-3 * 2/3) = x⁻² = 1/x². Hence the simplified form is 4/x².
正确分步：8^(2/3) = (8^(1/3))² = 2² = 4。x^(-3 * 2/3) = x⁻² = 1/x²。因此化简结果是 4/x²。
Remember: a negative index means reciprocal, and a fractional index m/n means the nth root raised to power m. Keep them separate and apply them in either order.
记住：负指数表示倒数；分数指数 m/n 表示先开 n 次方再 m 次乘方，顺序可交换。分步操作就不易出错。
3. Solving Quadratic Inequalities without Sketching | 解二次不等式不画草图
Solve x² – 4x + 3 > 0. Too often students find the roots and immediately write x < 1 or x > 3, but fail to check the middle region.
解 x² – 4x + 3 > 0。太多学生求出根后直接写 x < 1 或 x > 3，却忽略了中间区间的检验。
Wrong reasoning: Factorises to (x – 1)(x – 3) > 0. Since the product is positive, “both factors must be positive or both negative” – so x > 3 or x < 1. This is correct for this particular inequality, but if the problem were x² - 4x + 3 < 0, writing x < 1 or x > 3 would be a disastrous slip. Many candidates forget to test a value between the critical points.
错误推理：因式分解得 (x-1)(x-3) > 0。因为乘积为正，“两因式同号”，于是得到 x > 3 或 x < 1。这道题这样写是碰巧对了，但换成 x² - 4x + 3 < 0 而仍写成 x < 1 或 x > 3 就会大错特错。学生常忘记在临界值之间选取一个测试值。
Safe method: Draw a rough parabola. For a positive quadratic, the graph is ∪-shaped. The inequality > 0 means the graph is above the x-axis, which occurs outside the roots. So x < 1 or x > 3. For < 0, the solution would be between the roots: 1 < x < 3. Always sketch!
安全方法：画出抛物线草图。对于开口向上的二次函数，图像呈 ∪ 状。不等式 > 0 对应图像在 x 轴上方，也就是根的外侧。因此 x < 1 或 x > 3。若 < 0，解集在两根之间：1 < x < 3。一定要画图！
4. Logarithm Domain and Combining Logs Errors | 对数定义域与合并错误
Solve log₂(x + 2) + log₂(x – 2) = 5. A typical mistake is to combine logs and solve algebraically, forgetting domain restrictions.
解方程 log₂(x + 2) + log₂(x – 2) = 5。典型错误是把对数合并后直接代数求解，忘记检查定义域。
Incorrect approach: log₂[(x+2)(x-2)] = 5 → x² – 4 = 2⁵ = 32 → x² = 36 → x = ±6. Both answers are taken as valid, but x = -6 is impossible because it would make the arguments of the original logs negative.
错误解法：log₂[(x+2)(x-2)] = 5 → x² – 4 = 2⁵ = 32 → x² = 36 → x = ±6。两个解都被当作有效，但 x = -6 会使原对数的真数变为负数，不合定义。
Correct procedure: After obtaining x = 6 or x = -6, check the original domain: x + 2 > 0 and x – 2 > 0 ⇒ x > 2. Hence only x = 6 is acceptable. Always state the domain first: here x > 2.
正确步骤：得出 x = 6 或 x = -6 后，验证原始定义域：x + 2 > 0 且 x – 2 > 0 ⇒ x > 2。因此仅 x = 6 有效。解题前应先写出定义域：x > 2。
5. Trigonometric Expansion: Forgetting the Middle Term | 三角展开：遗漏中间项
Expand (sin θ + cos θ)². Many students incorrectly write sin² θ + cos² θ = 1, missing the 2 sin θ cos θ term.
展开 (sin θ + cos θ)²。不少学生错误地写成 sin² θ + cos² θ = 1，漏掉了 2 sin θ cos θ 项。
Flawed memory: “Square of a sum is the sum of squares.” That is only true for sums under a square root! (sin θ + cos θ)² = sin² θ + 2 sin θ cos θ + cos² θ = 1 + sin 2θ.
错误记忆：“和的平方等于平方的和”，这只在根号下才成立！(sin θ + cos θ)² = sin² θ + 2 sin θ cos θ + cos² θ = 1 + sin 2θ。
This mistake also appears when students try to simplify expressions like √(sin² θ + cos² θ) – they cancel the root and square incorrectly. Avoid carelessness by always using (a+b)² = a² + 2ab + b².
类似错误还出现在化简 √(sin² θ + cos² θ) 时，错误地直接消去根号与平方。避免粗心的最好办法是坚持使用完全平方公式 (a+b)² = a² + 2ab + b²。
6. Differentiation: Dropping the Chain Rule | 微分：链式法则的遗漏
Differentiate y = (3x² + 1)⁴. The slip is to write dy/dx = 4(3x² + 1)³ and stop, forgetting the derivative of the inner function.
对 y = (3x² + 1)⁴ 求导。最常见的差错是写出 dy/dx = 4(3x² + 1)³ 就停笔，忘记乘以内层函数的导数。
Incomplete answer: dy/dx = 4(3x²+1)³. This is only the outer derivative.
不完整的答案：dy/dx = 4(3x²+1)³。这只是外层的导数。
Correct application: Let u = 3x² + 1, so y = u⁴. Then dy/dx = dy/du · du/dx = 4u³ · 6x = 24x(3x²+1)³. The chain rule is non-negotiable for composite functions.
正确应用：设 u = 3x² + 1，则 y = u⁴。dy/dx = dy/du · du/dx = 4u³ · 6x = 24x(3x²+1)³。面对复合函数，链式法则不可省略。
7. Integration: Omitting the Constant and Mishandling Limits | 积分：遗漏常数与积分限处理不当
A mark-losing habit is forgetting the constant of integration for indefinite integrals, or substituting limits incorrectly in definite integrals.
一个扣分习惯是，求不定积分时忘记加积分常数，或者在定积分中代入限值时算错。
Example for indefinite: ∫ 6x dx. Some write 3x² and stop. The correct answer is 3x² + C. For definite, consider ∫₀² (2x+1) dx. Students might integrate to x² + x, then treat it as x² + x |₀² = (4+2) – (0+0) = 6, which is correct. However, when the integrand is more complex, like ∫₁² (3x-2)⁻² dx, they neglect to divide by the derivative of the inner function when reversing the chain rule.
以不定积分为例：∫ 6x dx。有人写成 3x² 就完事。正确答案是 3x² + C。定积分方面，看 ∫₀² (2x+1) dx，学生积分得 x² + x，然后代入得 (4+2) – (0) = 6，这道简单题对，但一旦被积函数复杂，如 ∫₁² (3x-2)⁻² dx，反向链式法则时经常忘记除以内层导数。
To integrate (3x-2)⁻², write as (3x-2)⁻² and increase the power: (3x-2)⁻¹ / (-1) * (1/3) = -1/[3(3x-2)]. Never skip the factor 1/3 from the inner derivative.
积分 (3x-2)⁻² 时，写成 (3x-2)⁻²，幂次加1再除： (3x-2)⁻¹ / (-1) × (1/3) = -1/[3(3x-2)]。永远别忘了乘上内层导数的倒数 1/3。
8. Probability: Mixing Up Mutually Exclusive and Independent Events | 概率：混淆互斥事件与独立事件
Many IB and WJEC students use the formula P(A ∩ B) = P(A) × P(B) when they should be checking independence, or treat disjoint events as independent. For instance, when flipping a fair coin twice, let A be “first toss is heads” and B be “exactly one head”. Then A and B are not independent, and P(A ∩ B) ≠ P(A)P(B).
很多 IB 和 WJEC 考生在不经检验独立性的情况下直接套用 P(A ∩ B) = P(A) × P(B)，或者误以为互不相容就是独立。例如掷一枚公平硬币两次，设事件 A 为“第一次正面”，事件 B 为“恰好一个正面”。那么 A 与 B 并不独立，P(A ∩ B) ≠ P(A)P(B)。
Common mistake: claiming independence because the coin tosses are independent; but the definition of B involves the whole experiment. Actually P(A∩B) = P(first head and exactly one head) = P(HT) = 1/4. P(A)=1/2, P(B)=2/4=1/2, product = 1/4. Here they coincidentally equal, so they are independent in this specific case! Let’s use a better example: A = first head, B = at most one head. Then P(A∩B)=1/4 (only HT), but P(A)P(B)=1/2 × 3/4 = 3/8, so not independent. The confusion remains: students must compute explicitly.
常见误区：因为硬币投掷相互独立就断言 A 与 B 独立，但事件 B 的定义涉及整个实验的结果。实际 P(A∩B) = P(第一次正面且恰好一个正面) = P(HT) = 1/4，P(A)=1/2，P(B)=1/2，乘积也是 1/4，在此特例下碰巧相等，学生们却经常不验证就默认。要避免错误，必须分别计算再比较，不靠直觉。
Always test independence with P(A ∩ B) versus P(A)P(B), and for mutual exclusivity check whether A ∩ B is empty. They are different concepts.
始终通过比较 P(A ∩ B) 与 P(A)P(B) 验证独立性，通过 A ∩ B 是否为空判断互斥。二者是不同的概念。
9. Statistics: Correlation Does Not Imply Causation | 统计：相关不等于因果
Given data showing a strong positive correlation coefficient r = 0.92 between hours spent watching TV and test scores, a student might conclude that more TV causes higher scores. This is a classic fallacy that examiners love to target.
给定数据表明看电视时长与考试分数之间的相关系数 r = 0.92 呈强正相关，学生可能会得出“多看 TV 导致分数提高”的结论。这是考官最爱考查的经典谬误。
Pitfall reasoning: High correlation means one variable directly affects the other. In reality, there could be a lurking variable – for example, students who watch educational programmes might do better, or the dataset is small and biased. Correlation quantifies linear association, not causation.
错误推理：高相关系数意味着一变量直接影响另一变量。实际上，可能存在潜在变量——例如看教育节目的学生成绩更好，或样本量小且有偏。相关系数量化的是线性关联，不是因果。
In IB and WJEC questions, always comment that correlation indicates a linear relationship; establishing causation requires a controlled experiment. Never write “x causes y” merely from a regression analysis.
在 IB 和 WJEC 试题中，一定要说明相关仅表明线性关系；确立因果关系需要进行对照实验。切勿仅凭回归分析就写“x 导致 y”。
10. Vectors: Direction and Scalar Product Sign Slips | 向量：方向与点积符号错误
When finding the angle between vectors a = 3i + 4j and b = -i + 2j, a common miscalculation is to drop the sign on the first component of b.
求向量 a = 3i + 4j 与 b = -i + 2j 之间的夹角时，常见的计算错误是弄丢 b 的第一分量的负号。
Wrong a · b: 3 × 1 + 4 × 2 = 3 + 8 = 11. This gives cos θ positive and an acute angle, but the correct dot product is 3×(-1) + 4×2 = -3 + 8 = 5. The angle is still acute? Let’s check: using correct dot = 5, |a|=5, |b|=√5 ≈2.236, cos θ = 5/(5×2.236) ≈0.4472, θ≈63.4°, acute. If we made the error and got 11/(5×2.236)≈0.983, θ≈10°, which is wrong. The sign matters greatly when components are negative.
错误点乘：3 × 1 + 4 × 2 = 3 + 8 = 11。这样得 cos θ 为正且角度很小，但正确点乘是 3×(-1) + 4×2 = 5，角度仍为锐角，但数值截然不同。如果碰到更复杂的负分量，符号错误会直接导致角度钝锐误判。
Always pair components carefully, keep the minus signs, and double-check before computing the angle. In WJEC mechanics, direction errors also plague vector equations of motion; treat vectors as whole entities.
务必将分量对应相乘，保留负号，计算角度前仔细核对。在 WJEC 力学中，方向错误同样困扰运动学向量方程；要把向量当作整体处理。
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