📚 Common Mistakes in International Mathematics A (MA04-QP-13Jun23) | 国际数学A卷常见易错点总结
This article reviews the most frequent errors students make under exam pressure, drawn from the style and challenge of the MA04 International Mathematics A paper. By examining these pitfalls, you can improve accuracy and secure marks that are all too often lost to simple slip‑ups.
本文梳理了学生在考试压力下最常犯的错误，这些错误来自 MA04 国际数学 A 卷的典型挑战。通过审视这些陷阱，你可以提高准确率，稳稳拿到那些因小失误而常常丢失的分数。
1. Algebraic Sign Errors When Expanding Brackets | 去括号时的符号错误
A classic mistake occurs when a negative sign sits outside a bracket, e.g. –(3x – 4). Students often write –3x – 4 instead of – + 4. Every term inside the bracket must have its sign reversed. Rewrite the expression as –1 × (3x – 4) if it helps.
当括号外有负号时，比如 –(3x – 4)，经典错误是写成 –3x – 4 而不是 –3x + 4。括号内的每一项符号都必须变号。如果有助于理解，可以先把式子写成 –1 × (3x – 4)。
Another sign slip occurs when subtracting a multi‑term bracket: in a – (b + c), students correctly handle the first term but forget to change the sign of c. Always distribute the minus sign across every term, even if you are rushing through rearrangements.
另一个符号疏漏发生在减去多占括号时：在 a – (b + c) 中，学生往往正确处理第一项却忘了改变 c 的符号。无论多着急整理式子，都要将减号分配到每一项。
2. Misapplying the Order of Operations (BIDMAS) | 运算顺序（BIDMAS）的误用
8. The same issue appears in fractional forms: (2 + 8)/2 ÷ 2 is often misinterpreted. Always bracket the numerator if needed.
在计算 6 + 4 ÷ 2 时，很多同学当作 (6 + 4) ÷ 2 = 5，但除法优先，正确结果是 6 + 2 = 8。同样的问题在分数形式中也会出现：(2 + 8)/2 ÷ 2 常被误读。必要时请把分子用括号括起来。
substitution, students forget to apply powers before multiplication. For example, when x = –2, 3x² is 3 × (–2)² = 12, not (3 × –2)². Always square first, then multiply.
在代数代入时，学生会忘记先算幂再算乘法。例如当 x = –2 时，3x² 是 3 × (–2)² = 12，而不是 (3 × –2)²。一定要先平方，再乘系数。
3. Confusing the 'Difference of Two Squares' | 混淆平方差公式
The identity x² – y² = (x + y)(x – y) is frequently used backwards incorrectly. A common error is writing x² – 25 = (x – t is a perfect square, not a difference of squares. The correct factorisation is (x + 5)(x – 5).
恒等式 x² – y² = (x + y)(x – y) 经常被反向使用错误。常见错误是把 x² – 25 写成 (x – 5)²，那是完全平方，并非平方差。正确分解因式是 (x + 5)(x – 5)。
Also watch out for expressions like 16a² – 9b². Students may treat it as (16a – 9b)(16a + 9b), forgetting to take the square root of the coefficients. The correct form is (4a – 3b)(4a + 3b).
还要留意如 16a² – 9b² 的式子。学生可能会处理成 (16a – 9b)(16a + 9b)，忘记了系数也要开方。正确形式是 (4a – 3b)(4a + 3b)。
4. Mistakes with Negative and Fractional Indices | 负指数与分数指数错误
rather than its reciprocal 1/aⁿ is a foundational slip. For instance, 3⁻² equals 1/9, not –9 or –1/9. Similarly, x⁻¹ = 1/x, not –x.
把 a⁻ⁿ 理解为负数而非它的倒数 1/aⁿ 是一个基础性错误。例如 3⁻² 等于 1/9，而不是 –9 或 –1/9。同样，x⁻¹ = 1/x，而不是 –x。
Fractional indices cause two mistakes at once. a1/n is the nth root, so 41/2 = √4 = 2. But am/n = (n√a)m or n√am. Students often apply powers in the wrong order, writing 82/3 as (8²)^⅓ but then miscalculating. Correct: 82/3 = (∛8)² = 4.
分数指数会同时引发两种错误。a1/n 是 n 次根，所以 41/2 = √4 = 2。而 am/n = (ⁿ√a)m 或 ⁿ√am。学生常常弄错运算顺序，例如把 82/3 写成 (8²)^⅓ 但又算错。正确：82/3 = (∛8)² = 4。
5. Solving Linear Equations: Losing the Constant | 解一次方程：丢失常数项
When solving an equation like 3x + 5 = 2x – subtract 2x from the left only, and forget to bring the constant along. They write x + 5 = –7, then x = – ever many skip steps and make sign errors. Always perform operations on both entire sides.
在解 3x + 5 = 2x – 7 这类方程时，学生经常只从左边减去 2x，而忘记带着常数项一起移，写成 x + 5 = –7 然后 x = –12。这里碰巧方法算对，但很多时候跳过步骤就会出现符号错误。一定要对等式的整个两边同时进行运算。
Another common slip: dividing by a negative coefficient without changing signs. For –2x = 10, some hastily state x = 5. The correct answer is x = –5. Always double‑check division by negative numbers.
另一个常见疏忽：除以负数系数时没有变号。对于 –2x = 10，有人草率地得出 x = 5，正确答案是 x = –5。一定要仔细复查负数除法。
The formula x = [–b ± √b² – 4ac] / (2a) requires careful substitution. A typical error is writing –b² inside the square root instead of (–b)², which would make the discriminant incorrect. Always compute b² first, then subtract 4ac.
公式 x = [–b ± √b² – 4ac] / (2a) 必须仔细代入。一个典型错误是在根号内写成 –b² 而不是 (–b)²，导致判别式出错。务必先算 b²，再减去 4ac。
When a question asks about the number of real roots, students confuse the discriminant's sign. b² – < 0 no real roots. Reversing these leads to completely wrong conclusions about the graph.
当题目询问实根个数时，学生会混淆判别式的符号。b² – 4ac > 0 表示两个不等实根，= 0 一个实根，< 0 无实根。弄反这些就会得出关于图像的完全错误结论。
7. Trigonometric Ratios Beyond Acute Angles | 超出锐角的三角比
Many learners memorize only the acute angle ratios (0° to 90°) and fail to apply CAST or the unit circle for angles like 150°. For example, sin 150° is 1/2, not –1/2, because sine is positive in the second quadrant. They mistakenly think sin(180° – θ) is –sin θ.
很多同学只背了锐角的三角比（0° 到 90°），在遇到 150° 这样的角度时不会用 CAST 图或单位圆。例如 sin 150° 是 1/2 而不是 –1/2，因为正弦在第二象限为正。他们误以为 sin(180° – θ) = –sin θ。
For cosine, cos 120° = –1/2 because cosine is negative in quadrant II. Remember: All Sinners Take Care (or CAST) – l exact values for 0°, 30°, 45°, 60°, 90° and their corresponding obtuse or reflex angles.
对于余弦，cos 120° = –1/2，因为余弦在第二象限为负。记住 CAST 规则：第二象限只有正弦为正，第四象限只有余弦为正，等等。熟练掌握 0°、30°、45°、60°、90° 的精确值及其对应的钝角和反射角。
8. Differentiation: Power Rule and Constant Terms | 微分：幂法则与常数项
² + 4, carrying the constant unchanged. The correct derivative is 15x². The constant vanishes.
一个非常基本的错误是忘了常数的导数为零。对于 y = 5x³ + 4，有人写成 dy/dx = 15x² + 4，把常数原样保留。正确导数是 15x²，常数项消失。
still applies. Given y = 3/x² = 3x⁻², students often treat it as 3x⁻² and then get dy/dx = –6x⁻¹, but then fail to rewrite it as –6/x³ correctly. Practice converting between rational and negative exponent forms fluently.
对于负指数，幂法则同样适用。给出 y = 3/x² = 3x⁻²，学生往往算成 dy/dx = –6x⁻¹，但不能正确改写回 –6/x³。要熟练在分式和负指数形式间切换。
9. Integration: The Missing '+C' | 积分：遗漏常数 +C
Omitting the constant of integration is one of the most penalised errors in indefinite integrals. Every time you integrate without limits, you must write '+ C'. Even if it seems trivial, examiners deduct marks.
不定积分中遗漏积分常数是被扣分最严重的错误之一。每次做不定积分，都必须写上 '+ C'。即使看起来微不足道，阅卷人也会扣分。
4) + C = x⁴ + C. A frequent mistake is to add the constant only at the very end of a multi‑ cceptable as long as one overall constant appears, but missing it entirely loses the mark.
在积分 4x³ 这样的幂函数时，结果是 4 × (x⁴/4) + C = x⁴ + C。常见错误是在多占领分项的式子中，只在最后加常数，却忘了为整式加 C；只要最终有一个总体常数即可，但完全缺失就会丢分。
10. Coordinate Geometry: Gradient and Distance Formulae Mix‑ups | 解析几何：斜率与距离公式混淆
The gradient formula m = (y₂ – y₁) / (x₂ – x₁) is often confused with the distance formula d = √[(x₂ – x₁)² + (y₂ – tudents sometimes subtract instead of average. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
斜率公式 m = (y₂ – y₁) / (x₂ – x₁) 经常与距离公式 d = √[(x₂ – x₁)² + (y₂ – y₁)²] 相混淆。在中点问题中，学生有时会相减而不是求平均。中点坐标是 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
n of a line, pupils correctly find the gradient but then substitute the wrong point into y – y₁ = m(x – given points consistently; mixing coordinates from two points creates an invalid equation.
在利用这些公式求直线方程时，学生能正确求出斜率，却把错误的点代入 y – y₁ = m(x – x₁)。务必始终使用同一个给定点；如果把两个点的坐标混用，就会得到错误的方程。
Published by TutorHao | International Mathematics Revision Series | aleveler.com
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