IGCSE AQA Maths: Common Mistakes and How to Fix Them | IGCSE AQA 数学：易错题精讲

📚 IGCSE AQA Maths: Common Mistakes and How to Fix Them | IGCSE AQA 数学：易错题精讲

The leap from KS3 to IGCSE often reveals gaps in fundamental understanding, and certain topics consistently trip students up in AQA exams. Percentages, indices, algebraic fraction manipulation, and circle theorems are among the areas where even strong candidates lose easy marks. This article breaks down ten of the most common pitfalls with clear, matched explanations so you can spot and correct these errors before the big day.

从 KS3 到 IGCSE 的跨越往往会暴露基础理解的漏洞，而有些知识点在 AQA 考试中总是让学生栽跟头。百分比、指数、代数分式运算以及圆定理等领域，就连程度很好的考生也常轻易丢分。本文拆解了十个最常见的易错点，并配有清晰的对照讲解，帮助你在考前提早发现并纠正这些错误。

1. Reverse Percentage Problems | 逆百分比问题

A classic mistake is thinking that if a price increases by 15% to £460, the original price is simply 460 × 0.85. This fails because the original amount is not 100% of 460; 460 represents 115% of the original.

一个典型错误是：若价格上涨15%后变成460英镑，就以为原价是 460 × 0.85。这之所以不对，是因为原价并不对应460的100%；460 代表的是原价的115%。

Correct approach: Let original be x. Then x × 1.15 = 460, so x = 460 ÷ 1.15 = £400. Always identify whether the given value represents more or less than 100% of the unknown.

正确方法：设原价为 x。则 x × 1.15 = 460，因此 x = 460 ÷ 1.15 = 400 英镑。始终要判别已知值对应的是未知量的百分之多少。

2. Misusing Negative and Fractional Indices | 负指数与分数指数的误用

Many learners incorrectly rewrite 16⁻³⁄² as 16^(3/2) with a positive sign, or forget that the denominator of a fractional index is the root. A negative index means reciprocal, and m/n means nth root raised to the m.

不少学生错误地把 16⁻³⁄² 改写为正指数的 16^(3/2)，或忘记分数指数的分母是根指数。负指数表示倒数，m/n 指数表示先开 n 次方再 m 次方。

16⁻³⁄² = 1 / (16³⁄²) = 1 / (√16)³ = 1 / 4³ = 1/64

Similarly, a term like 8⁻²⁄³ equals 1/(³√8)² = 1/2² = 1/4. Practise separating the sign, the root, and the power.

类似地，8⁻²⁄³ 等于 1/(³√8)² = 1/2² = 1/4。要多练习分解符号、根号和幂次。

3. Expanding Double Brackets with Negatives | 含负号的双括号展开

When expanding (x − 3)(2x − 5), students often miss a negative sign, writing a middle term of −5x − 6x instead of −5x − 6x? Actually the correct product is 2x² −5x −6x +15, and the error is in forgetting that −3 × −5 gives +15.

展开 (x − 3)(2x − 5) 时，学生常漏掉负号，错误地写出中间项 −5x − 6x，或者忘记 −3 × −5 得 +15。正确乘积应为 2x² −5x −6x +15。

A systematic grid method helps: multiply each term, then collect like terms carefully. Check the constant term sign intensely.

系统的表格法有助于：逐个相乘，然后仔细合并同类项。尤其要注意常数项的符号。

(x − 3)(2x − 5) = 2x² − 5x − 6x + 15 = 2x² − 11x + 15

4. Solving Equations with Fractions | 含分式方程的解法

In equations like x/3 + (x−1)/4 = 2, a frequent mistake is to combine numerators incorrectly without using a common denominator, or to multiply only part of the equation.

对于 x/3 + (x−1)/4 = 2 这样的方程，常见错误是不通分就直接错误合并分子，或者只乘方程的一部分。

Correct method: Multiply every term by the lowest common multiple (12) to clear fractions: 12×(x/3) + 12×((x−1)/4) = 12×2, giving 4x + 3(x−1) = 24. Then 4x + 3x − 3 = 24, 7x = 27, x = 27/7.

正确方法：每一项都乘以最小公倍数（12）来消去分母：12×(x/3) + 12×((x−1)/4) = 12×2，得到 4x + 3(x−1) = 24。然后 4x + 3x − 3 = 24，7x = 27，x = 27/7。

5. Simultaneous Equations – Elimination Errors | 联立方程——消元法错误

When solving 3x + 2y = 7 and 5x − 3y = 18, pupils often multiply one equation but forget to multiply the entire equation, or they subtract when they should add to eliminate.

解方程组 3x + 2y = 7 和 5x − 3y = 18 时，学生经常只乘了方程的一部分，或者该相加时却相减，导致消元失败。

To eliminate y, multiply first by 3 and second by 2: 9x + 6y = 21 and 10x − 6y = 36. Adding eliminates y: 19x = 57, x = 3. Then substitute back. Always double-check whether you need to add or subtract the new equations.

要消去 y，第一式乘3得 9x + 6y = 21，第二式乘2得 10x − 6y = 36。两式相加消去 y：19x = 57，x = 3。再代回求解。始终要确认合并时是加还是减。

6. Histograms and Frequency Density | 直方图与频率密度

IGCSE candidates often confuse frequency density with frequency and draw bars where height equals frequency. In a histogram, area is proportional to frequency, so height = frequency ÷ class width.

IGCSE 考生常混淆频率密度与频数，把柱子的高度直接画成了频数。在直方图中，面积与频数成正比，因此高度 = 频数 ÷ 组距。

For example, a class 10 ≤ x < 20 with frequency 30 has width 10, so frequency density = 30 ÷ 10 = 3. The bar height must be 3, not 30. When analysing a histogram, find frequency by calculating area of each bar.

例如，组 10 ≤ x < 20，频数为30，组距为10，频率密度 = 30 ÷ 10 = 3。柱高应为3，而非30。分析直方图时，要通过计算柱条面积来求频数。

7. Probability Tree Diagrams – Replacement Awareness | 概率树图——放回与否的判别

In a “without replacement” scenario, probabilities on the second set of branches must change to reflect the reduced total. A common blunder is to keep denominators the same, e.g., picking two reds from 5 red and 3 blue without replacement: first red = 5/8, but second red given first red should be 4/7, not 5/8 again.

在“不放回”情形中，第二层分支的概率必须相应调整以反映总数减少。常见错误是分母不变，比如从5红3蓝中无放回地取两次红球：第一个红球概率为 5/8，但在已取一个红球条件下第二次取红概率应为 4/7，而非还是 5/8。

Always pause and ask: has the object been replaced? If not, adjust numerator and denominator. Multiply along branches for combined probability.

务必停顿并自问：物体是否放回了？若没有，分子分母都要调整。求组合概率时沿分支相乘。

8. Choosing the Correct Trigonometric Ratio | 选择正确的三角比

Given a right-angled triangle with an angle, opposite 6 cm and hypotenuse 10 cm, many students hastily use tan or cos. The safe route: label sides O, A, H relative to the known angle, then decide which ratio uses the two known sides and the unknown.

给定一个直角三角形，已知一角，其对边6 cm，斜边10 cm，很多学生轻率地使用正切或余弦。稳妥的做法是：先标出相对已知角的 O、A、H，然后确定哪个比值联系了已知两边和未知量。

Here we know O and H, so sin θ = 6/10, θ = sin⁻¹(0.6). Writing SOH CAH TOA next to the question prevents rushed mistakes. For non-right triangles, use sine rule or cosine rule with correct pairing of sides and angles.

这里已知 O 和 H，因此 sin θ = 6/10，θ = sin⁻¹(0.6)。在题目旁写下 SOH CAH TOA 可有效避免草率错误。对于非直角三角形，要正确配对边角来使用正弦定理或余弦定理。

9. Quadratic Factorisation When the Leading Coefficient ≠ 1 | 二次项系数不为1的因式分解

Factorising 2x² + 7x + 3 often causes confusion. Trial-and-error methods become messy; many write (2x + 1)(x + 3), but expanding gives 2x² + 7x + 3, which is actually correct! Wait, correct example: 2x² + 7x + 3 = (2x + 1)(x + 3). But a common error is mixing up signs when constant is negative, like 3x² − 7x − 6.

分解 2x² + 7x + 3 时常让人混乱。试错法常常混乱；许多人写成 (2x + 1)(x + 3)，展开后得 2x² + 7x + 3，这其实正确。但常数项为负时，比如 3x² − 7x − 6，学生就容易把符号搞混。

A structured method: For 3x² − 7x − 6, find factors of 3×(−6) = −18 that add to −7, i.e., −9 and 2. Split the middle term: 3x² − 9x + 2x − 6, then factor by grouping: 3x(x − 3) + 2(x − 3) = (3x + 2)(x − 3).

可采用结构化方法：对 3x² − 7x − 6，找 3×(−6) = −18 的两个因子，使之相加为 −7，即 −9 和 2。分解中间项：3x² − 9x + 2x − 6，然后分组提取公因式：3x(x − 3) + 2(x − 3) = (3x + 2)(x − 3)。

10. Circle Theorems – Angles in the Same Segment | 圆定理——同弧上的圆周角

Students frequently confuse “angle at the centre is twice the angle at the circumference” with “angles in the same segment are equal”. When a diagram shows two angles subtended by the same chord on the same side of the chord, they are equal, even if one looks smaller due to drawing scale.

学生常将“圆心角是圆周角的两倍”与“同弦上的圆周角相等”混淆。如果图上两个圆周角由同一条弦所对且位于弦的同侧，那么这两个角就相等，不能因为画图比例看起来不一样就判断错误。

Always trace the chord: if the endpoints match and the vertices lie on the circumference on the same side, they are in the same segment and the angles are equal. Use this when solving for missing angles in circle geometry questions.

始终寻找弦：如果角的端点一致且顶点都在圆周上并位于弦的同一侧，那么它们就在同一个弓形内，角度相等。解答圆几何的求角问题时切记运用这一定理。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE AQA Maths: Common Mistakes and How to Fix Them | IGCSE AQA 数学：易错题精讲