GCSE AQA Maths: Common Mistakes Explained | GCSE AQA 数学：易错题精讲

📚 GCSE AQA Maths: Common Mistakes Explained | GCSE AQA 数学：易错题精讲

Even the most prepared students can lose marks in GCSE AQA Maths because of small, repeated errors. This article breaks down the most common mistakes seen in past papers, explains exactly why they happen, and shows you how to avoid them. By fixing these weak spots, you can quickly boost your grade without learning completely new topics.

即使是准备最充分的学生，也可能因为一些反复出现的小错误在 GCSE AQA 数学考试中丢分。本文深入分析往年试卷中最常见的错误，解释错误原因，并告诉你如何避开它们。修复这些薄弱环节，你无需学习全新专题，就能快速提升分数。

1. Confusing Perimeter and Area | 混淆周长与面积

Students often calculate perimeter when the question asks for area, or they mix up the units. A classic error is giving the perimeter of a rectangle as length × width instead of adding all sides.

学生常常在题目要求计算面积时却算了周长，或者搞混单位。一个经典错误是把长方形的周长错误地写成长 × 宽，而不是把四条边相加。

Perimeter = sum of all sides, Area of rectangle = length × width

周长 = 所有边的和，长方形面积 = 长 × 宽

Always circle the keyword ‘perimeter’ or ‘area’ in the question. After finding your answer, check the units: perimeter has units like cm or m, area has units like cm² or m².

一定要在题目中圈出关键词“周长”或“面积”。得到答案后检查单位：周长的单位是 cm 或 m，面积的单位是 cm² 或 m²。

2. Fraction Division Inversion Error | 分数除法忘记取倒数

When dividing by a fraction, many students multiply straight across instead of flipping the second fraction. For example, 2/3 ÷ 1/4 is wrongly computed as (2×1)/(3×4) = 2/12 = 1/6, but the correct method is 2/3 × 4/1 = 8/3.

除以一个分数时，很多学生直接相乘而不把第二个分数取倒数。比如 2/3 ÷ 1/4 被错误地计算为 (2×1)/(3×4) = 2/12 = 1/6，但正确做法是 2/3 × 4/1 = 8/3。

a/b ÷ c/d = a/b × d/c

Remember the KFC rule: Keep the first fraction, Flip the second, Change the division sign to multiplication. Practise with mixed numbers by converting them to improper fractions first.

记住 KFC 法则：保留第一个分数，翻转第二个，把除号变成乘号。遇到带分数时，先化成假分数再计算。

3. Expanding Brackets with Negative Signs | 去括号时忽略负号

A very common slip is forgetting to multiply the negative sign with every term inside the bracket. For 3 − 2(x + 4), students often write 3 − 2x + 8, missing that −2 × 4 = −8, so it should be 3 − 2x − 8 = −2x − 5.

一个常见失误是忘记把负号乘进括号里的每一项。对于 3 − 2(x + 4)，学生常写成 3 − 2x + 8，漏掉了 −2 × 4 = −8，正确结果应是 3 − 2x − 8 = −2x − 5。

−a(b + c) = −ab − ac

Treat the minus sign as part of the multiplier. Write −2 × (x + 4) explicitly before expanding: (−2) × x + (−2) × 4 = −2x − 8.

把负号看作乘数的一部分。展开前先明确写出 −2 × (x + 4)，即 (−2) × x + (−2) × 4 = −2x − 8。

4. Unbalanced Equations When Solving | 解方程时两边操作不一致

When adding 5 to isolate a term, students sometimes only add 5 to one side, or they divide only part of a side. For 2x + 5 = 13, they might write 2x = 18 instead of 2x = 8, or from 2x = 8 they write x = 8/2 = 4 correctly, but in more complex equations they forget to divide every term.

当加 5 分离某一项时，学生有时只加在等式的一边，或者只除以某一边的一部分。对于 2x + 5 = 13，他们可能写成 2x = 18 而不是 2x = 8；或者从 2x = 8 正确得到 x = 4，但在更复杂的方程中却忘记每一项都除以系数。

Whatever you do to one side, do exactly the same to the other side.

你对等式一边做了什么，对另一边也必须做完全相同的事。

Show all steps clearly: write ‘−5′ below both sides, or draw an arrow. When dividing, place a large division bar under the entire side to remind yourself to divide every term.

清晰地展示所有步骤：在两边下面写上“−5”，或画一个箭头。当除以某数时，用大分数线覆盖整个一侧，提醒自己每一项都要被除。

5. Direct and Inverse Proportion Mix-up | 正比与反比混淆

In direct proportion, y = kx; in inverse proportion, y = k/x. Students often swap the formulas or use the wrong constant calculation. For example, if y is inversely proportional to x, and y = 6 when x = 2, the correct k = 12, but a typical error is to write y = 12x instead of y = 12/x.

正比关系中，y = kx；反比关系中，y = k/x。学生经常混淆公式，或者计算常数时出错。例如，若 y 与 x 成反比，且 x = 2 时 y = 6，正确的 k = 12，但常见错误是写成 y = 12x 而不是 y = 12/x。

Direct: y ∝ x ⇒ y = kx. Inverse: y ∝ 1/x ⇒ y = k/x.

正比：y ∝ x ⇒ y = kx。反比：y ∝ 1/x ⇒ y = k/x。

Read the wording carefully: ‘directly proportional’ vs ‘inversely proportional’. After finding k, substitute back and test with the given values to see if the equation makes sense.

仔细阅读题干：“成正比”还是“成反比”。求出 k 后，代回原值检验，看方程是否合理。

6. Percentage Increase and Decrease Malpractice | 百分数增减计算错误

A frequent mistake is to find a percentage of the original amount and then add or subtract, but using the wrong multiplier. For a 15% increase, the multiplier is 1.15, not 0.15. Some students also apply a decrease by 20% as subtracting 20% of the new amount, leading to confusion in compound changes.

常见错误是找出原值的某百分比后加减，但用了错误的乘数。15% 的增长，乘数是 1.15，而不是 0.15。有些学生计算减少 20% 时减去新值的 20%，导致复合变化时混乱。

Increase by r% → Multiply by (1 + r/100). Decrease by r% → Multiply by (1 − r/100).

增加 r% → 乘以 (1 + r/100)。减少 r% → 乘以 (1 − r/100)。

For repeated changes, convert each change to a decimal multiplier and multiply them together. Do not just add or subtract the percentages.

对于重复变化，把每一次变化转化为小数乘数，然后相乘。不要简单地把百分数相加或相减。

7. Unit Conversion Errors | 单位换算错误

GCSE questions frequently include mixed units, and students forget to convert everything to the same unit before calculating. For example, finding the volume of a cuboid with dimensions 0.5 m, 20 cm and 150 mm often leads to wrong answers if cm, m and mm are used interchangeably.

GCSE 题目经常包含混合单位，学生常忘记在计算前将所有单位化统一。例如，计算长宽高分别为 0.5 m、20 cm 和 150 mm 的长方体体积时，如果混用 cm、m 和 mm，往往会得出错误答案。

1 m = 100 cm = 1000 mm, 1 litre = 1000 cm³

1 m = 100 cm = 1000 mm，1 升 = 1000 cm³

Underline all given units in the question. Convert to the smallest unit or the one requested in the answer line before doing any multiplication or division.

在题目中给所有给定单位画下划线。在做任何乘法或除法之前，将所有量转换为最小单位或答案要求的单位。

8. Frequency Density and Histogram Misunderstanding | 频率密度与直方图理解偏差

In histograms, the area of the bar represents frequency, so the vertical axis is frequency density. A common mistake is to read the frequency directly from the vertical scale, or to calculate frequency density as frequency × class width instead of frequency ÷ class width.

在直方图中，条形面积代表频数，因此纵轴是频率密度。常见错误是直接从纵轴刻度读取频数，或者把频率密度错误计算为频数 × 组距，而不是频数 ÷ 组距。

Frequency density = Frequency ÷ Class width

频率密度 = 频数 ÷ 组距

Always check the axis label. If it says ‘Frequency density’, you must multiply the height by the class width to get the frequency. When drawing a histogram, calculate the correct density for each bar.

始终检查坐标轴标签。如果写着“频率密度”，你必须把高度乘以组距才能得到频数。绘制直方图时，要为每个条形计算正确的密度。

9. Probability: With and Without Replacement | 概率：放回与不放回混淆

When working with tree diagrams, students often forget to adjust the denominator for the second event if there is no replacement. For example, picking two red balls from a bag of 3 red and 5 blue without replacement: the first probability is 3/8, the second should be 2/7, but many write 3/8 again.

在使用树状图时，学生经常忘记在不放回的情况下调整第二次事件的分母。例如，从装有 3 个红球和 5 个蓝球的袋中不放回地取两个红球：第一次概率是 3/8，第二次应该是 2/7，但很多人再次写成 3/8。

Without replacement: total outcomes decrease by 1 each time.

不放回：每次总结果数减少 1。

Read the question: ‘replaced’ or ‘not replaced’. Draw the tree with clear branches and label the changing denominators. Multiply along the branches for combined probabilities.

读清题意：“放回”还是“不放回”。画出树状图，清晰标出变化的分母。沿着分支相乘得到组合概率。

10. Misusing Pythagoras’ Theorem for Non-right Triangles | 对非直角三角形误用毕达哥拉斯定理

Pythagoras’ theorem (a² + b² = c²) only works for right-angled triangles. A common mistake is to apply it to any triangle when finding a missing side, or to identify the hypotenuse incorrectly. Some students also forget that c must be the longest side.

毕达哥拉斯定理（a² + b² = c²）只适用于直角三角形。常见错误是求任意三角形的未知边时直接使用该定理，或者错误地识别斜边。有些学生还忘记 c 必须是最长的那条边。

For a right-angled triangle, c is always the side opposite the right angle.

对于直角三角形，c 总是直角所对的边。

Check for the right-angle symbol. If there isn’t one, use the sine rule or cosine rule. To find the hypotenuse, square the two shorter sides, add them, then square root; to find a shorter side, square the hypotenuse minus the square of the other side.

检查是否有直角标记。如果没有，就使用正弦定理或余弦定理。求斜边时，两短边平方和再开方；求直角边时，用斜边平方减去另一直角边平方再开方。

11. Estimation and Rounding Over-Approximation | 估算与四舍五入的过度近似

When estimating the answer to a calculation, students often round numbers to too many decimal places or to values that make the approximation much harder. For example, to estimate 487 × 0.52, a good approach is 500 × 0.5 = 250, but some might round 487 to 490, which doesn’t simplify the multiplication enough.

在估算计算结果时，学生往往把数字舍入到过多的小数位，或者舍入到使近似计算更困难的数值。例如，估算 487 × 0.52，较好的方式是 500 × 0.5 = 250，但有人可能把 487 舍入到 490，这并没有充分简化乘法。

In estimation, round to one significant figure wherever possible.

估算时尽可能把数字舍入到一位有效数字。

Practise rounding to 1 significant figure: 487 → 500, 0.52 → 0.5. Then perform the simple calculation. Don’t forget to state clearly that it is an estimated value.

练习舍入到 1 位有效数字：487 → 500，0.52 → 0.5。然后进行简单计算。别忘了明确说明这是估计值。

12. Index Law Misapplications (Negative and Fractional Powers) | 指数法则误用（负指数与分数指数）

Many students treat negative indices as negative numbers, for instance thinking 3⁻² = −9. Fractional indices like 27^{1/3} are often mistaken for 27 ÷ 3. The laws of indices for multiplication and powers also get muddled: a² × a³ = a⁶ is a frequent error, when it should be a⁵.

许多学生把负指数当成负数，比如认为 3⁻² = −9。像 27^{1/3} 这样的分数指数常被误解为 27 ÷ 3。指数相乘和幂的乘方法则也经常混乱：a² × a³ = a⁶ 是常见错误，正确答案应为 a⁵。

a⁻ⁿ = 1/aⁿ, a^{m/n} = (ⁿ√a)ᵐ, aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ

a⁻ⁿ = 1/aⁿ，a^{m/n} = (ⁿ√a)ᵐ，aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ

Rewrite negative powers as reciprocals immediately. For fractional powers, the denominator gives the root, the numerator gives the power. Always write the base clearly and add the exponents, never multiply them when multiplying like bases.

立即把负指数改写为倒数形式。对于分数指数，分母表示开方，分子表示乘方。同底数相乘时，指数相加，不要相乘。

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GCSE AQA Maths: Common Mistakes Explained | GCSE AQA 数学：易错题精讲