📚 GCSE WJEC Maths: Common Mistakes & How to Avoid Them | GCSE WJEC 数学：易错题精讲
In the GCSE WJEC Mathematics exam, students often lose marks not because they do not understand the concepts, but because they make small, avoidable errors under pressure. These mistakes can appear in basic arithmetic, algebra, geometry, and data handling. This article takes you through the most common pitfalls and shows you exactly how to sidestep them. Each section includes a typical error, a clear explanation, and the correct method, so you can build confidence and accuracy before your exam.
在 GCSE WJEC 数学考试中，学生失分往往不是因为不懂概念，而是因为在压力下犯了细小的、可避免的错误。这些错误可能出现在基础算术、代数、几何和数据处理中。本文带你逐一剖析最常见的易错点，并展示如何准确避开它们。每个小节都包含典型错误、清晰的解释和正确做法，帮助你在考前建立信心和精确度。
1. Adding and Subtracting Fractions | 分数加减易错点
A very common mistake is to add the numerators and the denominators separately, for example ½ + ⅓ = (1+1)/(2+3) = 2/5. This is wrong because fractions must have a common denominator before they can be added or subtracted.
一个非常常见的错误是分别将分子和分母相加，例如 ½ + ⅓ = (1+1)/(2+3) = 2/5。这是错误的，因为分数必须先化为同分母才能进行加减。
The correct approach is to find the lowest common multiple of 2 and 3, which is 6. Convert each fraction: ½ becomes 3/6, and ⅓ becomes 2/6. Then add the numerators: 3/6 + 2/6 = 5/6.
正确的方法是找出 2 和 3 的最小公倍数，即 6。将每个分数转换：½ 变成 3/6，⅓ 变成 2/6。然后分子相加：3/6 + 2/6 = 5/6。
½ + ⅓ = 3/6 + 2/6 = 5/6
Remember that the same logic applies to subtraction and to mixed numbers – always change mixed numbers to improper fractions first.
记住，同样的逻辑也适用于减法和带分数——始终先将带分数化为假分数。
2. Negative Numbers and Order of Operations | 负数与运算顺序
Many students stumble when dealing with signs and powers. A classic error is to evaluate −3² as 9. In fact, the exponent applies only to the 3, not to the minus sign, so −3² means −(3²) = −9. When we want the negative to be squared as well, we must use brackets: (−3)² = 9.
很多学生在处理符号和幂时出错。一个经典错误是把 −3² 算成 9 。实际上指数只作用于 3，而不作用于负号，所以 −3² 表示 −(3²) = −9。如果希望负号也被平方，必须使用括号：(−3)² = 9。
−3² = −9, (−3)² = 9
Similarly, when combining negative numbers with subtraction, −2 − (−5) is often simplified incorrectly as −2 − 5 = −7. The correct simplification is −2 + 5 = 3, because subtracting a negative is equivalent to adding a positive.
类似地，当负数与减法结合时，−2 − (−5) 经常被错误简化为 −2 − 5 = −7。正确的简化是 −2 + 5 = 3，因为减去一个负数等同于加上一个正数。
Always use BIDMAS/BODMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
请始终使用运算顺序：括号、指数、乘除、加减。
3. Expanding Brackets with Negatives | 带负号括号的展开
When a negative sign sits in front of a bracket, it changes the sign of every term inside. A frequent mistake is to write −(2x − 5) = −2x − 5, forgetting that −(−5) becomes +5. The correct expansion is −2x + 5.
当括号前面有一个负号时，它会改变括号内每一项的符号。一个常见错误是把 −(2x − 5) 写成 −2x − 5，忘记了 −(−5) 应该变成 +5。正确的展开是 −2x + 5。
−(2x − 5) = −2x + 5
Similarly, for −3(4 − 2x), the mistake is to write −12 − 6x. The correct product is −3 × 4 = −12 and −3 × (−2x) = +6x, giving the result −12 + 6x.
类似地，对于 −3(4 − 2x)，错误是写成 −12 − 6x。正确的乘积是 −3 × 4 = −12，而 −3 × (−2x) = +6x，结果为 −12 + 6x。
Take extra care when there is more than one bracket and the second requires distributing a negative coefficient.
当存在多个括号且第二个需要分配负系数时，要格外小心。
4. Solving Equations with Variables on Both Sides | 解两边含变量的方程
Consider the equation 2x + 3 = x − 1. A common error is to move terms incorrectly and end up with x = −2. Let’s see the right steps. Subtract x from both sides: x + 3 = −1. Then subtract 3: x = −4.
考虑方程 2x + 3 = x − 1。常见错误是移项不当，得出 x = −2。我们来看正确步骤。两边同时减去 x：x + 3 = −1。然后再减去 3：x = −4。
2x + 3 = x − 1 → x + 3 = −1 → x = −4
Another pitfall is forgetting to apply the inverse operation to both sides, for example dividing only part of the left side. Always perform the same operation on the entire side.
另一个陷阱是忘记对等式两边同时进行逆运算，例如只除以左边的一部分。务必对整边进行相同的操作。
To check your answer, substitute it back into the original equation. Here, 2(−4) + 3 = −8 + 3 = −5, and −4 − 1 = −5, so it works.
要检查答案，将其代回原方程。这里，2(−4) + 3 = −8 + 3 = −5，而 −4 − 1 = −5，验证成立。
5. Sharing in a Given Ratio | 按比例分配易错题
A typical mistake when sharing £120 in the ratio 3:5 is to divide £120 by 3 and by 5, giving £40 and £24, which do not add up to £120. The key is to find the total number of parts first: 3 + 5 = 8 parts.
在按 3:5 的比例分配 120 英镑时，一个典型错误是将 120 英镑除以 3 和除以 5，得出 40 英镑和 24 英镑，而这两个数加起来不等于 120 英镑。关键在于先求出总份数：3 + 5 = 8 份。
One part is worth £120 ÷ 8 = £15. Then the shares are 3 × £15 = £45 and 5 × £15 = £75. Always check that the sum equals the original amount.
每份价值 120 ÷ 8 = 15 英镑。那么分配金额分别为 3 × 15 = 45 英镑和 5 × 15 = 75 英镑。务必检查总和是否等于原始金额。
Total parts = 3 + 5 = 8, one part = £120 ÷ 8 = £15, shares: £45 and £75
This method works for any ratio, including those with three terms.
这个方法适用于任何比例，包括三项比例。
6. Percentage Increase and Decrease | 百分比增减
To increase £80 by 15%, many students simply add £15 to get £95. This is incorrect because the percentage must be calculated relative to the original amount. The correct method is to find 15% of £80 = 0.15 × £80 = £12, then add: £80 + £12 = £92.
要将 80 英镑增加 15%，很多学生直接加上 15 得到 95 英镑。这是错误的，因为百分比必须相对于原始金额计算。正确的方法是先求 80 英镑的 15% = 0.15 × 80 = 12 英镑，然后相加：80 + 12 = 92 英镑。
Using a multiplier is faster and less prone to error: an increase of 15% corresponds to multiplying by 1.15. So £80 × 1.15 = £92. For a decrease of 15%, the multiplier would be 1 − 0.15 = 0.85, giving £80 × 0.85 = £68.
使用乘数更快捷且不易出错：增加 15% 相当于乘以 1.15。因此 £80 × 1.15 = £92。对于减少 15%，乘数是 1 − 0.15 = 0.85，得到 £80 × 0.85 = £68。
Increase: new value = original × (1 + percentage as decimal)
For successive percentage changes, multiply the multipliers in sequence. For example, a 10% increase followed by a 20% decrease gives an overall multiplier of 1.10 × 0.80 = 0.88, which is a 12% decrease overall.
对于连续的百分比变化，按顺序乘以乘数。例如，先增加 10% 再减少 20%，总乘数为 1.10 × 0.80 = 0.88，即总体上减少了 12%。
7. Area and Volume Unit Conversions | 面积与体积单位换算
One of the most frequent errors is to treat area and volume conversions like linear conversions. Students often think 1 m² = 100 cm² because 1 m = 100 cm. In reality, 1 m² = 1 m × 1 m = 100 cm × 100 cm = 10 000 cm².
最常见的错误之一是将面积和体积换算当作线性换算。学生常认为 1 m² = 100 cm²，因为 1 m = 100 cm。实际上，1 m² = 1 m × 1 m = 100 cm × 100 cm = 10 000 cm²。
1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³
For volume, 1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³. Remember: square conversion factors and cube conversion factors yourself rather than guessing.
对于体积，1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³。记住：要自己对换算因子进行平方或立方，而不是猜测。
When converting from a smaller unit to a larger one, divide by the conversion factor (e.g., 5000 cm² = 5000 ÷ 10 000 = 0.5 m²). Always double-check by visualizing the size.
当从小单位换算成大单位时，除以换算因子（例如 5000 cm² = 5000 ÷ 10 000 = 0.5 m²）。务必通过想象物体大小进行复核。
8. Pythagoras’ Theorem | 毕达哥拉斯定理应用题
A common blunder in Pythagoras’ Theorem is mixing up the hypotenuse and the legs. The formula a² + b² = c² only works when c is the longest side, opposite the right angle. Some students instinctively add the two given sides, e.g., for sides 3 and 4 they might write c = 3 + 4 = 7.
毕达哥拉斯定理的一个常见错误是混淆斜边和直角边。公式 a² + b² = c² 仅当 c 是最长边、直角对边时才成立。有些学生会本能地将给定的两边相加，例如对于边长 3 和 4，可能写成 c = 3 + 4 = 7。
The correct calculation: c = √(3² + 4²) = √(9 + 16) = √25 = 5. If you are given the hypotenuse and one leg, you must subtract: a = √(c² − b²).
正确的计算是：c = √(3² + 4²) = √(9 + 16) = √25 = 5。如果已知斜边和一条直角边，则必须用减法：a = √(c² − b²)。
c = √(a² + b²), a = √(c² − b²)
Always identify the hypotenuse first and label the triangle clearly before substituting values.
在代入数值之前，一定要先标出斜边并清楚地标记三角形。
9. Trigonometry: Choosing the Correct Ratio | 三角函数：选择正确的比率
In right-angled triangles, students often use the wrong trigonometric ratio. The mnemonic SOH CAH TOA helps: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Yet many misinterpret which sides are opposite and adjacent relative to the given angle.
在直角三角形中，学生经常用错三角函数比率。记忆口诀 SOH CAH TOA 很有用：Sin = 对边/斜边，Cos = 邻边/斜边，Tan = 对边/邻边。但很多学生会弄错相对于给定角的对边和邻边。
For example, given an angle of 35°, the opposite side of 5 cm, and needing to find the hypotenuse, some might set up cos(35) = 5/hyp. That is wrong because 5 is opposite, not adjacent. The correct ratio is sin(35) = 5/hyp, so hyp = 5/sin(35).
例如，给定角 35°，对边长 5 cm，需要求斜边。有人可能会列出 cos(35) = 5/斜边。这是错误的，因为 5 是对边而不是邻边。正确的比率是 sin(35) = 5/斜边，因此斜边 = 5/sin(35)。
Also, always check that your calculator is in degree mode (not radians). A quick sanity check: sin(30°) = 0.5; if you get a different value, you are in the wrong mode.
此外，一定要检查计算器是否处于度数模式（而非弧度模式）。一个快速检验：sin(30°) = 0.5；如果得到不同的数值，说明模式错了。
10. Probability Tree Diagrams | 概率树图常见错误
When drawing probability tree diagrams, a common slip is to forget that the probabilities on each set of branches must sum to 1. For instance, if the probability of a rainy day is 0.3, the probability of no rain is not simply 1 − 0.3 = 0.7 if there are more than two outcomes, but for two complementary events, it works. However, students sometimes label both branches with the same probability.
在画概率树图时，一个常见疏漏是忘记每一组分支的概率之和必须为 1。例如，如果雨天的概率是 0.3，那么非雨天的概率在互补事件中确实是 1 − 0.3 = 0.7。但学生有时会在两条分支上标上相同的概率。
Another error is to multiply along branches incorrectly or add outcomes that are not mutually exclusive without checking. To find the probability of at least one success, for example, it is often easier to use the complement rule: 1 − P(none).
另一个错误是错误地沿分支相乘，或者在未检查是否互斥的情况下直接相加结果。例如，要计算至少一次成功的概率，通常使用补集规则更简单：1 − P(零次成功)。
When items are not replaced, the probabilities on the second set of branches change. Always adjust denominators accordingly and double-check that the branch probabilities still sum to 1.
当物品不被放回时，第二组分支上的概率会改变。务必相应地调整分母，并再次确认分支概率之和仍为 1。
11. Standard Form Calculations | 标准形式计算
Standard form (A × 10ⁿ, where 1 ≤ A < 10) often causes mistakes when multiplying or dividing. A typical error is to add the coefficients together and keep the power the same: (4 × 10³) × (2 × 10²) = 6 × 10⁵. The correct method is to multiply the number parts and add the exponents: (4 × 2) × 10³⁺² = 8 × 10⁵.
标准形式（A × 10ⁿ，其中 1 ≤ A < 10）在乘除时常常出错。典型错误是将系数相加而保留相同的幂：(4 × 10³) × (2 × 10²) = 6 × 10⁵。正确的方法是乘以数字部分并将指数相加：(4 × 2) × 10³⁺² = 8 × 10⁵。
For division, divide the coefficients and subtract the exponents. For example, (6 × 10⁷) ÷ (2 × 10³) = 3 × 10⁴.
对于除法，系数相除，指数相减。例如 (6 × 10⁷) ÷ (2 × 10³) = 3 × 10⁴。
After calculating, ensure the answer is still in standard form: if the coefficient becomes 10 or more, adjust it, e.g., 12 × 10⁴ becomes 1.2 × 10⁵. If it is less than 1, adjust the exponent downwards.
计算后，确保答案仍是标准形式：如果系数大于或等于 10，进行调整，例如 12 × 10⁴ 变为 1.2 × 10⁵。如果系数小于 1，则向下调整指数。
12. Histograms and Frequency Density | 直方图与频率密度
In WJEC GCSE, histograms use frequency density = frequency ÷ class width. A common mistake is to plot frequency on the vertical axis instead of frequency density, which produces a distorted graph. Always calculate frequency density for each bar before drawing.
在 WJEC GCSE 中，直方图使用频率密度 = 频率 ÷ 组距。一个常见错误是在纵轴上绘制频率而不是频率密度，这会产生一个扭曲的图形。在绘制之前，务必为每个条形计算频率密度。
When given a histogram and asked to complete a frequency table, students sometimes multiply frequency density by the class width incorrectly if the class boundaries contain gaps. Ensure you use the correct class width (upper bound − lower bound).
给定直方图并要求补全频率表时，如果组边界有间隙，学生有时会在频率密度乘以组距时出错。确保使用正确的组距（上限 − 下限）。
Frequency = frequency density × class width
Also, remember that in a histogram the area of each bar is proportional to the frequency, not the height alone. This is why bars of unequal width must use frequency density.
同时记住，在直方图中，每个条形的面积与频率成正比，而不仅仅是高度。这就是为什么不等宽条形必须使用频率密度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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