📚 Cambridge Lower Secondary Mathematics Stage 9: Common Mistakes | 剑桥初中数学第九阶段易错点总结
Stage 9 of the Cambridge Lower Secondary Mathematics curriculum deepens students’ understanding of number, algebra, geometry and statistics. Many learners, however, repeat the same errors when tackling workbook exercises. This article highlights the most common mistakes, explains why they happen, and shows how to avoid them. By recognising these pitfalls, you can strengthen your problem‑solving skills and boost your confidence.
剑桥初中数学第九阶段加深了学生对数、代数、几何和统计的理解。然而，许多学习者在做练习册题目时会重复犯相同的错误。本文梳理了最常见的易错点，解释错误原因，并给出正确解法。认清这些陷阱，能帮助你提升解题能力，增强自信。
1. Negative Number Operations | 负数运算错误
A very frequent mistake is mishandling subtraction of negative numbers. For example, students often calculate −5 − (−3) as −8 instead of realising that subtracting a negative is equivalent to adding the positive.
最常出现的错误是处理负数的减法。例如，学生经常把 −5 − (−3) 算成 −8，而没有意识到减去一个负数等于加上它的相反数。
Another common error involves multiplying or dividing with negatives. Some learners think −2 × −3 equals −6, forgetting that the product of two negative numbers is positive. A quick way to remember is: same signs give positive, different signs give negative.
另一个常见错误是负数的乘除运算。有些学生认为 −2 × −3 等于 −6，忘记两个负数相乘结果为正。快速记忆法：同号得正，异号得负。
| Incorrect (错误) | Correct (正确) |
| −3 − (−5) = −8 | −3 − (−5) = −3 + 5 = 2 |
| −4 × (−2) = −8 | −4 × (−2) = 8 |
2. Fractions, Decimals and Recurring Conversions | 分数、小数与循环小数转换错误
When converting fractions to decimals, pupils often truncate recurring decimals too early. For instance, ⅓ is frequently written as 0.33 instead of 0.3̅ or kept as an exact fraction. In multi‑step problems, this rounding error leads to inaccurate final answers.
将分数化为小数时，学生常常过早截断循环小数。比如，⅓ 经常被写成 0.33 而不是 0.3̅ 或保留精确分数。在多步运算中，这种舍入错误会导致最终答案不准确。
Comparing fractions is another tricky area. Without finding a common denominator, learners might guess that ⅜ is larger than ⅖ simply because the numbers look bigger. The correct method is to rewrite both fractions with a common denominator or convert them to decimals accurately.
比较分数大小是另一个易错点。不通分时，学生可能仅凭数字表面大小猜测 ⅜ 大于 ⅖。正确做法是先通分或精确转换为小数再比较。
⅜ = 0.375, ⅖ = 0.4, so ⅜ < ⅖
3. Algebraic Expansion and Factorisation | 代数展开与因式分解错误
The most common slip in expanding brackets is forgetting to multiply the negative sign across all terms. For −2(x − 3), students often write −2x − 6, missing that −2 × (−3) = +6.
去括号时最常见的失误是忘记把负号乘遍每一项。对于 −2(x − 3)，学生常写成 −2x − 6，漏掉了 −2 × (−3) = +6。
When factorising, learners may not extract the highest common factor. For example, 6x + 9 is sometimes factorised as 3(2x + 9), when the correct factorisation is 3(2x + 3). Always check by expanding back.
因式分解时，学生可能没有提取最大公因式。比如 6x + 9 有时被分解成 3(2x + 9)，正确的应为 3(2x + 3)。务必通过展开验算。
−2(x − 3) = −2x + 6
4. Solving Linear Equations | 解一元一次方程错误
Moving terms to the opposite side without changing the sign is a classic error. When solving 3x + 7 = 22, some wrongly write 3x = 22 + 7, giving x = 29/3 instead of correctly rewriting as 3x = 22 − 7.
移项不变号是经典错误。解 3x + 7 = 22 时，有人错误地写成 3x = 22 + 7，得到 x = 29/3，而正确做法是移项得 3x = 22 − 7。
Another pitfall is mishandling coefficients. In 5x = 20, writing x = 20 + 5 rather than dividing both sides by 5 shows a misunderstanding of inverse operations. Always perform the same operation on both sides.
另一个陷阱是处理系数。在 5x = 20 中，把 x 写成 20 + 5 而不是两边除以 5，这表明对逆运算理解有误。要始终对等式两边执行相同的运算。
3x = 22 − 7 → 3x = 15 → x = 5
5. Ratio and Proportion | 比和比例错误
Simplifying ratios incompletely is very common. A ratio like 12:8 is often simplified to 6:4 instead of the fully reduced 3:2. Leaving a ratio with a common factor makes further proportion work harder.
没有彻底化简比是很常见的。12:8 这样的比常被化简为 6:4，而不是最简的 3:2。保留公因数会让后续的比例计算更困难。
In proportion problems, students sometimes mix up direct and inverse relationships. If 3 pens cost £2.10, to find the cost of 5 pens they may multiply incorrectly rather than finding the unit rate first. The reliable method is to calculate the value of one item, then scale up.
在比例问题中，学生有时会混淆正比和反比关系。如果 3 支笔 £2.10，求 5 支笔的价钱，他们可能错误相乘而不是先求单价。可靠方法是先求出单一物品的价格，再扩大。
Unit cost: £2.10 ÷ 3 = £0.70 → 5 pens: 5 × 0.70 = £3.50
6. Percentage Increase and Decrease | 百分比增减错误
A widespread misunderstanding is that a percentage increase followed by the same percentage decrease returns to the original value. For example, £100 increased by 20% becomes £120, but a 20% decrease on £120 gives £96, not £100. The decrease is taken on a larger amount, so the pound change is larger.
一个普遍的误解是，先增加一个百分比再减少相同的百分比会回到原值。比如 £100 增加 20% 变成 £120，但对 £120 减少 20% 得到 £96，而不是 £100。减少是基于更大的量，所以实际减去的金额更大。
Another error is adding or subtracting percentages directly without converting to a multiplier. To decrease £80 by 15%, some calculate £80 − 15 = £65, which ignores that 15% is a proportion, not an absolute number. The correct method uses the multiplier 0.85.
另一个错误是直接加减百分号而不转换为乘数。计算 £80 减少 15%，有人写成 £80 − 15 = £65，这忽略了 15% 是一个比例而非具体的数。正确做法是使用乘数 0.85。
£80 × 0.85 = £68
7. Linear Graphs and Equations of Lines | 直线图像与直线方程错误
When plotting y = mx + c, the gradient m is sometimes read upside down. Between (1,3) and (4,9), the gradient is (9−3)/(4−1) = 2, but students might write (4−1)/(9−3) = ½. Always check: rise over run means change in y over change in x.
绘制 y = mx + c 的图像时，斜率 m 有时会被颠倒理解。在点 (1,3) 和 (4,9) 之间，斜率应为 (9−3)/(4−1) = 2，但学生可能写成 (4−1)/(9−3) = ½。务必牢记：斜率是 y 的变化量除以 x 的变化量。
Identifying the y‑intercept from an equation like y = 2x − 5 can also cause confusion. Some think the intercept is 5 instead of −5. The constant c is the value of y when x = 0, so it includes the sign.
从方程 y = 2x − 5 中找出 y 轴截距也容易混淆。有人以为截距是 5 而不是 −5。常数项 c 是当 x = 0 时 y 的值，符号需要包含在内。
Gradient = (y₂ − y₁) ÷ (x₂ − x₁)
8. Perimeter, Area and Volume Unit Conversions | 周长、面积和体积的单位转换错误
Converting between units of area is a frequent source of error because the scale factor must be squared. Many learners incorrectly state that 1 m² = 100 cm², but in fact 1 m = 100 cm, so 1 m² = (100 × 100) cm² = 10 000 cm². The same applies to volume: 1 m³ = 1 000 000 cm³, not 1000 cm³.
面积单位换算是常见的错误来源，因为比例因子必须平方。许多学生错误地认为 1 m² = 100 cm²，但实际上 1 m = 100 cm，因此 1 m² = (100 × 100) cm² = 10 000 cm²。体积换算同理：1 m³ = 1 000 000 cm³，而不是 1000 cm³。
Confusion also arises when mixing units within a calculation. Adding a length in metres to one in centimetres without converting first gives a meaningless result. Always convert all measurements to the same unit before performing operations.
计算中混用单位也会引起混淆。不先把米换算成厘米就直接相加，会得到无意义的结果。进行运算前，务必将所有测量值转换为相同单位。
| 1 m² = ? cm² | Common mistake | Correct |
| 1 m² | 100 cm² | 10 000 cm² |
| 1 m³ | 1000 cm³ | 1 000 000 cm³ |
9. Statistics: Mean, Median and Mode | 统计：平均数、中位数和众数错误
Finding the median without first ordering the data is a routine mistake. Given 9, 2, 5, learners may simply pick the middle number in the unsorted list and write 2 or 5. The correct median can only be found after arranging values from smallest to largest.
找中位数时不先排序是一个常规错误。给定 9, 2, 5，学生可能直接从无序列表中选中间的数，得到 2 或 5。正确的中位数只有在数值从小到大排列后才能求出。
When there is an even number of data points, some forget to average the two middle numbers. For the dataset 4, 6, 8, 10, the median is (6+8)/2 = 7, not 6 or 8. Writing only one of the middle values loses marks.
当数据个数为偶数时，有些人忘记求中间两个数的平均值。对数据集 4, 6, 8, 10，中位数应为 (6+8)/2 = 7，而不是 6 或 8。只写一个中间值会丢分。
The mean is sometimes calculated by adding all numbers and dividing by a wrong count. Double‑check the total number of items. For 2, 4, 8, the mean is (2+4+8)/3 = 14/3 ≈ 4.67, not 14/4.
计算平均数时，有时会用错误的数据个数去除总和。务必核验总数。对于 2, 4, 8，平均数为 (2+4+8)/3 = 14/3 ≈ 4.67，而不是 14/4。
10. Probability Basics | 概率基础错误
Probability must always be a number between 0 and 1 inclusive, but pupils sometimes write probabilities as whole numbers (e.g., probability of heads = 50). The correct form is ½, 0.5 or 50%, but the numeric value must be between 0 and 1.
概率必须是介于 0 和 1 之间（含端点）的数，但学生有时会把概率写成整数（比如，正面朝上的概率 = 50）。正确形式是 ½、0.5 或 50%，但数值必须在 0 到 1 之间。
Misunderstanding independent events is another common error. If a fair coin shows heads five times in a row, students may think tails is “due” on the next toss. Each toss is independent; the probability of tails remains ½, regardless of previous outcomes.
误解独立事件是另一个常见错误。如果一枚公平硬币连续五次出现正面，学生可能会认为下一次“该”出反面了。每次抛掷都是独立的；反面的概率始终是 ½，与之前的结果无关。
P(event) = number of favourable outcomes ÷ total number of outcomes
11. Pythagoras’ Theorem Application | 毕达哥拉斯定理应用错误
The most serious mistake is misidentifying the hypotenuse. In a right‑angled triangle, the hypotenuse is always the longest side, opposite the right angle. Students sometimes label a shorter side as c and apply c² = a² + b², ending with an incorrect equation.
最严重的错误是弄错斜边。在直角三角形中，斜边一定是最长边，正对直角。学生有时会把较短的边标为 c，并用 c² = a² + b²，得出错误的等式。
When one leg is unknown, the correct rearrangement is a² = c² − b². Instead, some write a² = c² + b², mistakenly adding. For a right triangle with hypotenuse 13 cm and one leg 5 cm, the correct working is: other leg = √(13² − 5²) = √(169 − 25) = √144 = 12 cm, not √(169 + 25).
当一条直角边未知时，正确的变形是 a² = c² − b²。而有的人会写成 a² = c² + b²，错误地用加法。对于斜边 13 cm，一直角边 5 cm 的直角三角形，正确计算为：另一直角边 = √(13² − 5²) = √144 = 12 cm，而不是 √(169 + 25)。
c² = a² + b², where c is the hypotenuse
12. Inequalities and Sign Reversal | 不等式与符号变向错误
When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. Failing to do so is a persistent error. For −3x > 9, dividing by −3 gives x < −3, not x > −3.
当不等式两边乘或除以一个负数时，不等号方向必须改变。忽略这一点是一个顽固的错误。对于 −3x > 9，两边除以 −3 得到 x < −3，而不是 x > −3。
Graphing inequalities on a number line often causes confusion with open and closed circles. Strict inequalities (<, >) use an open circle to show the endpoint is not included, while ≤ and ≥ use a closed circle. Mixing these up changes the meaning of the solution set.
在数轴上表示不等式时，空心圆和实心圆也常引起混淆。严格不等号 (<, >) 用空心圆表示端点不包含在内，而 ≤ 和 ≥ 用实心圆。混淆这两者会改变解集的含义。
If you multiply or divide by a negative, flip the sign: −2x ≤ 6 → x ≥ −3
Published by TutorHao | Mathematics Revision Series | aleveler.com
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