Common Mistakes and Corrections for Year 11 CIE Mathematics | Year 11 CIE 数学：常见误区与纠正方法

📚 Common Mistakes and Corrections for Year 11 CIE Mathematics | Year 11 CIE 数学：常见误区与纠正方法

Many Year 11 students preparing for CIE IGCSE Mathematics lose marks not because they lack understanding, but because they fall into predictable traps. This article highlights the most common mistakes across algebra, number, geometry and exam technique, and shows clear corrections so you can avoid them in your exams.

许多 Year 11 学生在备考 CIE IGCSE 数学时丢分，并非因为理解不足，而是因为掉进了可预测的陷阱。本文梳理代数、数、几何与考试技巧中最常见的误区，并给出清晰的纠正方法，帮助你在考试中避开这些错误。

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1. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)

and write (3 + 4) × 2 = 14, assuming addition comes first simply because it appears first.

一个经典错误是忽略运算的优先级。计算 3 + 4 × 2 时，学生常常从左往右算，写成 (3 + 4) × 2 = 14，误以为加法因为先出现就先算。

ltiplication has higher priority than addition, so 4 × 2 = 8 comes first, then 3 + 8 = 11.

正确的方法遵循 BIDMAS：括号、指数、乘除、加减。乘法优先级高于加法，所以先算 4 × 2 = 8，再算 3 + 8 = 11。

3 + 4 × 2 = 3 + 8 = 11

2, division must be done before subtraction, so the answer is 20 − 4 = 16, not 12 ÷ 2 = 6.

同样的原则适用于减法与除法。在 20 − 8 ÷ 2 中，必须先算除法，所以答案是 20 − 4 = 16，而不是 12 ÷ 2 = 6。

Common Mistake | 常见错误	Correction | 正确做法
3 + 4 × 2 = 7 × 2 = 14	3 + 4 × 2 = 3 + 8 = 11
20 − 8 ÷ 2 = 12 ÷ 2 = 6	20 − 8 ÷ 2 = 20 − 4 = 16
2 + 3² = 5² = 25	2 + 3² = 2 + 9 = 11

Always write out intermediate steps and underline the operation you are doing first. This slows you down slightly but prevents careless errors under exam pressure.

务必写出中间步骤，并把优先计算的部分下划线。这会稍微放慢速度，却能避免考试压力下的粗心错误。

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2. Negative Numbers: Signs, Powers and Subtraction | 负数：符号、乘方与减法

uals 9, but the index applies only to 3, so it is −(3²) = −9. To square negative three, you must write (−3)² = 9.

减去负数和对负数平方是两大失分来源。例如，许多学生认为 −3² 等于 9，但指数只作用于 3，所以是 −(3²) = −9。要计算负三的平方，必须写成 (−3)² = 9。

2 = 3, but the correct simplification is 5 + 2 = 7. Two minus signs together make a plus.

同样，减去一个负数也常被算错。5 − (−2) 经常被算成 5 − 2 = 3，但正确的简化是 5 + 2 = 7。两个负号相遇变成加号。

5 − (−2) = 5 + 2 = 7

positive × negative = negative. A common error is writing (−2) × (−3) = −6 instead of +6.

乘除时要记住符号法则：正 × 正 = 正，负 × 负 = 正，正 × 负 = 负。常见错误是把 (−2) × (−3) 写成 −6，而不是 +6。

Common Mistake | 常见错误	Correction | 正确做法
−4² = 16	−4² = −(4²) = −16; (−4)² = 16
−7 − 3 = −4	−7 − 3 = −10
5 + (−2) = 7	5 + (−2) = 5 − 2 = 3

A reliable habit is to put negative numbers in brackets whenever you substitute them into a formula or expression. This makes the sign explicit and stops you from accidentally ignoring it.

一个可靠的习惯是：把负数代入公式或表达式时，始终加上括号。这样符号一目了然，不会不小心忽略它。

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3. Adding and Subtracting Fractions | 分数的加减法

s incorrect unless the denominators are already the same. You must find a common denominator first.

最常见的分数错误是把分子分母直接相加：a/b + c/d = (a + c)/(b + d)。除非分母本来就相同，否则这是错误的。必须先通分。

t 2/5 is less than 1/2, yet the sum of 1/2 and 1/3 must be greater than 1/2, so an estimate immediately reveals the error.

例子：1/2 + 1/3。错误做法得 2/5。正确做法是化成六分之一：3/6 + 2/6 = 5/6。注意 2/5 小于 1/2，而 1/2 与 1/3 的和必然大于 1/2，所以估算一下就能立刻发现错误。

1/2 + 1/3 = 3/6 + 2/6 = 5/6

mbers, convert to improper fractions first, or handle the whole numbers separately and be careful when borrowing.

减法同理。3/4 − 1/3 不是 2/1。化成十二分之一：9/12 − 4/12 = 5/12。带分数则先化为假分数，或者把整数部分单独处理，并小心借位。

Common Mistake | 常见错误	Correction | 正确做法
1/2 + 1/3 = 2/5	1/2 + 1/3 = 3/6 + 2/6 = 5/6
3/4 − 1/3 = 2/1 = 2	3/4 − 1/3 = 9/12 − 4/12 = 5/12
2 1/2 + 1 1/3 = 3 2/5	2 1/2 + 1 1/3 = 2 3/6 + 1 2/6 = 3 5/6

After finding the answer, always check whether it is reasonable. If you add two positive fractions, the result must be larger than either one. If you subtract a smaller fraction from a larger one, the result must be positive and smaller than the original.

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得到答案后，一定要检查是否合理。两个正分数相加，结果必须比其中任何一个都大。用较大分数减去较小分数，结果必须是正数且小于原来的数。

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4. Multiplying and Dividing Fractions | 分数的乘除法

6/12 = 1/2. A common error is to cross-multiply, which is the method for solving equations, not for multiplying fractions.

分数乘法很直接：分子乘分子，分母乘分母。所以 2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2。常见错误是交叉相乘，那是解方程的方法，不是分数乘法的方法。

tudents often forget to flip the second fraction, or they flip the first fraction instead.

除以一个分数等于乘以它的倒数。所以 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。学生常忘记翻转第二个分数，或者错误地翻转了第一个分数。

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

b = a/b + 1. Cancelling is only valid when the numerator and denominator are entirely multiplied.

另一个严重错误是在加号两侧约分。在 (a + b)/b 中，不能约去 b 得到 a。应拆开分数：(a + b)/b = a/b + 1。只有当分子和分母整体都是乘积时，约分才成立。

Common Mistake | 常见错误	Correction | 正确做法
2/3 × 3/4 = 8/9 (cross-multiplying)	2/3 × 3/4 = 6/12 = 1/2
2/3 ÷ 4/5 = 8/15	2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6
(x + 2)/2 = x	(x + 2)/2 = x/2 + 1

ample, 4/9 × 3/8 can be simplified to 1/3 × 1/2 = 1/6 by cancelling 4 with 8 and 3 with 9.

相乘之前，先找公因数提前约分。这样数字更小，减少算术错误。例如 4/9 × 3/8 可以先约成 1/3 × 1/2 = 1/6，方法是把 4 与 8、3 与 9 分别约去。

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5. Solving Linear Equations | 解一元一次方程

x = 5. A common mistake is to subtract 5 instead of adding it, giving 3x = 5 and x = 5/3.

解 3x − 5 = 10 时，目标是按相反顺序做逆运算来孤立 x。两边加 5：3x = 15。再除以 3：x = 5。常见错误是减 5 而不是加 5，得到 3x = 5 和 x = 5/3。

3x − 5 = 10 → 3x = 15 → x = 5

rror is to multiply only the first term: 2x − 3 = 8, which gives the wrong answer x = 5.5.

带括号时先小心展开。2(x − 3) = 8 展开为 2x − 6 = 8，则 2x = 14，x = 7。常见错误是只乘第一项：2x − 3 = 8，得到错误答案 x = 5.5。

0, so 5x = 30 and x = 6. Forgetting to multiply the right-hand side is a very common slip.

如果出现分数，就把每一项都乘以分母的最小公倍数。对于 x/2 + x/3 = 5，乘以 6 得到 3x + 2x = 30，所以 5x = 30，x = 6。忘记乘右边是非常常见的失误。

Common Mistake | 常见错误	Correction | 正确做法
3x − 5 = 10 → 3x = 5	3x − 5 = 10 → 3x = 15 → x = 5
2(x − 3) = 8 → 2x − 3 = 8	2(x − 3) = 8 → 2x − 6 = 8 → x = 7
x/2 + x/3 = 5 → 3x + 2x = 5	x/2 + x/3 = 5 → 3x + 2x = 30 → x = 6

Always substitute your answer back into the original equation to check. This takes only a few seconds and catches most sign and arithmetic errors immediately.

务必把答案代回原方程检验。这只需要几秒钟，却能立刻发现大多数符号和算术错误。

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6. Expanding Brackets and Factorising | 展开括号与因式分解

A very common error is to write −6x − 12, forgetting that a negative times a negative is positive.

展开就是用括号外的项乘以括号内的每一项。对于 −3(2x − 4)，正确展开是 −6x + 12，因为 −3 × −4 = +12。非常常见的错误是写成 −6x − 12，忘记负负得正。

−3(2x − 4) = −6x + 12

s error in all of algebra is (x + 3)² = x² + 9. It should be (x + 3)(x + 3) = x² + 6x + 9.

双括号可用 FOIL 或表格法。(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6。代数中最著名的错误是 (x + 3)² = x² + 9。它应该是 (x + 3)(x + 3) = x² + 6x + 9。

+ 6 = (x + 2)(x + 3). For x² − 9, recognise the difference of two squares: (x − 3)(x + 3).

因式分解是逆过程。分解 x² + 5x + 6，找两个数相乘得 6、相加得 5：2 和 3。所以 x² + 5x + 6 = (x + 2)(x + 3)。对于 x² − 9，识别平方差：(x − 3)(x + 3)。

Common Mistake | 常见错误	Correction | 正确做法
−3(2x − 4) = −6x − 12	−3(2x − 4) = −6x + 12
(x + 3)² = x² + 9	(x + 3)² = x² + 6x + 9
x² − 9 = (x − 3)²	x² − 9 = (x − 3)(x + 3)

You can always check a factorisation by expanding it back. If the expansion does not match the original expression, the factorisation is wrong. This simple check should become automatic.

你总可以把因式分解展开回去检验。如果展开结果与原式不符，分解就是错的。这个简单检查应该成为自动习惯。

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7. Laws of Indices and Surds | 指数法则与根式

The index laws state that aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ

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