Math Practice Animation 5: Common Pitfalls Summary | 数学练习动画5：常见易错点总结

📚 Math Practice Animation 5: Common Pitfalls Summary | 数学练习动画5：常见易错点总结

This article highlights the ten most frequent mistakes students make when working through the Math Practice Animation 5 exercises. Understanding these pitfalls will help you avoid losing marks in exams and build stronger mathematical foundations.

本文总结了学生在完成数学练习动画5习题时最常见的十个错误。理解这些易错点有助于避免考试丢分，并巩固数学基础。

1. Misusing Order of Operations | 误用运算顺序

When faced with a calculation like 8 + 2 × 3, many learners add first, obtaining 10 × 3 = 30. According to the BIDMAS/BODMAS rule, multiplication takes precedence over addition, so the correct step is 2 × 3 = 6, then 8 + 6 = 14.

面对像 8 + 2 × 3 这样的计算时，许多学习者会先相加，得到 10 × 3 = 30。根据 BIDMAS/BODMAS 运算顺序规则，乘除优先于加减，正确的步骤是 2 × 3 = 6，再加 8 得到 14。

Another classic example is 5² × 3. Some misinterpret it as (5 × 3)² = 225, because they add brackets incorrectly. The correct order evaluates the exponent first: 5² = 25, then 25 × 3 = 75.

另一个典型例子是 5² × 3。有人错误地将其理解为 (5 × 3)² = 225，因为他们随意添加了括号。正确的顺序是先计算指数：5² = 25，再乘以 3 得 75。

❌ (8 + 2) × 3 = 30 ✅ 8 + 2 × 3 = 14

2. Sign Errors When Expanding Brackets | 去括号时的符号错误

When expanding –2(x – 3), a common mistake is to write –2x – 6, forgetting that the negative sign must be multiplied with –3 to give +6. The correct expansion is –2x + 6.

在展开 –2(x – 3) 时，常见的错误是写成 –2x – 6，忘记了负号与 –3 相乘应得 +6。正确的展开是 –2x + 6。

In the product (x – 4)(2x + 1), many students only multiply the first and last terms: x·2x = 2x², –4·1 = –4, and incorrectly write 2x² – 4. The full expansion must include all four products: x·2x = 2x², x·1 = x, –4·2x = –8x, –4·1 = –4, which sum to 2x² – 7x – 4.

在计算乘积 (x – 4)(2x + 1) 时，许多学生只乘首项和尾项：x·2x = 2x²，–4·1 = –4，然后错误地写成 2x² – 4。完整的展开必须包含全部四项乘积：x·2x = 2x²，x·1 = x，–4·2x = –8x，–4·1 = –4，合并得 2x² – 7x – 4。

❌ (x – 4)(2x + 1) = 2x² – 4 ✅ = 2x² – 7x – 4

3. Incorrect Transposition in Linear Equations | 线性方程移项错误

When solving 3x + 5 = 20, a typical error is to divide by 3 first, yielding x + 5 = 6.67, or to subtract 5 but then forget to divide. The reliable two-step process: subtract 5 from both sides to get 3x = 15, then divide by 3 to obtain x = 5.

解方程 3x + 5 = 20 时，一个典型错误是先除以 3，得到 x + 5 = 6.67，或者减去 5 后忘记再除以 3。可靠的两步法是：两边同减 5 得 3x = 15，再除以 3 得 x = 5。

For equations with variables on both sides such as 4x – 3 = 2x + 7, students sometimes collect terms incorrectly. After subtracting 2x they obtain 2x – 3 = 7, but then they add 3 to 7 instead of adding 3 to both sides, writing 2x = 10 incorrectly? Actually 2x = 10 is correct if they added 3 to both sides. The mistake is when they move –3 and change +3 but keep the sign wrong: e.g. 2x = 7 – 3 = 4. Always perform the same operation on both sides: add 3 to give 2x = 10, so x = 5.

对于两边都含变量的方程，如 4x – 3 = 2x + 7，学生有时在移项时出错。他们减去 2x 后得到 2x – 3 = 7，接着可能错误地写成 2x = 7 – 3 = 4。正确的做法是始终在两边执行相同运算：两边加 3 得 2x = 10，故 x = 5。

❌ 4x – 3 = 2x + 7 → 2x = 4 ✅ → 2x = 10, x = 5

4. Factorising Quadratics: Missing Factors | 二次因式分解遗漏因子

For x² – x – 6, the correct factors need two numbers that multiply to –6 and add to –1: –3 and 2. A frequent error is to write (x – 3)(x – 2), which expands to x² – 5x + 6. The right factorisation is (x – 3)(x + 2).

对于 x² – x – 6，正确的因式需要两个乘积为 –6、和为 –1 的数，即 –3 和 2。一个常见错误是写成 (x – 3)(x – 2)，它展开后得到 x² – 5x + 6。正确的因式分解是 (x – 3)(x + 2)。

When the coefficient of x² is not 1, say 3x² + 10x + 8, learners often guess (3x + 4)(x + 2) and stop checking. That combination is actually correct (3x·x + 3x·2 + 4·x + 8 = 3x² + 6x + 4x + 8 = 3x² + 10x + 8), but a more typical slip is to mismanage signs or to miss the cross terms. Always verify by expanding.

当 x² 的系数不为 1 时，例如 3x² + 10x + 8，学生常猜 (3x + 4)(x + 2) 后就不再验证。这个组合碰巧是对的，但更常见的滑漏是弄错符号或漏掉交叉项。一定要通过展开来验证。

❌ x² – x – 6 = (x – 3)(x – 2) ✅ = (x – 3)(x + 2)

5. Misapplying Index Laws | 误用指数运算法则

Multiplying a² × a³ often leads to the error a⁶ instead of adding the indices to

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