Maths: G-k Maths Practice Animation – 6 Common Mistakes | G-k 数学练习动画-6 易错点总结

📚 Maths: G-k Maths Practice Animation – 6 Common Mistakes | G-k 数学练习动画-6 易错点总结

In the G-k series of maths practice animations, students often encounter recurring pitfalls that can be avoided with targeted awareness. This summary highlights six (and more) typical errors observed across topics from algebra to statistics. By studying these mistakes, you can strengthen your problem-solving accuracy and boost exam confidence.

在 G-k 数学练习动画系列中，学生常常遇到一些反复出现的易错点。本文总结了从代数到统计等多个主题的典型错误。通过分析这些误解，你可以提高解题的准确性并增强考试信心。

1. Misusing Negative Signs | 负号误用

Many learners forget that a negative sign in front of a number raised to a power applies only if the base is in parentheses. For example, −3² is often misinterpreted as (−3)², yielding 9 instead of the correct −9.

许多学生忘记，幂运算前的负号只有底数带括号时才适用。例如 −3² 常被误解为 (−3)²，得到 9，而正确答案是 −9。

−3² = −(3²) = −9, while (−3)² = 9

The same confusion appears when substituting values into formulas like −x² at x = −4. Without brackets, the expression becomes −(−4)² = −16, not +16.

同样的问题出现在代入公式时，如当 x = −4 计算 −x²，不带括号时变成 −(−4)² = −16，而不是 +16。

Common Mistake	Correct Approach
−5² = 25	−5² = −25 (−5)² = 25

2. Expanding Brackets Incorrectly | 括号展开错误

When multiplying a bracket by a negative term, it is easy to forget to change the sign of every term inside. For instance, −2(x − 3) should become −2x + 6, but many write −2x − 6.

当用负数乘以括号时，容易忘记改变括号内每一项的符号。例如 −2(x − 3) 应化为 −2x + 6，但很多学生写成 −2x − 6。

Double brackets are another source of error: (x + 2)(x − 5) must be expanded by multiplying each term in the first bracket by each term in the second, not just the first and last.

双括号是另一个错误来源：(x + 2)(x − 5) 必须将第一个括号中的每一项与第二个括号中的每一项相乘，而不能只乘首尾两项。

Typical Error	Correct Expansion
−3(2y − 1) = −6y − 3	−3(2y − 1) = −6y + 3
(x + 3)(x − 4) = x² − 12	(x + 3)(x − 4) = x² − x − 12

3. Confusions in Fraction Operations | 分数运算混乱

Adding and subtracting fractions often causes mistakes when students add denominators directly. They may compute 1/2 + 1/3 as 2/5, ignoring the need for a common denominator.

分数加减时，学生常直接将分母相加。他们可能会把 1/2 + 1/3 算成 2/5，而忽略了需要通分。

Multiplication and division also bring pitfalls: dividing by a fraction means multiplying by its reciprocal. Yet many treat 3/4 ÷ 2/5 as simply 3 ÷ 2 over 4 ÷ 5.

乘法和除法也有陷阱：除以一个分数等于乘以其倒数，但许多人将 3/4 ÷ 2/5 当作分子分母分别相除。

Addition: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, not 2/5.

加法：1/2 + 1/3 = 3/6 + 2/6 = 5/6，而不是 2/5。
Division: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

除法：3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8。

4. Misunderstanding Function Notation | 误解函数记号

Confusion between f(x) and ‘f times x’ is common. When asked to evaluate f(2) for f(x) = 3x + 1, some students write 3 × 2 + 1 = 7 but then treat f as a variable instead of the function’s output.

混淆 f(x) 和 ‘f 乘以 x’ 很常见。对于 f(x) = 3x + 1，要求计算 f(2) 时，有些学生虽然算出 3×2+1=7，却把 f 当作变量而非函数输出。

Composite functions can be mishandled: f(g(x)) means substitute g(x) into f, not multiply f(x) and g(x).

复合函数也可能出错：f(g(x)) 表示把 g(x) 代入 f，而不是将 f(x) 和 g(x) 相乘。

Misinterpretation	Correct Reading
fg(x) = f(x) × g(x)	fg(x) means f(g(x))

5. Losing Solutions When Solving Equations | 解方程时漏解

When solving quadratic equations by factorising, learners sometimes divide away a common factor instead of setting factors to zero. For x² = 5x, dividing by x gives x = 5, losing the solution x = 0.

用因式分解解二次方程时，学习者有时会约去公因式而不是令因式等于零。对于 x² = 5x，两边除以 x 得到 x = 5，丢失了 x = 0 这个解。

Similarly, taking the square root without considering both positive and negative branches leads to missing solutions. If x² = 9, then x = ±3, not just 3.

同样，开平方时不考虑正负两个分支会导致漏解。如果 x² = 9，那么 x = ±3，而不仅是 3。

x² = 5x → x² − 5x = 0 → x(x − 5) = 0 → x = 0 or x = 5

6. Scale and Unit Conversion Errors | 比例尺与单位转换错误

In map scale problems, students often confuse the ratio. A scale of 1:50 000 means 1 cm on the map represents 50 000 cm in reality, which equals 0.5 km. Misplacing the decimal point or forgetting to convert cm to km is a frequent mistake.

在地图比例尺问题中，学生常混淆比例关系。比例尺 1:50000 表示地图上 1 cm 代表实际 50000 cm，即 0.5 km。点错小数点或忘记把厘米转换为千米是常见错误。

Unit conversions within compound measures, such as converting km/h to m/s, also trip up candidates. The correct conversion factor is ÷3.6, but many multiply instead.

复合单位转换，比如将千米/小时转换为米/秒，也常让学生犯错。正确的转换系数是除以 3.6，但很多人反而相乘。

1 km/h = 1000 m / 3600 s = 1/3.6 m/s ≈ 0.278 m/s.

1 km/h = 1000 m / 3600 s = 1/3.6 m/s ≈ 0.278 m/s。

7. Misreading Statistical Diagrams | 统计图表误读

Interpreting histograms requires careful attention to frequency density, not just bar height. A common error is to compare the heights of bars with unequal class widths directly, as if they were bar charts.

解读直方图需要关注频率密度，而不仅仅是条形高度。一个常见错误是直接比较不等组距的条形高度，就像看条形图一样。

Cumulative frequency graphs are sometimes misread: students may take the x-value directly as the median without using the ‘half of total frequency’ rule on the y-axis.

累积频数图有时被误读：学生可能直接把 x 轴的值当作中位数，而没有用 y 轴上“总频数的一半”来定位。

Error	Correct Interpretation
Tallest bar = highest frequency	Frequency = frequency density × class width; tallest bar may not be most frequent.

8. Mixing Up Trigonometric Ratios | 三角函数混用

Students often mislabel opposite, adjacent, and hypotenuse in a right-angled triangle, especially when the triangle is rotated. This leads to applying the wrong ratio: using sin instead of cos, for instance.

学生们常常分不清直角三角形中的对边、邻边和斜边，尤其是当三角形旋转后。这会导致使用错误的比值，例如本该用 cos 却用了 sin。

Another mistake is forgetting to use the inverse trig functions when finding an angle. If sin θ = 0.5, θ = sin⁻¹(0.5) = 30°, but some write θ = 0.5°.

另一个错误是求角度时忘记用反三角函数。如果 sin θ = 0.5，θ = sin⁻¹(0.5) = 30°，但有人却写成 θ = 0.5°。

SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent

9. Ignoring the Domain | 忽略定义域

When working with functions or equations, domain restrictions are frequently overlooked. For example, in the function f(x) = √(x − 2), the expression under the square root must be ≥ 0, so x ≥ 2. Substituting x = 1 gives a mathematical error.

处理函数或方程时，定义域的限制常被忽略。例如对于 f(x) = √(x − 2)，根号下的表达式必须 ≥ 0，因此 x ≥ 2。代入 x = 1 会导致数学错误。

Rational functions also demand attention: in 1/(x − 3), x cannot be 3, or the denominator becomes zero. Students may give a solution set that includes the restricted value.

有理函数也需要留意：在 1/(x − 3) 中，x 不能等于 3，否则分母为零。学生可能会给出包含这个限制值的解集。

f(x) = √(x+4) → domain: x ≥ −4.

f(x) = √(x+4) → 定义域：x ≥ −4。
g(x) = 2/(x² − 1) → x ≠ ±1.

g(x) = 2/(x² − 1) → x ≠ ±1。

10. Calculator Mode Mistakes | 计算器模式错误

Using the wrong angle mode (degrees vs. radians) can distort all trigonometric answers. If a problem expects degrees but the calculator is in radians, sin 30 will not equal 0.5.

使用错误的角度模式（角度制与弧度制）会使所有三角函数答案出错。如果题目要求角度制而计算器处于弧度模式，sin 30 就不会等于 0.5。

Additionally, many students rely on calculators for simple fraction arithmetic without understanding the steps, leading to entry errors like 2/3+1 being interpreted as 2/(3+1).

此外，许多学生依赖计算器进行简单的分数运算却不理解步骤，导致输入错误，如 2/3+1 被理解为 2/(3+1)。

Always check the mode indicator and use parentheses to group denominators and numerators.

务必检查模式指示符，并用括号将分母和分子分组。

sin 30° in degree mode: 0.5; in radian mode: sin 30 ≈ −0.988 (wrong)

11. Algebraic Fraction Simplification Slips | 代数分式化简失误

Cancelling terms instead of factors is a classic error. In (x + 2)/(x + 3), students sometimes cross out the ‘x’ or the numbers, thinking the fraction simplifies, but the whole expression cannot be reduced further.

约项而不是约因式是一个经典错误。在 (x + 2)/(x + 3) 中，学生有时会划掉 ‘x’ 或数字，以为能化简，但实际上整个表达式无法再约分。

Similarly, when simplifying (x² − 4)/(x − 2), the numerator must be factorised to (x − 2)(x + 2) before cancellation, not by subtracting exponents.

同样，化简 (x² − 4)/(x − 2) 时，分子必须因式分解为 (x − 2)(x + 2) 后才能约分，而不是通过指数相减。

Wrong Simplification	Correct Process
(x² + 3x)/(x) = x² + 3	= x + 3 (after factoring x)
(x + 5)/(x + 2) = 5/2	Cannot be simplified; stays as is.

12. Rounding and Significant Figures Errors | 四舍五入与有效数字错误

Rounding too early in multi-step calculations can cause a significant drift in the final answer. Answers should be carried to several extra decimal places during working, then rounded at the end as specified.

在多步计算中过早四舍五入会导致最终答案明显偏离。解题过程中应多保留几位小数，最后按题目要求舍入。

Significant figure rules are also confused: 0.00456 to 2 significant figures is 0.0046, not 0.00 because leading zeros are not significant.

有效数字规则也常被混淆：0.00456 保留 2 位有效数字是 0.0046，而不是 0.00，因为前导零不算有效数字。

3.14159 to 3 s.f. = 3.14; but always use unrounded values for intermediate steps.

3.14159 保留 3 位有效数字 = 3.14；但中间步骤始终用未舍入的值。

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Maths: G-k Maths Practice Animation – 6 Common Mistakes | G-k 数学练习动画-6 易错点总结