📚 Year 11 OCR Maths: Common Mistakes and How to Fix Them | Year 11 OCR 数学：常见误区与纠正方法
Year 11 students preparing for OCR GCSE Maths often lose marks not because they don't know the material, but because of common, avoidable mistakes. This article identifies frequent errors across algebra, geometry, statistics and more, and shows you how to correct them with clear explanations and examples.
准备 OCR GCSE 数学的 Year 11 学生常常不是因为不懂知识而丢分，而是因为一些常见且可以避免的错误。本文找出代数、几何、统计等领域的常见错误，并通过清晰的解释和示例告诉你如何纠正它们。
1. Misunderstanding BIDMAS (Order of Operations) | 误解运算顺序
d division have equal priority and are performed left to right, just as addition and subtraction are equal. A typical error is evaluating 20 ÷ 5 × 2 as 20 ÷ 10 = 2, instead of correctly working (20 ÷ 5) × 2 = 8.
很多学生以为乘法总是在除法之前，加法总是在减法之前。实际上乘法和除法的优先级相同，从左到右依次运算，加法和减法也一样。一个典型的错误是把 20 ÷ 5 × 2 算成 20 ÷ 10 = 2，而正确的做法是 (20 ÷ 5) × 2 = 8。
indices first give 4² = 16, then multiplication gives 2 × 3 = 6, leaving 3 + 16 − 6 = 13.
为避免这个陷阱，可以给同级运算加上括号，或在脑子里分组。对于 3 + 4² − 2 × 3，先算指数 4² = 16，再算乘法 2 × 3 = 6，得到 3 + 16 − 6 = 13。
2. Area and Perimeter Confusion | 面积与周长的混淆
th, while the perimeter is (length + width) × 2. A common mistake is to calculate area using the perimeter formula or to forget to double the lengths.
学生经常混淆面积和周长的公式，尤其是在有复合形状的时候。对于长方形，面积是长 × 宽，而周长是 (长 + 宽) × 2。一个常见错误是用周长公式算面积，或者忘记把长宽加倍。
When dealing with composite shapes, break them into known rectangles, triangles or circles, find individual areas and add them. For perimeter, add the outer sides only – do not include internal dividing lines.
处理复合图形时，可以把它们分解成已知的长方形、三角形或圆形，分别求出面积再加起来。计算周长时只加外边的长度，不要包含内部的分隔线。
3. Sign Errors When Expanding Brackets | 去括号时的符号错误
When multiplying a negative term outside a bracket, every sign inside must change. Students often forget to distribute the minus sign to every term. For example, −2(x − 3) becomes −2x + 6, not −2x − 6. Missing the second sign change is a frequent slip.
当括号外是负号时，括号内每一项的符号都要改变。学生经常忘记把负号分配给每一项。例如 −2(x − 3) 应得 −2x + 6，而不是 −2x − 6。漏掉第二个符号变化是常见的失误。
a minus sign as multiplying by −1, so you can write − (3x − 4) = −1 × (3x − 4) = −3x + 4.
仔细展开来检查： −2 × x = −2x，−2 × (−3) = +6。始终把减号看作乘以 −1，因此 −(3x − 4) = −1 × (3x − 4) = −3x + 4。
4. Incorrectly Rearranging Equations | 错误地移项
3x = 15, then divide both sides by 3 to get x = 5. Skipping the balance step costs marks.
解方程 3x + 5 = 20 时，一个典型的错误是只在左边减去 5，忘了右边。正确的移项是两边同时减 5：3x = 15，然后两边除以 3 得 x = 5。省略“两边同做”的步骤会导致丢分。
)/m. Always perform the same operation on both sides, and write the division line clearly.
对于 y = mx + c 这样的公式，要表示成 x = 的形式需要细心：y − c = mx，然后 x = (y − c)/m。始终在等号两边做相同的运算，并清晰地写出分数线。
5. Misapplying Index Laws | 指数定律的误用
³⁺⁴ = 2⁷ = 128. Confusing addition of exponents with multiplication of bases is very common.
学生常常在应该将指数相加时去乘底数，例如把 2³ × 2⁴ 写成 4⁷，而正确答案是 2⁷。正确的法则是 aᵐ × aⁿ = aᵐ⁺ⁿ，因此 2³ × 2⁴ = 2⁷ = 128。把指数相加与底数相乘搞混是非常常见的。
ⁿ. So (x⁵)² = x¹⁰, not x⁷. When dividing powers, remember aᵐ ÷ aⁿ = aᵐ⁻ⁿ, so x⁸ ÷ x² = x⁶.
另一个错误出现在 (aᵐ)ⁿ 中：它等于 aᵐⁿ，而不是 aᵐ⁺ⁿ。所以 (x⁵)² = x¹⁰，而不是 x⁷。当幂相除时，记住 aᵐ ÷ aⁿ = aᵐ⁻ⁿ，因此 x⁸ ÷ x² = x⁶。
6. Trigonometry: SOHCAHTOA Mistakes | 三角学：SOHCAHTOA 错误
In right‑angled trigonometry, picking the wrong pair of sides is a frequent mistake. Students label the opposite and adjacent sides relative to the given angle, but then substitute into the wrong ratio. For example, using sin when tan is needed because they mix up opposite/adjacent.
在直角三角形中，选错边对是常见错误。学生根据给定角标出对边和邻边，却代入了错误的比值。例如需要正切时却用了正弦，因为混淆了对边与邻边。
te/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. After labelling, double‑check which two sides you know and which ratio connects them.
始终写下 SOH CAH TOA：Sin = 对边/斜边，Cos = 邻边/斜边，Tan = 对边/邻边。标好边后，再检查一次已知哪两条边，以及哪一组比值把它们联系起来。
sin θ = O/H, cos θ = A/H, tan θ = O/A
7. Inequalities: When to Flip the Sign | 不等式：何时需要变号
A dangerous mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. For example, solving −2x > 6, dividing by −2 gives x < −3, not x > −3. Many students keep the direction unchanged and lose the mark.
在乘以或除以一个负数时，忘记把不等号方向反过来是一个危险的错误。例如解 −2x > 6，除以 −2 得到 x < −3，而不是 x > −3。很多学生保持方向不变而丢掉分数。
ties on a number line correctly: use open circles for < and >, closed circles for ≤ and ≥.
这条规则只适用于乘或除以负数；加减法不用改变方向。此外，记得在数轴上正确表示不等式：< 和 > 用空心圆，≤ 和 ≥ 用实心圆。
8. Probability: Adding Instead of Multiplying | 概率：加法与乘法的混淆
example, the chance of rolling a 5 on a fair die and then a 5 again is 1/6 × 1/6 = 1/36, not 1/6 + 1/6 = 1/3.
计算两个独立事件同时发生的概率时应相乘。常见的失误是做加法。例如，掷一个均匀骰子得到 5，再掷一次又得到 5 的概率是 1/6 × 1/6 = 1/36，而不是 1/6 + 1/6 = 1/3。
on is used for mutually exclusive events where either one outcome or the other occurs. For 'rolling a 2 or a 3' on one die, P(2 or 3) = 1/6 + 1/6 = 2/6. Always identify whether it is an 'AND' situation or an 'OR' situation.
加法适用于互斥事件，即其中一个结果或另一个结果发生的情况。掷一次骰子，得到 2 或 3 的概率 P(2 或 3) = 1/6 + 1/6 = 2/6。始终要判断这是“且”还是“或”的情形。
9. Percentage Increase/Decrease Errors | 百分比增减的错误
Students often use the wrong multiplier for percentage change. For a 15% increase, the correct multiplier is 1.15, not 0.15. For a 20% decrease, the multiplier is 0.80, not 0.20. Using just the percentage as a decimal gives only the change, not the final amount.
学生常常用错百分比变化的乘数。增加 15%，正确的乘数是 1.15，而不是 0.15。减少 20%，乘数是 0.80，而不是 0.20。只用百分数当小数算出的只是变化量，不是最终值。
as compound interest, raise the multiplier to the power of the number of years. After a 4% increase for 3 years, the overall multiplier is 1.04³, not 1.04 × 3.
计算重复百分比变化，例如复利时，要把乘数乘上几次方。每年增长 4%，三年后总乘数是 1.04³，而不是 1.04 × 3。
10. Misreading Cumulative Frequency and Histograms | 误读累积频率图和直方图
In cumulative frequency graphs, students sometimes read the frequency directly from the vertical axis instead of finding the difference between two points. The median is at half the total frequency; the interquartile range is the difference between the upper and lower quartiles found on the curve.
在累积频率图中，学生有时直接从纵轴上读取频数，而忘了应计算两点之间的差值。中位数在总频数的一半位置；四分位距是由曲线上找到的上四分位数与下四分位数之差。
quency density), not the height. A common error is to assume the height alone represents frequency. Check the frequency density scale and work out area.
在直方图中，频数是由条块的面积（组距 × 频率密度）表示的，而不是高度。常见的错误是以为高度就代表频数。务必查看频率密度刻度并计算面积。
11. Ratio and Proportion Pitfalls | 比例陷阱
a given ratio, students often forget to find the total number of parts first. For dividing £80 in the ratio 3 : 5, the total parts are 3 + 5 = 8, so one part is £10. The two shares are 3 × £10 = £30 and 5 × £10 = £50. A mistake is to split £80 into £30 and £50 without checking that they sum to £80.
按给定比例分配时，学生经常忘记先求出总份数。把 £80 按 3 : 5 分配，总份数是 3 + 5 = 8，所以一份为 £10。两份分别是 3 × £10 = £30 和 5 × £10 = £50。常见的错误是直接拆成 £30 和 £50 而不检查总和。
k/x, gives a curved graph. Students often mix up the two by using the wrong formula for k.
在正比例中，y = kx，图像是一条过原点的直线。反比例，y = k/x，图像是一条曲线。学生经常混淆这两种关系，用错求 k 的公式。
12. Algebraic Fractions Simplification Errors | 代数分式化简错误
ole numerator. Correct cancellation is possible only with factors: (3x)/3 = x, or (x(x + 2))/(x) = x + 2, provided x ≠ 0.
一个危险的化简是约掉项而不是因式。例如 (x + 3)/3 不能简化成 x，因为 3 不是整个分子的因式。正确的约分只能针对因式： (3x)/3 = x，或 (x(x + 2))/(x) = x + 2，前提是 x ≠ 0。
ly, e.g. a/b + c/d = (a + c)/(b + d), which is wrong. Instead, a/b + c/d = (ad + bc)/(bd).
对代数分式进行加减时，先找到公分母。学生常常直接把分子和分母分别相加，如 a/b + c/d = (a + c)/(b + d)，这是错误的。正确的做法是 a/b + c/d = (ad + bc)/(bd)。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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