📚 Common Mistakes in Essential Maths Book 9i Compressed | KS3 数学易错点总结（Essential Maths Book 9i 压缩版）
This article highlights the most frequent errors students make when working through the Essential Maths Book 9i Compressed, a KS3 curriculum resource. By understanding these pitfalls, you can build a stronger foundation in mathematics and avoid losing marks in assessments.
本文总结了学生在使用 KS3 课程资源《Essential Maths Book 9i 压缩版》时最常犯的错误。理解这些易错点能帮助你夯实数学基础，避免在考试中无谓失分。
1. Operations with Negative Numbers | 负数的运算
Many pupils forget that subtracting a negative number is equivalent to addition. For example, 3 − (−5) becomes 3 + 5 = 8, not −2.
许多学生忘记减去一个负数等价于加上它的相反数。例如，3 − (−5) 等于 3 + 5 = 8，而不是 −2。
A common slip occurs when multiplying or dividing: (−4) × (−6) = 24, but (−4) × 6 = −24. Always check the sign rules.
在乘除运算中也常出现疏忽：(−4) × (−6) = 24，但 (−4) × 6 = −24。务必牢记符号法则。
When adding a string of negatives, such as −2 − 7 + 4, work step‑by‑step from left to right: (−2 − 7) = −9, then −9 + 4 = −5.
遇到含多个负数的加减运算，如 −2 − 7 + 4，应按从左到右分步计算：(−2 − 7) = −9，然后 −9 + 4 = −5。
2. Fraction Arithmetic and Simplification | 分数的运算与化简
Students often add fractions incorrectly by adding numerators and denominators directly: ½ + ⅓ ≠ ⅖. Instead, find a common denominator (6) to get 3/6 + 2/6 = 5/6.
学生常错误地将分子分母直接相加：½ + ⅓ ≠ ⅖。正确做法是先通分（公分母6），得到 3/6 + 2/6 = 5/6。
When multiplying fractions, remember to multiply numerators together and denominators together: ⅔ × ⅘ = 8/15, then simplify if possible (it is already in simplest form).
乘法时，分子相乘、分母相乘：⅔ × ⅘ = 8/15，能约分则约分（此处已最简）。
Dividing by a fraction means multiplying by its reciprocal. For ⅞ ÷ ¾, rewrite as ⅞ × 4/3 = 28/24 = 7/6 = 1⅙. Forgetting to flip the second fraction is a classic error.
除以一个分数等于乘以它的倒数。如 ⅞ ÷ ¾，应改为 ⅞ × 4/3 = 28/24 = 7/6 = 1⅙。忘记翻转第二个分数是经典错误。
3. Algebraic Expansion and Factorisation | 代数展开与因式分解
When expanding brackets like 3(x + 4), don’t just write 3x + 4. Multiply every term inside: 3 × x and 3 × 4 give 3x + 12.
展开括号如 3(x + 4) 时，不能只写成 3x + 4。应将括号内每一项都乘以 3：3 × x 和 3 × 4，得到 3x + 12。
With double brackets, e.g. (x + 2)(x + 5), use FOIL: First (x × x = x²), Outer (x × 5 = 5x), Inner (2 × x = 2x), Last (2 × 5 = 10), then collect like terms: x² + 7x + 10.
遇到双重括号如 (x + 2)(x + 5)，运用 FOIL 法则：首项 x²，外项 5x，内项 2x，末项 10，再合并同类项得 x² + 7x + 10。
In factorisation, always look for the highest common factor first. For 4x² + 8x, both terms share 4x, so factor to 4x(x + 2). A common mistake is to stop at 2x(2x + 4), leaving a further common factor.
因式分解时务必先提取最大公因数。如 4x² + 8x，两项的公因数是 4x，分解为 4x(x + 2)。常见错误是止于 2x(2x + 4)，留下还可提取的公因数。
4. Solving Linear Equations | 解一元一次方程
When solving 2x + 3 = 11, the aim is to isolate x. Subtract 3 from both sides to get 2x = 8, then divide by 2. Many students mistakenly divide by 2 before subtracting, leading to x + 1.5 = 5.5 and confusion.
解方程 2x + 3 = 11 时，目标是分离出 x。先两边减 3 得到 2x = 8，再除以 2。许多学生错误地先除以 2，导致 x + 1.5 = 5.5，徒增困扰。
Equations with unknowns on both sides, e.g. 5x − 7 = 3x + 9, require collecting like terms. Bring 3x to the left as −3x: 2x − 7 = 9, then solve. Forgetting to change signs when moving terms is a frequent slip.
含两侧未知数的方程如 5x − 7 = 3x + 9，需要移项合并。将 3x 移至左边变为 −3x，得 2x − 7 = 9 再求解。移项时忘变号是常见失误。
Always check your solution by substituting back into the original equation. For x = 8 in the above, 5(8) − 7 = 33 and 3(8) + 9 = 33 ✓.
养成将解代入原方程检验的习惯。如上例 x = 8，5(8) − 7 = 33，3(8) + 9 = 33 ✓。
5. Percentages, Increase and Decrease | 百分数增减
To increase £80 by 15%, don’t just find 15% of £80 and add. A multiplier method is safer: 100% + 15% = 115% = 1.15, so new amount = 80 × 1.15 = £92.
将 80 英镑增加 15%，不要仅算出 15% 再相加。用乘数法更安全：100% + 15% = 115% = 1.15，新金额 = 80 × 1.15 = 92 英镑。
For repeated percentage changes, multiply successively. A 20% decrease followed by a 20% increase does not return to the original. A £50 item reduced by 20% becomes £40, then increased by 20% becomes 40 × 1.2 = £48.
连续百分比变化需逐次相乘。先减少 20% 再增加 20% 不能 回到原值。50 英镑的商品减价 20% 变为 40 英镑，再提价 20% 得 40 × 1.2 = 48 英镑。
When calculating the original price after a discount, divide by the multiplier. If a £54 price includes 20% VAT, the original (ex‑VAT) price is 54 ÷ 1.2 = £45, not 54 × 0.8.
计算打折前的原价时，应用除法。若 54 英镑为含 20% 增值税的价格，不含税原价为 54 ÷ 1.2 = 45 英镑，而非 54 × 0.8。
6. Ratio and Proportion Problems | 比与比例问题
When sharing £60 in the ratio 3 : 5, first find the total number of parts (3 + 5 = 8). Each part is £60 ÷ 8 = £7.50, so the shares are 3 × £7.50 = £22.50 and 5 × £7.50 = £37.50.
将 60 英镑按 3 : 5 分配时，先求总份数 (3 + 5 = 8)。每份为 £60 ÷ 8 = £7.50，因此两份额分别为 3 × £7.50 = £22.50 和 5 × £7.50 = £37.50。
Do not confuse ratio with fraction. A ratio of 1 : 4 means the first part is 1/5 of the whole, not 1/4. In a 1 : 4 mix of cordial to water, the total mixture has 5 parts.
切勿混淆比与分数。比 1 : 4 表示第一部分占总体的 1/5，而非 1/4。按 1 : 4 调配浓浆与水时，混合物共有 5 份。
In scale maps, if the scale is 1 : 50 000, 3 cm on the map represents 3 × 50 000 cm = 150 000 cm = 1500 m = 1.5 km. Convert units consistently.
地图比例尺 1 : 50 000 意味着图上 3 cm 代表 3 × 50 000 cm = 150 000 cm = 1500 m = 1.5 km。单位换算需保持一致。
7. Angle Reasoning in Geometry | 几何中的角度推理
Angles on a straight line add up to 180°. If one angle is given as 135°, the adjacent angle is 180° − 135° = 45°, not 135° itself. Label your diagram to avoid confusion.
平角等于 180°。若已知一角为 135°，其邻角为 180° − 135° = 45°，而不是 135°。在图上标注可避免混淆。
Vertically opposite angles are equal, but many students incorrectly assume that all angles around intersecting lines look equal. Only the ones directly opposite each other are equal, not the adjacent pair.
对顶角相等，但很多学生错误地认为相交线周围所有角都相等。只有直接对顶的角相等，相邻角并不相等。
In parallel lines, alternate angles are equal (Z‑shape), corresponding angles are equal (F‑shape), and co‑interior angles add to 180° (C‑shape). Mixing up these rules is a top mistake.
在平行线中，内错角相等（Z 形），同位角相等（F 形），同旁内角互补（C 形，和为 180°）。混淆这些规则是首要错误。
8. Area and Volume Calculations | 面积与体积的计算
For the area of a triangle, the formula is ½ × base × vertical height. If the triangle is slanted, the height is the perpendicular distance from the base to the opposite vertex, not the slant side.
三角形面积公式为 ½ × 底 × 高。若三角形倾斜，高是指从底边到对顶点的垂直距离，而非斜边长度。
When finding the area of a compound shape, split it into rectangles and triangles, calculate each area, then add or subtract. Forgetting to divide by 2 for triangular parts is common.
计算组合图形面积时，先分割成矩形和三角形，分别求面积再相加或相减。常忘记三角形部分要除以 2。
Volume of a prism = area of cross‑section × length. For a cylinder, that’s πr²h. Students often use the diameter instead of the radius in πr², giving a value 4 times too large.
柱体体积 = 横截面积 × 长。对于圆柱，即 πr²h。学生常误用直径代替半径代入 πr²，导致结果大了 4 倍。
9. Interpreting Statistical Graphs | 统计图表的解读
When reading a bar chart with grouped data, the class width may be unequal. The frequency is given by the height of the bar, but if widths differ, frequency density must be used (frequency ÷ class width).
阅读分组数据条形图时，组距可能不等。频数由柱高表示，但当组距不同时，应使用频数密度（频数 ÷ 组距）。
Pie charts: the angle for a sector = (frequency ÷ total frequency) × 360°. A common error is to divide by 100 instead of 360 or to forget to multiply by 360 after finding the fraction.
饼图：扇形角度 = (频数 ÷ 总频数) × 360°。常见错误是除以 100 而非 360，或求出占比后忘记乘以 360°。
In scatter graphs, correlation does not imply causation. A line of best fit should be drawn roughly through the middle of the points, splitting them evenly on both sides; avoid joining dot‑to‑dot.
散点图中，相关性不代表因果关系。最佳拟合线应大致穿过点群中央，使两侧点数均匀；避免逐点连线。
10. Coordinates and Straight Line Graphs | 坐标与直线图像
The gradient of a line is rise ÷ run. From (1, 2) to (5, 10), rise = 8, run = 4, gradient = 2. A negative gradient slopes downwards from left to right, such as −1/2.
直线的斜率 = 纵向变化 ÷ 横向变化。从 (1, 2) 到 (5, 10)，纵向变化 8，横向 4，斜率为 2。负斜率从左向右下降，例如 −½。
When plotting y = 3x − 1, use a table of values: x = −1, 0, 1, 2 to find corresponding y. Then plot the points and draw a straight line through them. Join with a ruler – freehand curves are a mark‑loser.
画 y = 3x − 1 的图像时，先用表格取值：x = −1, 0, 1, 2 求出对应 y。描点后用直尺连成直线。徒手画弯线会导致扣分。
The y‑intercept is the constant term. In y = 2x + 5, the line crosses the y‑axis at (0, 5). Parallel lines have the same gradient, e.g. y = 2x + 5 and y = 2x − 3 are parallel.
直线与 y 轴的截距即常数项。y = 2x + 5 中直线交 y 轴于 (0, 5)。平行线斜率相同，如 y = 2x + 5 与 y = 2x − 3 平行。
11. BIDMAS / BODMAS Order of Operations | 运算顺序 BIDMAS / BODMAS
The order is Brackets, Indices (powers), Division & Multiplication (left to right), Addition & Subtraction (left to right). For 3 + 5 × 2, multiplication comes first: 5 × 2 = 10, then 3 + 10 = 13, not 8 × 2.
顺序为：括号、指数、除与乘（从左到右）、加与减（从左到右）。计算 3 + 5 × 2 时，先乘：5 × 2 = 10，再 3 + 10 = 13，而不是 8 × 2。
When powers and roots appear, treat them after brackets. √(25) + 3² = 5 + 9 = 14. Do not add before taking the root: √(25 + 9) is wrong here.
出现乘方和根号时，在括号之后处理。√25 + 3² = 5 + 9 = 14。不应先相加再开方，此处 √(25 + 9) 是错误的。
If only addition and subtraction remain, work left to right: 10 − 3 + 2 = 7 + 2 = 9, not 10 − 5 = 5.
若只剩加减，从左到右计算：10 − 3 + 2 = 7 + 2 = 9，而非 10 − 5 = 5。
12. Rounding, Estimation and Standard Form | 四舍五入、估算与标准形式
When rounding 4.376 to 2 decimal places, look at the third decimal digit (6), so round up: 4.38. For 1 decimal place, look at the hundredths digit: 4.376 → 4.4.
将 4.376 保留两位小数，看第三位小数 6，因此向上舍入得 4.38。保留一位小数则看百分位：4.376 → 4.4。
Estimating before calculating helps spot mistakes: 298 × 5.1 ≈ 300 × 5 = 1500 (actual 1519.8, close). Round each number to one significant figure first.
先估算后计算有助于发现错误：298 × 5.1 ≈ 300 × 5 = 1500（实际为 1519.8，很接近）。先将每个数保留一位有效数字。
Standard form is A × 10ⁿ where 1 ≤ A < 10. 45 600 is 4.56 × 10⁴. A common error is writing 45.6 × 10³, which is not standard form.
标准形式为 A × 10ⁿ，其中 1 ≤ A < 10。45 600 写为 4.56 × 10⁴。常见错误是写成 45.6 × 10³，这并非标准形式。
When multiplying standard form numbers, multiply the A numbers and add the exponents: (3 × 10⁵) × (2 × 10³) = 6 × 10⁸.
标准形式相乘时，A 部分相乘、指数相加：(3 × 10⁵) × (2 × 10³) = 6 × 10⁸。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导