📚 Common Mistakes in Essential Maths 7H Homework Book | Essential Maths 7H 作业本易错点总结
The Essential Maths 7H Homework Book is designed to reinforce the key skills that Year 7 higher-level students. However, many pupils need before they move on to KS4. In practice, however, the same handful of mistakes reappear in homework after homework: operations done in the wrong order, sign slips with negative numbers, denominators added instead of made common, brackets expanded carelessly and answers left unrounded or unsimplified. Each of these costs marks that the pupil already knows how to earn. This article collects the most frequent errors found in the 7H book, explains exactly why each one is wrong, and gives a short, repeatable fix for every case.
Essential Maths 7H 作业本的作用，是帮助七年级高水平学生巩固进入 KS4 之前必须具备的核心技能。但在实际使用中，同一批错误会在一次次作业里反复出现：运算顺序做反、负数符号出错、分母直接相加而不通分、去括号时漏乘漏号、结果不约分或不符合四舍五入要求。这些错误丢掉的分数，其实是学生本来完全有能力拿到的。本文汇总 7H 作业本中最常见的错误，逐条说明错在哪里，并给出可以反复使用的纠正办法。
1. Order of Operations (BIDMAS) Mistakes | 运算顺序错误
n from left to right. The most common slip is doing addition before multiplication. In 3 + 4 × 2 many pupils write (3 + 4) × 2 = 14, but the multiplication must happen first, so 3 + 4 × 2 = 3 + 8 = 11. A second classic error appears when division and multiplication sit side by side: 24 ÷ 6 × 2 is often treated as 24 ÷ (6 × 2) = 2, yet the rule demands left-to-right evaluation, giving (24 ÷ 6) × 2 = 4 × 2 = 8. Indices are just as fragile: 2 + 3² = 2 + 9 = 11, not (2 + 3)² = 25. Underline the operation you must do first before you start calculating.
BIDMAS 规定了固定顺序：括号、指数、乘除（从左到右）、加减（从左到右）。最常见的错误是先算加法再算乘法。对于 3 + 4 × 2，许多学生会写成 (3 + 4) × 2 = 14，但乘法必须先算，所以 3 + 4 × 2 = 3 + 8 = 11。第二个典型错误出现在除法和乘法并列时：24 ÷ 6 × 2 常被当成 24 ÷ (6 × 2) = 2，但规则要求从左到右计算，即 (24 ÷ 6) × 2 = 4 × 2 = 8。指数同样容易出错：2 + 3² = 2 + 9 = 11，而不是 (2 + 3)² = 25。动笔计算之前，先用铅笔标出必须先算的那一步。
2. Negative Number Arithmetic Errors | 负数运算错误
ore steps to the left, so -5 - 3 = -8, not -2. A useful check is to ask whether the answer should be further from zero than the starting number; if you are subtracting a positive, it always should be. In multiplication and division, apply the sign rules: (-4) × (-3) = 12 because negative × negative = positive, while (-4) × 3 = -12 and 12 ÷ (-4) = -3. When two signs meet, replace them before calculating: 7 - (-2) = 7 + 2 = 9, and -6 + (-4) = -6 - 4 = -10.
负数容易在两个地方出错：数轴上的减法，以及乘除中的符号。从 -5 出发再减去 3，意味着继续向左移动三步，所以 -5 - 3 = -8，而不是 -2。一个实用的检查方法是问自己：结果是不是比起点离零更远？如果减的是一个正数，答案一定离零更远。乘除法要套用符号法则：(-4) × (-3) = 12，因为负负得正；而 (-4) × 3 = -12，12 ÷ (-4) = -3。当两个符号相邻时，先化简再计算：7 - (-2) = 7 + 2 = 9，-6 + (-4) = -6 - 4 = -10。
plies only to the 3. Read the brackets before you square anything, and remember that a negative base raised to an even power is positive while an odd power stays negative.
另一个独立陷阱是负数的乘方。(-3)² = 9，因为括号表示对整个 -3 平方；而 -3² = -(3²) = -9，因为指数只作用于 3。平方之前先看清有没有括号，并记住：负数的偶次幂为正，奇次幂仍为负。
3. Adding and Subtracting Fractions Mistakes | 分数加减法常见错误
ors and the denominators. The denominator describes the size of the pieces, so the pieces must be made the same size first. Since 6 is a common multiple of 2 and 3, write 1/2 = 3/6 and 1/3 = 2/6, then add only the numerators: 3/6 + 2/6 = 5/6. Subtraction follows exactly the same rule: 3/4 - 1/6 = 9/12 - 2/12 = 7/12. Always ask whether the answer can be simplified, and give the final answer in the form the question asks for.
分母不同的分数加法是最容易出错的地方。很多学生把 1/2 + 1/3 直接答成 2/5，方法是分子和分母分别相加。分母表示每一份的大小，所以必须先把两份的大小变成一样。因为 6 是 2 和 3 的公倍数，就把 1/2 写成 3/6、1/3 写成 2/6，然后只把分子相加：3/6 + 2/6 = 5/6。减法的规则完全相同：3/4 - 1/6 = 9/12 - 2/12 = 7/12。算完后一定要检查结果能否约分，并按题目要求的形式给出最终答案。
| Question / 题目 | Common wrong answer / 常见错解 | Correct answer / 正确答案 |
|---|---|---|
| 1/2 + 1/3 | 2/5 | 5/6 |
| 3/4 - 1/6 | 2/2 = 1 | 7/12 |
| 5/8 + 1/8 | 6/16 | 6/8 = 3/4 |
4. Mixed Numbers and Improper Fractions | 带分数与假分数互化错误
the whole number by the denominator and add the numerator: (1 × 3 + 2)/3 = 5/3. A very common error is to multiply the whole number by the numerator instead, giving 3/3 or 2/4. Going the other way, divide the numerator by the denominator: 5/3 = 1 remainder 2, so 5/3 = 1 2/3. The remainder always becomes the new numerator, and the denominator never changes.
带分数与假分数的互化会稳定地丢分。把 1 2/3 化成假分数时，要用整数部分乘分母再加上分子：(1 × 3 + 2)/3 = 5/3。非常常见的错误是把整数部分乘成了分子，结果写成 3/3 或 2/4。反方向转换时，用分子除以分母：5/3 = 1 余 2，所以 5/3 = 1 2/3。余数永远作为新的分子，分母始终不变。
rying to multiply the whole number and the fraction separately gives the wrong answer every time.
在做乘除法之前，也必须先把带分数化成假分数。计算 1 1/2 × 2/3 时，先写 1 1/2 = 3/2，再相乘：3/2 × 2/3 = 6/6 = 1。若把整数部分和分数部分分开去乘，答案一定出错。
5. Multiplying and Dividing Fractions | 分数乘除与约分错误
The error to avoid is hunting for a common denominator, which is only needed for addition and subtraction. Dividing is where most marks disappear: keep the first fraction, change ÷ to ×, and flip the second fraction. So 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. A frequent wrong answer is 1/8, produced by multiplying straight across instead of inverting.
乘法最简单：分子乘分子、分母乘分母，然后约分。所以 1/2 × 3/4 = 3/8，2/3 × 3/4 = 6/12 = 1/2。要避免的错误是去通分——只有加减法才需要通分。除法才是丢分的重灾区：第一个分数保持不变，把 ÷ 改成 ×，第二个分数上下颠倒。所以 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2。常见的错误答案 1/8，就是因为没有颠倒而是直接分子分母相乘。
Dividing by a fraction can be checked with whole numbers: 3 ÷ 1/4 asks how many quarters fit into 3, and the answer is 12, because there are four quarters in every whole. If your answer is smaller than 3, you have almost certainly divided when you should have multiplied by the reciprocal.
除以分数可以用整数来检验：3 ÷ 1/4 问的是 3 里面有多少个四分之一，答案是 12，因为每个整数里有四个四分之一。如果你的答案比 3 还小，那基本可以确定你该乘倒数却做成了除。
6. Decimals, Percentages and Rounding | 小数、百分数与四舍五入错误
s are decimals multiplied by 100, and the place value trips pupils up constantly. 0.4 = 40%, 0.04 = 4%, and 0.004 = 0.4%. To find 15% of a quantity, multiply by 0.15, or find 10% and 5% and add them. A specific trap is 0.5% nd is fifty percent, and the two answers differ by a factor of 100. When the question says "increase by 15%", multiply by 1.15 rather than adding 15 to the number; when it says "decrease by 15%", multiply by 0.85.
百分数就是小数乘以 100，而数位常常让学生栽跟头。0.4 = 40%，0.04 = 4%，0.004 = 0.4%。求某数的 15%，就乘 0.15，或者分别求出 10% 和 5% 再相加。一个专门的陷阱是 0.5% 和 0.5 的区别：前者是百分之一的二分之一，后者是百分之五十，两个答案相差 100 倍。题目说"增加 15%"时，要乘 1.15，而不是在原数上加 15；说"减少 15%"时，要乘 0.85。
Rounding errors are more subtle. Rounding 3.476 to two decimal places gives 3.48 because the third decimal digit is 6, which is 5 or more. Simply chopping digits off gives 3.47 and loses the mark. When rounding 2.5 to the nearest whole number, follow the convention your teacher uses and state it; when a question asks for an estimate, round every number first and then calculate, rather than calculating exactly and rounding at the end.
四舍五入的错误更隐蔽。把 3.476 保留两位小数得 3.48，因为第三位小数是 6，大于或等于 5。直接把后面的数位砍掉会得到 3.47，从而丢分。把 2.5 取整到最接近的整数时，要遵循老师规定的约定并写清楚；题目要求估算时，应先把每个数取整再计算，而不是精确算完再取整。
7. Simplifying Algebraic Expressions | 代数式化简与合并同类项错误
n the same way, 5y - 3y = 2y, and a + a = 2a, but a × a = a². A frequent confusion is 2x × 3x = 6x², which many pupils write as 6x; remember that x × x produces x², and only x + x produces 2x. Coefficients multiply, indices add when the bases are the same.
只有同类项才能合并。3a + 5b 中的两项不是同类项，所以该式保持不变；写成 8ab 是错的，因为 8ab 表示 8 × a × b。同样，5y - 3y = 2y，a + a = 2a，但 a × a = a²。一个常见混淆是 2x × 3x = 6x²，很多学生会写成 6x；要记住 x × x 得到 x²，只有 x + x 才得到 2x。系数相乘，底数相同时指数相加。
3 × -2)² = 36. Write the brackets in first, substitute the number, and only then evaluate.
代入求值是另一个薄弱环节。如果 a = -2，那么 3a² 表示 3 × (-2)² = 3 × 4 = 12，而不是 3 × (-2) × 2 = -12，也不是 (3 × -2)² = 36。先把括号写上，再代入数值，最后才计算。
8. Expanding Brackets | 展开括号错误
only the first term and writing 3x + 4. Negative multipliers cause even more damage: -2(x - 5) = -2x + 10, because -2 × -5 = +10; writing -2x - 10 loses both the sign and the mark. When two brackets are multiplied, use the grid method: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6, and never forget the middle terms.
括号内的每一项都必须与括号外的项相乘。所以 3(x + 4) = 3x + 12，2(4y - 5) = 8y - 10。最大的错误是只乘第一项，写成 3x + 4。负数乘数造成的错误更多：-2(x - 5) = -2x + 10，因为 -2 × -5 = +10；写成 -2x - 10 既错了符号也丢了分。两个括号相乘时，用网格法：(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6，千万不要漏掉中间的项。
Expanding is also reversible, and factorising back is the best check: if
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