KS3 Maths: Essential Maths 7 Higher – Common Mistakes Summary | KS3 数学：Essential Maths 7 Higher 易错点总结

📚 KS3 Maths: Essential Maths 7 Higher – Common Mistakes Summary | KS3 数学：Essential Maths 7 Higher 易错点总结

In Essential Maths 7 Higher, students build on foundational skills, but certain pitfalls repeatedly trip them up. This article collects the most common errors from this textbook and shows how to avoid them, covering order of operations, negative numbers, fractions, algebra, and more. By understanding these typical mistakes, you can sharpen your problem-solving and boost your confidence.

在《Essential Maths 7 Higher》中，学生在基础技能上不断提升，但某些陷阱会反复绊倒他们。本文总结了该教材中最常见的错误，并展示如何避免，涵盖运算顺序、负数、分数、代数等。通过理解这些典型错误，你可以提高解题能力并增强信心。

1. Order of Operations (BIDMAS/BODMAS) | 运算顺序（先乘除后加减）

Many students forget that multiplication and division have the same priority and are performed left to right, same for addition and subtraction.

许多学生忘记乘法和除法优先级相同，从左到右计算，加法和减法也相同。

A classic mistake: 4 + 2 × 3 is wrongly calculated as 6 × 3 = 18. The correct approach: multiply first, 2 × 3 = 6, then add 4, giving 10.

经典错误：4 + 2 × 3 被错误地计算为 6 × 3 = 18。正确做法：先乘，2 × 3 = 6，再加 4，得 10。

When brackets are present, always resolve inside them first. Example: (7 – 2)² + 1. Mistake: 7 – 2² + 1 = 7 – 4 + 1 = 4. Correct: (5)² + 1 = 25 + 1 = 26.

当有括号时，总是先计算括号内的值。例如：(7 – 2)² + 1。错误：7 – 2² + 1 = 7 – 4 + 1 = 4。正确：(5)² + 1 = 25 + 1 = 26。

Even with multiple brackets, work from the innermost outwards. Always write intermediate steps to avoid skipping operations.

即使有多层括号，也要从内向外计算。始终写出中间步骤，避免遗漏运算。

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2. Negative Numbers | 负数

Adding and subtracting negatives causes confusion: subtracting a negative is the same as adding a positive.

正负数加减法容易混淆：减去一个负数等于加上它的相反数（正数）。

Error: –5 – (–3) = –5 – 3 = –8. Correct: –5 – (–3) = –5 + 3 = –2.

错误：–5 – (–3) = –5 – 3 = –8。正确：–5 – (–3) = –5 + 3 = –2。

When multiplying or dividing: two negatives make a positive, one negative stays negative. Common slip: –4 × –2 = –8. Right answer: +8.

乘除法：负负得正，一正一负得负。常见失误：–4 × –2 = –8。正确答案：+8。

Using number lines can help visualise movement: start at first number, move left for subtraction or negative addition, right for addition or negative subtraction.

使用数轴有助于形象化移动：从第一个数开始，减或加负数时向左移动，加正数或减负数时向右移动。

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3. Fractions, Decimals and Percentages | 分数、小数和百分比

Converting between these forms often goes wrong when denominators are not simplified or division is performed incorrectly.

在这些形式之间转换时，如果分母未化简或除法计算错误，常常会出错。

Example: Write 3/8 as a decimal. Incorrect: 0.375? Some will misplace decimal point. Correct division: 3 ÷ 8 = 0.375. But care: 3/8 = 0.375 exactly.

例如：将 3/8 写成小数。错误：小数点位置错。正确除法：3 ÷ 8 = 0.375。

When adding fractions, always find a common denominator. Mistake: 1/2 + 1/3 = 2/5. Correct: 3/6 + 2/6 = 5/6.

分数加法时，一定要找到公分母。错误：1/2 + 1/3 = 2/5。正确：3/6 + 2/6 = 5/6。

Percentages: ‘of’ means multiply. 10% of 250: Some do 10 × 250 = 2500. Correct: 10/100 × 250 = 25.

百分比：”的”表示乘。250 的 10%：有人算 10 × 250 = 2500。正确：10/100 × 250 = 25。

In reverse, to find what percentage 20 is of 80: 20/80 × 100 = 25%.

反过来，求 20 是 80 的百分之几：20/80 × 100 = 25%。

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4. Algebraic Expressions | 代数表达式

Collecting like terms: only terms with exactly the same variable and power can be combined. Students often add unlike terms: 3x + 2y = 5xy (false).

合并同类项：只有变量和指数完全相同的项才能合并。学生常把不同类项相加：3x + 2y = 5xy（错）。

When simplifying 2x + 3x, the correct answer is 5x, not 5x². Remember the variable does not change exponent.

化简 2x + 3x 时，正确答案是 5x，而不是 5x²。记住变量指数不变。

Substitution errors: Evaluate 2a + 3b when a = –1, b = 2. Mistake: 2 – 1 + 3 × 2 = 1 + 6 = 7? Actually: 2(–1) + 3(2) = –2 + 6 = 4.

代入错误：当 a = –1, b = 2 时，求 2a + 3b 的值。错误：2 – 1 + 3 × 2 = 1 + 6 = 7？正确：2(–1) + 3(2) = –2 + 6 = 4。

Expanding brackets: 3(x + 2) = 3x + 6, not 3x + 2. Always multiply every term inside.

去括号：3(x + 2) = 3x + 6，而不是 3x + 2。始终乘括号内的每一项。

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5. Solving One-Step and Two-Step Equations | 解一步和两步方程

Pupils often forget to do the same operation on both sides or misuse inverse operations.

学生经常忘记在等号两边进行相同运算，或误用逆运算。

For x + 5 = 12, subtract 5 from both sides: x = 7. Mistake: moving 5 to the other side as –5 without changing sign: x = 12 – 5 = 7 is correct, but some write x = 12 + 5 = 17.

对于 x + 5 = 12，两边减5：x = 7。错误：把5移项时符号变错，写成 x = 12 + 5 = 17。

Two-step: 2x – 3 = 7. Common error: add 3 then divide: 2x = 10 → x = 5. That’s correct. But many do: 2x = 7 – 3 = 4 → x = 2. Always reverse operations: undo subtraction first, then division.

两步方程：2x – 3 = 7。常见错误：先加3再除，2x = 10 → x = 5。这正确。但有人：2x = 7 – 3 = 4 → x = 2。一定要逆运算：先消去减法，再消去乘法。

When the variable is on the right, e.g., 10 = 2x + 2, treat symmetrically. Subtract 2: 8 = 2x, then divide by 2: 4 = x, so x = 4.

当变量在右侧时，例如 10 = 2x + 2，同样处理。两边减2：8 = 2x，再除以2：4 = x，所以 x = 4。

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6. Ratio and Proportion | 比与比例

Simplifying ratios: divide all parts by their highest common factor. Mistake: simplifying 6:9 to 3:4.5? Must use integers: 2:3.

化简比：将所有部分除以最大公因数。错误：将 6:9 化简为 3:4.5？比的前后项必须是整数，正确为 2:3。

When sharing in a ratio, find total parts first

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