Complete Mathematics for Cambridge Secondary 1 Book 2: Year 8 Core Revision | 剑桥初中数学第二册：八年级核心复习

📚 Complete Mathematics for Cambridge Secondary 1 Book 2: Year 8 Core Revision | 剑桥初中数学第二册：八年级核心复习

This revision guide covers the main topics in Complete Mathematics for Cambridge Secondary 1 Book 2, designed for Year 8 learners following the CIE Lower Secondary Mathematics curriculum. Use it to consolidate key skills, spot common mistakes, and build confidence before assessments.

本复习指南涵盖《Complete Mathematics for Cambridge Secondary 1 Book 2》的主要专题，适用于学习剑桥初中数学课程的八年级学生。用它来巩固关键技能、发现常见错误，并在考前建立信心。

1. Integers, Powers and Roots | 整数、幂与平方根

In Book 2, students extend whole-number arithmetic to negative numbers, powers and roots. Always apply the order of operations: brackets, indices, division and multiplication, addition and subtraction (BIDMAS). For example, 5 + (−3)² × 2 = 5 + 9 × 2 = 5 + 18 = 23.

在第二册中，学生将整数运算扩展到负数、幂和平方根。必须始终遵循运算顺序：括号、指数、乘除、加减（BIDMAS）。例如 5 + (−3)² × 2 = 5 + 9 × 2 = 5 + 18 = 23。

A negative base raised to an even power becomes positive, but an odd power stays negative: (−2)² = 4 and (−2)³ = −8. This often causes sign errors, so check carefully.

负底数的偶次幂为正，奇次幂仍为负：(−2)² = 4，(−2)³ = −8。这一点经常引起符号错误，请仔细检查。

Square roots and cube roots undo powers: √81 = 9 because 9² = 81; ∛64 = 4 because 4³ = 64. Estimating roots without a calculator is also tested, for example √50 lies between 7 and 8.

平方根和立方根是幂的逆运算：√81 = 9，因为 9² = 81；∛64 = 4，因为 4³ = 64。不借助计算器估算根号也是考点，例如 √50 介于 7 和 8 之间。

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

You must be able to convert between fractions, decimals and percentages fluently. For example, 3/8 = 0.375 = 37.5%. Use division to change a fraction to a decimal, and multiply by 100 to change a decimal to a percentage.

你必须能够熟练地在分数、小数和百分数之间转换。例如 3/8 = 0.375 = 37.5%。用除法将分数化为小数，再将小数乘以 100 化为百分数。

Adding and subtracting fractions requires a common denominator. To calculate 2/3 + 1/4, rewrite as 8/12 + 3/12 = 11/12. For mixed numbers, convert to improper fractions first.

分数的加减需要公分母。计算 2/3 + 1/4 时，先化为 8/12 + 3/12 = 11/12。带分数要先化为假分数。

Percentage increase and decrease are common in real-life problems. A 15% increase on $60 is found by multiplying 60 × 1.15 = $69. A 20% decrease is 60 × 0.80 = $48.

百分数增加和减少在现实问题中很常见。在 60 美元基础上增加 15%，用 60 × 1.15 = 69 美元；减少 20%，则 60 × 0.80 = 48 美元。

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3. Ratio, Proportion and Rates | 比、比例与速率

Ratios compare quantities and can be simplified by dividing all parts by the same number. The ratio 24:36 simplifies to 2:3 because both divide by 12.

比用于比较数量，可通过将各部分同时除以同一个数来化简。24:36 化简为 2:3，因为两者都可以除以 12。

To divide a quantity in a given ratio, add the parts first. Divide $90 in the ratio 2:3: the total parts are 2 + 3 = 5, so each part is $18. The amounts are 2 × $18 = $36 and 3 × $18 = $54.

按给定比分配数量时，先求总份数。将 90 美元按 2:3 分配：总份数为 2 + 3 = 5，每份为 18 美元；因此两部分分别为 2 × 18 = 36 美元和 3 × 18 = 54 美元。

Direct proportion means two quantities increase or decrease at the same rate. Rates such as speed use the formula speed = distance ÷ time. A car travelling 180 km in 2 hours has speed 180 ÷ 2 = 90 km/h.

正比例意味着两个量以相同的速率增加或减少。速率（如速度）使用公式：速度 = 路程 ÷ 时间。一辆汽车 2 小时行驶 180 千米，速度为 180 ÷ 2 = 90 千米/小时。

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4. Algebraic Expressions and Formulae | 代数表达式与公式

Algebra in Book 2 focuses on simplifying expressions by collecting like terms. For instance, 3x + 5y − 2x + y = x + 6y. Only terms with exactly the same variable part can be combined.

第二册的代数重点是合并同类项来化简表达式。例如 3x + 5y − 2x + y = x + 6y。只有变量部分完全相同的项才能合并。

Expanding brackets uses the distributive law: a(b + c) = ab + ac. For example, 5(2x − 3) = 10x − 15. Be careful with signs when the bracket is negative: −2(x + 4) = −2x − 8.

去括号使用分配律：a(b + c) = ab + ac。例如 5(2x − 3) = 10x − 15。括号前为负号时要格外小心：−2(x + 4) = −2x − 8。

Substitution means replacing letters with numbers. If a = 3 and b = −2, then 4a − b² = 4(3) − (−2)² = 12 − 4 = 8. Always use brackets when substituting negative values.

代入法指用数字替换字母。若 a = 3、b = −2，则 4a − b² = 4(3) − (−2)² = 12 − 4 = 8。代入负值时请始终加括号。

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5. Linear Equations and Inequalities | 线性方程与不等式

Solving linear equations means finding the unknown value that makes the equation true. Use inverse operations, and keep the equation balanced by doing the same to both sides. For 3x + 5 = 20, subtract 5: 3x = 15, then divide by 3: x = 5.

解线性方程就是求出使等式成立的未知数值。使用逆运算，并在等式两边同时进行相同操作以保持平衡。例如 3x + 5 = 20，先减 5 得 3x = 15，再除以 3 得 x = 5。

Equations with unknowns on both sides require collecting terms. Solve 5x − 4 = 2x + 8: subtract 2x to get 3x − 4 = 8, add 4 to get 3x = 12, so x = 4.

未知数在等号两边的方程需要先移项。解 5x − 4 = 2x + 8：两边同时减去 2x 得 3x − 4 = 8，再加 4 得 3x = 12，因此 x = 4。

Inequalities are solved like equations, but remember that multiplying or dividing by a negative number reverses the sign. For −2x ≤ 6, divide by −2 to get x ≥ −3.

不等式的解法与方程类似，但乘以或除以负数时不等号方向要改变。例如

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