📚 Cambridge Lower Secondary Mathematics WB 8 Answers: Key Concepts Explained | 剑桥初中数学练习册8答案解析：核心知识点精讲
This article provides a detailed walkthrough of the key topics covered in the Cambridge Lower Secondary Mathematics Workbook 8, explaining the reasoning behind common answer types. Stage 8 learners consolidate their understanding of number, algebra, geometry, and statistics, moving towards more formal problem-solving. By studying these solutions, students can deepen their grasp of essential methods and avoid frequent mistakes.
本文深入讲解剑桥初中数学练习册8中的核心知识点，详细剖析常见题型的解答思路。八年级学生在数的运算、代数、几何和统计方面打牢基础，逐步学习规范的问题解决方法。通过研读这些解析，同学们能够加深对关键方法的理解，避免典型错误。
1. Integer Operations and Order of Operations | 整数运算与运算顺序
In Workbook 8, integer calculations involve negative numbers and the rules for addition, subtraction, multiplication, and division. When evaluating expressions like −15 + 8 × (−2), we must follow the order of operations: perform multiplication before addition. 8 × (−2) = −16, then −15 + (−16) = −31. A common error is adding −15 and 8 first, which ignores the priority of multiplication.
练习册8中，整数运算涉及负数加减乘除的规则。计算诸如 −15 + 8 × (−2) 的表达式时，必须遵循运算顺序：先乘除，后加减。8 × (−2) = −16，然后 −15 + (−16) = −31。常见的错误是先计算 −15 + 8，这忽略了乘法的优先级。
The product of two negative numbers is positive, while the product of a positive and a negative number is negative. For example, (−4) × (−6) = 24, but (−4) × 6 = −24. Division follows the same sign rules.
两个负数相乘得正数，一正一负相乘得负数。例如 (−4) × (−6) = 24，而 (−4) × 6 = −24。除法遵循相同的符号规则。
Word problems involving temperature change or bank account balances often require subtracting a negative number, which is equivalent to adding the opposite. A drop from 5 °C to −8 °C is a change of −8 − 5 = −13 °C.
涉及温度变化或银行余额的文字题常需要减去一个负数，相当于加上它的相反数。从 5 °C 降到 −8 °C，变化为 −8 − 5 = −13 °C。
2. Fractions, Decimals, and Percentages | 分数、小数与百分数
Converting between fractions, decimals, and percentages is a central skill. To express 3/8 as a decimal, divide 3 by 8 to get 0.375; then multiply by 100 to obtain 37.5%. When ordering mixed sets, it is often easiest to convert all numbers to decimals for direct comparison.
分数、小数和百分数之间的互化是一项核心技能。将 3/8 化成小数，用 3 除以 8 得 0.375；再乘以 100 得到 37.5%。在排列含有不同形式数字的混合组时，通常先把所有数都化为小数以便直接比较。
Adding and subtracting fractions require a common denominator. For 2/3 + 1/4, the least common denominator is 12, giving 8/12 + 3/12 = 11/12. With mixed numbers, convert to improper fractions first: 1½ + 2⅓ = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6.
分数加减需要通分。计算 2/3 + 1/4 时，最小公分母为 12，得到 8/12 + 3/12 = 11/12。涉及带分数时，先转化为假分数：1½ + 2⅓ = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6。
Percentage increase and decrease problems use the formula: new value = original × (1 ± percentage as a decimal). A 15% discount on £80 gives £80 × 0.85 = £68. Finding the original price after a 20% reduction to £64 requires solving x × 0.80 = 64, so x = 64 ÷ 0.80 = £80.
百分数增减运用公式：新值 = 原值 × (1 ± 百分数的小数形式)。£80 打 15% 折，应付 £80 × 0.85 = £68。若折扣 20% 后售价为 £64，求原价则解方程 x × 0.80 = 64，得 x = 64 ÷ 0.80 = £80。
3. Algebraic Expressions and Simplification | 代数表达式与化简
Workbook 8 introduces combining like terms and using the distributive property. In the expression 4a + 7b − 2a + 3b, we group the a terms and the b terms to get (4 − 2)a + (7 + 3)b = 2a + 10b. Remember that terms must have exactly the same variable part to be combined.
练习册8引入了合并同类项和分配律。在表达式 4a + 7b − 2a + 3b 中，将含 a 的项和含 b 的项分别合并，得 (4 − 2)a + (7 + 3)b = 2a + 10b。要记住，只有所含字母部分完全相同的项才能合并。
Expanding brackets uses the distributive law: 5(3x − 2) = 15x − 10. When a negative sign is in front of the bracket, all signs inside change, e.g. −2(4y − 7) = −8y + 14.
去括号运用分配律：5(3x − 2) = 15x − 10。当括号前是负号时，括号内所有符号都要改变，如 −2(4y − 7) = −8y + 14。
Simplifying expressions with powers requires careful attention to the base and exponents. x³ × x² = x⁵, while (x³)² = x⁶. The expression 2x² + 3x² simply gives 5x², because the powers are identical.
含有幂的表达式化简要特别注意底数和指数。x³ × x² = x⁵，而 (x³)² = x⁶。2x² + 3x² 直接得 5x²，因为指数相同。
4. Solving Linear Equations | 解一元一次方程
The golden rule of equation solving is to do the same thing to both sides. For 3x − 7 = 14, add 7 to both sides: 3x = 21, then divide by 3: x = 7. Checking by substitution is essential: 3(7) − 7 = 21 − 7 = 14, which matches the right side.
解方程的黄金法则是对等式两边同时进行相同的操作。对于 3x − 7 = 14，两边先加 7：3x = 21，再除以 3：x = 7。代入检验至关重要：3(7) − 7 = 21 − 7 = 14，与右边吻合。
Equations with variables on both sides require collecting x terms on one side first. 5x + 9 = 2x + 24 → 5x − 2x = 24 − 9 → 3x = 15 → x = 5.
若未知数出现在等号两边，应先将含 x 的项移到一边。5x + 9 = 2x + 24 → 5x − 2x = 24 − 9 → 3x = 15 → x = 5。
Word problems are translated into equations by defining a variable. “Three more than twice a number is nineteen” becomes 2n + 3 = 19, giving n = 8. Always define the variable clearly in your answer.
应用问题需设未知数列方程。“一个数的两倍加三等于十九”写成 2n + 3 = 19，解得 n = 8。在答案中要明确写出所设变量。
5. Sequences and Patterns | 数列与规律
Linear sequences have a constant term-to-term difference. For the sequence 7, 11, 15, 19, … the common difference is 4. The nth term rule is given by: first term + (n − 1) × difference, so here 7 + (n − 1) × 4 = 4n + 3. Checking for n = 2 gives 4(2) + 3 = 11, which matches.
线性数列的相邻两项之差为常数。数列 7, 11, 15, 19, … 的公差为 4。第 n 项公式为：首项 + (n − 1) × 公差，因此得 7 + (n − 1) × 4 = 4n + 3。检验 n = 2 时，4(2) + 3 = 11，符合。
Non-linear sequences, such as square numbers (1, 4, 9, 16, …) or triangular numbers, follow quadratic rules. Recognising whether a sequence is arithmetic or not helps in formulating the correct expression.
非线性数列，如平方数 (1, 4, 9, 16, …) 或三角形数，遵循二次式规则。识别数列是否为等差数列，有助于写出正确的通项表达式。
Workbook 8 also introduces simple spatial patterns, such as matchstick diagrams, where the number of sticks follows a linear pattern. A row of n squares may require 3n + 1 sticks.
练习册8还介绍了简单的空间规律，如火柴棍图形，其数量通常遵循线性规律。一排 n 个正方形可能需要 3n + 1 根火柴棍。
6. Geometry: Angles, Triangles, and Quadrilaterals | 几何：角度、三角形与四边形
Angle facts are heavily tested: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In a triangle, the interior angles sum to 180°. A scalene triangle has all sides and angles different; an isosceles triangle has two equal angles opposite two equal sides.
角度知识是考查重点：平角为 180°，周角为 360°，对顶角相等。三角形内角和为 180°。不等边三角形的各边各角均不相等；等腰三角形有两个底角相等，对应两腰相等。
When finding unknown angles, label the diagram and work stepwise. In a triangle with angles x, 2x, and 3x, the equation x + 2x + 3x = 180 → 6x = 180 → x = 30°, so the angles are 30°, 60°, 90°. This systematic approach prevents omission.
求未知角时，先在图上做好标注，再逐步计算。在一个内角为 x、2x 和 3x 的三角形中，列出方程 x + 2x + 3x = 180 → 6x = 180 → x = 30°，因此三个角分别为 30°、60°、90°。这种系统方法可以避免遗漏。
Properties of quadrilaterals: a parallelogram has opposite sides parallel and equal, opposite angles equal, and adjacent angles supplementary (sum to 180°). The sum of interior angles in any quadrilateral is 360°.
四边形的性质：平行四边形对边平行且相等，对角相等，邻角互补（和为 180°）。任何四边形的内角和都是 360°。
7. Perimeter, Area, and Volume | 周长、面积与体积
For compound shapes, perimeter is found by summing all outer side lengths, often deducing missing lengths from given dimensions. Area of a rectangle is length × width; area of a triangle is ½ × base × height. A trapezium’s area uses ½ × (a + b) × h, where a and b are the parallel sides.
计算组合图形的周长，需将所有外围边长相加，通常要通过已知尺寸推断缺失的长度。长方形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高。梯形面积公式为 ½ × (a + b) × h，其中 a 和 b 为两平行边。
Surface area of a cuboid is the total area of all faces: 2(lw + lh + wh). Given a cuboid with length 5 cm, width 3 cm, height 4 cm, the surface area is 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 94 cm². Volume is l × w × h = 60 cm³.
长方体的表面积是所有面的面积之和：2(lw + lh + wh)。一个长 5 cm、宽 3 cm、高 4 cm 的长方体，表面积为 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 94 cm²。体积为长 × 宽 × 高 = 60 cm³。
Circles: circumference = π × d or 2πr, and area = πr². At Stage 8, answers are often left in terms of π or approximated using 3.14. A semicircle’s perimeter includes the curved part πr plus the diameter 2r.
圆：周长 = π × d 或 2πr，面积 = πr²。在八年级阶段，答案常保留 π，或用 3.14 取近似值。半圆的周长包含曲线部分 πr 加上直径 2r。
8. Data Handling: Mean, Median, Mode, and Range | 数据处理：平均数、中位数、众数与极差
The mean is the sum of values divided by the number of values. For the set 7, 9, 12, 8, the mean is (7+9+12+8)/4 = 9. The median is found by ordering the data: 7, 8, 9, 12; with an even number, median = (8+9)/2 = 8.5. Mode is the most frequent value; range = highest − lowest.
平均数 = 所有数值之和 ÷ 数据个数。数据集 7, 9, 12, 8 的平均数为 (7+9+12+8)/4 = 9。中位数需先将数据排序：7, 8, 9, 12；数据个数为偶数时，中位数 = (8+9)/2 = 8.5。众数是出现次数最多的值；极差 = 最大值 − 最小值。
A frequency table includes the count of each data point. To find the mean from a frequency table, multiply each value by its frequency, sum these products, then divide by the total frequency.
频数表记录每个数据点出现的次数。根据频数表求平均数时，先将每个数值乘以对应频数，把这些乘积求和，再除以频数总和。
Interpreting graphs such as bar charts and pie charts is essential. In a pie chart, the fraction of the circle for a category equals the category’s frequency divided by the total. An angle of 90° represents ¼ of the total data.
解读条形图和饼图至关重要。在饼图中，一个类别所占的圆比例等于该类别频数除以总数。90° 的圆心角代表总数据的 ¼。
9. Ratio and Proportion | 比与比例
Ratios compare quantities of the same kind. The ratio 3:4 can be scaled up or down by multiplying both sides by the same number. A recipe using sugar and flour in ratio 2:5 means that for every 2 parts of sugar there are 5 parts of flour. If 300 g of flour is used, the sugar amount is (300 ÷ 5) × 2 = 120 g.
比用于同类量的比较。比 3:4 可通过将两项同乘或同除以一个数来扩分或约分。某食谱中糖和面粉的比例为 2:5，意为每 2 份糖对应 5 份面粉。若用了 300 g 面粉，则糖的用量为 (300 ÷ 5) × 2 = 120 g。
Direct proportion means two quantities increase at the same rate. If 5 pens cost £3.50, then 8 pens cost (3.50/5) × 8 = £5.60. Setting up a proportion equation avoids confusion: cost₁/quantity₁ = cost₂/quantity₂.
正比例意味着两个量以相同速率增加。若 5 支笔价值 £3.50，则 8 支笔价值 (3.50/5) × 8 = £5.60。建立比例式可以避免混淆：价格₁/数量₁ = 价格₂/数量₂。
Dividing a quantity in a given ratio involves finding how many total parts there are. To divide £240 in the ratio 3:5:2, total parts = 3+5+2=10. Each part = £24. The shares are 3×24=£72, 5×24=£120, and 2×24=£48.
按给定比例分配数量，需先计算总份数。将 £240 按 3:5:2 分配，总份数 = 3+5+2=10，每份为 £24。各份分别为 3×24=£72、5×24=£120 和 2×24=£48。
10. Probability Basics | 概率基础
Probability is measured on a scale from 0 (impossible) to 1 (certain). For a fair six-sided die, the probability of rolling a 4 is 1/6. Probability is calculated as (number of favourable outcomes) / (total number of possible outcomes), assuming all outcomes are equally likely.
概率的取值范围从 0（不可能）到 1（必然）。掷一枚均匀的六面骰子，掷出 4 的概率为 1/6。概率的计算公式为（有利结果数）/（所有可能结果总数），并假设所有结果等可能。
The probability of an event not happening is 1 − P(event). If the chance of rain on a given day is 0.3, the chance it does not rain is 0.7. Experiment probability may be estimated from relative frequency after many trials.
某事件不发生的概率为 1 − P(事件)。若某天下雨的概率是 0.3，则不下雨的概率为 0.7。实验概率可以通过大量试验后的相对频率来估算。
Workbook 8 includes listing outcomes using sample space diagrams or two-way tables for two combined events, such as flipping a coin and rolling a die. The total number of outcomes is the product of individual numbers: 2 × 6 = 12.
练习册8包括用样本空间图表或双向表格列出两个组合事件的结果，例如抛一枚硬币和掷一次骰子。结果总数为各个事件可能结果数的乘积：2 × 6 = 12。
11. Negative Numbers in Context | 现实情境中的负数
Negative numbers appear in temperatures, elevations below sea level, and financial debt. Describing a change from −4 to 3 as an increase of 7 units reinforces the idea of movement on a number line. Subtracting a negative, such as −3 − (−8), is the same as adding the positive: −3 + 8 = 5.
负数出现在温度、海平面以下的高度以及债务中。将 −4 变为 3 描述为增加了 7 个单位，能强化数轴上的移动概念。减去一个负数，如 −3 − (−8)，等同于加上正数：−3 + 8 = 5。
Ordering directed numbers becomes intuitive when students visualise the number line. The smallest number is the furthest left. −12 is smaller than −5, even though 12 is larger than 5.
当学生想象数轴时，有向数的排序就变得直观了。最小的数在最左边。−12 比 −5 小，尽管 12 比 5 大。
Worked examples involving bank balances: if an account has −£25 and a deposit of £80 is made, the new balance is −25 + 80 = £55. Withdrawing more than the current balance results in negative numbers, forming a foundation for real-world financial literacy.
与银行余额相关的例题：如果账户中有 −£25，存入 £80 后，新余额为 −25 + 80 = £55。取款超过当前余额会导致负数，这就为现实中的理财知识打下了基础。
12. Transformations and Symmetry | 变换与对称
Reflection, rotation, and translation are introduced as rigid motions that preserve shape and size. Reflecting a point across the x-axis changes the sign of its y-coordinate; across the y-axis changes the sign of the x-coordinate. Rotation by 90° around the origin uses the rule (x, y) → (−y, x).
轴对称、旋转和平移作为刚体运动被引入，它们保持形状和大小不变。一个点关于 x 轴对称，y 坐标变号；关于 y 轴对称，x 坐标变号。绕原点旋转 90° 的变换规则为 (x, y) → (−y, x)。
Lines of symmetry divide a shape into two mirror-image halves. Rotational symmetry order is the number of times a shape matches itself in a full 360° turn. A square has rotational symmetry of order 4.
对称轴将图形分成互为镜像的两半。旋转对称的阶数是指一个图形在旋转一周 (360°) 的过程中与自身重合的次数。正方形有 4 阶旋转对称。
Describing a translation involves specifying a column vector (x above y) indicating the horizontal and vertical movement. For example, moving 3 units right and 2 units down is written as (3, −2) in vector form.
描述平移需要用一个列向量（x 在上，y 在下）来表示水平和垂直方向上的移动。例如，右移 3 个单位，下移 2 个单位用向量形式表示为 (3, −2)。
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