📚 Essential Maths Book 8 Answers: Key Topics Explained | KS3数学：关键知识点答案解析
Essential Maths Book 8 is a core resource for KS3 students, covering topics such as number operations, algebra, geometry, ratio, and probability. This article explains key concepts from the book and provides worked answers to typical questions. Use these explanations to reinforce your understanding, check your homework, and prepare for tests with confidence.
Essential Maths Book 8 是 KS3 阶段的核心教材，涵盖数字运算、代数、几何、比和概率等主题。本文梳理书中关键知识点，并对典型题目提供详细解答。通过精讲，你可以巩固概念、核对作业，更有信心地应对测验。
1. Negative Numbers: Addition and Subtraction | 负数的加减法
When adding or subtracting negative numbers, think of the number line. Adding a negative is the same as subtracting its positive. Subtracting a negative is the same as adding its positive. For example, 5 + (−3) = 5 − 3 = 2, and 5 − (−3) = 5 + 3 = 8. Always simplify double signs first.
进行负数的加减法时，可以借助数轴来理解。加上一个负数等于减去它的相反数；减去一个负数等于加上它的相反数。例如 5 + (−3) = 5 − 3 = 2，而 5 − (−3) = 5 + 3 = 8。解题时优先简化双重符号。
- Worked example: Calculate −7 − (−4) + 2. First, −7 − (−4) becomes −7 + 4 = −3. Then −3 + 2 = −1.
- 解析示例：计算 −7 − (−4) + 2。首先 −7 − (−4) 化为 −7 + 4 = −3，然后 −3 + 2 = −1。
- Common mistake: Treating −5 − 3 as −2. The correct answer is −8, because you move further left on the number line.
- 常见错误：把 −5 − 3 误算为 −2。正确答案是 −8，因为要在数轴上继续向左移动。
2. Multiplying and Dividing with Negatives | 负数的乘除法
The rules for multiplying and dividing negative numbers are simple: same signs give a positive, different signs give a negative. For example, (−6) × (−3) = 18, and 12 ÷ (−4) = −3. Apply the sign rule first, then perform the operation with the absolute values.
负数乘除法的规则很简单：同号得正，异号得负。例如 (−6) × (−3) = 18，12 ÷ (−4) = −3。先确定符号，再用绝对值进行运算。
When several factors are involved, count the number of negative signs. An odd number of negatives gives a negative product; an even number gives a positive product. For instance, (−2) × 3 × (−5) = 30 (two negatives, even).
多个因数相乘时，统计负号的个数。奇数个负号结果为负，偶数个负号结果为正。例如 (−2) × 3 × (−5) = 30（两个负号，偶数）。
Example: (−4)² = (−4) × (−4) = 16, but −4² = −(4 × 4) = −16. The exponent applies only to the number immediately before it.
注意区分： (−4)² = 16，而 −4² = −16。指数只作用于紧邻它的数。
3. Order of Operations: BIDMAS | 运算顺序 BIDMAS
BIDMAS stands for Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Always follow this hierarchy to avoid mistakes. For 3 + 4 × 2, multiplication comes first: 4 × 2 = 8, then 3 + 8 = 11.
BIDMAS 代表括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。严格按照优先级计算，可以避免错误。例如 3 + 4 × 2，乘法优先：4 × 2 = 8，再加 3 得 11。
When division and multiplication appear together, work from left to right. In 24 ÷ 6 × 2, do 24 ÷ 6 = 4 first, then 4 × 2 = 8. If you multiply first, you get 24 ÷ 12 = 2, which is incorrect.
当除法和乘法同时出现时，从左到右依次计算。例如 24 ÷ 6 × 2，先算 24 ÷ 6 = 4，再算 4 × 2 = 8。如果先乘后除，会得到 24 ÷ 12 = 2，这是错误的。
| Expression | Correct Step-by-step | Answer |
| (5 + 3)² ÷ 4 − 2 | Brackets: 8² ÷ 4 − 2 → 64 ÷ 4 − 2 → 16 − 2 | 14 |
| 表达式 | 正确步骤 | 答案 |
| (5 + 3)² ÷ 4 − 2 | 括号：8² ÷ 4 − 2 → 64 ÷ 4 − 2 → 16 − 2 | 14 |
4. Simplifying Algebraic Expressions | 化简代数表达式
Combine like terms (same variable and same power) by adding or subtracting their coefficients. For 3a + 2b − a + 4b, group the a terms: 3a − a = 2a, and the b terms: 2b + 4b = 6b. The simplified expression is 2a + 6b. Constants without variables can only be combined with each other.
通过合并同类项（相同字母且相同指数）来化简代数式。例如 3a + 2b − a + 4b，合并 a 项：3a − a = 2a，合并 b 项：2b + 4b = 6b，化简得 2a + 6b。纯数字只能与纯数字合并。
When multiplying, multiply the coefficients and the variables separately: 2x × 3x = 6x². Remember that x means 1x, and x × x = x². In expressions like 4p × (−2p), the product is −8p² because a positive times a negative gives a negative.
乘法运算时，系数和变量分别相乘：2x × 3x = 6x²。注意 x 即 1x，x × x = x²。对于 4p × (−2p)，结果为 −8p²，因为正乘负得负。
5. Solving One-step Equations | 解一步方程
To solve an equation like x + 7 = 12, perform the inverse operation on both sides. Subtract 7 from both sides to isolate x: x = 12 − 7, so x = 5. For multiplication equations such as 4x = 20, divide both sides by 4: x = 20 ÷ 4 = 5.
解 x + 7 = 12 这样的方程，需在等式两边同时进行逆运算。两边同减 7，得 x = 12 − 7，即 x = 5。对于乘法方程如 4x = 20，两边同除以 4，x = 5。
Always check your solution by substituting it back into the original equation. If the left side equals the right side, the answer is correct. For x/3 = 9, multiply both sides by 3 to get x = 27. Check: 27 ÷ 3 = 9, correct.
解出答案后一定要代回原方程检验。若左边等于右边，答案即为正确。例如 x/3 = 9，两边同乘 3，得 x = 27。检验：27 ÷ 3 = 9，正确。
Example: y − 5 = −3 → y = −3 + 5 → y = 2
例：y − 5 = −3 → y = −3 + 5 → y = 2
6. Solving Two-step Equations | 解两步方程
For equations like 2x + 3 = 11, undo the operations in reverse order of BIDMAS. First subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4. Always start with addition/subtraction, then deal with multiplication/division.
解 2x + 3 = 11 这种两步方程，按 BIDMAS 的逆顺序运算。首先两边减 3：2x = 8，然后两边除以 2：x = 4。务必先处理加减，再处理乘除。
When the variable appears with a coefficient as a fraction, it is often easier to eliminate the denominator first. In (x/4) − 6 = 2, add 6 to both sides: x/4 = 8, then multiply by 4: x = 32.
当变量带有分数系数时，通常先消去分母更简便。例如 (x/4) − 6 = 2，两边加 6 得 x/4 = 8，再两边乘 4，x = 32。
Negative solutions are possible and should be treated just like positive ones. For 10 − 3y = 22, subtract 10: −3y = 12, divide by −3: y = −4.
方程的解也可以是负数，处理方法与正数相同。例如 10 − 3y = 22，两边减 10：−3y = 12，再除以 −3，y = −4。
7. Ratio and Proportion | 比和比例
A ratio compares two or more quantities. It can be simplified like a fraction by dividing all terms by their highest common factor. The ratio 10:15 simplifies to 2:3 (divide both by 5). Ratios have no units and can be written in different forms, such as 2:3 or 2 to 3.
比用来比较两个或多个数量。类似于分数，比可以通过除以各项的最大公约数来化简。10:15 化简为 2:3（两项同除以 5）。比不带单位，可写成 2:3 或 2 to 3 等形式。
When sharing a quantity in a given ratio, first find the total number of parts. To share £60 in the ratio 3:2, total parts = 3 + 2 = 5. Each part is worth £60 ÷ 5 = £12. The first share is 3 × £12 = £36, and the second share is 2 × £12 = £24.
按比例分配数量时，先求总份数。将 £60 按 3:2 分配，总份数 = 5，每份 £12。第一份得 3 × £12 = £36，第二份得 2 × £12 = £24。
Proportion problems often involve direct scaling. If 5 pens cost £3.50, then 1 pen costs £3.50 ÷ 5 = £0.70, so 8 pens cost 8 × £0.70 = £5.60. The key is finding the value for one unit first (the unitary method).
比例问题常涉及按倍数缩放。若 5 支笔价格 £3.50，则 1 支笔价格为 £3.50 ÷ 5 = £0.70，8 支笔则为 8 × £0.70 = £5.60。关键先求出单一量（单位法）。
8. Angles in Parallel Lines | 平行线中的角
When a transversal crosses two parallel lines, several equal angle pairs are formed. Corresponding angles are equal (same position at each intersection). Alternate angles are equal (Z-shape). Interior (co-interior) angles sum to 180° (C-shape).
当一条截线穿过两条平行线时，会产生几组相等的角。同位角相等（位于截线同侧且同位置）；内错角相等（Z 形）；同旁内角互补，和为 180°（C 形）。
To find unknown angles, label the given angle and use the angle facts systematically. If one angle is 65°, its corresponding angle is also 65°, the alternate angle is 65°, and the co-interior angle is 180° − 65° = 115°.
求未知角时，先标记已知角，再系统运用角度规律。若一个角为 65°，则它的同位角也是 65°，内错角也是 65°，同旁内角为 180° − 65° = 115°。
Vertically opposite angles are always equal, whether lines are parallel or not. They form when two straight lines cross. This fact often helps in angle chase problems.
对顶角始终相等，与被截直线是否平行无关。两条直线交叉时形成对顶角，这一性质常协助解决求角问题。
9. Area of Compound Shapes | 复合图形的面积
A compound shape can be split into simpler rectangles, triangles, or parallelograms. Find the area of each part separately and add them together. Alternatively, subtract the area of a missing piece from a larger rectangle.
复合图形可以分割成简单的矩形、三角形或平行四边形。分别计算各部分的面积，再相加求和。也可以用大矩形面积减去空缺部分的面积。
Area of a rectangle = length × width. Area of a triangle = ½ × base × vertical height. Always ensure the height is perpendicular to the base. When side lengths are given in different units, convert to the same unit first.
矩形面积 = 长 × 宽。三角形面积 = ½ × 底 × 垂直高。确保高与底垂直。若边长单位不同，要先统一单位再计算。
Example: L-shape split into two rectangles 6 × 4 and 3 × 2 → Area = 24 + 6 = 30 cm²
例子：L 形拆分为两个矩形 6×4 和 3×2 → 面积 = 24 + 6 = 30 cm²
10. Volume of Prisms | 棱柱的体积
The volume of any prism is found by multiplying the area of its cross-section by its length: Volume = cross-sectional area × length. The cross-section is the shape you see when cutting straight through the prism. Units are cubic, e.g., cm³.
任何棱柱的体积等于其横截面积乘以长度：体积 = 横截面积 × 长度。横截面是沿垂直方向切开后看到的形状。单位为立方，如 cm³。
For a cube of side 5 cm, the cross-section is a square of area 25 cm². Length = 5 cm, so volume = 125 cm³. For a triangular prism, work out the area of the triangle (½ × base × height) and multiply by the prism length.
边长为 5 cm 的正方体，横截面为正方形，面积 25 cm²。长度 5 cm，体积为 125 cm³。对于三棱柱，先计算三角形面积（½ × 底 × 高），再乘以棱柱的长度。
A common error is mixing up the height of the triangle and the length of the prism. Label your sketch clearly and remember: the triangle’s base and height are perpendicular, and the prism length connects the two identical cross-sections.
常见错误是混淆三角形的高和棱柱的长。画图时要清晰标注，记住：三角形的底和高互相垂直，棱柱的长是连接两个相同横截面的距离。
11. Probability Scale and Simple Events | 概率尺度与简单事件
Probability is a number between 0 and 1 that describes the chance of an event happening. 0 means impossible, 1 means certain. Probability = number of successful outcomes ÷ total number of possible outcomes, provided all outcomes are equally likely.
概率是 0 到 1 之间的数，描述事件发生的可能性。0 表示不可能，1 表示必然。若所有可能结果等可能发生，概率 = 成功结果数 ÷ 所有可能结果总数。
If a fair 6-sided die is rolled, the probability of rolling a 4 is 1/6. The probability of rolling an even number is 3/6 = 1/2. Probabilities can be written as fractions, decimals, or percentages.
掷一枚均匀的六面骰子，掷出 4 的概率是 1/6。掷出偶数的概率为 3/6 = 1/2。概率可用分数、小数或百分数表示。
The probability of an event not happening = 1 − probability that it does happen. If the chance of rain tomorrow is 0.3, the chance of no rain is 0.7.
事件不发生的概率 = 1 − 事件发生的概率。若明天下雨的概率是 0.3，则不下雨的概率是 0.7。
12. Interpreting Charts and Averages | 图表的解读与平均值
Bar charts and pictograms display frequencies. Read values carefully from the scale; each small interval on a bar chart may represent more than one unit. The mode is the most frequent value, the median is the middle value when data are ordered, and the mean is sum of values ÷ number of items.
条形图和象形图展示频数。读图时要注意坐标尺度，柱状图的每个小格可能代表多于一个单位。众数是出现次数最多的值，中位数是排序后位于中间的值，平均数 = 数据总和 ÷ 数据个数。
To find the median of an even-sized list, take the mean of the two middle numbers. For the data set 3, 7, 8, 12, the median is (7+8)÷2 = 7.5.
若数据个数为偶数，中位数是中间两个数的平均数。数据集 3, 7, 8, 12 的中位数为 (7+8)÷2 = 7.5。
Interpreting a pie chart requires understanding that each sector angle is proportional to the frequency. Since the whole circle is 360°, a sector of 90° represents 90/360 = 1/4 of the total.
解读饼图需要明白每个扇形的角度与频数成比例。整个圆为 360°，所以 90° 的扇形代表 90/360 = 1/4 的整体。
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