Essential Maths Book 9H Answers – Core Topics Explained | KS3 数学 9H 练习答案精讲

📚 Essential Maths Book 9H Answers – Core Topics Explained | KS3 数学 9H 练习答案精讲

This article unpacks the key mathematical ideas behind the Essential Maths Book 9H exercises, providing clear explanations and worked examples for students aiming to master Year 9 Higher level content. By going beyond just the final answers, we highlight the reasoning and methods that build deep understanding.

本文深入讲解《Essential Maths Book 9H》练习中涉及的核心数学知识，为目标是掌握九年级高阶内容的学生提供清晰的解释和解题示范。我们不仅给出最终答案，更强调推理过程和方法，帮助建立扎实的理解。

1. Number Systems and Place Value | 数系与位值

Understanding place value up to billions and down to thousandths is essential for higher-level work with decimals, standard form, and estimation. The 9H exercises test ordering numbers, rounding, and identifying significant figures.

理解从十亿到千分位的位值，是处理小数、标准形式和估算等高阶内容的基础。9H 的练习题考查数字排序、四舍五入和确定有效数字的能力。

• Significant figures: All non‑zero digits are significant; zeros between non‑zero digits are significant; trailing zeros in a decimal are significant. For example, 0.00508 has three significant figures (5, 0, 8).
• 有效数字：所有非零数字都是有效数字；非零数字之间的零是有效数字；小数末尾的零也是有效数字。例如，0.00508 有三个有效数字（5、0、8）。
• Rounding: To round 45.736 to two decimal places, look at the third decimal digit (6). Since it is 5 or more, round up the second decimal from 3 to 4, giving 45.74.
• 四舍五入：将 45.736 保留两位小数，看第三位小数（6）。因为大于等于 5，所以第二位小数 3 进 1 变为 4，得到 45.74。

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

Book 9H revises fraction operations and extends to recurring decimals and percentage change. You need to be fluent converting between all three forms, including mixed numbers and improper fractions.

9H 教材复习分数运算，并扩展到循环小数和百分比变化。你需要能够熟练地在三种形式之间转换，包括带分数和假分数。

Converting recurring decimals: Let r = 0.636363… . Multiply by 100 to shift two digits: 100r = 63.6363… . Subtract the original: 99r = 63, so r = 63/99 = 7/11.

循环小数化分数：设 r = 0.636363…，乘以 100 移动两位：100r = 63.6363…，相减得 99r = 63，所以 r = 63/99 = 7/11。

Percentage increase and decrease: To increase £80 by 15%, multiply by 1.15: £80 × 1.15 = £92. For a 20% decrease, multiply by 0.80.

百分比的增减：将 80 英镑增加 15%，乘以 1.15：80 × 1.15 = 92。减少 20% 则乘以 0.80。

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3. Ratio and Proportion | 比与比例

Ratio problems in 9H often involve sharing amounts, comparing parts, and linking ratios with fractions. Direct proportion and the unitary method are heavily tested.

9H 中的比与比例问题常涉及分配数量、比较部分以及将比与分数联系。正比例和单位量法是重点考查内容。

Sharing in a ratio: Divide £450 in the ratio 2:3:5. Total parts = 2 + 3 + 5 = 10. One part is £450 ÷ 10 = £45. So the shares are £90, £135, £225.

按比例分配：将 450 英镑按 2:3:5 分配。总份数 = 2 + 3 + 5 = 10。一份为 450 ÷ 10 = 45 英镑。因此分配额为 90 英镑、135 英镑、225 英镑。

Direct proportion: If 8 pens cost £5.60, then 20 pens cost £5.60 ÷ 8 × 20 = £0.70 × 20 = £14.00.

正比例：如果 8 支笔花费 5.60 英镑，那么 20 支笔花费 5.60 ÷ 8 × 20 = 0.70 × 20 = 14.00 英镑。

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4. Algebraic Expressions and Equations | 代数式与方程

This section covers simplifying expressions, expanding brackets, factorising, and solving linear equations. Higher tier students also encounter simple quadratic expansions.

这部分涵盖代数式的化简、去括号、因式分解以及解一元一次方程。高阶学生还会接触简单的二次展开。

Expanding and simplifying: 4(2x − 3) − 2(3x + 1) = 8x − 12 − 6x − 2 = 2x − 14.

展开并化简：4(2x − 3) − 2(3x + 1) = 8x − 12 − 6x − 2 = 2x − 14。

Solving equations: 5x + 7 = 3x − 9 → 5x − 3x = −9 − 7 → 2x = −16 → x = −8.

解方程：5x + 7 = 3x − 9 → 5x − 3x = −9 − 7 → 2x = −16 → x = −8。

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5. Linear Graphs and Coordinates | 线性图像与坐标

9H problems require plotting straight lines from equations, finding gradients and intercepts, and interpreting real-life graphs. The form y = mx + c is fundamental.

9H 的题目要求根据方程绘制直线、求斜率和截距，以及解读实际问题的图像。y = mx + c 的形式是基础。

The gradient m is rise over run. In y = 3x − 4, the gradient is 3 and the y-intercept is (0, −4). To find the x-intercept, set y = 0: 3x − 4 = 0 → x = 4/3.

斜率 m 是纵变除以横变。在 y = 3x − 4 中，斜率为 3，y 截距为 (0, −4)。求 x 截距时令 y = 0：3x − 4 = 0 → x = 4/3。

Parallel lines have the same gradient. y = 2x + 5 is parallel to y = 2x − 1. Perpendicular lines have gradients that multiply to −1 (e.g. 2 and −1/2).

平行线斜率相同。y = 2x + 5 与 y = 2x − 1 平行。垂直线的斜率乘积为 −1（如 2 和 −1/2）。

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6. Sequences and the nth Term | 数列与通项公式

Students must generate terms from a given rule and find the nth term for linear sequences. The 9H book also introduces simple quadratic sequences.

学生需要根据给定规则写出数列的若干项，并求出线性数列的通项公式。9H 教材还引入了简单的二次数列。

The nth term of 5, 8, 11, 14, … is 3n + 2 (common difference 3, adjust by finding the zero term). For a quadratic sequence 2, 5, 10, 17, …, the second differences are constant (2), so nth term involves n², leading to n² + 1.

数列 5, 8, 11, 14, … 的通项公式是 3n + 2（公差为 3，通过零项调整）。二次数列 2, 5, 10, 17, … 的二次差为常数 2，因此通项包含 n²，得出 n² + 1。

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7. Perimeter, Area and Volume | 周长、面积与体积

9H extends area calculations to parallelograms, trapeziums, and compound shapes. Volume of prisms and cylinders is also covered, alongside surface area.

9H 将面积计算扩展到平行四边形、梯形和组合图形。棱柱和圆柱的体积，以及表面积也是学习内容。

Area of a trapezium = ½(a + b)h, where a and b are the parallel sides, h is the perpendicular height.

梯形面积 = ½(a + b)h，其中 a 和 b 为平行边，h 为垂直高度。

Volume of a prism = area of cross-section × length. For a cylinder, V = πr²h. Surface area of a cylinder = 2πr² + 2πrh.

棱柱体积 = 截面积 × 长度。圆柱体积 V = πr²h。圆柱表面积 = 2πr² + 2πrh。

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8. Angles and Shapes | 角与图形

Key angle facts, including those in parallel lines, triangles, and quadrilaterals, are revisited. Students also use angle sum of polygons and solve problems with bearings.

复习关键的角度定理，包括平行线中的角、三角形和四边形中的角关系。学生还将使用多边形内角和公式并解决方位角问题。

Sum of interior angles of an n-sided polygon = (n − 2) × 180°. A hexagon has 720°. Each interior angle of a regular hexagon = 720° ÷ 6 = 120°.

多边形内角和 = (n − 2) × 180°。六边形内角和为 720°。正六边形每个内角 = 720° ÷ 6 = 120°。

Alternate angles on parallel lines are equal. Corresponding angles are equal. Co-interior angles sum to 180°.

平行线中的内错角相等。同位角相等。同旁内角之和为 180°。

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9. Transformations and Symmetry | 变换与对称

Questions involve reflections, rotations, translations, and enlargements on a coordinate grid. Scale factors and centres of enlargement are specified, including fractional scale factors.

题目涉及坐标系中的反射、旋转、平移和位似。需要确定缩放因子和位似中心，包括分数缩放因子。

Enlargement with scale factor 2 from centre (0,0): Point (3,1) maps to (6,2). With scale factor 1/2, (4,6) maps to (2,3).

位似：以 (0,0) 为中心、缩放因子 2，点 (3,1) 变换为 (6,2)。缩放因子 1/2 时，(4,6) 变为 (2,3)。

Reflection in the line y = x swaps coordinates: (a,b) → (b,a).

关于直线 y = x 的反射交换坐标：(a,b) → (b,a)。

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10. Statistics and Averages | 统计与平均数

The 9H book covers mean, median, mode, range, and introduces grouped frequency tables. Students also calculate estimated means and draw stem-and-leaf diagrams.

9H 教材涵盖平均数、中位数、众数和极差，并引入分组频数表。学生还要计算估计平均数并绘制茎叶图。

Mean from a frequency table: sum of (value × frequency) ÷ total frequency. For grouped data, use midpoints of intervals.

由频数表求平均数：（值 × 频数）的总和 ÷ 总频数。对于分组数据，使用组中值。

Stem-and-leaf diagram orders data and shows shape. Key: 4|5 means 45. Back-to-back stem plots compare two sets.

茎叶图将数据排序并显示分布形状。关键标示 4|5 表示 45。背靠背茎叶图用于比较两组数据。

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11. Probability | 概率

Probability skills include listing outcomes, using sample space diagrams, and calculating combined probabilities for independent events. 9H also introduces relative frequency.

概率技能包括列出结果、使用样本空间图以及计算独立事件的组合概率。9H 还引入了相对频率。

Probability of an event = number of favourable outcomes ÷ total number of possible outcomes. P(rolling a prime on a die) = 3/6 = 1/2.

事件概率 = 有利结果数 ÷ 所有可能结果总数。掷骰子得到质数的概率 P = 3/6 = 1/2。

Combined events: P(A and B) = P(A) × P(B) if events are independent. Flipping two heads: 1/2 × 1/2 = 1/4.

组合事件：若事件独立，P(A 与 B) = P(A) × P(B)。两次抛硬币都得正面：1/2 × 1/2 = 1/4。

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12. Problem-Solving and Applying Mathematics | 问题解决与数学应用

The final strand of 9H integrates topics into multi-step word problems, often requiring estimation, trial and improvement, or constructing and solving equations from a scenario.

9H 的最后一部分将各个主题综合为多步应用题，往往需要估算、尝试改进或者根据实际情境建立并求解方程。

Example: A rectangle’s length is 3 cm more than its width. The perimeter is 26 cm. Let width = w. Then length = w + 3. Equation: 2(w + w + 3) = 26 → 2(2w + 3) = 26 → 4w + 6 = 26 → 4w = 20 → w = 5. Length = 8 cm.

示例：长方形的长比宽多 3 cm，周长为 26 cm。设宽为 w，长则为 w + 3。方程：2(w + w + 3) = 26 → 2(2w + 3) = 26 → 4w + 6 = 26 → 4w = 20 → w = 5。长为 8 cm。

These skills mirror the type of reasoning required to use the compressed answers effectively: not just checking if the answer is correct, but understanding the path to get there.

这些技能正好反映了有效使用压缩答案所需的那种推理：不仅是核对答案是否正确，更是理解得出答案的路径。

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