📚 Cambridge Lower Secondary Mathematics Workbook 9 Answers: Key Concept Walkthrough | 剑桥初中数学练习册9答案解析：知识点精讲
This article unpacks the key topics in the Cambridge Lower Secondary Mathematics Workbook 9 and explains the reasoning behind common answer types. Whether you are checking solutions or revising for assessments, understanding these concepts will strengthen your foundation. We walk through number skills, algebra, geometry, data handling and probability, connecting each idea to real workbook-style questions.
本文梳理剑桥初中数学练习册9中的核心知识点，并讲解常见题型背后的解题思路。无论是核对答案还是备考复习，理解这些概念都能帮助你夯实基础。我们将依次梳理数系运算、代数、几何、数据统计和概率，把每个知识点与练习册中的典型题目联系起来。
1. Integers, Powers and Roots | 整数、乘方与方根
aluate (−3)³ or find the value of 5⁻². Remember that a negative base raised to an odd power remains negative, and that a negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ. Square roots and cube roots of perfect squares and cubes also appear frequently, such as √64 or ³√−27.
练习册9要求熟练掌握负数的运算、运算顺序（先括号、指数、乘除、加减）以及指数的运算律。例如，题目可能要求计算 (−3)³ 或求出 5⁻² 的值。记住负数的奇数次幂仍为负数，而负指数表示倒数：a⁻ⁿ = 1/aⁿ。完全平方数和完全立方数的平方根与立方根也经常出现，例如 √64 或 ³√−27。
- Any number to the power of 0 equals 1 (except 0⁰).
- 任何非零数的0次幂都等于1。
- Product of powers rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- 同底数幂的乘法：aᵐ × aⁿ = aᵐ⁺ⁿ
- Power of a power: (aᵐ)ⁿ = aᵐⁿ
- 幂的乘方：(aᵐ)ⁿ = aᵐⁿ
When checking answers, watch for sign errors with roots: √−4² is 4, not −4.
核对答案时，注意根号下负号的处理：√−4² 等于 4，而不是 −4。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
percentage increase/decrease problems, and using multipliers. A typical workbook question: “Increase £340 by 15%”. The multiplier method gives 340 × 1.15 = £391. Reversing percentages is trickier: “A price after 20% reduction is £48, find the original price.” The original is 48 ÷ 0.8 = £60.
本章综合考查分数、小数和百分数的互化，以及增减百分比和乘数法的应用。一道典型的练习册题目是：“将 £340 增加 15%”。乘数法的计算为 340 × 1.15 = £391。反向求原价则更需小心：“降价20%后价格为£48，求原价。”原价应为 48 ÷ 0.8 = £60。
Recurring decimals also feature: convert 0.3̇6̇ act to get 99x = 36, so x = 36/99 = 4/11. These steps often appear in the answers section.
循环小数也会出现：将 0.3̇6̇ 化为分数。设 x = 0.363636...，则 100x = 36.3636...，相减得 99x = 36，所以 x = 36/99 = 4/11。这些步骤常在答案中出现。
| Decimal | 小数 | Fraction | 分数 | Percentage | 百分数 |
|---|---|---|
| 0.45 | 9/20 | 45% |
| 0.8 | 4/5 | 80% |
3. Algebraic Expressions and Equations | 代数表达式与方程
ble brackets, e.g., (x + 5)(x − 2) = x² + 3x − 10. Factorising the reverse: x² − 9 = (x + 3)(x − 3), recognising the difference of two squares. Checking answers: always re‑expand to verify.
代数表达式的核心包括合并同类项、展开括号和因式分解。练习册9常出现展开双括号的题目，例如 (x + 5)(x − 2) = x² + 3x − 10。逆过程因式分解则要熟悉平方差公式：x² − 9 = (x + 3)(x − 3)。核对答案时，务必重新展开以验证结果。
Solving two‑step and multi‑ lways perform the inverse operation to isolate the variable, and substitute back to check.
解两步或多步方程也会考查：3x − 7 = 2x + 5 → x = 12。始终使用逆运算逐步分离变量，并将解代入原方程验算。
4. Linear Equations and Inequalities | 一次方程与不等式
11. Subtract 5: −2x > 6. Divide by −2: x < −3. Number line representation is often requested; use an open circle for < or >, closed for ≤ or ≥.
解不等式时，当两边同时乘以或除以负数，不等号方向要改变。练习中有题目：解 5 − 2x > 11。减5得 −2x > 6；除以 −2 得 x < −3。题目常要求在数轴上表示解集；注意开区间用空心圆，闭区间用实心圆。
10 and x − y = 2, add to eliminate y: 3x = 12 → x = 4, y = 2. Always check both equations.
解联立方程组通常使用消元法或代入法。对于 2x + y = 10 和 x − y = 2，两式相加消去 y 得 3x = 12，所以 x = 4, y = 2。务必代入两个方程验算。
5. Ratio and Proportion | 比与比例
reasoning to dividing quantities, map scales, and direct proportion. A question might say: “Share £120 in the ratio 3:5.” Total parts = 8, so each part = £15; amounts are £45 and £75. For direct proportion, if y ∝ x, write y = kx; find k from given values, then use it to find unknown.
练习册9将比的应用扩展到分配问题、地图比例尺和正比例。题目如：“将 £120 按 3:5 的比例分配。”总份数为 8，每份 £15，所以金额为 £45 和 £75。对于正比例关系 y ∝ x，写出 y = kx；先用已知值求出 k，再代入求未知量。
Best buys use unitary method: find price per unit or weight to compare. 300 g for £1.50 (0.5p/g) versus 500 g for £2.40 (0.48p/g) – the larger pack is cheaper per gram.
最佳购买问题使用单位化方法：比较每克价格。300克售价£1.50（0.5便士/克），500克售价£2.40（0.48便士/克）——大包装更便宜。
6. Sequences and Functions | 数列与函数
nces like 2, 6, 12, 20... second difference constant 2, leading to nth term = n² + n. Working backwards from a function machine to find the inverse function also features.
求一次和二次数列的通项公式是一项核心技能。一次数列如 5, 9, 13, 17... 公差为4，通项公式为 4n + 1。对于二次数列如 2, 6, 12, 20...，二阶差为常数2，最终通项为 n² + n。根据函数机器反推逆函数也是常见题型。
2, find f(4) = 10, or solve f(x) = 7 to get x = 3. Answers must show substitution clearly.
函数记法：若 f(x) = 3x − 2，求 f(4) = 10，或解方程 f(x) = 7 得 x = 3。作答时应清晰展示代入过程。
7. Geometry: Angles, Constructions | 几何：角与作图
ines, triangles, and polygons are examined. Corresponding angles, alternate angles, and co‑ terior angles sum to 360°. Constructions using a pair of compasses and a straight edge include angle bisectors and perpendicular bisectors – the answer booklet often shows step‑by‑step arcs.
本部分考查平行线、三角形和多边形的角性质。同位角、内错角和同旁内角出现在练习册图形题中。多边形内角和公式为 (n − 2) × 180°，外角和恒为 360°。使用圆规和直尺作图包括角平分线和垂直平分线——答案册中通常逐步画出弧线。
Bearings are measured clockwise from north; a bearing of 120° means turn 120° from north towards east. Ensure your protractor is correctly aligned.
方位角是从正北顺时针方向量度的角；方位角 120° 表示从北向东转120°。使用量角器时务必正确对齐。
8. Pythagoras’ Theorem | 勾股定理
Right‑ hypotenuse. Workbook questions ask to find a missing side or check if a triangle is right‑angled. Example: sides 9 cm, 12 cm, hypotenuse = √9² + 12² = √225 = 15 cm. When given hypotenuse and one leg, subtract: leg = √c² − a².
直角三角形中，a² + b² = c²，其中 c 为斜边。练习册题目要求求未知边长或判断三角形是否为直角三角形。例如：两直角边分别为9 cm和12 cm，斜边 = √9² + 12² = √225 = 15 cm。若已知斜边和一直角边，则另一直角边 = √c² − a²。
Applications include finding the distance between two points on a coordinate grid and solving real‑life problems like ladder lengths. Diagrams are essential; always sketch the right angle.
应用包括求坐标网格上两点之间的距离，以及解决实际生活中的梯子长度等问题。画图至关重要；务必标出直角。
9. Perimeter, Area and Volume | 周长、面积与体积
ten given in terms of π or rounded to 3 significant figures. Volume of prisms = area of cross‑section × length. Surface area of cylinders also appears.
组合图形的面积、圆的面积 (πr²) 和梯形的面积 ((a+b)h/2) 是常见考点。圆的周长为 2πr。当答案中含有 π，通常保留 π 或四舍五入至三位有效数字。棱柱的体积 = 横截面积 × 长度。圆柱体的表面积也会出现。
m², 1 L = 1000 cm³. A workbook problem might ask to convert cm³ to litres for a fish tank.
注意单位换算：1 m² = 10 000 cm²，1 L = 1000 cm³。练习册题目可能要求将鱼缸的体积从 cm³ 换算为升。
10. Transformations and Symmetry | 变换与对称
Reflections, rotations, translations, and enlargements are tested. Describing a transformation completely means stating: centre of rotation and angle, vector of translation, mirror line equation, or scale factor and centre of enlargement. Enlargement negative scale factors (e.g., −2) appear in Workbook 9.
反射、旋转、平移和放大是考查重点。完整描述一种变换需要指明：旋转的旋转中心和角度、平移的位移向量、反射的镜面直线方程，或者放大的比例系数和中心。练习册9中还会涉及负比例系数的放大（如 −2）。
Symmetry: line symmetry (reflection) and rotational symmetry of order n. A regular pentagon has 5 lines of symmetry and rotational symmetry of order 5.
对称性：线对称（反射对称）和旋转对称的阶数。正五边形有5条对称轴，旋转对称的阶为5。
11. Statistics: Charts and Averages | 统计：图表与平均数
Calculating mean, median, mode and range from frequency tables, grouped data, and stem‑and‑ al frequency, using midpoints. When the workbook answer gives a median from a frequency table, it uses the (n+1)/2 term position, then cumulative frequency to locate the group.
从频数表、分组数据和茎叶图计算平均数、中位数、众数和极差是重要考点。分组数据的平均数使用组中值估算，公式为估计总和 ÷ 总频数。练习册答案从频数表中求中位数时，先用 (n+1)/2 确定位置，再通过累积频数找到所在组。
Interpreting scatter graphs: correlation (positive, negative, none), line of best fit to make predictions. Outliers should be mentioned when present.
解读散点图：相关性（正相关、负相关、无相关），利用最佳拟合线进行预测。出现异常值时需进行说明。
12. Probability | 概率
lity is always between 0 and 1. The sum of probabilities of all outcomes is 1. For mutually exclusive events, P(A or B) = P(A) + P(B).
练习册9涵盖单一事件和组合事件的概率，包括样本空间图和可放回情况的树状图。概率值始终在 0 到 1 之间。所有可能结果的概率之和为 1。对于互斥事件，P(A 或 B) = P(A) + P(B)。
ind probability of at least one red. Use complementary probability 1 − P(no red) = 1 − (5/8)².
树状图中，沿分支相乘，各结果相加。典型题目：袋中有3红5蓝的弹珠，有放回抽取两次。求至少抽到一次红色的概率。使用互补概率：1 − P(无红色) = 1 − (5/8)²。
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