📚 Cambridge Lower Secondary Mathematics Workbook 9 Answers Guide | CIE 数学练习册9答案与解析指南
Cambridge Lower Secondary Mathematics Workbook 9 supports learners in Stage 9 of the Cambridge Lower Secondary curriculum, the final stage before IGCSE or O Level. This guide explains how to work through the answers, interpret marking points, apply the correct methods for every strand, and avoid the mistakes that cost marks most often. It does not replace the official mark scheme, but it gives structured worked examples and revision notes matched to typical Workbook 9 topics.
剑桥初中数学练习册9适用于剑桥初中课程第9阶段，也就是衔接 IGCSE 或 O Level 之前的最后一年。本指南讲解如何正确使用答案、理解得分点、掌握各知识板块的规范解法，并避开最常见的失分错误。它不会替代官方评分标准，但会提供结构化的例题解析和复习笔记，匹配练习册9的常见主题。
1. What Workbook 9 Covers and How Answers Are Structured | 练习册9的内容范围与答案结构
Workbook 9 mirrors the Stage 9 learner's book and provides extra practice across six strands: number, algebra, geometry and measure, statistics and probability, and mathematical thinking and working. Answers in the back of the book are usually short - often a single value, a simplified fraction or a coordinate pair - because space is limited. That means the answer key tells you the destination, not the route.
练习册9与第9阶段学生用书配套，在六个板块提供额外练习：数与运算、代数、几何与度量、统计与概率，以及数学思维与过程能力。书末答案通常很简短——往往只是一个数值、一个最简分数或一组坐标——因为篇幅有限。这意味着答案只是"终点"，并不展示"路线"。
Use this table to see which strand each chapter belongs to and what the answers usually look like:
可用下表判断每一章属于哪个板块，以及答案通常以什么形式出现：
| Strand 板块 | Typical answer form 典型答案形式 | Check you show 需要写出的过程 |
|---|---|---|
| Number 数与运算 | Value, index form, ratio 2 : 3 | Index laws, conversion steps 指数法则、换算步骤 |
| Algebra 代数 | Simplified expression, x = 4 | Each balancing step 每一步等式变形 |
| Geometry 几何 | Angle, length, area | Reason in brackets 括号内的理由 |
| Statistics 统计 | Mean, median, probability | Working and units 算式与单位 |
2. How to Use Workbook 9 Answers Effectively | 如何有效使用练习册9答案
Workbook 9 answers should be treated as a learning tool, not a shortcut. For every exercise, cover the answer column first and write your own solution. Then compare your working, not just the final value. If your final answer matches but the method is wrong, the mark scheme may still deduct marks because Cambridge assessment often rewards correct method.
练习册9答案应视为学习工具，而非捷径。每道练习先遮住答案栏，写下自己的解答。然后对比你的解题过程，而不仅仅是最终结果。若最终答案正确但方法错误，评分标准仍可能扣分，因为剑桥评估通常奖励正确的方法。
Use these habits when checking Workbook 9 answers:
- Attempt every question fully before you look at the key. 看答案之前先把每道题完整做完。
- Mark your own work in a different colour and annotate the reason for each correction. 用另一种颜色批改，并在每处订正旁写清原因。
- If your answer differs, ask whether the key used a different but valid method before assuming you are wrong. 若答案不同，先判断参考答案是否只是用了另一种同样正确的方法。
- Re-do incorrect questions after three days and again after two weeks. 做错的题在三天后和两周后各重做一次。
- Keep a one-page error log listing the topic, the slip and the fix. 用一页纸记录错题：主题、错因、改正方法。
A useful routine is the "three-pass method": first pass with books closed, second pass checking only the final answer, third pass comparing method line by line. The third pass is where most of the improvement happens, because it exposes the reasoning gaps that a quick answer check hides.
一个实用的做法是"三遍法"：第一遍合上书本独立完成；第二遍只核对最终答案；第三遍逐行对比解题方法。第三遍带来的提升最大，因为快速对答案会掩盖的推理漏洞，会在这一遍暴露出来。
3. Number: Indices, Roots and Standard Form | 数与运算：指数、方根与科学计数法
answers here are unforgiving: an index that is not fully simplified or a standard form value written as 32 × 10⁴ instead of 3.2 × 10⁵ is simply wrong, even if the underlying arithmetic is sound.
第9阶段的数与运算巩固指数法则，并延伸到负指数与零指数，同时结合科学计数法与估算。这一部分的答案容错率很低：指数没有化到最简，或把科学计数法写成 32 × 10⁴ 而不是 3.2 × 10⁵，即便算术本身没错，答案也是错的。
Learn these rules as a single family rather than four separate facts:
要把下面几条当作一个整体来记，而不是孤立地背四条规则：
bsp; aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1 ÷ aⁿ
7, so the answer is 4.7 × 10⁻⁴. Multiplying numbers in standard form means multiplying the A values and adding the indices, then re-adjusting if the new A value is 10 or more.
科学计数法始终写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。把 0.00047 转成科学计数法，小数点向右移动四位得到 4.7，因此答案是 4.7 × 10⁻⁴。两个科学计数法形式的数相乘时，A 值相乘、指数相加，若新的 A 值达到或超过 10，还要再调整一次。
Common errors to check against the answer key:
- Writing 3.2 × 10⁻⁵ as 3.2 × 10⁵ for small numbers. 小数写成负数指数时漏掉负号。
- Applying the power rule incorrectly, e.g. (2x³)² = 2x⁶ instead of 4x⁶. 幂的乘方漏平方系数。
- Rounding too early and then giving a final answer that is one unit out. 过早取近似导致最终结果偏差。
4. Fractions, Decimals, Percentages and Ratio | 分数、小数、百分数与比
This topic appears in almost every Workbook 9 chapter, usually as a supporting skill rather than a topic in itself. Answers must be given in the form requested: a fraction in lowest terms, a percentage to one decimal place, or a ratio with whole-number parts. Always re-read the instruction line before checking the key, because a correct value in the wrong form loses the mark.
这一主题几乎出现在练习册9的每一章中，通常作为支撑技能而非独立主题。答案必须按题目要求的形式给出：最简分数、保留一位小数的百分数，或整数项的比。核对答案前先重读指令句，因为数值正确但形式不符同样不得分。
Worked example: a jacket costs $80 and is reduced by 15%, then a further 10% off the sale price. Do not add the percentages.
例题：一件外套原价 80 美元，先降价 15%，再在折后价基础上再打九折。不要把两个百分数相加。
80 × 0.85 = 68 68 × 0.90 = 61.20
So the final price is $61.20, a total reduction of 23.5%, not 25% mpound percentage changes must always be applied in sequence using multipliers, where a 15% decrease means × 0.85 and a 10% decrease means × 0.90.
因此最终价格是 61.20 美元，总降幅为 23.5%，而不是 25%。复合百分数变化必须按顺序用乘数计算，降价 15% 对应 × 0.85，降价 10% 对应 × 0.90。
the form 1 : n or n : 1 to make comparison easy. If 3 : 5 is shared between two people for $240, one part is 240 ÷ 8 = $30, so the shares are $90 and $ ion uses y = k ÷ x. Identify which one the question describes before touching the numbers.
比的题目建议化为 1 : n 或 n : 1 的形式，便于比较。若 240 美元按 3 : 5 分配，则一份为 240 ÷ 8 = 30 美元，两方分别得 90 美元和 150 美元。正比例用 y = kx，反比例用 y = k ÷ x。动笔之前先判断题目属于哪一种关系。
5. Algebra: Expressions, Formulae and Rearranging | 代数：表达式、公式与移项
Stage 9 algebra answers are checked term by term. When the key gives 5x² − 7x + 2, it is testing whether you collected like terms correctly, not whether your arithmetic was quick. Expanding double brackets, factorising into a single bracket, and substituting into formulae are the three skills that appear most often.
第9阶段的代数答案要逐项核对。当参考答案给出 5x² − 7x + 2 时，它考查的是同类项合并是否正确，而不是你算得有多快。展开双括号、提取公因式分解，以及代入公式，是出现频率最高的三项技能。
Expanding (2x − 3)(x + 4) gives 2x² + 8x − 3x − 12, which simplifies to 2x² + 5x − 12. A frequent error is losing the sign on the negative term, producing 2x² − 11x − 12. Write out all four products before simplifying, even when you can do it mentally, until the accuracy is secure.
展开 (2x − 3)(x + 4) 得到 2x² + 8x − 3x − 12，化简为 2x² + 5x − 12。常见错误是丢掉负项的符号，写成 2x² − 11x − 12。在准确率稳定之前，即使能心算，也要把四项乘积完整写出来再化简。
ing rules as solving equations. To make r the subject of C = 2πr, divide both sides by 2π:
公式移项与解方程遵循相同的平衡规则。若要把 C = 2πr 变形为以 r 为主项，两边同时除以 2π：
r = C ÷ 2π
ubstitution questions should always be re-checked by reading your own expression aloud with the numbers in place.
若主项在式子中出现两次，先把它集中到一边，再提取公因式。例如由 ax + bx = y 提取得 x(a + b) = y，所以 x = y ÷ (a + b)。代入求值题做完后，把数字代回原式读一遍，是最有效的自查方法。
6. Equations, Inequalities and Simultaneous Equations | 方程、不等式与联立方程
Linear equations with brackets and fractions on both sides are standard in Workbook 9. The answer key usually shows only the final value of x, so your check is simple: substitute back into the original equation and confirm both sides are equal. Do this even when you are confident, because sign errors in the expansion step are extremely common.
两边都带括号或分数的线性方程是练习册9的标准题型。答案通常只给出 x 的最终值，所以自查很简单：代回原方程，确认两边相等。即使很有把握也要做这一步，因为展开环节的符号错误极为常见。
x² − 5x + 6 = 0 requires factorising into two brackets, then setting each factor to zero:
解 x² − 5x + 6 = 0 需要先分解为两个括号，再令每个因式为 0：
(x − 2)(x − 3) = 0 → x = 2 or x = 3
Always give both solutions. A single root in a quadratic answer is treated as incomplete. Inequalities behave like equations except that multiplying or dividing by a negative number reverses the inequality sign, so 3 − 2x > 7 becomes x < −2.
一定要写出两个解。二次方程只写一个根会被视为不完整。解不等式的步骤与解方程相同，但乘以或除以负数时不等号方向要反过来，因此 3 − 2x > 7 化简后是 x < −2。
For simultaneous equations, the elimination method is usually faster and less error-prone than substitution. Line up the two equations, scale one or both so a variable has matching coefficients, then add or subtract. Check the final pair in both original equations - if it satisfies only one, an arithmetic slip has occurred in the elimination step.
解联立方程时，加减消元法通常比代入法更快、更不易出错。把两个方程对齐，将其中一个或两个方程放大，使某个变量的系数相同，然后相加或相减。最后要把答案代回两个原方程检验——若只满足其中一个，说明消元过程中出现了计算失误。
7. Sequences, Functions and Graphs | 数列、函数与图像
Sequence questions in Stage 9 go beyond "add 3 each time". Learners must find the nth term of a linear sequence, recognise a simple quadratic sequence, and explain the rule in words. The coefficient of n is the common difference, and the constant term is adjusted from the first term.
第9阶段的数列题不再只是"每次加 3"。学生需要求线性数列的通项、识别简单的二次数列，并用语言说明规律。n 的系数就是公差，常数项则由首项调整得出。
the nth term starts as 4n. Substituting n = 1 gives 4, but the first term is 7, so add 3:
对于数列 7, 11, 15, 19, …，公差为 4，所以通项先写成 4n。代入 n = 1 得 4，但首项是 7，因此再加 3：
nth term = 4n + 3
al sequence. For quadratic sequences, the second difference is constant; halve it to get the coefficient of n², then subtract n² terms to find the remaining linear part.
求出的通项务必用 n = 1、n = 2、n = 3 代回与原数列核对。二次数列的二阶差分为常数，将其除以 2 即得 n² 的系数，再减去各 n² 项，剩下的就是线性部分。
he gradient and c is the y-intercept; the gradient of a line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁) ÷ (x₂ − x₁). Parallel lines share the same gradient, while perpendicular gradients multiply to −1. When a question asks you to "use your graph" ng the correct value from your own line, so a misplotted point can cost two marks at once.
图像题要求描点和标注准确。对于 y = mx + c，m 是斜率，c 是 y 轴截距；过点 (x₁, y₁) 与 (x₂, y₂) 的直线斜率为 (y₂ − y₁) ÷ (x₂ − x₁)。平行直线斜率相同，垂直直线斜率之积为 −1。题目若要求"利用你的图像求解"，分数是给从你自己画的直线上读取的正确数值，因此一个描错的点可能一次性丢掉两分。
8. Geometry: Angles, Polygons and Circles | 几何：角、多边形与圆
In geometry, the answer key gives the number, but the mark scheme also requires a reason. Answers written without brackets such as "(angles on a straight line)" may lose the reasoning mark even when the value is correct. Train yourself to write the reason at the same moment you write the number.
几何题中，答案给出的是数值，但评分标准还要求写出理由。若像"(同一直线上的角)"这样的括号理由缺失，即使数值正确也可能丢掉推理分。要养成写数值的同时就写理由的习惯。
Key angle facts to quote accurately:
- Angles on a straight line add to 180°. 同一直线上的角之和为 180°。
- Angles at a point add to 360°. 同一点的角之和为 360°。
- Alternate and corresponding angles are equal for parallel lines; co-interior angles add to 180°. 平行线中内错角与同位角相等，同旁内角之和为 180°。
- Exterior angle of a triangle equals the sum of the two opposite interior angles. 三角形的一个外角等于两个不相邻内角之和。
e exterior angles of any convex polygon always add to 360°, which makes the exterior route faster: for a regular octagon, each exterior angle is 360° ÷ 8 = 45°, so each interior angle is 135°.
对于多边形，n 边形的内角和为 (n − 2) × 180°；正多边形的每个内角等于内角和除以 n。任何凸多边形的外角和都是 360°，用外角计算更快：正八边形的每个外角为 360° ÷ 8 = 45°，因此每个内角为 135°。
Circle work at this stage focuses on vocabulary and measurement rather than theorems: radius, diameter, chord, arc, sector and tangent, together with circumference and area. Use π throughout and only round at the end, unless the question states otherwise.
这一阶段的圆主要考查术语与度量，而非定理：半径、直径、弦、弧、扇形和切线，以及周长与面积。计算时全程保留 π，除非题目另有要求，最后一步才取近似。
Circumference = πd = 2πr Area = πr²
9. Transformations, Constructions and Bearings | 变换、作图与方位角
Transformation questions are marked on the image, not on the description, so precision matters. Reflection must be the same distance behind the mirror line; rotation must preserve orientation with the centre and angle exactly as stated; translation uses a column vector; enlargement uses a scale factor from a stated centre and, at this stage, may include fractional or negative scale factors.
变换题是给图打分的，不是给文字描述打分的，所以精确度至关重要。轴对称的像与镜面线的距离必须相等；旋转必须保持方向，旋转中心与角度要与题目完全一致；平移用列向量表示；放大以给定中心按比例因子作图，这一阶段还可能涉及分数比例因子或负比例因子。
5. A common Workbook 9 error is drawing the correct shape but from the wrong centre, which usually scores zero for the whole question.
检查放大图时，比较两条对应边。若原图边长为 3 cm，像的对应边为 7.5 cm，则比例因子为 7.5 ÷ 3 = 2.5。练习册9中常见的错误是图形画对了但中心取错，这种情况通常整题不得分。
Constructions must be done with a sharp pencil and a pair of compasses, leaving all arc lines visible. Perpendicular bisectors and angle bisectors are the two constructions examined most often, and the arcs themselves carry the marks.
作图必须用削尖的铅笔和圆规完成，所有弧线都要保留。垂直平分线与角平分线是考查最多的两种作图，分数正是给在这些弧线上的。
Bearings are always three figures, measured clockwise from north, and written with a degree symbol, for example 047°. Back bearings are found by adding or subtracting 180°: the bearing of B from A is 047°, so the bearing of A from B is 227°. Scale drawing questions require you to state the scale used, such as 1 cm : 5 km, before measuring.
方位角一律写成三位数，从正北顺时针量取，并带度符号，例如 047°。反方位角通过加减 180° 得到：由 A 测 B 的方位角是 047°，则由 B 测 A 是 227°。比例尺作图题在测量前必须写明所用比例尺，例如 1 cm : 5 km。
10. Measure: Area, Volume, Surface Area and Compound Measures | 度量：面积、体积、表面积与复合单位
Measure questions are easy to lose marks on through unit slips rather than mathematical difficulty. Check that every length is in the same unit before substituting, and always write the correct unit on the final line. Area answers carry squared units such as cm², and volume answers carry cubed units such as m³.
度量题失分往往不是因为数学难度，而是单位疏漏。代入前先确认所有长度单位一致，最后一行务必写对单位。面积单位带平方，如 cm²；体积单位带立方，如 m³。
Formulae you should be able to quote without a formula sheet:
- Area of a triangle = ½ × base × perpendicular height. 三角形面积 = ½ × 底 × 垂直高。
- Area of a trapezium = ½ × (a + b) × h. 梯形面积 = ½ × (a + b) × h。
- Volume of a prism = cross-sectional area × length. 棱柱体积 = 横截面积 × 长度。
- Volume of a cylinder = πr²h. 圆柱体积 = πr²h。
Compound measures combine two quantities and are examined heavily in Stage 9. Speed, density and rates are all handled with the same triangle structure, so a single method covers all of them:
复合单位由两个量组合而成，是第9阶段的重点考查内容。速度、密度和各种比率都可以用同一个"三角形"结构处理，因此一种方法就能覆盖全部：
Speed = Distance ÷ Time Density = Mass ÷ Volume
ert between area units, square the linear conversion factor, so 1 m² = 10 000 cm², not 100 cm². For volume, cube it: 1 m³ = 1 000 000 cm³.
单位换算是丢分最集中的地方。把 km/h 换成 m/s，要乘以 1000 再除以 3600；因此 72 km/h 等于 20 m/s。面积单位换算要平方线性换算系数，所以 1 m² = 10 000 cm²，而不是 100 cm²。体积单位要立方：1 m³ = 1 000 000 cm³。
11. Statistics and Probability | 统计与概率
Statistics in Workbook 9 covers data collection, tables and charts, and the three averages with the range. The most common slip is confusing the mean with the median when data are unordered; always sort the list first. For an even number of values, the median is the mean of the two middle values.
练习册9的统计内容包括数据收集、图表，以及三种平均数和极差。最常见的失误是在数据未排序时把平均数与中位数混淆；务必先把数据排序。当数据个数为偶数时，中位数是中间两个数的平均值。
Compare averages using the three-step phrase that examiners expect: say which average is higher, give the values, then interpret what that means in context. For example: "Class A has a higher median score (72 compared with 65), so Class A performed better overall,
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