📚 Book 9 Answer Key: Preliminary Review | 第九册答案初步解析
This article provides a preliminary review of the main answer types in Book 9 mathematics. It covers core number, algebra, geometry, trigonometry, statistics, and probability skills, with worked examples and concise explanations.
本文提供第九册数学初步答案的复习指南。内容涵盖核心的数、代数、几何、三角、统计与概率技能，并配有例题和简要解析。
1. Number Operations and Estimation | 数的运算与估算
Book 9 preliminary exercises test your understanding of order of operations, directed numbers, percentages, and estimation. You should be able to convert between fractions, decimals, and percentages with confidence.
第九册初步练习考查运算顺序、有向数、百分数和估算。你应当能够熟练地在分数、小数和百分数之间进行转换。
1/3) × 12. First work out 3/4 − 1/3 = 9/12 − 4/12 = 5/12. Then multiplying by 12 gives 5.
示例：计算 (3/4 − 1/3) × 12。先算 3/4 − 1/3 = 9/12 − 4/12 = 5/12，再乘以 12 得到 5。
Estimation is also common: estimate 482 ÷ 0.49 to one significant figure. Round to 500 ÷ 0.5, giving an estimate of 1000.
估算也经常出现：将 482 ÷ 0.49 估算到一位有效数字。把式子近似为 500 ÷ 0.5，得到估算值 1000。
2. Algebraic Expressions and Simplification | 代数表达式与化简
You will meet questions that require expanding brackets and collecting like terms. For example, simplify 3(2x − 4) + 5x.
你会遇到要求展开括号并合并同类项的题目。例如，化简 3(2x − 4) + 5x。
The solution expands to 6x − 12 + 5x, which simplifies to 11x − 12.
解答过程为展开得到 6x − 12 + 5x，再合并同类项得到 11x − 12。
Another common skill is factorising linear expressions, such as writing 12ab − 8a as 4a(3b − 2).
另一个常见技能是分解线性表达式，例如将 12ab − 8a 写成 4a(3b − 2)。
3. Linear Equations and Inequalities | 线性方程与不等式
iables on one side. For the equation 2(x + 3) = 10, expanding gives 2x + 6 = 10, so x = 2.
解线性方程通常需要去括号并将变量移到一边。对于方程 2(x + 3) = 10，展开得 2x + 6 = 10，因此 x = 2。
ive number. For example, solving −2x ≤ 6 requires dividing by −2, so the answer is x ≥ −3.
对于不等式，请记住当乘以或除以负数时要反转不等号。例如，解 −2x ≤ 6 需要除以 −2，因此答案为 x ≥ −3。
4. Quadratic Equations | 二次方程
can be factorised. Solve x² − 5x + 6 = 0 by writing (x − 2)(x − 3) = 0, so x = 2 or x = 3.
初步练习可能包括可以因式分解的二次方程。解 x² − 5x + 6 = 0，写成 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
If factorisation is not straightforward, use the quadratic formula:
如果因式分解不容易，可以使用求根公式：
x = (−b ± √b² − 4ac) ÷ 2a
k that the expression is in the standard form ax² + bx + c = 0 before substituting values.
在代入数值之前，一定要先确认表达式为标准形式 ax² + bx + c = 0。
5. Coordinate Geometry and Graphs | 坐标几何与图像
dient of a straight line between two points and write the equation in the form y = mx + c.
你应当能够求出两点间直线的斜率，并写出 y = mx + c 形式的方程。
(3, 8), the gradient is m = (8 − 2)/(3 − 1) = 3. Substituting one point gives y = 3x − 1.
示例：对于点 (1, 2) 和 (3, 8)，斜率为 m = (8 − 2)/(3 − 1) = 3。代入其中一个点可得 y = 3x − 1。
You may also need to interpret graphs such as distance–time or speed–time graphs by finding gradients and areas under the curve.
你可能还需要解释距离–时间图或速度–时间图，通过求斜率和曲线下方的面积来解决问题。
6. Geometry: Angles and Polygons | 几何：角与多边形
umming to 360°, and the sum of interior angles of an n-sided polygon being (n − 2) × 180°.
关于角的基本事实包括：平角为 180°，周角为 360°，n 边形的内角和为 (n − 2) × 180°。
ntagon has an interior angle sum of (5 − 2) × 180° = 540°, so each interior angle is 108°.
示例：正五边形的内角和为 (5 − 2) × 180° = 540°，因此每个内角为 108°。
Exterior angles of any convex polygon always sum to 360°, which is useful for finding missing angles in regular polygons.
任何凸多边形的外角和总是 360°，这一性质对于求正多边形中的未知角非常有用。
7. Trigonometry Basics | 三角学基础
s sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.
直角三角形三角学需要掌握以下比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
iangle has an angle of 30° and hypotenuse 10 cm, the opposite side is 10 × sin 30° = 5 cm.
示例：若直角三角形一个角为 30°，斜边为 10 cm，则对边为 10 × sin 30° = 5 cm。
You should also know exact trigonometric values for 30°, 45°, and 60°, such as sin 45° = √2/2 and cos 60° = 1/2.
你还应当记住 30°、45° 和 60° 的精确三角函数值，例如 sin 45° = √2/2，cos 60° = 1/2。
8. Mensuration: Area and Volume | 测量：面积与体积
nclude area of a circle A = πr², circumference C = 2πr, and volume of a cylinder V = πr²h.
关键公式包括圆面积 A = πr²，圆周长 C = 2πr，圆柱体积 V = πr²h。
Example: A cylinder with radius 3 cm and height 5 cm has volume π × 3² × 5 = 45π cm³.
示例：一个底面半径为 3 cm、高为 5 cm 的圆柱，体积为 π × 3² × 5 = 45π cm³。
You may also need to work with surface area and composite shapes, so break down the shape into simpler parts before applying formulas.
你可能还需要处理表面积和组合图形，因此要先将其分解为简单图形再套用公式。
9. Probability | 概率
Probability is calculated as the number of favourable outcomes divided by the total number of outcomes. The sum of probabilities of all possible outcomes is always 1.
概率的计算方法是有利结果数除以总结果数。所有可能结果的概率之和始终为 1。
Example: A bag has 3 red and 2 blue marbles. P(red) = 3/5 and P(blue) = 2/5.
示例：袋中有 3 个红球和 2 个蓝球。P(红) = 3/5，P(蓝) = 2/5。
For combined events, use tree diagrams or sample space diagrams to list outcomes and find probabilities systematically.
对于组合事件，可使用树状图或样本空间图系统地列出结果并求出概率。
10. Statistics and Data Handling | 统计与数据处理
You may be asked to calculate the mean, median, mode, and range for a small data set, and to interpret cumulative frequency or box plots.
你可能需要计算小数据集的平均数、中位数、众数和极差，并解释累积频率图或箱线图。
12, the mean is (4+7+7+9+12)/5 = 7.8, the median is 7, the mode is 7, and the range is 8.
示例：对于数据集 4, 7, 7, 9, 12，平均数为 (4+7+7+9+12)/5 = 7.8，中位数为 7，众数为 7，极差为 8。
When working with grouped data, use the midpoint of each class interval to estimate the mean, and be careful to define modal class rather than a single mode.
在处理分组数据时，使用每个组区间的中值来估算平均数，并注意要确定众数组而不是单一众数。
11. Functions and Sequences | 函数与数列
an input and an output. For f(x) = 2x + 3, substituting x = 4 gives f(4) = 2(4) + 3 = 11.
函数描述输入与输出之间的关系。对于 f(x) = 2x + 3，代入 x = 4 可得 f(4) = 2(4) + 3 = 11。
Sequences include arithmetic progressions with a common difference. For the sequence 5, 9, 13, 17, the common difference is 4, so the nth term is 4n + 1.
数列包括具有公差的等差数列。对于数列 5, 9, 13, 17，公差为 4，因此第 n 项为 4n + 1。
substituting n = 1, n = 2, and n = 3 to make sure they generate the given terms correctly.
可以通过代入 n = 1、n = 2 和 n = 3 来检验第 n 项公式，确保公式能正确生成给定的项。
12. Transformation Geometry | 变换几何
Transformations include translations, reflections, rotations, and enlargements. You should describe them fully using vectors, mirror lines, centres, angles, or scale factors.
变换包括平移、反射、旋转和放大。你应当用向量、对称轴、中心、角度或比例因子完整地描述它们。
Example: A translation by vector (3, −2) moves every point 3 units right and 2 units down.
示例：按向量 (3, −2) 平移，将每个点向右移动 3 个单位、向下移动 2 个单位。
For enlargements, remember that if the scale factor is greater than 1 the shape becomes larger, and if it is between 0 and 1 the shape becomes smaller.
对于放大变换，请记住：比例因子大于 1 时图形变大，比例因子在 0 与 1 之间时图形变小。
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