📚 Essential Maths Book 9H Answer Guide | 《Essential Maths 9H》答案解析与考点复习
This guide helps Year 9 students use the Essential Maths Book 9H answer key effectively. It focuses on the main topics, common question types and the working methods behind each answer rather than simply copying solutions.
本指南帮助 Year 9 学生有效使用《Essential Maths 9H》答案，重点讲解主要考点、常见题型以及每道答案背后的解题方法，而不是简单地抄写答案。
1. How to Use the Answer Key | 如何正确使用答案
First, attempt every question without looking at the answers. This builds recall and shows exactly which topics need more revision.
首先，尝试独立完成每道题，不要查看答案。这能训练记忆并准确显示哪些知识点需要更多复习。
After checking your work, mark any errors and redo those questions from scratch before reading the printed solution. Writing out the correct steps fixes errors in long-term memory.
检查答案后，标记所有错误，并在阅读印刷解答之前从头重新做这些题。写出正确步骤能帮助长期记忆纠错。
Finally, keep a short error log. Record the topic, the mistake and the correct method so you can review it before a test.
最后，保持一个简短的错题记录。记下知识点、错误原因和正确方法，方便考试前复习。
2. Number: Indices and Standard Form | 数与指数：标准形式
Year 9 higher work expects you to apply the laws of indices fluently. When multiplying powers with the same base, add the exponents.
Year 9 高阶数学要求你熟练运用指数法则。同底数幂相乘时，指数相加。
am × an = am+n
ividing powers with the same base, subtract the exponents. For example, 2⁷ ÷ 2³ = 2⁴ = 16.
同底数幂相除时，指数相减。例如，2⁷ ÷ 2³ = 2⁴ = 16。
as 5.6 × 10³, and 0.00042 is 4.2 × 10⁻⁴. Check that the first factor is between 1 and 10.
标准形式也是一项关键技能。例如 5,600 写作 5.6 × 10³，0.00042 写作 4.2 × 10⁻⁴。检查第一个因数是否在 1 到 10 之间。
3. Fractions, Decimals and Percentages | 分数、小数与百分比
een fractions, decimals and percentages without a calculator. For example, 3/4 = 0.75 = 75%.
你应该能够不借助计算器在分数、小数和百分比之间转换。例如，3/4 = 0.75 = 75%.
divide by a fraction, multiply by its reciprocal. So 5 ÷ 2/3 becomes 5 × 3/2 = 15/2 = 7.5.
除以一个分数时，乘以它的倒数。因此 5 ÷ 2/3 变成 5 × 3/2 = 15/2 = 7.5。
Percentage increase and decrease questions often appear in 9H answers. A 20% increase means multiplying by 1.2, while a 15% decrease means multiplying by 0.85.
百分比增加和减少的题目经常出现在 9H 答案中。增加 20% 表示乘以 1.2，减少 15% 表示乘以 0.85。
4. Ratio and Proportion | 比与比例
Simplify ratios by dividing all parts by the highest common factor. For example, 12:18 simplifies to 2:3.
通过将所有部分除以最大公因数来化简比。例如，12:18 化简为 2:3。
To share an amount in a ratio, find the total number of parts first. If £60 is shared in the ratio 2:3, the total parts are 5, so one part is £12, giving £24 and £36.
按比例分配金额时，先求出总份数。如果 £60 按 2:3 分配，总份数为 5，每份 £12，因此得到 £24 和 £36。
Direct proportion means that as one quantity increases, the other increases at the same rate. If 4 pens cost £3, then 10 pens cost £7.50.
正比例意味着一个量增加时，另一个量以相同速率增加。如果 4 支笔 £3，那么 10 支笔 £7.50。
5. Algebraic Expressions and Formulae | 代数表达式与公式
e 9H skill. Multiply each term inside the bracket by the term outside: 3(x + 4) = 3x + 12.
展开括号是 9H 的核心技能。将括号外的一项乘以括号内的每一项：3(x + 4) = 3x + 12。
highest common factor first, then recognise special cases such as x² − 9 = (x + 3)(x − 3).
因式分解是展开的逆运算。先找最高公因数，再识别特殊形式，如 x² − 9 = (x + 3)(x − 3)。
e order of operations. If v = u + at with u = 2, a = 3 and t = 4, then v = 2 + 3 × 4 = 14.
代入公式时需要仔细遵循运算顺序。如果 v = u + at，其中 u = 2、a = 3、t = 4，则 v = 2 + 3 × 4 = 14。
6. Linear Equations and Inequalities | 线性方程与不等式
For 2x + 3 = 11, subtract 3 from both sides to get 2x = 8, then divide by 2 to get x = 4.
解线性方程时，两边进行相同运算。对于 2x + 3 = 11，两边减 3 得 2x = 8，再除以 2 得 x = 4。
When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number. For example, −2x < 6 becomes x > −3.
解不等式时，如果乘以或除以负数，记得反转不等号。例如，−2x < 6 变为 x > −3。
Always check your solution by substituting it back into the original equation to make sure both sides balance.
始终将解代回原方程检验，确保两边相等。
7. Sequences and Graphs | 数列与图像
For an arithmetic sequence, find the nth term by identifying the common difference and the zero term. The sequence 5, 8, 11, 14 has nth term 3n + 2.
对于等差数列，通过确定公差和第零项来求第 n 项。数列 5, 8, 11, 14 的第 n 项是 3n + 2。
e y-intercept. In y = 2x + 1, the gradient is 2 and the line crosses the y-axis at (0, 1).
线性图像遵循 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。在 y = 2x + 1 中，斜率是 2，直线与 y 轴交于 (0, 1)。
To find the equation of a straight line from two points, first find the gradient using (y₂ − y₁)/(x₂ − x₁), then substitute one point to find c.
通过两点求直线方程时，先用 (y₂ − y₁)/(x₂ − x₁) 求斜率，再代入一点求 c。
8. Geometry: Angles and Shapes | 几何：角与图形
of interior angles of a polygon with n sides is (n − 2) × 180°. A quadrilateral has 360°.
n 边形内角和为 (n − 2) × 180°。四边形内角和为 360°。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°.
两条平行线被一条横截线所截时，同位角相等，内错角相等，同旁内角之和为 180°。
The exterior angles of any convex polygon always add up to 360°, which is useful for finding missing angles in regular polygons.
任何凸多边形的外角和始终为 360°，这对求正多边形中的未知角很有用。
9. Area, Volume and Pythagoras | 面积、体积与勾股定理
riangle area = ½ × base × height, trapezium area = ½ × (a + b) × h, and circle area = πr².
必须掌握常见面积公式：三角形面积 = ½ × 底 × 高，梯形面积 = ½ × (a + b) × 高，圆面积 = πr²。
is found by multiplying the cross-sectional area by the length. For a cylinder, V = πr²h.
棱柱体积等于横截面积乘以长度。对于圆柱，V = πr²h。
Pythagoras' ed triangles. If a and b are the shorter sides and c is the hypotenuse, then a² + b² = c².
勾股定理只适用于直角三角形。如果 a 和 b 是较短的边，c 是斜边，则 a² + b² = c²。
10. Transformations and Coordinates | 变换与坐标
A translation moves every point by the same vector. A vector (3, −2) means 3 units right and 2 units down.
平移使每个点按相同向量移动。向量 (3, −2) 表示向右 3 个单位、向下 2 个单位。
Reflections need a mirror line. A reflection in the x-axis changes the sign of the y-coordinate, and a reflection in the y-axis changes the sign of the x-coordinate.
反射需要一条对称轴。关于 x 轴反射会改变 y 坐标的正负号，关于 y 轴反射会改变 x 坐标的正负号。
Rotations require a centre, angle and direction. The most common is 90° clockwise or anticlockwise about the origin, which swaps coordinates and changes one sign depending on direction.
旋转需要旋转中心、角度和方向。最常见的是绕原点顺时针或逆时针旋转 90°，这会交换坐标并根据方向改变一个符号。
11. Probability | 概率
Probability is always between 0 and 1, inclusive. A probability of 0 means impossible, and 1 means certain.
概率始终在 0 到 1 之间，包括端点。概率为 0 表示不可能，1 表示必然发生。
the probabilities. The probability of getting two heads with two fair coins is ½ × ½ = ¼.
对于独立事件，要相乘概率。两枚均匀硬币同时出现正面的概率为 ½ × ½ = ¼。
Tree diagrams help when there are two or more stages. Write the probabilities on each branch and multiply along the path to find the total probability of that outcome.
树形图在有两个或多个阶段时很有帮助。在每条分支上写上概率，并沿着路径相乘得到该结果的总概率。
12. Statistics and Averages | 统计与平均数
and dividing by the number of values. For 3, 5, 5, 7, 10 the mean is (3+5+5+7+10) ÷ 5 = 6.
平均数是所有数值之和除以数值个数。对于 3, 5, 5, 7, 10，平均数为 (3+5+5+7+10) ÷ 5 = 6。
The median is the middle value when data is ordered. If there are two middle values, take their mean. The mode is the most frequent value, and the range is the largest minus the smallest.
中位数是数据排序后的中间值。如果有两个中间值，则取它们的平均数。众数是最常出现的数值，极差是最大值减最小值。
For grouped frequency tables, estimate the mean by using the midpoint of each class interval and multiplying by the frequency.
对于分组频率表，使用每个组距的中点乘以频数来估算平均数。
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