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  • Year 9 Essential Maths Book 9S: Core Topics and Exam Practice | 九年级 Essential Maths Book 9S：核心主题与考试练习

    📚 Year 9 Essential Maths Book 9S: Core Topics and Exam Practice | 九年级 Essential Maths Book 9S：核心主题与考试练习

    This guide summarises the core skills in Essential Maths Book 9S. It focuses on the key Year 9 topics that build a strong foundation for GCSE Mathematics, including number work, algebra, geometry, statistics and probability.

    本指南总结了 Essential Maths Book 9S 中的核心技能。它聚焦九年级的关键主题，为 GCSE 数学奠定坚实基础，涵盖数字运算、代数、几何、统计与概率。

    1. Number Skills: Place Value, Decimals and Rounding | 数字技能：位值、小数与四舍五入

    Place value tells you the value of a digit by its position. In 4.37, the 3 is worth 3/10 and the 7 is worth 7/100.

    位值表示一个数字因其位置而具有的值。在 4.37 中，3 表示十分之三，7 表示百分之七。

    Rounding to decimal places means keeping a fixed number of digits after the decimal point. To round 3.467 to 2 d.p., look at the third decimal digit: it is 7, so round up to 3.47.

    四舍五入到小数位表示保留小数点后固定位数。将 3.467 保留到 2 位小数时，看第三位小数：它是 7，因此进位为 3.47。

    Rounding to significant figures focuses on the first non-zero digit. 3.467 to 2 s.f. is 3.5 because the third significant digit is 6, so the 4 rounds up.

    四舍五入到有效数字关注第一个非零数字。3.467 保留到 2 位有效数字为 3.5，因为第三位有效数字是 6，所以 4 进位。

    3.467 ≈ 3.47 (2 d.p.) and 3.467 ≈ 3.5 (2 s.f.)

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

    To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.

    要将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。

    Percentage change compares the actual change to the original amount. A percentage increase or decrease is always calculated using the original value.

    百分比变化将实际变化量与原始量进行比较。百分比增加或减少始终使用原始值进行计算。

    Percentage change = (new value − original value) ÷ original value × 100%

    For reverse percentage problems, divide by the original multiplier. If a price increases by 20% to £60, the multiplier is 1.2, so the original price was £60 ÷ 1.2 = £50.

    对于逆向百分比问题，除以原始乘数。如果价格增加 20% 后为 60 英镑，则乘数为 1.2，因此原始价格为 60 ÷ 1.2 = 50 英镑。

    3. Directed Numbers and Order of Operations | 有向数与运算顺序

    When adding or subtracting directed numbers, remember that subtracting a negative is the same as adding. For example, 5 − (−2) = 5 + 2 = 7.

    在加减有向数时，记住减去一个负数等同于加上一个正数。例如，5 − (−2) = 5 + 2 = 7。

    For multiplication and division, two same signs give a positive answer, and two different signs give a negative answer.

    对于乘法和除法，两个相同符号得正，两个不同符号得负。

    Always apply BIDMAS or BODMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. So 2 + 3 × 4 = 14, because multiplication comes before addition.

    始终应用 BIDMAS 或 BODMAS：括号、指数、除法和乘法、加法和减法。因此 2 + 3 × 4 = 14，因为乘法先于加法。

    −4 × −6 = 24 and 2 + 3 × 4 = 2 + 12 = 14

    4. Algebraic Expressions and Simplification | 代数表达式与化简

    Collect like terms by adding or subtracting the coefficients of the same variable. For example, 3x + 2y − x + 4y = 2x + 6y.

    通过加减相同变量的系数来合并同类项。例如，3x + 2y − x + 4y = 2x + 6y。

    Expanding brackets means multiplying each term inside the bracket by the term outside. To expand double brackets, multiply every term in the first bracket by every term in the second.

    展开括号意味着将括号内的每一项乘以括号外的项。展开双括号时，将第一个括号中的每一项乘以第二个括号中的每一项。

    a(b + c) = ab + ac and (x + 2)(x + 3) = x² + 5x + 6

    Factorising is the reverse of expanding. To factorise x² + 5x + 6, find two numbers that multiply to 6 and add to 5: these are 2 and 3.

    因式分解是展开的逆运算。要对 x² + 5x + 6 进行因式分解，找到两个相乘为 6 且相加为 5 的数：它们是 2 和 3。

    5. Solving Linear Equations | 解一元一次方程

    To solve a linear equation, use the balance method: do the same operation to both sides until the unknown is isolated.

    要解一元一次方程，使用平衡法：对方程两边执行相同的运算，直到未知数被单独分离出来。

    For 2x + 3 = 11, first subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.

    对于 2x + 3 = 11，首先两边同时减去 3 得到 2x = 8，然后两边同时除以 2 得到 x = 4。

    2x + 3 = 11 → 2x = 8 → x = 4

    If the equation has brackets, expand them first. If the unknown appears on both sides, collect the variable terms on one side before solving.

    如果方程含有括号，请先展开括号。如果未知数出现在两边，先将含变量的项移到一边再求解。

    6. Ratio, Proportion and Rates | 比、比例与速率

    To simplify a ratio, divide all parts by their highest common factor. The ratio 12:18 simplifies to 2:3.

    要化简比，将各部分除以它们的最大公因数。比 12:18 化简为 2:3。

    To share a quantity in a given ratio, find the total number of parts, divide the quantity by that total, then multiply by each part.

    按给定比例分配数量时，先求出总份数，将数量除以总份数，再乘以每份对应的份数。

    Ratio 2:3 with total £60 → 5 parts → one part = £12 → shares £24 and £36

    Direct proportion means that as one quantity increases, the other increases at the same rate. Use the unitary method: find the value of one unit first.

    正比例意味着一个量增加时，另一个量以相同速率增加。使用单位法：先求出一个单位的值。

    7. Angles and Properties of Shapes | 角与图形性质

    Key angle rules include: angles on a straight line sum to 180°, angles around a point sum to 360°, and the angles in a triangle sum to 180°.

    关键角度规则包括：直线上的角之和为 180°，一点周围的角之和为 360°，三角形内角之和为 180°。

    When parallel lines are cut by a transversal, alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°.

    当平行线被一条截线所截时，内错角相等，同位角相等，同旁内角之和为 180°。

    • Angles on a straight line: 直线上的角：sum = 180°
    • Angles around a point: 一点周围的角：sum = 360°
    • Triangle angle sum: 三角形内角和：sum = 180°
    • Quadrilateral angle sum: 四边形内角和：sum = 360°

    For any polygon, the sum of interior angles is (n − 2) × 180°, where n is the number of sides. The sum of exterior angles is always 360°.

    对于任意多边形，内角和为 (n − 2) × 180°，其中 n 是边数。外角和始终为 360°。

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

    The perimeter is the distance around the outside of a 2D shape. The area of a rectangle is length × width, and the area of a triangle is half the base times the vertical height.

    周长是二维图形外边缘的总长度。矩形的面积等于长 × 宽，三角形的面积等于底乘垂直高的一半。

    Rectangle area: A = l × w Triangle area: A = ½ × b × h

    For circles, the circumference is C = 2πr or πd, and the area is A = πr², where r is the radius.

    对于圆，周长为 C = 2πr 或 πd，面积为 A = πr²，其中 r 是半径。

    The volume of a cuboid is length × width × height. The volume of any prism is the area of the cross-section multiplied by the length.

    长方体的体积等于长 × 宽 × 高。任何棱柱的体积等于横截面积乘以长度。

    V = l × w × h and V = cross-sectional area × length

    9. Statistics: Averages and Charts | 统计：平均数与图表

    The mean is found by adding all values and dividing by the number of values. The median is the middle value, the mode is the most frequent value, and the range is the difference between the largest and smallest values.

    平均数的计算方法是将所有数值相加后除以数值的个数。中位数是中间值，众数是出现频率最高的值，极差是最大值与最小值之差。

    Mean = (sum of values) ÷ (number of values)

    Bar charts show frequency by the height of bars. Pie charts show proportions as angles, where each angle = (frequency ÷ total) × 360°.

    条形图通过条形的高度表示频数。饼图通过角度表示比例，每个角度 = (频数 ÷ 总数) × 360°。

    Scatter graphs show the relationship between two variables. If points slope upwards, there is a positive correlation; if they slope downwards, the correlation is negative.

    散点图显示两个变量之间的关系。如果点总体向上倾斜，则为正相关；如果向下倾斜，则为负相关。

    10. Probability Basics | 概率基础

    Probability is a number between 0 and 1, where 0 means impossible and 1 means certain. It can also be written as a fraction, decimal or percentage.

    概率是介于 0 和 1 之间的数，0 表示不可能，1 表示必然发生。它也可以写成分数、小数或百分比。

    P(event) = number of favourable outcomes ÷ total number of outcomes

    If all outcomes are equally likely, the probability of rolling a 4 on a fair six-sided die is 1/6.

    如果所有结果等可能，掷一枚公平的六面骰子得到 4 的概率是 1/6。

    The probabilities of all mutually exclusive outcomes add up to 1. So P(event not happening) = 1 − P(event happening).

    所有互斥事件的概率之和为 1。因此 P(事件不发生) = 1 − P(事件发生)。

    11. Coordinates and Linear Graphs | 坐标与线性图像

    Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. The origin is (0, 0).

    坐标写作 (x, y)，其中 x 是水平位置，y 是垂直位置。原点是 (0, 0)。

    A linear graph has the equation y = mx + c. m is the gradient, which measures steepness, and c is the y-intercept, where the line crosses the y-axis.

    线性图像具有方程 y = mx + c。m 是斜率，表示倾斜程度；c 是 y 轴截距，即直线与 y 轴的交点。

    y = 2x + 1 gradient = 2 y-intercept = 1

    To plot a linear graph, choose a table of x values, substitute them into the equation to find y, then plot the points and join them with a straight line.

    要绘制线性图像，先选定一组 x 值，将它们代入方程求出 y，然后描点并用直线连接。

    12. Transformations | 图形变换

    The four transformations are translation, reflection, rotation and enlargement. A translation moves a shape without turning or flipping it.

    四种图形变换是平移、反射、旋转和放大。平移是移动图形而不旋转或翻转它。

    A reflection flips a shape over a mirror line. A rotation turns a shape around a centre point by a given angle and direction.

    反射是使图形沿对称轴翻转。旋转是使图形绕中心点按给定角度和方向转动。

    An enlargement changes the size of a shape by a scale factor from a centre of enlargement. If the scale factor is greater than 1, the shape gets bigger; if it is between 0 and 1, the shape gets smaller.

    放大是使图形按比例因子相对于放大中心改变大小。如果比例因子大于 1，图形变大；如果在 0 和 1 之间，图形变小。

    Enlargement scale factor k: new length = k

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  • Essential Maths Book 9S Answers Compressed | Year 9 数学精华答案解析

    📚 Essential Maths Book 9S Answers Compressed | Year 9 数学精华答案解析

    This compressed revision guide supports independent study with the Year 9 Essential Maths Book 9S. The answers and worked solutions below are parallel examples, not page-by-page reproductions, so you can check the same skills and correct your own working.

    本精华复习指南配合 Year 9 Essential Maths Book 9S 自主学习使用。下列答案与详解为平行例题，并非原书逐页复制，目的是帮助你核查同类技能并纠正自己的解题过程。

    1. Using the 9S Answers as a Revision Tool | 如何将 9S 答案用作复习工具

    Always attempt the exercise before looking at the answer. Write down your working, not just the final number, because method marks matter in school tests and later IGCSE/GCSE papers.

    在查看答案之前，一定要先独立完成练习。把解题过程写下来，而不只是最终答案，因为在学校测验和之后的 IGCSE/GCSE 考试中，步骤分同样重要。

    When you check an answer, mark the question right or wrong, then diagnose the error. If you made a sign mistake, redo the last three lines; if you misunderstood a word problem, underline the key information and try a similar question.

    核对答案时，把题目判为对或错，然后诊断错误原因。如果出现符号错误，重做最后三行；如果没读懂文字题，就圈出关键信息并再练一道同类题。

    2. Number Skills: Fractions and Decimals | 数字技能：分数与小数

    Key skills include adding and subtracting fractions with different denominators, multiplying and dividing mixed numbers, and converting terminating decimals to fractions.

    核心技能包括不同分母分数的加减、带分数的乘除，以及有限小数化成分数。

    Worked example: Calculate 2/3 + 1/4, giving your answer as a fraction.

    例题：计算 2/3 + 1/4，答案写成分数。

    Solution: The lowest common denominator of 3 and 4 is 12. Write 2/3 = 8/12 and

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  • Year 9 Higher Homework Compressed: Core Mathematics Revision | 九年级高阶作业浓缩：核心数学复习

    📚 Year 9 Higher Homework Compressed: Core Mathematics Revision | 九年级高阶作业浓缩：核心数学复习

    This revision guide compresses the key topics from a typical Year 9 Higher homework pack into one focused resource. You will revisit algebra, equations, fractions, ratio, graphs, indices, geometry and probability, with straightforward methods and examples.

    本复习指南将九年级高阶作业包中的核心主题压缩为一份重点资源。你将复习代数、方程、分数、比、图像、指数、几何和概率，并配有简洁的方法和示例。

    1. Algebraic Simplification | 代数化简

    Simplify by collecting like terms. Terms with the same variable and power can be added or subtracted. For example, 5x + 3y − 2x + 7y = 3x + 10y.

    通过合并同类项进行化简。具有相同变量和次数的项可以相加或相减。例如 5x + 3y − 2x + 7y = 3x + 10y。

    Always handle signs carefully. Multiplying terms uses index laws: x × x² = x³.

    始终小心处理符号。项与项相乘使用指数法则：x × x² = x³。

    5x + 3y − 2x + 7y = 3x + 10y

    2. Solving Linear Equations | 解一元一次方程

    To solve an equation, do the same operation to both sides. For 3x − 5 = 16, add 5 then divide by 3 to get x = 7.

    解方程时，对方程两边进行相同运算。对于 3x − 5 = 16，先加 5 再除以 3，得到 x = 7。

    If variables appear on both sides, collect them first. For 5x + 2 = 2x + 11, subtract 2x and subtract 2 to find 3x = 9, so x = 3.

    如果变量出现在两边，先合并变量。对于 5x + 2 = 2x + 11，减去 2x 并减去 2，得到 3x = 9，所以 x = 3。

    3x − 5 = 16 → x = 7

    3. Expanding and Factorising | 展开与因式分解

    Expand brackets by multiplying each term inside by the factor outside: 3(x + 4) = 3x + 12.

    展开括号时，将括号内的每一项乘以外面的因数：3(x + 4) = 3x + 12。

    Factorising reverses the process. Look for the highest common factor: 6x + 9 = 3(2x + 3).

    因式分解是展开的逆过程。找出最高公因数：6x + 9 = 3(2x + 3)。

    Double brackets expand to three terms: (x + 2)(x + 5) = x² + 7x + 10.

    双括号展开得到三项：(x + 2)(x + 5) = x² + 7x + 10。

    (x + 2)(x + 5) = x² + 7x + 10

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

    Convert between forms by remembering key equivalences: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%.

    通过记住关键等值关系在形式之间转换：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%。

    To add fractions, use a common denominator: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

    分数相加时使用公分母：1/3 + 1/4 = 4/12 + 3/12 = 7/12。

    Find a percentage of a quantity by multiplying by the decimal or fraction equivalent: 15% of 60 = 0.15 × 60 = 9.

    求一个量的百分比，乘以对应的小数或分数：60 的 15% = 0.15 × 60 = 9。

    1/3 + 1/4 = 4/12 + 3/12 = 7/12

    5. Ratio and Proportion | 比与比例

    Simplify ratios by dividing all parts by a common factor, e.g. 12:18 = 2:3.

    通过将各部分除以公因数来化简比，例如 12:18 = 2:3。

    Share a quantity in a ratio by finding the total number of parts. For £50 in a 2:3 ratio, total parts = 5, so each part = £10, giving £20 and £30.

    按比例分配数量时，先求总份数。将 50 英镑按 2:3 分配，总份数 = 5，每份 = 10 英镑，因此得到 20 英镑和 30 英镑。

    Direct proportion means when one quantity doubles, the other doubles too. Use a multiplier method to solve problems.

    正比例意味着一个量翻倍，另一个量也翻倍。使用倍率法解决问题。

    12:18 = 2:3

    6. Linear Graphs and Gradient | 直线图像与斜率

    The equation of a straight line is often y = mx + c, where m is the gradient and c is the y-intercept.

    直线方程通常为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。

    Gradient measures steepness: gradient = change in y ÷ change in x. A positive gradient slopes upward.

    斜率衡量倾斜程度：斜率 = y 的变化量 ÷ x 的变化量。正斜率向上倾斜。

    Parallel lines have the same gradient, for example y = 2x + 1 and y = 2x − 4.

    平行线具有相同的斜率，例如 y = 2x + 1 和 y = 2x − 4。

    y = mx + c

    7. Indices and Standard Form | 指数与标准形式

    Index laws simplify products and powers: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.

    指数法则可简化乘积和幂：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。

    Any non-zero number to the power 0 is 1: 5⁰ = 1. A negative index gives a reciprocal: 2⁻³ = 1/2³ = 1/8.

    任何非零数的 0 次方为 1：5⁰ = 1。负指数表示倒数：2⁻³ = 1/2³ = 1/8。

    Standard form writes numbers as a × 10ⁿ where 1 ≤ a < 10. For example, 4500 = 4.5 × 10³.

    标准形式将数字写作 a × 10ⁿ，其中 1 ≤ a < 10。例如 4500 = 4.5 × 10³。

    aᵐ × aⁿ = aᵐ⁺ⁿ

    8. Angles in Polygons | 多边形内角

    The sum of interior angles of an n-sided polygon is (n − 2) × 180°. A pentagon has sum 540°.

    n 边形内角和为 (n − 2) × 180°。五边形内角和为 540°。

    The sum of exterior angles of any polygon is always 360°. Each exterior angle of a regular polygon is 360° ÷ n.

    任何多边形外角和始终为 360°。正多边形每个外角为 360° ÷ n。

    Angles on a straight line add to 180°, and angles around a point add to 360°.

    直线上的角相加为 180°，围绕一点的角相加为 360°。

    Sum of interior angles = (n − 2) × 180°

    9. Area, Perimeter and Volume | 面积、周长与体积

    Area of a rectangle is length × width; triangle is 1/2 × base × height; trapezium is 1/2 × (a + b) × h.

    矩形面积 = 长 × 宽；三角形面积 = 1/2 × 底 × 高；梯形面积 = 1/2 × (a + b) × 高。

    Volume of a cube is side³; cuboid is length × width × height; prism is area of cross-section × length.

    正方体体积 = 边长³；长方体体积 = 长 × 宽 × 高；棱柱体积 = 横截面积 × 长度。

    Circumference of a circle = 2πr and area = πr². Use π ≈ 3.14 or leave answers in terms of π.

    圆的周长 = 2πr，面积 = πr²。可使用 π ≈ 3.14 或用 π 表示答案。

    A = πr²

    10. Probability of Combined Events | 组合事件概率

    Probability of an event = number of favourable outcomes ÷ total number of outcomes. Probabilities always range from 0 to 1.

    事件的概率 = 有利结果数 ÷ 总结果数。概率总是在 0 到 1 之间。

    For independent events, multiply probabilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add probabilities.

    对于独立事件，概率相乘：P(A 与 B) = P(A) × P(B)。对于互斥事件，概率相加。

    A tree diagram helps list outcomes for two or more events. The sum of probabilities on branches from a point is 1.

    树状图有助于列出两个或多个事件的结果。从一点出发的各分支概率之和为 1。

    P(A and B) = P(A) × P(B)

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  • Year 9 Essential Maths Book 9i: Core Skills | Year 9 基础数学核心技能

    📚 Year 9 Essential Maths Book 9i: Core Skills | Year 9 基础数学核心技能

    This revision guide reviews the key topics in the Year 9 Essential Maths course, with a focus on the skills students need for confident problem solving. Each section includes a short explanation and a worked example.

    本复习指南回顾 Year 9 基础数学课程的关键主题，重点放在学生解题所需的核心技能。每节都包含简短说明和解题示例。

    1. Number Skills and BIDMAS | 数的运算与 BIDMAS 规则

    Always apply operations in the correct order: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).

    始终按照正确顺序进行运算：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。

    Example: 3 + 4 × 2 = 3 + 8 = 11, not 14.

    示例：3 + 4 × 2 = 3 + 8 = 11，而不是 14。

    Negative numbers follow clear rules: adding a negative is subtraction; subtracting a negative is addition; multiplying or dividing two negatives gives a positive.

    负数遵循明确规则：加上负数等于减法；减去负数等于加法；两个负数相乘或相除得正数。

    Worked example: (-2) × (-3) + 5 = 6 + 5 = 11.

    解答示例：(-2) × (-3) + 5 = 6 + 5 = 11。

    2. Fractions, Decimals and Percentages | 分数、小数和百分数

    To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.

    将分数转换为小数，用分子除以分母；将小数转换为百分数，乘以 100。

    Example: 3/4 = 0.75 = 75%. In reverse, 45% = 45/100 = 9/20.

    示例：3/4 = 0.75 = 75%。反过来，45% = 45/100 = 9/20。

    When adding or subtracting fractions, find a common denominator first. For multiplication, multiply the numerators and denominators separately.

    分数加减时，先找到公分母；分数相乘时，分子乘分子，分母乘分母。

    Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.

    示例：2/3 + 1/4 = 8/12 + 3/12 = 11/12。

    3. Ratio and Proportion | 比与比例

    A ratio compares parts of a whole. To share an amount in a ratio, add the parts to find the total number of shares, then divide the amount by this total.

    比用于比较整体的各部分。按比例分配数量时，先将各部分相加得到总份数，再用总量除以总份数。

    Example: Divide £120 in the ratio 3:5. Total parts = 8, one part = £15, so the shares are £45 and £75.

    示例：将 £120 按 3:5 分配。总份数 = 8，每份 = £15，因此份额为 £45 和 £75。

    Proportion problems often involve scaling up or down. If 3 pens cost £2.10, one pen costs £0.70 and 5 pens cost £3.50.

    比例问题常涉及放大或缩小。如果 3 支笔售价 £2.10，一支笔为 £0.70，5 支笔为 £3.50。

    4. Algebra: Simplifying and Expanding | 代数：化简与展开

    Collect like terms by combining terms that have the same variable and power. Constants combine separately.

    合并同类项时，将具有相同变量和幂的项合并，常数项单独合并。

    Example: 4a + 3b – 2a + 5b simplifies to 2a + 8b.

    示例：4a + 3b – 2a + 5b 化简为 2a + 8b。

    Expand brackets using the distributive law: a(b + c) = ab + ac. Double brackets follow: (x + a)(x + b) = x² + (a + b)x + ab.

    用分配律展开括号：a(b + c) = ab + ac。双括号展开遵循：(x + a)(x + b) = x² + (a + b)x + ab。

    Example: 3(x + 4) = 3x + 12; (x + 2)(x + 5) = x² + 7x + 10.

    示例：3(x + 4) = 3x + 12；(x + 2)(x + 5) = x² + 7x + 10。

    5. Solving Linear Equations | 解一元一次方程

    Use inverse operations to isolate the unknown. Always keep the equation balanced by doing the same operation to both sides.

    使用逆运算求未知数。始终对方程两边进行相同操作以保持平衡。

    Example: 2x + 5 = 13. Subtract 5: 2x = 8. Divide by 2: x = 4.

    示例：2x + 5 = 13。两边减 5：2x = 8。除以 2：x = 4。

    When the unknown appears on both sides, collect like terms first. 5x – 3 = 2x + 6 becomes 3x = 9, so x = 3.

    当未知数出现在方程两边时，先合并同类项。5x – 3 = 2x + 6 变为 3x = 9，所以 x = 3。

    6. Sequences and the nth Term | 数列与第 n 项

    An arithmetic sequence has a constant difference between consecutive terms. The nth term can be written as an + b, where a is the common difference.

    等差数列的相邻项之差为常数。第 n 项可以写成 an + b，其中 a 是公差。

    Example: 4, 7, 10, 13 has common difference 3. The nth term is 3n + 1 because when n = 1, 3(1) + 1 = 4.

    示例：数列 4, 7, 10, 13 的公差为 3。第 n 项为 3n + 1，因为当 n = 1 时，3(1) + 1 = 4。

    Use the nth term to find any term, such as the 20th term: 3(20) + 1 = 61.

    利用第 n 项可求任意项，如第 20 项：3(20) + 1 = 61。

    7. Angles and Polygons | 角与多边形

    Angles on a straight line sum to 180°, angles around a point sum to 360°, and the three angles in a triangle sum to 180°.

    直线上的角之和为 180°，围绕一点的角之和为 360°，三角形内角和为 180°。

    The sum of interior angles of a polygon with n sides is (n – 2) × 180°. For example, a pentagon has 5 sides, so its interior angles sum to (5 – 2)

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  • Essential Maths Book 9i Answers | Year 9 数学 Essential Maths Book 9i 答案与解析

    📚 Essential Maths Book 9i Answers | Year 9 数学 Essential Maths Book 9i 答案与解析

    For many Year 9 students, the Essential Maths Book 9i is a core practice book that builds confidence across number, algebra, geometry and statistics. Using the answer pages well is not about copying; it is about checking your method, locating the exact step where you went wrong, and learning to self-correct before your next assessment.

    对于许多 Year 9 学生来说，Essential Maths Book 9i 是一本核心练习册，帮助大家在数、代数、几何和统计方面建立信心。合理使用答案页面不是为了抄答案，而是检查你的解题方法，找到出错的具体步骤，并在下一次评估前学会自我纠正。

    1. Understanding the Book 9i Structure | 了解 Book 9i 的结构

    Book 9i is typically organised into units that mix fluency, reasoning and problem-solving questions. Each chapter ends with a review exercise, and the answers section at the back is usually split by exercise number. Before checking answers, write down the page and question number so you can revisit weak topics later.

    Book 9i 通常按单元编排，混合了流利度、推理和问题解决题型。每一章末尾有复习练习，书后的答案部分通常按练习编号划分。在核对答案之前，写下页码和题号，这样你之后可以重新复习薄弱知识点。

    • Always attempt the question fully before looking at the answer. / 在查看答案之前，一定要先完整地尝试题目。
    • Mark your own work with a tick or cross, then correct errors in a different colour. / 用勾或叉批改自己的作业，然后用不同颜色订正错误。
    • If your answer differs, compare the method, not just the final number. / 如果答案不同，要比较方法，而不仅仅是最终数字。

    2. Integers, Decimals and Negative Numbers | 整数、小数与负数

    This unit covers the four operations with integers and decimals, plus directed numbers. A common task is to evaluate expressions such as −7 + 12 or (−3) × 4. Remember that adding a negative is the same as subtracting: a + (−b) = a − b. Multiplying two negative numbers gives a positive product, while a negative times a positive gives a negative product.

    本单元涵盖整数和小数的四则运算，以及有向数。常见题目是计算如 −7 + 12 或 (−3) × 4 这样的表达式。记住加一个负数等于减去它的相反数：a + (−b) = a − b。两个负数相乘得正，负数乘正数得负。

    Example: Evaluate 8 − (−5). Since subtracting a negative is the same as adding, 8 − (−5) = 8 + 5 = 13. / 示例：计算 8 − (−5)。因为减去负数等于加上相反数，所以 8 − (−5) = 8 + 5 = 13。

    For decimals, keep place value aligned when adding or subtracting, and count decimal places when multiplying. / 对于小数，加减时要对齐数位，乘法时要数小数位数。

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

    Book 9i expects you to convert freely between fractions, decimals and percentages. For example, ¾ = 0.75 = 75%. To find a percentage of an amount, convert the percentage to a decimal and multiply: 15% of 240 = 0.15 × 240 = 36. For fraction addition and subtraction, find a common denominator first.

    Book 9i 要求能够在分数、小数和百分比之间自由转换。例如，¾ = 0.75 = 75%。要求一个数量的百分比，先把百分比转为小数再相乘：240 的 15% = 0.15 × 240 = 36。分数加减时，先找公分母。

    Worked example: Convert 0.35 to a fraction in its simplest form. 0.35 = 35/100 = 7/20. / 解题示例：将 0.35 化为最简分数。0.35 = 35/100 = 7/20。

    When comparing quantities, change them all to the same form, such as decimals, to avoid mistakes. / 在比较数量时，把它们都化为同一种形式，例如小数，以避免出错。

    4. Ratio and Proportion | 比率与比例

    Ratio questions in Book 9i often ask you to simplify, share an amount, or use a scale factor. To simplify 12:18, divide both sides by the highest common factor, 6, giving 2:3. For sharing, add the parts: sharing £60 in the ratio 3:2 gives 3 + 2 = 5 parts, so one part is £12, and the shares are £36 and £24.

    Book 9i 中的比率题常要求化简、按比例分配或使用比例因子。化简 12:18 时，两边同时除以最大公因数 6，得到 2:3。分配时先把份数相加：按 3:2 分 60 英镑，总份数为 3 + 2 = 5 份，每份是 12 英镑，所以分得 36 英镑和 24 英镑。

    • Check that the units are the same before simplifying a ratio. / 化简比率前，确保单位相同。
    • For proportion, use a multiplier or the unitary method to find a missing value. / 对于比例，使用倍数法或单位法求未知值。

    5. Algebraic Expressions and Equations | 代数表达式与方程

    Algebra work in Year 9 includes expanding brackets, collecting like terms, and solving linear equations. For expanding: 3(x + 4) = 3x + 12. Collect like terms: 5a + 2b + 3a = 8a + 2b. To solve 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2 to get x = 4.

    Year 9 的代数学习包括去括号、合并同类项和解一元一次方程。去括号：3(x + 4) = 3x + 12。合并同类项：5a + 2b + 3a = 8a + 2b。解方程 2x + 5 = 13 时，两边同时减 5 得到 2x = 8，再除以 2 得到 x = 4。

    Worked example: Solve 4(y − 2) = 12. Expand the bracket: 4y − 8 = 12, add 8: 4y = 20, divide by 4: y = 5. / 解题示例：解方程 4(y − 2) = 12。去括号：4y − 8 = 12，加 8：4y = 20，除以 4：y = 5。

    Always write each step on a new line so that the examiner or your teacher can follow your logic. / 每一步换行书写，这样考官或老师能跟上你的逻辑。

    6. Sequences and Functions | 数列与函数

    For linear sequences, find the common difference and use it to write the nth term. The sequence 5, 9, 13, 17, … has a common difference of 4, so the nth term is 4n + 1. For a function machine, apply the operations in the correct order: input 3 into x → 2x + 1 gives output 7.

    对于等差数列，找出公差并写成第 n 项。数列 5, 9, 13, 17, … 的公差是 4，因此第 n 项为 4n + 1。对于函数机器，按正确顺序执行运算：输入 3 到 x → 2x + 1 得到输出 7。

    To check a sequence formula, substitute n = 1, 2, 3 and compare with the given terms. / 检查数列公式时，代入 n = 1、2、3 并与给定项比较。

    Some Book 9i questions ask you to find the rule from a table of values; draw a small table and look for the difference between consecutive outputs. / Book 9i 中的一些题目要求从数值表中找出规律；画一个小表，观察相邻输出值之间的差。

    7. Geometry: Angles, Area and Volume | 几何：角、面积与体积

    Key angle facts include: angles on a straight line add to 180°, angles around a point add to 360°, and the interior angles of a triangle add to 180°. For area, use A = l × w for a rectangle, A = ½ × b × h for a triangle, and A = πr² for a circle. Volume of a cuboid is V = l × w × h, and volume of a prism is area of cross-section × length.

    关键角度事实包括：直线上的角之和为 180°，点周围的角之和为 360°，三角形内角和为 180°。面积方面，矩形 A = 长 × 宽，三角形 A = ½ × 底 × 高，圆 A = πr²。长方体体积 V = 长 × 宽 × 高，棱柱体积 = 截面面积 × 长度。

    Worked example: Find the area of a triangle with base 8 cm and height 5 cm. A = ½ × 8 × 5 = 20 cm². / 解题示例：求底为 8 cm、高为 5 cm 的三角形面积。A = ½ × 8 × 5 = 20 cm²。

    Remember to include units in your final answer and to square or cube the unit for area or volume. / 最终答案要带单位，面积用平方单位，体积用立方单位。

    8. Statistics and Probability | 统计与概率

    Statistics questions ask for the mean, median, mode and range. The mean is the sum of values divided by the number of values. The median is the middle value when data are ordered; if there are two middle numbers, take their mean. Probability is calculated as number of favourable outcomes ÷ total number of outcomes, and it always lies between 0 and 1.

    统计题要求计算平均数、中位数、众数和极差。平均数等于数值之和除以数值个数。中位数是数据排序后的中间值；如果中间有两个数，取它们的平均数。概率计算为有利结果数 ÷ 总结果数，其值始终在 0 到 1 之间。

    When data are grouped, read the question carefully to see whether you need to estimate the mean using midpoints. / 当数据是分组数据时，仔细读题，看是否需要使用组中点来估算平均数。

    For probability, write answers as a fraction, decimal or percentage but simplify fractions where possible. / 概率可以写成分数、小数或百分比，但分数要尽量化简。

    9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

    One frequent error is dropping the negative sign when solving equations or working with directed numbers. Another is mixing up area and perimeter formulas. A third mistake is forgetting to simplify fractions or ratios in the final answer. To catch these

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  • Essential Maths Book 9H Answer Guide | 《Essential Maths 9H》答案解析与考点复习

    📚 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

    When dividing 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。

    Standard form is also a key skill. A number such as 5,600 is written 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

    You should be able to convert between 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%.

    To 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

    Expanding brackets is a core 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。

    Factorising reverses expansion. Look for the 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)。

    Substituting into formulae requires careful use of the 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

    To solve a linear equation, perform the same operation on both sides. 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。

    Linear graphs follow the form y = mx + c, where m is the gradient and c is the 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

    The sum 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

    Common area formulas must be known: triangle 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。

    Volume of a prism 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’ theorem applies only to right-angled 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。

    For independent events, multiply 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

    The mean is found by adding all values 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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  • Essential Maths Book 9F: Year 9 Mathematics Core Topics and Practice | 《Essential Maths 9F》Year 9 数学核心考点与练习精讲

    📚 Essential Maths Book 9F: Year 9 Mathematics Core Topics and Practice | 《Essential Maths 9F》Year 9 数学核心考点与练习精讲

    Welcome to this Essential Maths Book 9F revision guide. It brings together the key skills that Year 9 students need to master: number work, algebra, ratio, geometry, graphs and probability. Use it for quick review, homework support and end-of-topic tests.

    欢迎使用本《Essential Maths 9F》复习指南。它汇总了 Year 9 学生必须掌握的核心技能：数与运算、代数、比与比例、几何、图像以及概率。可用于快速复习、作业辅导和单元测验备考。

    1. Number and Place Value | 数系与位值

    Year 9 builds on earlier number work by extending place value to very large and very small numbers using standard form. You must be able to round to a given number of decimal places or significant figures, and use estimation to check answers.

    Year 9 在之前的基础上进一步扩展位值，使用标准形式表示极大和极小的数。你需要会将数精确到指定的小数位或有效数字，并利用估算来检查答案。

    Example: Write 4 500 000 in standard form as 4.5 × 10⁶. Round 3.276 to 2 decimal places gives 3.28. Estimate 48.7 × 9.2 by using 50 × 9 = 450.

    例如：将 4 500 000 写成标准形式为 4.5 × 10⁶。将 3.276 精确到两位小数为 3.28。估算 48.7 × 9.2 时可使用 50 × 9 = 450。

    Standard form also uses negative powers for very small numbers. Write 0.00042 as 4.2 × 10⁻⁴. Significant figures count from the first non-zero digit, so 0.00476 rounded to 2 significant figures is 0.0048.

    标准形式也使用负指数表示非常小的数。将 0.00042 写作 4.2 × 10⁻⁴。有效数字从第一个非零数字算起，所以 0.00476 精确到两位有效数字为 0.0048。

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

    Converting between fractions, decimals and percentages is essential. You also need to find a percentage of an amount, increase or decrease by a percentage, and solve reverse percentage problems.

    分数、小数和百分数之间的转换至关重要。你还需要求一个数的百分之几、按百分比增加或减少，并解决逆向百分数问题。

    Fraction Decimal Percentage
    1/2 0.5 50%
    3/4 0.75 75%
    1/3 0.333… 33⅓%

    For percentage change, use the formula: percentage change = (change ÷ original amount) × 100%. For example, a price rising from £40 to £48 is an increase of 20%.

    对于百分比变化，使用公式：百分比变化 = (变化量 ÷ 原量) × 100%。例如，价格从 £40 上涨到 £48，涨幅为 20%。

    Reverse percentage problems work backwards. If a coat is reduced by 20% and now costs £48, the original price was £48 ÷ 0.8 = £60. Always identify whether you are working forward or backward before calculating.

    逆向百分数问题需要倒推。如果一件外套降价 20% 后现价为 £48，那么原价是 £48 ÷ 0.8 = £

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  • Essential Maths Book 9C Answers Compressed: Year 9 Key Topic Guide | Year 9 数学 Essential Maths 9C 答案压缩精讲

    📚 Essential Maths Book 9C Answers Compressed: Year 9 Key Topic Guide | Year 9 数学 Essential Maths 9C 答案压缩精讲

    This guide breaks down the main topics covered in Essential Maths Book 9C so that checking answers becomes a genuine revision task, not just copying numbers.

    本指南拆解 Essential Maths Book 9C 的主要知识模块，让核对答案成为真正的复习过程，而不是简单抄写数字。

    1. Understanding the 9C Syllabus and Answer Book | 了解 9C 大纲与答案册

    Essential Maths Book 9C is designed for Year 9 students who are working at the higher end of Key Stage 3 and preparing for the transition to GCSE.

    Essential Maths Book 9C 面向 Year 9 中较高水平的学生，旨在帮助他们从 Key Stage 3 平稳过渡到 GCSE。

    The answer section usually gives final values only, so you should always show the steps that lead to those values in your exercise book.

    答案部分通常只给出最终结果，因此你应当在练习本中完整写出得到这些结果的过程。

    A compressed answer file simply collects these results in a shorter form; it is useful for quick checking but should never replace worked solutions.

    压缩答案文件只是把这些结果以更简短的形式汇总，适合快速核对，但不能代替完整解题步骤。

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

    To convert a fraction to a decimal, divide the numerator by the denominator; for example, 3/8 = 3 ÷ 8 = 0.375.

    把分数化为小数，用分子除以分母即可；例如 3/8 = 3 ÷ 8 = 0.375。

    To convert a decimal to a percentage, multiply by 100; so 0.375 becomes 37.5%.

    把小数化为百分数，乘以 100；因此 0.375 变为 37.5%。

    For percentage increase, multiply the original amount by 1 plus the rate: a 15% increase gives new value = original × 1.15.

    计算百分数增加时，用原数乘以 1 加百分率：增加 15% 后，新值 = 原数 × 1.15。

    For percentage decrease, multiply by 1 minus the rate: a 20% decrease gives new value = original × 0.80.

    计算百分数减少时，用原数乘以 1 减百分率：减少 20% 后，新值 = 原数 × 0.80。

    Fraction → Decimal → Percentage: 3/8 = 0.375 = 37.5%

    百分数增减公式：新值 = 原值 × (1 ± 百分率)

    3. Ratio and Proportion | 比与比例

    A ratio compares parts of a whole; for example, a ratio 2:3 means the whole is split into 2 + 3 = 5 equal parts.

    比用来表示整体中各部分的关系；例如 2:3 表示整体被分成 2 + 3 = 5 等份。

    To share £60 in the ratio 2:3, one part is 60 ÷ 5 = £12, so the shares are 2 × 12 = £24 and 3 × 12 = £36.

    按 2:3 分配 £60，一份是 60 ÷ 5 = £12，因此两部分分别为 2 × 12 = £24 和 3 × 12 = £36。

    Direct proportion problems often use the unitary method: find the value of one unit first, then multiply by the required number of units.

    正比例问题常用单位法：先求出一个单位的值，再乘以所需单位数。

    For example, if 5 pens cost £3.50, one pen costs £0.70, so 8 pens cost 8 × £0.70 = £5.60.

    例如，5 支笔售价 £3.50，一支笔为 £0.70，因此 8 支笔为 8 × £0.70 = £5.60。

    4. Algebra: Expressions and Equations | 代数：表达式与方程

    Simplify expressions by collecting like terms: 3x + 2y + 5x – y = 8x + y.

    化简代数式要合并同类项：3x + 2y + 5x – y = 8x + y。

    To solve an equation, perform the same operation on both sides; for 3x + 5 = 20, subtract 5 to get 3x = 15, then divide by 3 to get x = 5.

    解方程时，两边进行相同运算：对于 3x + 5 = 20，先减 5 得 3x = 15，再除以 3 得 x = 5。

    Expand brackets by multiplying each term inside: (x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6.

    展开括号要把括号内每一项分别相乘：(x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6。

    Factorising reverses expanding; for x² + 5x + 6, find two numbers that multiply to 6 and add to 5, giving (x + 2)(x + 3).

    因式分解是展开的逆运算；对于 x² + 5x + 6，找到乘积为 6、和为 5 的两个数，得到 (x + 2)(

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  • Year 9 Essential Maths 9H Homework Companion | 九年级 Essential Maths 9H 作业书精要

    📚 Year 9 Essential Maths 9H Homework Companion | 九年级 Essential Maths 9H 作业书精要

    This revision guide summarises the key topics found in the Year 9 Essential Maths 9H Homework Book. It is designed to help you consolidate higher-tier skills through clear explanations, worked examples and useful formulas. Each section pairs an English explanation with a Chinese translation, so you can study in either language or check your understanding of the terminology.

    本复习指南总结了九年级 Essential Maths 9H 作业书中的核心主题。它通过清晰的讲解、例题和实用公式帮助你巩固高阶技能。每个小节都提供中英文对照，方便你使用任一语言学习，或检查自己对术语的理解。

    1. Number Skills and Index Laws | 数与指数法则

    In 9H, index laws are tested with negative and fractional powers, not just positive integers. You must be fluent in multiplying and dividing powers with the same base, raising a power to a power, and converting between root and fractional index forms. This underpins standard form, algebraic simplification and sequences.

    在 9H 课程中，指数法则不仅考查正整数幂，还涉及负指数和分数指数。你必须熟练进行同底数幂的乘除、幂的乘方，以及根式与分数指数之间的互化。这些技能是科学记数法、代数化简和数列部分的重要基础。

    aᵐ × aⁿ = a

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  • Year 9 Essential Maths 9H: Core Skills and Higher Applications | Year 9 数学 9H 核心技能与高阶应用

    📚 Year 9 Essential Maths 9H: Core Skills and Higher Applications | Year 9 数学 9H 核心技能与高阶应用

    This revision guide covers the core Year 9 Higher topics from Essential Maths 9H, building strong foundations for GCSE. Work through each section, practise the worked examples and check your understanding with past-paper style questions.

    本复习指南涵盖 Essential Maths 9H 的 Year 9 高阶核心主题，为 GCSE 打下坚实基础。逐节学习，练习示例，并用真题风格的问题检验理解。

    1. Integers, Powers and Roots | 整数、幂与根

    In Year 9 Higher, you must be confident with negative numbers and the rules of indices. When multiplying powers with the same base, add the indices; when raising a power to another power, multiply the indices; a negative index means a reciprocal.

    在 Year 9 高阶课程中，你必须熟练掌握负数运算和指数法则。同底数幂相乘时指数相加；幂的乘方时指数相乘；负指数表示取倒数。

    aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ a⁻ⁿ = 1/aⁿ

    Square roots and cube roots can be estimated between integer bounds. For example, √50 lies between 7 and 8 because 7² = 49 and 8² = 64. You should also know that √a × √b = √(ab) and be able to simplify surds such as √50 = 5√2.

    平方根和立方根可以在整数界限之间估算。例如 √50 介于 7 和 8 之间，因为 7² = 49 而 8² = 64。你还需要知道 √a × √b = √(ab)，并能化简根式，如 √50 = 5√2。

    √a × √b = √(ab) √50 = 5√2

    When working with standard form, a number is written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. This is essential for very large and very small numbers in science and maths.

    使用科学计数法时，一个数写作 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。这对表示非常大和非常小的数在科学和数学中至关重要。

    A × 10ⁿ 1 ≤ A < 10

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

    Adding or subtracting fractions requires a common denominator. For example, 2/3 + 1/4 = 8/12 + 3/12 = 11/12. For multiplication, multiply the numerators and denominators directly; for division, multiply by the reciprocal.

    分数加减需要通分。例如 2/3 + 1/4 = 8/12 + 3/12 = 11/12。分数乘法直接分子乘分子、分母乘分母；分数除法乘以倒数。

    2/3 + 1/4 = 8/12 + 3/12 = 11/12

    Converting between fractions, decimals and percentages is a core skill. A percentage increase or decrease uses a multiplier: increase by r% means multiply by (1 + r/100), and decrease by r% means multiply by (1 − r/100).

    分数、小数和百分比之间的转换是核心技能。百分比增减使用乘数：增加 r% 表示乘以 (1 + r/100)，减少 r% 表示乘以 (1 − r/100)。

    增加 r% → × (1 + r/100) 减少 r% → × (1 − r/100)

    Compound percentage changes, such as interest or repeated discounts, are calculated by applying the multiplier repeatedly. For example, increasing £200 by 5% for 3 years gives 200 × 1.05³.

    复合百分比变化，如利息或重复折扣，通过反复应用乘数来计算。例如，200 英镑以 5% 增长 3 年得到 200 × 1.05³。

    £200 × 1.05³

    3. Ratio and Proportion | 比与比例

    A ratio compares parts of a whole and can be simplified by dividing all parts by a common factor. If the ratio of boys to girls is 3:5, the total number of parts is 3 + 5 = 8.

    比率比较整体的各个部分，可以通过将所有部分除以公因数来化简。如果男生与女生的比是 3:5，则总份数为 3 + 5 = 8。

    3:5 → 总份数 = 3 + 5 = 8

    To divide a quantity in a given ratio, find the value of one part first. For example, to share £240 in the ratio 3:5, one part is £

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  • Year 9 Core Homework Compressed: Essential Skills and Worked Examples | 九年级核心作业精编：核心技能与例题解析

    📚 Year 9 Core Homework Compressed: Essential Skills and Worked Examples | 九年级核心作业精编：核心技能与例题解析

    This revision guide compresses the Year 9 core homework programme into ten essential skill areas. It gives clear rules, worked examples and common pitfalls so that you can complete homework accurately and revise efficiently.

    这份复习指南把九年级核心作业计划浓缩为十个核心技能领域。它给出清晰规则、例题和常见易错点，帮助你准确完成作业并高效复习。

    1. Number and Place Value | 数与位值

    In Year 9 core homework, you must be confident with positive and negative numbers, powers of 10, and standard form. For example, 3.2 × 10⁴ means 3.2 × 10,000, which equals 32,000. When multiplying by 10ⁿ, move the decimal point n places to the right; when dividing, move it n places to the left.

    在九年级核心作业中，你需要熟练掌握正负数、10 的幂和标准形式。例如，3.2 × 10⁴ 表示 3.2 × 10,000，等于 32,000。乘以 10ⁿ 时，小数点向右移动 n 位；除以 10ⁿ 时，小数点向左移动 n 位。

    A common error is writing 0.00056 as 5.6 × 10⁻³ instead of 5.6 × 10⁻⁴. Count the places from the first non-zero digit, not from the decimal point. With negative indices, the number is smaller than 1.

    一个常见错误是把 0.00056 写成 5.6 × 10⁻³，而正确应为 5.6 × 10⁻⁴。要从第一个非零数字开始数位数，而不是从小数点开始数。负指数表示该数小于 1。

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

    To compare fractions, decimals and percentages, convert them to the same form. For example, ⅗ = 0.6 = 60%. To find a percentage of an amount, multiply the amount by the percentage as a decimal: 15% of £80 = 0.15 × 80 = £12.

    比较分数、小数和百分比时，先把它们化成同一种形式。例如，⅗ = 0.6 = 60%。求一个数的百分比时，用该数乘以百分比化成的小数：80 英镑的 15% = 0.15 × 80 = 12 英镑。

    Percentage increase and decrease often appear in homework. A jumper priced at £40 with a 25% decrease becomes £40 × (1 − 0.25) = £30. To reverse a percentage change, divide by the multiplier.

    百分比的增加和减少经常出现在作业中。一件标价 40 英镑的毛衣降价 25%，变为 40 × (1 − 0.25) = 30 英镑。要逆向求原价，则除以对应的乘数。

    3. Ratio and Proportion | 比与比例

    A ratio compares parts of a whole. The ratio 3:2 means 3 + 2 = 5 equal parts in total. If a drink is mixed in the ratio 3:2 and the total volume is 600 ml, one part is 600 ÷ 5 = 120 ml, so the parts are 360 ml and 240 ml.

    比用来比较整体中的各个部分。3:2 表示总共有 3 + 2 = 5 等份。如果一种饮料按 3:2 混合，总体积为 600 毫升，则一份是 600 ÷ 5 = 120 毫升，因此两部分分别为 360 毫升和 240 毫升。

    Direct proportion means that as one quantity increases, the other increases at the same rate. If 5 pens cost £3, then 20 pens cost £3 × (20 ÷ 5) = £12 because the number of pens is multiplied by 4.

    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 5 支笔花费 3 英镑，那么 20 支笔花费 3 × (20 ÷ 5) = 12 英镑，因为笔的数量乘了 4。

    4. Algebraic Expressions | 代数表达式

    Simplify expressions by collecting like terms. For example, 5a + 3b − 2a + 7b = 3a + 10b. Only terms with exactly the same letter part can be combined; a and a² are not like terms.

    通过合并同类项来化简表达式。例如，5a + 3b − 2a + 7b = 3a + 10b。只有字母部分完全相同的项才能合并；a 和 a² 不是同类项。

    Expanding brackets uses the distributive law: 4(2x + 3) = 8x + 12. For double brackets, (x + 3)(x + 2) = x² + 5x + 6. Remember to multiply every term in the first bracket by every term in the second.

    去括号使用分配律：4(2x + 3) = 8x + 12。对于双括号，(x + 3)(x + 2) = x² + 5x + 6。记住要把第一个括号中的每一项与第二个括号中的每一项相乘。

    5. Solving Linear Equations | 解一元一次方程

    To solve an equation, keep it balanced by doing the same operation to both sides. For 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.

    解方程时，要对两边进行相同的运算以保持等式平衡。对于 3x + 5 = 20，两边先减 5 得到 3x = 15，然后两边除以 3 得到 x = 5。

    If the unknown appears on both sides, collect the variable terms on one side. Example: 5x − 2 = 2x + 7. Subtract 2x from both sides: 3x − 2 = 7. Add 2: 3x = 9, so x = 3.

    如果未知数出现在等号两边，就把含未知数的项移到一边。例如：5x − 2 = 2x + 7。两边减去 2x：3x − 2 = 7。再加 2：3x = 9，所以 x = 3。

    6. Coordinates and Linear Graphs | 坐标与线性图像

    The coordinate (x, y) gives the horizontal position first and the vertical position second. The midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). For (2, 3) and (6, 7), the midpoint is (4, 5).

    坐标 (x, y) 先给出水平位置，再给出竖直位置。两点 (x₁, y₁) 和 (x₂, y₂) 的中点坐标是 ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)。对于 (2, 3) 和 (6, 7)，中点为 (4, 5)。

    A straight line has equation y = mx + c, where m is the gradient and c is the y-intercept. If y = 2x + 3, the gradient is 2 and the line crosses the y-axis at (0, 3).

    直线的方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。如果 y = 2x + 3，则斜率为 2，图像与 y 轴交于 (0, 3)。

    The gradient between two points is the change in y divided by the change in x: m = (y₂ − y₁) ÷ (x₂ − x₁). This is often remembered as rise over run.

    两点间的斜率等于 y 的变化量除以 x 的变化量：m = (y₂ − y₁) ÷ (x₂ − x₁)。这通常记作铅垂升高除以水平距离。

    7. Geometry: Area and Perimeter | 几何：面积与周长

    Perimeter is the distance around a shape. For a rectangle with length l and width w, perimeter P = 2l + 2w. Area is the space inside: A = l × w. Always include units such as cm, m² or cm².

    周长是围绕图形一周的长度。长为 l、宽为 w 的长方形，周长 P = 2l + 2w。面积是图形内部的大小：A = l × w。始终要写单位，如 cm、m² 或 cm²。

    The area of a triangle is A = ½ × base × height. The height must be perpendicular to the base. For a base of 8 cm and height of 5 cm, A = ½ × 8 × 5 = 20 cm².

    三角形的面积公式为 A = ½ × 底 × 高。高必须与底垂直。底为 8 厘米、高为 5 厘米时，A = ½ × 8 × 5 = 20 平方厘米。

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

    梯形的面积公式为 A = ½ × (a + b) × h，其中 a 和 b 是两条平行边，h 是垂直高度。

    8. Angles and Properties of Shapes | 角与图形的性质

    On a straight line, angles add to 180°. Around a point, angles add to 360°. In a triangle, the interior angles add to 180°. These facts allow you to find missing angles without measuring.

    直线上的角相加为 180°。一个点周围的角相加为 360°。三角形内角相加为 180°。利用这些事实，你可以不测量就求出缺失的角。

    In a parallelogram, opposite angles are equal and co-interior angles add to 180°. If one angle of a parallelogram is 65°, the angle next to it is 180° − 65° = 115°.

    在平行四边形中，对角相等，同旁内角相加为 180°。如果平行四边形的一个角为 65°，与它相邻的角就是 180° − 65° = 115°。

    Alternate angles and corresponding angles are equal when two parallel lines are cut by a transversal. Recognising these angle pairs is useful for proving that lines are parallel.

    当两条平行线被一条截线所截时，内错角相等，同位角相等。识别这些角对有助于证明两条直线平行。

    9. Statistics and Probability | 统计与概率

    The mean is the sum of values divided by the number of values. For 4, 7, 8, 10, 11, the mean is (4 + 7 + 8 + 10 + 11) ÷ 5 = 40 ÷ 5 = 8. The median is the middle value when the data are ordered.

    平均数等于所有数值之和除以数值个数。对于 4、7、8、10、11，平均数为 (4 + 7 + 8 + 10 + 11) ÷ 5 = 40 ÷ 5 = 8。中位数是数据按顺序排列后位于中间的值。

    Probability is measured from 0 to 1. P(event) = number of favourable outcomes ÷ total number of outcomes. If a bag has 3 red, 2 blue and 5 green counters, P(red) = 3 ÷ 10 = 0.3.

    概率的取值范围是 0 到 1。P(事件) = 有利结果的数量 ÷ 所有结果的总数。如果一个袋子里有 3 个红球、2 个蓝球和 5 个绿球，则 P(红球) = 3 ÷ 10 = 0.3。

    The probability of an event not happening is 1 − P(event). For example, if P(rain) = 0.4, then P(no rain) = 1 − 0.4 = 0.6.

    某事件不发生的概率是 1 − P(事件)。例如，如果 P(下雨) = 0.4，那么 P(不下雨) = 1 − 0.4 = 0.6。

    10. Problem-Solving Strategies | 解题策略

    Read the problem twice and underline key numbers and units. Write down what you know, what you need to find, and a plan. For multi-step problems, break them into smaller, manageable steps.

    把题目读两遍，并在关键数字和单位下画线。写下已知条件、要求解的量以及求解计划。对于多步骤问题，把它拆分成若干较小、可处理的步骤。

    Estimate before calculating and check your answer makes sense. For example, if you calculate a speed of 1200 km/h for a bicycle, you have probably mixed up units or placed a decimal point incorrectly.

    计算前先估算，并检查答案是否合理。例如，如果你算出一辆自行车的速度为 1200 km/h，那你很可能把单位弄混了或小数点位置写错了。

    Use diagrams, tables or algebra to organise information. When stuck, try a simpler version of the problem with smaller numbers, then apply the same logic to the original question.

    用图表、表格或代数来组织信息。遇到困难时，可以先用更小的数字简化问题，然后把同样的逻辑应用到原题上。

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  • Essential Maths 9H Homework Answers: Topic Review and Worked Solutions | Essential Maths 9H 作业答案：主题复习与详细解答

    📚 Essential Maths 9H Homework Answers: Topic Review and Worked Solutions | Essential Maths 9H 作业答案：主题复习与详细解答

    This article is a topic-by-topic guide to the types of homework questions found in Essential Maths 9H. We do not reproduce complete answer keys, but we show how to solve representative problems and how to check your own work.

    本文按主题梳理 Essential Maths 9H 中的常见作业题型。我们不复制完整答案，但会示范如何解答典型题目，以及如何检查自己的作业。

    1. Number and Place Value | 数与位值

    In Year 9 Higher work, you must be confident with the order of operations (BIDMAS/BODMAS), powers and roots. A typical homework question asks you to evaluate an expression such as 3² × (4 + 5) ÷ 3.

    在 Year 9 高阶课程中，你必须熟练掌握运算顺序（BIDMAS/BODMAS）、幂与方根。常见的作业题目会要求你计算类似 3² × (4 + 5) ÷ 3 的表达式。

    Worked answer: 3² = 9, 4 + 5 = 9, so 9 × 9 = 81, then 81 ÷ 3 = 27.

    解答：3² = 9，4 + 5 = 9，所以 9 × 9 = 81，然后 81 ÷ 3 = 27。

    Also check rounding and significant figures: for example, 0.004738 to 2 significant figures is 0.0047 because the first non-zero digit is 4 and the next digit is 7.

    还要检查四舍五入与有效数字：例如 0.004738 保留 2 位有效数字是 0.0047，因为第一个非零数字是 4，下一位是 7。

    Standard form is often tested: 45000 = 4.5 × 10⁴ and 0.0032 = 3.2 × 10⁻³. To multiply numbers in standard form, multiply the front numbers and add the powers of 10.

    标准形式也常考：45000 = 4.5 × 10⁴，0.0032 = 3.2 × 10⁻³。计算标准形式乘法时，先乘前部的数，再把 10 的指数相加。

    2³ × 2² = 2⁵ = 32 and (3²)³ = 3⁶ = 729

    Remember that a negative power means a reciprocal: 10⁻² = 1/100 and 5⁻¹ = 1/5.

    请记住，负指数表示倒数：10⁻² = 1/100，5⁻¹ = 1/5。

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

    Fractions, decimals and percentages are tested together. To add 2/3 and 1/4, write both over a common denominator: 2/3 = 8/12 and 1/4 = 3/12, so the sum is 11/12.

    分数、小数和百分比经常综合考查。计算 2/3 + 1/4 时，先通分：2/3 = 8/12，1/4 = 3/12，所以和为 11/12。

    To multiply fractions, multiply the numerators and denominators separately: 2/5 × 3/4 = 6/20 = 3/10. To divide, flip the second fraction and multiply: 3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10.

    分数乘法：分子乘分子，分母乘分母：2/5 × 3/4 = 6/20 = 3/10。分数除法：把第二个分数取倒数后相乘：3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10。

    A common percentage increase question: increase £240 by 15%. Multiply by 1.15: 240 × 1.15 = 276, so the new amount is £276.

    常见的百分比增加题：将 240 英镑增加 15%。乘以 1.15：240 × 1.15 = 276，所以新金额为 276 英镑。

    To decrease by 20%, multiply by 0.80 because 100% − 20% = 80%. A £60 item after a 20% decrease costs 60 × 0.80 = £48.

    减少 20% 时乘以 0.80，因为 100% − 20% = 80%。原价 60 英镑的商品减少 20% 后价格为 60 × 0.80 = 48 英镑。

    Convert a recurring decimal to a fraction when needed: for example, 0.375 = 375/1000 = 3/8 after dividing by 125.

    需要时将小数化为分数：例如 0.375 = 375/1000，分子分母同除以 125 得 3/8。

    3. Ratio and Proportion | 比与比例

    Ratio questions often ask you to share an amount in a given ratio. To share £56 in the ratio 3 : 5, first add the parts: 3 + 5 = 8 parts. One part is 56 ÷ 8 = 7, so the shares are 3 × 7 = £21 and 5 × 7 = £35.

    比例题常要求按给定比例分配金额。将 56 英镑按 3 : 5 分配，先求总份数：3 + 5 = 8 份。一份是 56 ÷ 8 = 7，所以分配为 3 × 7 = 21 英镑和 5 × 7 = 35 英镑。

    Always simplify a ratio like 24 : 36 by dividing both sides by 12: the simplified ratio is 2 : 3.

    化简比例 24 : 36 时，两边同时除以 12：化简后为 2 : 3。

    A

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  • Year 9 Maths Support Answers: Worked Examples for Core Skills | Year 9 数学支持答案：核心技能的分步示例

    📚 Year 9 Maths Support Answers: Worked Examples for Core Skills | Year 9 数学支持答案：核心技能的分步示例

    This support-answers guide for Year 9 maths walks through common question types and the working you need to show. Use it to check method, not just final answers.

    本 Year 9 数学支持答案指南逐步讲解常见题型与需要展示的解题过程，用于检查方法而不仅仅是最终答案。

    1. Negative Numbers and Order of Operations | 负数与运算顺序

    Worked question: Evaluate (-3) × 4 + 12 ÷ (-2) – (-5).

    例题：计算 (-3) × 4 + 12 ÷ (-2) – (-5)。

    Support answer: Use BIDMAS. Multiplication gives -12; division gives -6. Now work left to right: -12 + (-6) = -18, then -18 – (-5) = -18 + 5 = -13.

    支持答案：使用 BIDMAS 规则。乘法得 -12；除法得 -6。然后从左到右计算：-12 + (-6) = -18，接着 -18 – (-5) = -18 + 5 = -13。

    (-3) × 4 + 12 ÷ (-2) – (-5) = -13

    Common mistake: writing -18 – (-5) as -23. Subtracting a negative is adding the positive value.

    常见错误：把 -18 – (-5) 写成 -23。减去一个负数等于加上它的相反数。

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

    Worked question: Write 7/8 as a decimal and a percentage. Support answer: 7 ÷ 8 = 0.875; percentage = 0.875 × 100 = 87.5%.

    例题：将 7/8 写成小数和百分比。支持答案：7 ÷ 8 = 0.875；百分比 = 0.875 × 100 = 87.5%。

    Worked question: Order 3/5, 0.58 and 62% from smallest to largest. Convert all to percentages: 3/5 = 60%, 0.58 = 58%, 62% stays 62%. Order: 0.58, 3/5, 62%.

    例题：将 3/5、0.58 和 62% 从小到大排列。全部转换为百分比：3/5 = 60%，0.58 = 58%，62% 不变。顺序为：0.58、3/5、62%。

    Remember to use a common form when comparing; decimals are often fastest for calculator use.

    比较时要使用统一形式；使用计算器时，小数通常最快。

    3. Algebraic Simplification and Expansion | 代数化简与展开

    Worked question: Expand and simplify 3(2x – 1) + 2(x + 4).

    例题：展开并化简 3(2x – 1) + 2(x + 4)。

    Support answer: Multiply out each bracket: 6x – 3 + 2x + 8. Collect like terms: 6x + 2x = 8x and -3 + 8 = 5, so the answer is 8x + 5.

    支持答案：逐项展开括号：6x – 3 + 2x + 8。合并同类项：6x + 2x = 8x，-3 + 8 = 5，因此答案为 8x + 5。

    3(2x – 1) + 2(x + 4) = 8x + 5

    Worked question: Factorise 4y + 10. The highest common factor of 4 and 10 is 2, so 4y + 10 = 2(2y + 5).

    例题：因式分解 4y + 10。4 和 10 的最大公因数是 2，因此 4y + 10 = 2(2y + 5)。

    4. Solving Linear Equations | 解一元一次方程

    Worked question: Solve 5x – 7 = 3x + 9.

    例题：解方程 5x – 7 = 3x + 9。

    Support answer: Add 7 to both sides: 5x = 3x + 16. Subtract 3x from both sides: 2x = 16. Divide by 2: x = 8.

    支持答案：两边加 7：5x = 3x + 16。两边减 3x：2x = 16。两边除以 2：x = 8。

    5x – 7 = 3x + 9 → x = 8

    Always check by substituting x = 8 into the original equation: left side 5(8) – 7 = 33, right side 3(8) + 9 = 33, so it is correct.

    务必把 x = 8 代入原方程检验：左边 5(8) – 7 = 33，右边 3(8) + 9 = 33，因此正确。

    5. Ratio and Proportion | 比例与比例问题

    Worked question: Divide £240 in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £240 ÷ 8 = £30. First amount = 3 × £30 = £90; second amount = 5 × £30 = £150.

    例题：按 3 : 5 的比例分配 £240。总份数 = 3 + 5 = 8。一份 = £240 ÷ 8 = £30。第一份 = 3 × £30 = £90；第二份 = 5 × £30 = £150。

    Worked question: A recipe for 6 people needs 300 g of flour. How much is needed for 10 people? Find flour per person: 300 g ÷ 6 = 50 g. For 10 people: 50 g × 10 = 500 g.

    例题：一份 6 人食谱需要 300 克面粉。10 人需要多少？先求每人用量：300 克 ÷ 6 = 50 克。10 人需要：50 克 × 10 = 500 克。

    For ratio problems, never add the parts incorrectly when one share is given; always identify the value of one part first.

    解决比例问题时，如果已知某一份量，切勿错误相加；始终先求出一份的值。

    6. Area and Perimeter of Compound Shapes | 复合图形的面积与周长

    Worked question: Find the area of an L-shape made by joining a 5 cm by 3 cm rectangle and a 2 cm by 2 cm square on one corner.

    例题：求一个由 5 厘米 × 3 厘米矩形和一个角上连接的 2 厘米 × 2 厘米正方形组成的 L 形的面积。

    Support answer: Split the shape into two rectangles. Area of first rectangle = 5 × 3 = 15 cm². Area of square = 2 × 2 = 4 cm². Total area = 15 + 4 = 19 cm².

    支持答案：将图形分成两个矩形。第一个矩形面积 = 5 × 3 = 15 平方厘米。正方形面积 = 2 × 2 = 4 平方厘米。总面积 = 15 + 4 = 19 平方厘米。

    Total area = 15 cm² + 4 cm² = 19 cm²

    For perimeter, add only the outer edges of the combined shape; do not include the internal sides where the two rectangles touch.

    计算周长时，只加组合图形的外边；不要包含两个矩形相接的内部边。

    7. Volume and Surface Area of Prisms | 棱柱体积与表面积

    Worked question: A triangular prism has a right-angled triangle cross-section with base 4 cm, height 3 cm and prism length 10 cm. Find the volume and total surface area.

    例题：一个三棱柱的横截面是底为 4 厘米、高为 3 厘米的直角三角形，棱柱长度为 10 厘米。求体积和总表面积。

    Support answer: Area of cross-section = 0.5 × 4 × 3 = 6 cm². Volume = cross-section area × length = 6 × 10 = 60 cm³.

    支持答案：横截面积 = 0.5 × 4 × 3 = 6 平方厘米。体积 = 横截面积 × 长度 = 6 × 10 = 60 立方厘米。

    The hypotenuse of the triangle is √(4² + 3²) = 5 cm, so the perimeter of the cross-section is 4 + 3 + 5 = 12 cm. Surface area = 2 × 6 + 12 × 10 = 12 + 120 = 132 cm².

    三角形的斜边为 √(4² + 3²) = 5 厘米，所以横截面周长为 4 + 3 + 5 = 12 厘米。表面积 = 2 × 6 + 12 × 10 = 12 + 120 = 132 平方厘米。

    Volume = 60 cm³, Surface area = 132 cm²

    Write the correct units: area in cm², volume in cm³, and surface area also in cm².

    写出正确单位：面积用 cm²，体积用 cm³，表面积也用 cm²。

    8. Pythagoras’ Theorem | 勾股定理

    Worked question: Find the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm.

    例题：求直角边分别为 6 厘米和 8 厘米的直角三角形的斜边。

    Support answer: Use c² = a² + b². Substitute: c² = 6² + 8² = 36 + 64 = 100. Therefore c = √100 = 10 cm.

    支持答案：使用 c² = a² + b²。代入：c² = 6² + 8² = 36 + 64 = 100。因此 c = √100 = 10 厘米。

    c = √(6² + 8²) = √100 = 10 cm

    Pythagoras’ theorem only applies to right-angled triangles. Check the right angle is between the two shorter sides.

    勾股定理只适用于直角三角形。请确认直角位于两条较短边之间。

    Worked question: Is a triangle with sides 5 cm, 12 cm and 13 cm right-angled? Check 5² + 12² = 25 + 144 = 169 = 13², so yes.

    例题：三边分别为 5 厘米、12 厘米和 13 厘米的三角形是直角三角形吗？检验 5² + 12² = 25 + 144 = 169 = 13²，所以是的。

    9. Probability of Combined Events | 组合事件概率

    Worked question: Two fair six-sided dice are rolled. Find the probability that the sum is 7.

    例题：掷两枚公平的六面骰子。求点数和为 7 的概率。

    Support answer: List the successful outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 successful outcomes out of 36 total outcomes.

    支持答案：列出成功结果：(1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)。共有 6 个成功结果，总结果数为 36。

    P(sum = 7) = 6/36 = 1/6

    For independent events A and B, multiply probabilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add their probabilities.

    对于相互独立的事件 A 和 B，概率相乘：P(A 且 B) = P(A) × P(B)。对于互斥事件，则概率相加。

    Always simplify fractions in probability answers unless the question asks for a decimal or percentage.

    除非题目要求小数或百分比，概率答案中的分数要化简。

    10. Averages and Range | 平均数与极差

    Worked question: Find the mean, median, mode and range of the data set: 5, 9, 12, 12, 16, 20.

    例题：求数据集 5、9、12、12、16、20 的平均数、中位数、众数和极差。

    Support answer: Mean = (5 + 9 + 12 + 12 + 16 + 20) ÷ 6 = 74 ÷ 6 ≈ 12.3. The ordered data is already given; median = (12 + 12) ÷ 2 = 12. Mode = 12. Range = 20 – 5 = 15.

    支持答案：平均数 = (5 + 9 + 12 + 12 + 16 + 20) ÷ 6 = 74 ÷ 6 ≈ 12.3。数据已按顺序排列；中位数 = (12 + 12) ÷ 2 = 12。众数 = 12。极差 = 20 – 5 = 15。

    Mean ≈ 12.3, Median = 12, Mode = 12, Range = 15

    If the data set has an even number of values, the median is the mean of the two middle values.

    如果数据个数为偶数，中位数就是中间两个数的平均数。

    11. Straight Line Graphs | 直线图像

    Worked question: Draw the graph of y = 2x + 1 for x values -2, -1, 0, 1, 2.

    例题：画出 y = 2x + 1 在 x = -2、-1、0、1、2 时的图像。

    Support answer: Substitute each x value: for x = -2, y = 2(-2) + 1 = -3; for x = -1, y = -1; for x = 0, y = 1; for x = 1, y = 3; for x = 2, y = 5.

    支持答案：逐个代入 x 值：当 x = -2，y = 2(-2) + 1 = -3；当 x = -1，y = -1；当 x = 0，y = 1；当 x = 1，y = 3；当 x = 2，y = 5。

    The graph is a straight line with gradient 2 and y-intercept 1. The gradient is the coefficient of x.

    图像是一条斜率为 2、y 截距为 1 的直线。斜率就是 x 的系数。

    y = 2x + 1: gradient = 2, y-intercept = 1

    12. Exam Technique and Checking Answers | 考试技巧与答案检查

    Always show each step of working, even when the final answer is wrong; method marks can still be awarded.

    始终展示每个解题步骤，即使最终答案错误；方法分仍然可以获得。

    Check units, simplify fractions, and reread the question to make sure the answer is in the required form.

    检查单位、化简分数，并重新读题，确保答案符合要求的形式。

    Use estimation to see if a numerical answer is sensible, and substitute solutions back into equations where possible.

    使用估算来判断数值答案是否合理，并在可能时将解代回方程检验。

    If you finish early, revisit the questions you found hardest and verify your calculations line by line.

    如果提前完成，重新检查你觉得最难的题目，并逐行验证计算。

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  • Essential Maths 9C Homework Answers: Compressed Year 9 Core Maths Guide | 9年级数学核心 9C 作业答案精要指南

    📚 Essential Maths 9C Homework Answers: Compressed Year 9 Core Maths Guide | 9年级数学核心 9C 作业答案精要指南

    This guide compresses the most common Year 9 Essential Maths 9C homework topics into worked examples and answer-checking methods. Use it to review key skills and check your own working, not to copy answers without understanding.

    本指南将 Year 9 Essential Maths 9C 作业中最常见的主题压缩为示例题和答案检查方法。请用它来复习关键技能并检查自己的解题过程，而不是在不理解的情况下抄写答案。

    1. Number and Place Value: Rounding and Estimation | 数字与位值：四舍五入与估算

    Round 0.008364 to 2 significant figures. The first two significant figures are 8 and 3, so the answer is 0.0084 because the next digit 6 is 5 or more.

    将 0.008364 四舍五入到 2 位有效数字。前两位有效数字是 8 和 3，因此答案是 0.0084，因为下一位数字 6 大于等于 5。

    0.008364 ≈ 0.0084 (2 s.f.)

    Estimate 4.2 × 7.8 by rounding to one significant figure: 4 × 8 = 32, so the estimate is 32. This is a fast way to check whether a calculator answer is sensible.

    将 4.2 × 7.8 四舍五入到一位有效数字来估算：4 × 8 = 32，因此估算值为 32。这是检查计算器答案是否合理的快速方法。

    Upper and lower bounds are also tested. A length given as 6.5 cm to 1 decimal place has a lower bound of 6.45 cm and an upper bound of 6.55 cm.

    上界和下界也会考查。一个长度写成 6.5 cm（精确到 1 位小数）的下界是 6.45 cm，上界是 6.55 cm。

    2. Fractions, Decimals and Percentages | 分数、小数和百分比

    Convert 3/8 to a decimal and a percentage. Divide 3 by 8 to get 0.375, then multiply by 100 to get 37.5%.

    将 3/8 转换为小数和百分比。用 3 除以 8 得到 0.375，再乘以 100 得到 37.5%。

    3/8 = 0.375 = 37.5%

    To increase £140 by 15%, first find 15% of 140: 0.15 × 140 = 21. Then add: 140 + 21 = 161. The final amount is £161.

    要将 £140 增加 15%，先求 140 的 15%：0.15 × 140 = 21。然后相加：140 + 21 = 161。最终金额为 £161。

    For a percentage decrease, subtract instead. Decreasing £80 by 10% gives 80 − 8 = 72, because 10% of 80 is 8.

    对于百分比减少，则用减法。将 £80 减少 10% 得到 80 − 8 = 72，因为 80 的 10% 是 8。

    3. Ratio and Proportion | 比与比例

    Simplify the ratio 24:36 by dividing both sides by their highest common factor, 12. The simplified ratio is 2:3.

    将比 24:36 两边同时除以它们的最大公因数 12，化简为 2:3。

    24:36 = 2:3

    To share £300 in the ratio 2:3, first add the parts: 2 + 3 = 5. One part is 300 ÷ 5 = 60. The shares are 2 × 60 = 120 and 3 × 60 = 180.

    按 2:3 的比例分配 £300，先把份数相加：2 + 3 = 5。每份是 300 ÷ 5 = 60。分配结果是 2 × 60 = 120 和 3 × 60 = 180。

    Proportion problems often ask for the value of one item. If 5 pens cost £3.50, then 1 pen costs 3.50 ÷ 5 = 0.70, so 8 pens cost 8 × 0.70 = 5.60.

    比例问题常求单个物品的价值。如果 5 支笔花费 £3.50，那么 1 支笔花费 3.50 ÷ 5 = 0.70，因此 8 支笔花费 8 × 0.70 = 5.60。

    4. Algebraic Expressions and Linear Equations | 代数表达式与线性方程

    Expand 5(2x − 3) by multiplying each term inside the bracket by 5. This gives 10x − 15.

    将 5(2x − 3) 展开，用 5 乘以括号内每一项，得到 10x − 15。

    5(2x − 3) = 10x − 15

    Solve the equation 3x + 7 = 22. First subtract 7 from both sides: 3x = 15. Then divide by 3: x = 5.

    解方程 3x + 7 = 22。先从两边减去 7：3x = 15。然后除以 3：x = 5。

    3x + 7 = 22 → x = 5

    When solving equations with brackets, expand first. For 2(x + 3) = 14, expand to 2x + 6 = 14, then subtract 6 and divide by 2 to get x = 4.

    解带括号的方程时，先展开。对于 2(x + 3) = 14，展开为 2x + 6 = 14，然后减去 6 再除以 2，得到 x = 4。

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

    For the line y = 2x + 1, the gradient is 2 and the y-intercept is 1. The gradient means y increases by 2 for every increase of 1 in x.

    对于直线 y = 2x + 1，斜率是 2，y 截距是 1。斜率表示 x 每增加 1，y 增加 2。

    To find the value of y when x = 3, substitute x = 3 into y = 2x + 1. This gives y = 2 × 3 + 1 = 7.

    要求当 x = 3 时 y 的值，将 x = 3 代入 y = 2x + 1。得到 y = 2 × 3 + 1 = 7。

    y = 2(3) + 1 = 7

    To plot a linear graph, choose three x-values, calculate the matching y-values, and plot the points. The points should lie on a straight line; if not, check your arithmetic.

    绘制线性图像时，选择三个 x 值，计算对应的 y 值，然后描点。这些点应在一条直线上；如果不是，检查计算。

    6. Angles and Polygons | 角与多边形

    The sum of interior angles of a polygon with n sides is (n − 2) × 180°. For a pentagon, n = 5, so the sum is (5 − 2) × 180° = 540°.

    n 边形内角和为 (n − 2) × 180°。对于五边形，n = 5，所以内角和为 (5 − 2) × 180° = 540°。

    Sum of interior angles = (n − 2) × 180°

    The sum of exterior angles of any polygon is always 360°. For a regular hexagon, each exterior angle is 360° ÷ 6 = 60°, so each interior angle is 180° − 60° = 120°.

    任意多边形的外角和总是 360°。对于正六边形，每个外角为 360° ÷ 6 = 60°，因此每个内角为 180° − 60° = 120°。

    Angles on a straight line add up to 180°, and angles around a point add up to 360°. These facts are often used in multi-step geometry questions.

    直线上的角相加为 180°，围绕一点的角相加为 360°。这些事实常用于多步几何题中。

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

    The area of a circle is A = πr². If the radius is 3 cm, then A = π × 3² = 9π ≈ 28.3 cm² to 1 decimal place.

    圆的面积为 A = πr²。如果半径为 3 cm，则 A = π × 3² = 9π ≈ 28.3 cm²（精确到 1 位小数）。

    The area of a trapezium is A = ½(a + b)h, where a and b are the parallel sides and h is the height. For a = 5 cm, b = 9 cm and h = 4 cm, A = ½ × (5 + 9) × 4 = 28 cm².

    梯形面积为 A = ½(a + b)h，其中 a 和 b 是平行边，h 是高。当 a = 5 cm、b = 9 cm、h = 4 cm 时，A = ½ × (5 + 9) × 4 = 28 cm²。

    Volume of a prism = area of cross-section × length. A cylinder with radius 2 cm and height 10 cm has volume V = π × 2² × 10 = 40π ≈ 125.7 cm³.

    棱柱体积 = 横截面积 × 长度。半径为 2 cm、高为 10 cm 的圆柱体体积为 V = π × 2² × 10 = 40π ≈ 125.7 cm³。

    8. Pythagoras’ Theorem | 勾股定理

    Pythagoras’ theorem states a² + b² = c², where c is the hypotenuse. For a right-angled triangle with legs 6 cm and 8 cm, c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.

    勾股定理为 a² + b² = c²，其中 c 是斜边。对于直角边分别为 6 cm 和 8 cm 的直角三角形，c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。

    a² + b² = c²

    To find a shorter side, rearrange the formula. If the hypotenuse is 13 cm and one leg is 5 cm, then the other leg is √(13² − 5²) = √(169 − 25) = √144 = 12 cm.

    求直角边时，重新整理公式。如果斜边为 13 cm，一条直角边为 5 cm，则另一条直角边为 √(13² − 5²) = √(169 − 25) = √144 = 12 cm。

    Always identify the hypotenuse first: it is the side opposite the right angle and is always the longest side.

    始终先找出斜边：它是直角对面的边，并且总是最长的边。

    9. Probability and Statistics | 概率与统计

    The probability of rolling a 5 on a fair six-sided die is 1/6, because there is one favourable outcome out of six possible outcomes.

    在一枚均匀的六面骰子上掷出 5 的概率是 1/6，因为六个可能结果中只有一个有利结果。

    P(5) = 1/6

    The mean is found by adding all values and dividing by the number of values. For 3, 7, 8, 10, 12, the mean is (3 + 7 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.

    平均数通过将所有数值相加再除以数值个数求得。对于 3、7、8、10、12，平均数为 (3 + 7 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8。

    The range is the difference between the largest and smallest values. For the same data, the range is 12 − 3 = 9. The median is the middle value, 8, when the data is ordered.

    极差是最大值与最小值之差。对于同一组数据，极差为 12 − 3 = 9。中位数是排序后位于中间的值，即 8。

    10. Transformations and Congruence | 变换与全等

    A translation moves a shape without turning or flipping it. A shape translated by the vector (3, −2) moves 3 units right and 2 units down.

    平移是移动图形而不旋转或翻转它。图形按向量 (3, −2) 平移，即向右移动 3 个单位、向下移动 2 个单位。

    A rotation of 90° clockwise around the origin turns a shape one quarter turn. A reflection in the x-axis flips the shape over the x-axis, changing the sign of each y-coordinate.

    绕原点顺时针旋转 90° 将图形转动四分之一圈。关于 x 轴反射将图形翻转到 x 轴另一侧，改变每个 y 坐标的符号。

    An enlargement with scale factor 2 doubles every side length. The shape remains similar, meaning angles stay the same and side lengths are multiplied by 2.

    比例因子为 2 的放大将每条边长加倍。图形保持相似，即角度不变，边长乘以 2。

    11. Indices and Standard Form | 指数与标准形式

    When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128.

    同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。例如，2³ × 2⁴ = 2⁷ = 128。

    2³ × 2⁴ = 2⁷

    When dividing powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 10⁵ ÷ 10² = 10³ = 1000.

    同底数幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。例如，10⁵ ÷ 10² = 10³ = 1000。

    Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10. The number 4500 in standard form is 4.5 × 10³, and 0.0062 is 6.2 × 10⁻³.

    标准形式将一个数写成 a × 10ⁿ，其中 1 ≤ a < 10。数字 4500 的标准形式为 4.5 × 10³，0.0062 为 6.2 × 10⁻³。

    12. Answer Checking and Common Errors | 答案检查与常见错误

    Always check units. If a length is in cm and a volume is in cm³, avoid mixing cm and m without converting first.

    始终检查单位。如果长度用 cm，体积用 cm³，在没有先换算的情况下不要混用 cm 和 m。

    Check negative signs carefully. In a calculation like −5 + 3, the result is −2, not −8. Writing each step clearly reduces sign errors.

    仔细检查负号。在 −5 + 3 这样的计算中，结果是 −2，而不是 −8。清晰写出每一步可以减少符号错误。

    When using a calculator, estimate first. For 29.8 × 3.05, estimate 30 × 3 = 90, then compare with the calculator answer to spot accidental button presses.

    使用计算器时，先估算。对于 29.8 × 3.05，估算为 30 × 3 = 90，然后与计算器答案比较，以发现误按按钮。

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  • Support and Compression: How Structures Carry Load | 支撑与压缩：结构如何承重

    📚 Support and Compression: How Structures Carry Load | 支撑与压缩：结构如何承重

    In Year 9 science, forces are not just about movement – they also act inside solid objects. Support and compression explain why bridges do not collapse, chairs do not snap, and bones carry body weight. This article explores how materials and shapes resist squeezing forces.

    在九年级科学中，力不仅与运动有关，它们也作用在固体内部。支撑与压缩解释了为什么桥梁不会坍塌、椅子不会断裂、骨骼能承受体重。本文将探讨材料与形状如何抵抗挤压作用力。

    1. What Is Support? | 什么是支撑？

    Support is the ability of a structure or material to hold a load without failing. A load can be a person on a chair, a car on a bridge, or a roof on a wall.

    支撑是结构或材料承受载荷而不失效的能力。载荷可以是椅子上的人、桥上的汽车或墙上的屋顶。

    In force diagrams, support is often shown by reaction forces pushing up from the ground while the load presses down. The structure keeps these forces in balance so it does not move or break.

    在力的示意图中，支撑通常由地面向上的反作用力表示，而载荷向下压。结构使这些力保持平衡，因此不会移动或断裂。

    2. Compression vs Tension | 压缩与拉伸

    Compression is a pushing or squeezing force that shortens an object. Tension is a pulling or stretching force that lengthens it. Most structures experience both at the same time.

    压缩是使物体缩短的推挤力。拉伸是使物体伸长的牵拉力。大多数结构同时承受这两种力。

    Force 中文 Direction Example
    Compression 压缩 pushes inward table leg under weight
    Tension 拉伸 pulls outward rope in tug-of-war

    Understanding the difference helps engineers choose the right material for each part of a structure.

    理解两者的区别有助于工程师为结构的每个部分选择合适的材料。

    3. Everyday Examples of Compression | 生活中的压缩实例

    A table leg is compressed by the weight of the tabletop. A car tyre is compressed by the car body. Even a piece of chalk is compressed when you press its ends together.

    桌腿被桌面的重量压缩。汽车轮胎被车身压缩。即使一支粉笔，当你挤压它的两端时也被压缩。

    • Pillars and foundations carry the compressive load of buildings.
    • 柱子和地基承受建筑物的压缩载荷。
    • Bones in the legs are compressed by the upper body.
    • 腿骨被上半身压缩。
    • Stacked books compress the book at the bottom.
    • 堆叠的书本压缩最底下的那本书。

    4. How Materials Resist Compression | 材料如何抵抗压缩

    Materials resist compression because their atoms or fibres push back when squeezed. The stiffness and strength of a material determine how much load it can bear before changing shape.

    材料能抵抗压缩，是因为原子或纤维在受挤压时会产生反向推力。材料的刚度和强度决定了它在变形前能承受多少载荷。

    Concrete is very strong in compression but weak in tension, so steel bars are often added to form reinforced concrete. Steel is strong in both compression and tension, which makes it useful for frames and cables.

    混凝土抗压很强但抗拉很弱，因此通常会加入钢筋形成钢筋混凝土。钢在压缩和拉伸方面都很强，因此适用于框架和缆索。

    5. Structural Design and Load Paths | 结构设计与力的传递路径

    Engineers design structures so that compressive loads travel along planned paths from the load to the ground. A column carries the load straight down; an arch spreads it sideways and then down.

    工程师设计结构时，让压缩载荷沿着规划的路径从载荷传递到地面。柱子将载荷垂直向下传递；拱则将载荷向两侧分散后再向下传递。

    If a load path is interrupted or too narrow, stress concentrates and the structure may fail. Good design spreads the load over a wide area and down to strong foundations.

    如果力的传递路径被打断或过窄，应力就会集中，结构可能失效。好的设计会把载荷分散到较大面积，并向下传到坚固的地基上。

    6. Beams, Columns and Arches | 梁、柱与拱

    A beam usually bends under load, so its top surface is compressed and its bottom surface is in tension. A column is mostly in compression along its whole length. An arch works mainly in compression, which suits stone and concrete.

    梁在载荷作用下通常发生弯曲，因此上表面受压，下表面受拉。柱子整体主要受压。拱主要承受压缩，适合石材和混凝土。

    Structure 中文 Main force Common material
    Beam 梁 bending: top compressed, bottom tension steel, timber
    Column 柱 compression concrete, steel
    Arch 拱 mainly compression stone, concrete

    7. The Role of Shape and Cross-Section | 形状与截面的作用

    A hollow tube can be lighter than a solid rod but almost as strong in compression, because material far from the centre resists buckling. I-beams use flanges to carry bending and a web to resist shear.

    空心管可以比实心杆更轻，但压缩强度几乎相同，因为远离中心的材料可以抵抗屈曲。工字梁用翼缘承受弯曲，用腹板抵抗剪切。

    Folding a flat sheet into a corrugated shape greatly increases its stiffness. This is why cardboard boxes and metal roofing are often corrugated.

    将平板折叠成波纹状可以大幅提高刚度。这就是纸箱和金属屋顶常采用波纹形状的原因。

    8. Calculating Pressure and Stress | 压强与应力计算

    Pressure and stress tell us how concentrated a force is. If a force F acts on area A, the pressure p is given by p = F ÷ A. The units are pascals (Pa) or newtons per square metre (N m⁻²).

    压强和应力表示力有多集中。若力 F 作用在面积 A 上，则压强 p = F ÷ A。单位是帕斯卡（Pa）或牛每平方米（N m⁻²）。

    p = F ÷ A

    Example: A 600 N person stands on one heel of area 0.0001 m². Pressure = 600 ÷ 0.0001 = 6,000,000 Pa. That is much higher than standing on two flat feet, which is why sharp heels can damage floors.

    示例：一个 600 N 的人站在面积为 0.0001 m² 的一只鞋跟上。压强 = 600 ÷ 0.0001 = 6,000,000 Pa。这远高于双脚平放时的压强，因此尖鞋跟会损坏地板。

    9. Safety Factors and Failure | 安全系数与失效

    Structures are designed with safety factors, meaning the maximum expected load is multiplied by a number greater than 1. This protects against unexpected forces, material flaws and wear.

    结构设计会引入安全系数，即将最大预期载荷乘以大于 1 的系数。这样可以防止意外受力、材料缺陷和磨损造成失效。

    • Buckling: a slender column suddenly bends sideways under compression.
    • 屈曲：细长柱在压缩作用下突然向侧面弯曲。
    • Crushing: the material itself is squeezed beyond its compressive strength.
    • 压碎：材料被挤压到超过其抗压强度。
    • Fatigue: repeated loading causes tiny cracks to grow over time.
    • 疲劳：反复加载使微小裂纹随时间扩展。

    10. Simple Experiments to Try | 简单实验

    You can test compression by stacking books on a sponge, or compare a flat paper sheet with a rolled paper tube under a small weight. The tube supports more because its shape resists buckling.

    你可以通过在海绵上堆放书本测试压缩，或将平展的纸张与卷成筒的纸管放在小重物下进行比较。纸管能承受更多重量，因为它的形状能抵抗屈曲。

    Another experiment is to press an empty eggshell from its two ends. It is surprisingly strong in compression because its curved shape spreads the force evenly.

    另一个实验是从两端挤压空蛋壳。它出人意料地坚固，因为弯曲的形状能均匀分散力。

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  • Year 9 Higher Maths: Compressed Answers and Methods | Year 9 高阶数学：压缩答案与方法速查

    📚 Year 9 Higher Maths: Compressed Answers and Methods | Year 9 高阶数学：压缩答案与方法速查

    This revision note condenses the most common Year 9 Higher tier answers into a quick-check format. Each section gives a worked example, the final answer, and a short method note. Use it after attempting questions yourself, so that the answers become a checking tool rather than a shortcut.

    本复习笔记将 Year 9 高阶数学卷中最常见的答案压缩为速查格式。每个小节提供一道例题、最终答案和简短的方法提示。请先自己尝试题目，再使用这些答案进行检查，而不是直接抄捷径。

    1. Simplifying Expressions | 化简代数式

    Worked example: Simplify 5x + 3y – 2x + 8y. Answer: 3x + 11y.

    例题：化简 5x + 3y – 2x + 8y。答案：3x + 11y。

    Method: Collect like terms. 5x – 2x = 3x and 3y + 8y = 11y. Keep the x terms and y terms separate because they are not like terms.

    方法：合并同类项。5x – 2x = 3x，3y + 8y = 11y。x 项和 y 项不是同类项，必须分开合并。

    Common error: Adding x and y terms together to make xy or a single number. Always group only identical variable parts.

    常见错误：把 x 项和 y 项相加变成 xy 或一个数。必须只对完全相同的变量部分进行合并。

    2. Solving Linear Equations | 解一元一次方程

    Worked example: Solve 4x + 7 = 2x + 19. Subtract 2x from both sides: 2x + 7 = 19. Subtract 7: 2x = 12. Divide by 2: x = 6.

    例题：解方程 4x + 7 = 2x + 19。两边同时减去 2x：2x + 7 = 19。再减去 7：2x = 12。除以 2：x = 6。

    x = 6

    Check: Substitute x = 6 back into the original equation. Left: 4(6) + 7 = 31. Right: 2(6) + 19 = 31. Both sides match.

    检验：把 x = 6 代回原方程。左边：4(6) + 7 = 31。右边：2(6) + 19 = 31。两边相等。

    Key rule: Do the same operation to both sides. If a term is added, subtract it. If a term is multiplied, divide by it.

    关键规则：等式两边必须同时进行相同的运算。有加上的项就减去，有乘上的项就除以。

    3. Expanding and Factorising | 展开与因式分解

    Worked example: Expand and simplify (x + 4)(x – 3). First: x². Outer: -3x. Inner: +4x. Last: -12. Combine the x terms: -3x + 4x = x. Final answer: x² + x – 12.

    例题：展开并化简 (x + 4)(x – 3)。首项：x²。外项：-3x。内项：+4x。末项：-12。合并 x 项：-3x + 4x = x。最终答案：x² + x – 12。

    Factorising is the reverse process. Example: Factorise x² + 7x + 10. Find two numbers that multiply to 10 and add to 7: 2 and 5. Answer: (x + 2)(x + 5).

    因式分解是展开的逆运算。例题：将 x² + 7x + 10 因式分解。找到两个数，乘积为 10，和为 7：2 和 5。答案：(x + 2)(x + 5)。

    Common error: Forgetting the negative sign in the outer or inner term. Double-check the signs after expanding.

    常见错误：展开时忘记外项或内项的负号。展开后要仔细检查符号。

    4. Inequalities | 不等式

    Worked example: Solve 5x – 3 > 12. Add 3 to both sides: 5x > 15. Divide by 5: x > 3.

    例题：解不等式 5x – 3 > 12。两边同时加 3：5x > 15。两边除以 5：x > 3。

    On a number line, draw an open circle at 3 and shade to the right. The open circle shows that 3 is not included.

    在数轴上，在 3 处画一个空心圆，并向右画阴影。空心圆表示 3 不包含在解集内。

    Important rule: If you multiply or divide both sides by a negative number, reverse the inequality sign. For example, -2x < 8 becomes x > -4.

    重要规则：如果两边同时乘以或除以一个负数，必须反转不等号方向。例如，-2x < 8 变为 x > -4。

    5. Linear Sequences | 线性数列

    Worked example: Find the nth term of the sequence 4, 7, 10, 13, 16. The common difference is +3, so the nth term starts with 3n. Compare 3n to the sequence: when n = 1, 3n = 3, but the first term is 4. So add 1. Answer: nth term = 3n + 1.

    例题：求数列 4, 7, 10, 13, 16 的第 n 项。公差是 +3，因此第 n 项从 3n 开始。将 3n 与数列比较：当 n = 1 时，3n = 3，但第一项是 4，所以要加 1。答案：第 n 项 = 3n + 1。

    Check: Substitute n = 2: 3(2) + 1 = 7, which matches the second term. Substitute n = 5: 3(5) + 1 = 16, which matches the fifth term.

    检验：代入 n = 2：3(2) + 1 = 7，与第二项相符。代入 n = 5：3(5) + 1 = 16，与第五项相符。

    Common error: Using the first term as the multiplier instead of the common difference. The common difference gives the coefficient of n.

    常见错误：用首项而不是公差作为 n 的系数。公差才是 n 的系数。

    6. Ratio and Proportion | 比例与分配

    Worked example: Divide £120 in the ratio 2:3. Total parts = 2 + 3 = 5. One part = £120 ÷ 5 = £24. Shares: 2 × £24 = £48 and 3 × £24 = £72.

    例题：将 £120 按 2:3 的比例分配。总份数 = 2 + 3 = 5。一份 = £120 ÷ 5 = £24。分配结果：2 × £24 = £48，3 × £24 = £72。

    Check: £48 + £72 = £120, so the whole amount is shared exactly.

    检验：£48 + £72 = £120，因此总金额被完全分配。

    For a ratio such as 1/2 : 1/4, multiply all parts by the lowest common denominator first. 1/2 : 1/4 becomes 2:1 after multiplying by 4.

    对于 1/2 : 1/4 这样的比例，先将所有部分乘以最小公分母。1/2 : 1/4 乘以 4 后变成 2:1。

    7. Pythagoras Theorem | 勾股定理

    Worked example: Find the hypotenuse when the two shorter sides are 6 cm and 8 cm. Use a² + b² = c², where c is the hypotenuse.

    例题：当两条较短边分别为 6 cm 和 8 cm 时，求斜边。使用 a² + b² = c²，其中 c 是斜边。

    6² + 8² = c²
    36 + 64 = 100
    c = √100 = 10 cm

    Answer: The hypotenuse is 10 cm.

    答案：斜边是 10 cm。

    If finding a shorter side, subtract the known square from the hypotenuse square. Example: hypotenuse 13 cm, one side 5 cm. Other side = √(13² – 5²) = √144 = 12 cm.

    如果求一条较短边，用斜边平方减去已知边的平方。例题：斜边 13 cm，一条直角边 5 cm。另一条边 = √(13² – 5²) = √144 = 12 cm。

    8. Area of 2D Shapes | 平面图形面积

    Worked example: Find the area of a trapezium with parallel sides 7 cm and 13 cm, and height 4 cm. Area = ½(a + b)h = ½(7 + 13) × 4 = 40 cm².

    例题：求梯形面积，平行边分别为 7 cm 和 13 cm，高为 4 cm。面积 = ½(a + b)h = ½(7 + 13) × 4 = 40 cm²。

    For a triangle: Area = ½ × base × height. Example: base 10 cm, height 6 cm. Area = ½ × 10 × 6 = 30 cm².

    三角形面积 = ½ × 底 × 高。例题：底 10 cm，高 6 cm。面积 = ½ × 10 × 6 = 30 cm²。

    Common error: Using the slant side instead of the perpendicular height. Height must always be at right angles to the base.

    常见错误：用斜边代替垂直高度。高必须与底边成直角。

    9. Angles in Polygons | 多边形内角

    Worked example: Find the sum of interior angles of a hexagon. Formula: (n – 2) × 180°. For a hexagon, n = 6, so (6 – 2) × 180° = 720°.

    例题：求六边形的内角和。公式：(n – 2) × 180°。六边形 n = 6，因此 (6 – 2) × 180° = 720°。

    For a regular hexagon, all interior angles are equal. Each interior angle = 720° ÷ 6 = 120°.

    对于正六边形，所有内角相等。每个内角 = 720° ÷ 6 = 120°。

    For exterior angles: The sum of exterior angles of any polygon is always 360°. For a regular hexagon, each exterior angle = 360° ÷ 6 = 60°.

    外角方面：任何多边形的外角和始终为 360°。对于正六边形，每个外角 = 360° ÷ 6 = 60°。

    10. Straight Line Graphs | 直线图像

    Worked example: Find the gradient of the line through (1, 2) and (4, 8). Gradient = (y₂ – y₁) / (x₂ – x₁) = (8 – 2) / (4 – 1) = 6 / 3 = 2.

    例题：求经过 (1, 2) 和 (4, 8) 两点的直线斜率。斜率 = (y₂ – y₁) / (x₂ – x₁) = (8 – 2) / (4 – 1) = 6 / 3 = 2。

    Equation: y = 2x + c. Substitute (1, 2): 2 = 2(1) + c, so c = 0. Final equation: y = 2x.

    方程：y = 2x + c。代入 (1, 2)：2 = 2(1) + c，所以 c = 0。最终方程：y = 2x。

    Key form: y = mx + c, where m is the gradient and c is the y-intercept. The y-intercept is where the line crosses the y-axis.

    关键形式：y = mx + c，其中 m 是斜率，c 是 y 轴截距。y 轴截距是直线与 y 轴相交的位置。

    11. Probability from Tables | 表格概率

    Worked example: A bag contains 3 red, 5 blue, and 2 green counters. One counter is taken at random. Total = 3 + 5 + 2 = 10.

    例题：一个袋子中有 3 个红色、5 个蓝色和 2 个绿色计数器。随机取出一个。总数 = 3 + 5 + 2 = 10。

    P(red) = 3/10. P(not blue) = (3 + 2)/10 = 5/10 = 1/2. P(blue or green) = (5 + 2)/10 = 7/10.

    P(红色) = 3/10。P(不是蓝色) = (3 + 2)/10 = 5/10 = 1/2。P(蓝色或绿色) = (5 + 2)/10 = 7/10。

    Probability always lies between 0 and 1. If all outcomes are equally likely, probability = number of favourable outcomes ÷ total number of outcomes.

    概率始终在 0 到 1 之间。如果所有结果等可能，概率 = 有利结果数 ÷ 总结果数。

    12. Averages and Range | 平均数与极差

    Worked example: Find the mean, median, and range of 4, 7, 9, 12, 15. Mean = (4 + 7 + 9 + 12 + 15) ÷ 5 = 47 ÷ 5 =

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  • Year 9 Higher Mathematics: Compressed Mastery Guide | 9年级高阶数学：压缩精讲掌握指南

    📚 Year 9 Higher Mathematics: Compressed Mastery Guide | 9年级高阶数学：压缩精讲掌握指南

    Year 9 Higher Mathematics brings together the skills that form the foundation for GCSE Higher tier success. This compressed guide focuses on the core topics most often examined: algebra, shape, ratio, probability and statistics.

    9年级高阶数学汇集了为 GCSE 高阶阶段成功奠定基础的技能。这份压缩指南聚焦于最常考查的核心主题：代数、图形、比、概率与统计。

    1. Algebraic Manipulation and Expanding Brackets | 代数运算与去括号

    Expanding brackets means multiplying each term inside the bracket by the term outside. For double brackets, apply the FOIL method: first, outer, inner, last.

    去括号是指将括号内的每一项都与括号外的项相乘。对于双重括号，应用 FOIL 方法：首项、外项、内项、末项。

    (a + b)(c + d) = ac + ad + bc + bd

    The three special products appear frequently in Year 9 Higher papers. Recognising them saves time and reduces sign errors.

    三个特殊乘积式在9年级高阶试卷中经常出现。识别它们可以节省时间并减少符号错误。

    (a + b)² = a² + 2ab + b²    (a – b)² = a² – 2ab + b²    (a + b)(a – b) = a² – b²

    Factorising is the reverse process. Always look for a common factor first, then check for a difference of two squares or a quadratic trinomial.

    因式分解是相反的运算。始终先寻找公因式，然后检查是否为平方差或二次三项式。

    2. Linear Equations and Inequalities | 线性方程与不等式

    To solve a linear equation, keep the equation balanced by doing the same operation to both sides. Collect like terms before isolating the unknown.

    解线性方程时，通过对等号两边进行相同的运算来保持方程平衡。先合并同类项，再分离未知数。

    ax + b = c  →  x = (c – b) ÷ a

    For inequalities, the solution set can be shown on a number line using open or closed circles. When multiplying or dividing by a negative number, reverse the inequality sign.

    对于不等式，解集可以用数轴上的空心圆或实心圆表示。当乘以或除以负数时，要改变不等号的方向。

    • English: x < 3 uses an open circle at 3; x ≤ 3 uses a closed circle.
    • 中文：x < 3 在 3 处使用空心圆；x ≤ 3 使用实心圆。

    Always check solutions by substituting back into the original equation or inequality.

    始终将解代回原方程或不等式进行检验。

    3. Sequences and the nth Term | 数列与第n项

    An arithmetic sequence has a constant difference between consecutive terms. The nth term formula links the term position n to its value.

    等差数列的相邻项之间差为常数。第n项公式将项的位置 n 与其数值联系起来。

    uₙ = a + (n – 1)d

    Here a is the first term and d is the common difference. If the sequence decreases, d is negative.

    其中 a 是首项，d 是公差。如果数列递减，d 为负数。

    For a quadratic sequence, the second difference is constant. Compare the sequence with n² to find the full nth term expression.

    对于二次数列，第二层差分为常数。将数列与 n² 比较，以求出完整的第n项表达式。

    4. Ratio, Proportion and Direct Proportion | 比、比例与正比例

    To divide a quantity in a given ratio, find the total number of parts first. Then multiply the value of one part by the required number of parts.

    按给定比例分配数量时，首先求出总份数。然后用一份的值乘以所需的份数。

    Direct proportion means two quantities increase or decrease at the same rate. The relationship can be written as y = kx, where k is the constant of proportionality.

    正比例意味着两个量以相同的速率增加或减少。这种关系可以写成 y = kx，其中 k 是比例常数。

    y ∝ x  →  y = kx

    Use the unitary method for proportion problems: find the value of one unit first, then scale up or down.

    使用单位法解决比例问题：先求出一个单位的值，然后按比例放大或缩小。

    5. Percentages and Compound Change | 百分数与复合变化

    Percentage change compares the actual change to the original value and multiplies by 100%.

    百分比变化将实际变化量与原始值进行比较，并乘以 100%。

    Percentage change = (change ÷ original) × 100%

    Compound interest calculates interest on both the original amount and the interest already earned. Use the multiplier method for repeated percentage changes.

    复利计算基于原始金额和已获得利息的总和。对于重复的百分比变化，使用乘数法。

    A = P(1 + r/100)ⁿ

    In this formula, A is the final amount, P is the principal, r is the percentage rate, and n is the number of periods.

    在此公式中，A 是最终金额，P 是本金，r 是百分比率，n 是期数。

    6. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数

    Pythagoras’ theorem applies only to right-angled triangles. The square of the hypotenuse equals the sum of the squares of the other two sides.

    勾股定理仅适用于直角三角形。斜边的平方等于另外两条边的平方之和。

    a² + b² = c²

    Trigonometry links an angle to the ratios of side lengths in a right-angled triangle.

    三角函数将角度与直角三角形边长的比联系起来。

    sin θ = opposite ÷ hypotenuse    cos θ = adjacent ÷ hypotenuse    tan θ = opposite ÷ adjacent

    English 中文
    Label the opposite side based on the angle θ, the adjacent side next to θ, and the hypotenuse opposite the right angle. 根据角 θ 标记对边、邻边和斜边：斜边是直角对面的边。

    Use SOH CAH TOA to remember the three ratios. Always show the substitution step clearly.

    使用 SOH CAH TOA 记忆三个比值。始终清晰地写出代入步骤。

    7. Area, Volume and Surface Area | 面积、体积与表面积

    The area of a circle is πr² and the circumference is 2πr. For a cylinder, the volume is the base area multiplied by height.

    圆的面积为 πr²，周长为 2πr。对于圆柱体，体积等于底面积乘以高。

    Area of circle = πr²    Circumference = 2πr    Volume of cylinder = πr²h

    The curved surface area of a cylinder is 2πrh, and the total surface area includes the two circular ends: 2πr(r + h).

    圆柱的侧面积为 2πrh，总表面积包括两个圆形底面：2πr(r + h)。

    For a cone, the volume is one third of the base area times the vertical height: V = ⅓πr²h.

    对于圆锥体，体积等于底面积乘以垂直高度的三分之一：V = ⅓πr²h。

    Always check whether the question asks for surface area or volume, and use consistent units.

    始终检查题目要求的是表面积还是体积，并使用一致的单位。

    8. Straight Line Graphs | 直线图像

    The equation of a straight line is usually written as y = mx + c. Here m is the gradient and c is the y-intercept.

    直线方程通常写作 y = mx + c。其中 m 是斜率，c 是 y 轴截距。

    y = mx + c

    The gradient between two points (x₁, y₁) and (x₂, y₂) is calculated using rise over run.

    两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率用纵差除以横差来计算。

    m = (y₂ – y₁) ÷ (x₂ – x₁)

    Parallel lines have the same gradient. Perpendicular lines have gradients whose product is -1.

    平行线具有相同的斜率。垂直线的斜率乘积为 -1。

    To sketch a straight line, plot the y-intercept first, then use the gradient to find another point.

    要画直线图，先标出 y 轴截距，然后利用斜率找到另一个点。

    9. Probability of Combined Events | 组合事件概率

    For independent events, the probability that both events occur is the product of their individual probabilities.

    对于独立事件，两个事件同时发生的概率等于各自概率的乘积。

    P(A and B) = P(A) × P(B)

    For mutually exclusive events, the probability that either event occurs is the sum of their probabilities.

    对于互斥事件，任一事件发生的概率等于各自概率之和。

    P(A or B) = P(A) + P(B)

    Tree diagrams help multiply along branches and add between branches. Label each branch with its probability, and check that the total probability is 1.

    树状图有助于沿分支相乘并在分支之间相加。在每个分支上标出概率，并检查总概率是否为 1。

    10. Averages and Statistical Diagrams | 平均数与统计图

    The mean is the sum of all values divided by the number of values. The median is the middle value, and the mode is the most frequent value.

    平均数是所有数值之和除以数值的个数。中位数是中间的数值，众数是出现频率最高的数值。

    Mean = Σx ÷ n

    Range measures spread and equals the largest value minus the smallest value. For grouped data, estimate the mean using midpoints of class intervals.

    极差衡量数据的离散程度，等于最大值减最小值。对于分组数据，使用组中值来估计平均数。

    • English: Cumulative frequency diagrams show the running total; box plots display minimum, lower quartile, median, upper quartile and maximum.
    • 中文：累积频率图显示累加总数；箱线图显示最小值、下四分位数、中位数、上四分位数和最大值。

    Choose the most appropriate average: mean is sensitive to extreme values, while median is robust against outliers.

    选择最合适的平均数：平均数对极端值敏感，而中位数对异常值具有稳健性。

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  • Year 9 Core Mathematics: Compressed Revision Guide | 九年级核心数学：压缩复习指南

    📚 Year 9 Core Mathematics: Compressed Revision Guide | 九年级核心数学：压缩复习指南

    This compressed revision guide covers the essential skills in the Year 9 Core Mathematics course. It is organised into ten focused sections that provide clear explanations, worked examples, and key formulas needed for success in end-of-year assessments.

    本压缩复习指南涵盖九年级核心数学课程的基本技能。全文分为十个重点小节，提供清晰讲解、例题和关键公式，帮助你在期末考试中取得成功。

    1. Number Sense and Place Value | 数感与位值

    In Year 9, place value underpins all number work. The digits in 3 456.78 represent 3×1000 + 4×100 + 5×10 + 6×1 + 7×(1/10) + 8×(1/100), which can also be written as 3.45678 × 10³ in standard form.

    九年级中，位值是所有数字计算的基础。数字 3 456.78 中各位数字代表 3×1000 + 4×100 + 5×10 + 6×1 + 7×(1/10) + 8×(1/100)，也可以写作标准形式 3.45678 × 10³。

    Always follow the order of operations: brackets first, then indices, then division and multiplication from left to right, then addition and subtraction from left to right. This is often remembered as BIDMAS or BODMAS.

    始终遵守运算顺序：先算括号，再算指数，然后从左到右计算乘除，最后从左到右计算加减。这通常被记为 BIDMAS 或 BODMAS。

    Brackets → Indices → Division/Multiplication → Addition/Subtraction

    Negative numbers follow consistent rules: adding a negative is the same as subtracting, and multiplying or dividing two negatives gives a positive. For example, (−3) × (−4) = 12 and (−5) + (−2) = −7.

    负数遵循一致规则：加上一个负数等同于减去一个正数；两个负数相乘或相除结果为正数。例如，(−3)

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  • Year 9 Core Homework Answers Compressed | Year 9 核心作业答案精华速览

    📚 Year 9 Core Homework Answers Compressed | Year 9 核心作业答案精华速览

    This compressed guide collects the most common Year 9 core homework questions and model answers into one place. It covers number, algebra, geometry, ratio, proportion, probability, statistics and sequences. Each section gives a typical question, a quick answer and a short method so you can check your work and revise efficiently.

    这份精华速览把 Year 9 核心课程最常见的作业题目与标准答案集中在一起。内容涵盖数字、代数、几何、比与比例、概率、统计和数列。每个小节给出一道典型题、快速答案和简要方法，方便你核对作业并高效复习。

    1. Number Operations and BIDMAS | 数字运算与运算顺序

    A typical Year 9 core homework question asks you to calculate 24 ÷ 6 + 2 × (5 − 3)². The key is to follow BIDMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. Work through each step in order and write down every line so mistakes are easy to find.

    一道典型的 Year 9 核心作业题是计算 24 ÷ 6 + 2 × (5 − 3)²。关键是遵循 BIDMAS 运算顺序：括号、指数、除法和乘法、加法和减法。每一步都按顺序写出，这样容易发现错误。

    Answer: 24 ÷ 6 = 4, 5 − 3 = 2, 2² = 4, 2 × 4 = 8, so 4 + 8 = 12. Final answer: 12.

    答案：24 ÷ 6 = 4，5 − 3 = 2，2² = 4，2 × 4 = 8，因此 4 + 8 = 12。最终答案为 12。

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

    To convert a fraction to a percentage, find an equivalent fraction with denominator 100. To find a percentage of an amount, change the percentage to a decimal and multiply. A common homework question is: Convert 7/20 to a percentage and find 15% of 240.

    要把分数转换成百分数，先找到一个分母为 100 的等价分数。要求一个数的百分之几，把百分数化成小数再相乘。常见的作业题是：把 7/20 转换成百分数，并求 240 的 15%。

    Answer: 7/20 = 35% because 20 × 5 = 100 and 7 × 5 = 35. Also 15% of 240 = 0.15 × 240 = 36.

    答案：7/20 = 35%，因为 20 × 5 = 100，7 × 5 = 35。另外 240 的 15% 等于 0.15 × 240 = 36。

    3. Algebraic Simplification and Indices | 代数化简与指数

    In algebra, you can only add or subtract like terms: terms that have exactly the same variable and the same power. For indices, when you multiply powers of the same base, keep the base and add the exponents.

    在代数中，你只能合并同类项：变量和指数完全相同的项。对于指数，同底数幂相乘时，底数不变，指数相加。

    Simplify 3x + 5y − x + 2y. Answer: 2x + 7y, because 3x − x = 2x and 5y + 2y = 7y. Also simplify x³ × x². Answer: x⁵, because 3 + 2 = 5.

    化简 3x + 5y − x + 2y。答案：2x + 7y，因为 3x − x = 2x，5y + 2y = 7y。再化简 x³ × x²。答案：x⁵，因为 3 + 2 = 5。

    4. Solving Linear Equations | 解一元一次方程

    To solve a linear equation, move all variable terms to one side and all constant terms to the other side. Do the same operation to both sides so the equation stays balanced. Then isolate the variable by division.

    解一元一次方程时，把所有含变量的项移到一边，把所有常数项移到另一边。等式两边同时进行相同运算，使等式保持平衡。然后通过除法求出变量的值。

    Solve 5x − 7 = 2x + 11. Subtract 2x from both sides: 3x − 7 = 11. Add 7 to both sides: 3x = 18. Divide by 3: x = 6.

    解方程 5x − 7 = 2x + 11。两边同时减去 2x：3x − 7 = 11。两边同时加 7：3x = 18。两边除以 3：x = 6。

    5. Expanding Brackets and Factorising | 去括号与因式分解

    Expanding brackets means multiplying each term inside the bracket by the term outside. Factorising is the reverse process: find the highest common factor and write it outside a bracket.

    去括号就是用括号外面的项乘括号里的每一项。因式分解是相反的过程：找到最高公因式，把它写在括号外面。

    Expand 4(2x − 3). Answer: 8x − 12, because 4 × 2x = 8x and 4 × (−3) = −12. Factorise 6x² + 9x. Answer: 3x(2x + 3), since 3x is the highest common factor and 6x² ÷ 3x = 2x, 9x ÷ 3x = 3.

    展开 4(2x − 3)。答案：8x − 12，因为 4 × 2x = 8x，4 × (−3) = −12。对 6x² + 9x 进行因式分解。答案：3x(2x + 3)，因为 3x 是最高公因式，且 6x² ÷ 3x = 2x，9x ÷ 3x = 3。

    6. Ratio and Proportion | 比与比例

    To divide an amount in a given ratio, add the parts to find the total number of parts, divide the amount by that total to find one part, then multiply by each ratio number. For proportion, use a scaling factor or find the unit rate.

    按给定比例分配一个量时，先把比例的份数相加得到总份数，用总量除以总份数得到一份的量，再分别乘以比例的每个数。对于比例应用题，可以使用倍数或先求单位量。

    Divide £56 in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £56 ÷ 8 = £7. The amounts are 3 × £7 = £21 and 5 × £7 = £35. A recipe uses 2 cups of flour for 5 cakes. For 20 cakes, the number of cakes is multiplied by 4, so flour needed = 2 × 4 = 8 cups.

    把 56 英镑按 3 : 5 分配。总份数 = 3 + 5 = 8。一份 = 56 ÷ 8 = 7 英镑。两份分别是 3 × 7 = 21 英镑和 5 × 7 = 35 英镑。一个食谱做 5 个蛋糕需要 2 杯面粉。做 20 个蛋糕时，蛋糕数量乘以 4，所以需要面粉 2 × 4 = 8 杯。

    7. Percentage Increase, Decrease and Reverse | 百分数增减与逆运算

    For a percentage decrease, find the percentage of the original amount and subtract it from the original. For a reverse percentage, divide the final amount by the decimal multiplier that represents the increase or decrease.

    计算百分数减少时，先求出原数的百分比，再从原数中减去。计算逆百分数时，用最终数量除以表示增加或减少的小数乘数。

    A jacket costing £80 is reduced by 15%. Decrease = 0.15 × 80 = £12. Sale price = 80 − 12 = £68. After a 20% increase, a value is 120. The multiplier is 1.20, so original value = 120 ÷ 1.20 = 100.

    一件夹克原价 80 英镑，降价 15%。降价额 = 0.15 × 80 = 12 英镑。售价 = 80 − 12 = 68 英镑。一个数增加 20% 后为 120。乘数是 1.20，所以原数 = 120 ÷ 1.20 = 100。

    8. Area, Perimeter and Volume | 面积、周长与体积

    The area of a triangle is half the base times the perpendicular height. The volume of a cuboid is length times width times height. Always include the correct unit: cm² for area and cm³ for volume.

    三角形的面积等于底乘高再除以 2。长方体的体积等于长乘宽乘高。始终要写明正确单位：面积用 cm²，体积用 cm³。

    Find the area of a triangle with base 12 cm and height 5 cm. Area = ½ × 12 × 5 = 30 cm². Find the volume of a cuboid with dimensions 4 cm, 3 cm and 2 cm. Volume = 4 × 3 × 2 = 24 cm³.

    求底为 12 cm、高为 5 cm 的三角形面积。面积 = ½ × 12 × 5 = 30 cm²。求长、宽、高分别为 4 cm、3 cm、2 cm 的长方体体积。体积 = 4 × 3 × 2 = 24 cm³。

    9. Pythagoras’ Theorem | 勾股定理

    Pythagoras’ theorem only applies to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is always the longest side and is opposite the right angle.

    勾股定理只适用于直角三角形。它指出斜边的平方等于另外两条边的平方和。斜边永远是最长的边，并且位于直角的对面。

    A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse. c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.

    一个直角三角形的两条较短边分别为 6 cm 和 8 cm。求斜边。c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。

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

    A linear graph has equation y = mx + c, where m is the gradient and c is the y-intercept. To plot the graph, substitute x-values into the equation to find matching y-values, then plot the coordinate pairs.

    线性图像的一般方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。画图时，把 x 值代入方程求出对应的 y 值，然后描出坐标点。

    Plot y = 2x + 1 for x = 0, 1, 2, 3. The points are (0, 1), (1, 3), (2, 5), (3, 7). Gradient = 2 and y-intercept = 1, because the line is in the form y = mx + c with m = 2 and c = 1.

    画出 y = 2x + 1 在 x = 0, 1, 2, 3 时的图像。坐标点为 (0, 1)、(1, 3)、(2, 5)、(3, 7)。斜率为 2，y 轴截距为 1，因为这条直线符合 y = mx + c 的形式，且 m = 2，c = 1。

    11. Probability | 概率

    Probability is a number between 0 and 1 that describes how likely an event is. To find the probability of an event, divide the number of favourable outcomes by the total number of possible outcomes. The probability of an event not happening is 1 minus the probability that it does happen.

    概率是介于 0 到 1 之间的数，用来描述一个事件发生的可能性。求一个事件的概率，用有利结果的数量除以所有可能结果的总数。一个事件不发生的概率等于 1 减去它发生的概率。

    A bag contains 3 red, 2 blue and 5 green counters. Total = 3 + 2 + 5 = 10. P(red) = 3/10. P(not green) = 1 − P(green) = 1 − 5/10 = 5/10 = 1/2.

    一个袋子里有 3 个红色、2 个蓝色和 5 个绿色筹码。总数 = 3 + 2 + 5 = 10。P(红) = 3/10。P(非绿) = 1 − P(绿) = 1 − 5/10 = 5/10 = 1/2。

    12. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数和极差

    The mean is the sum of all values divided by the number of values. The median is the middle value when the data are ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.

    平均数等于所有数值之和除以数值的个数。中位数是把数据排序后位于中间的数。众数是出现次数最多的数。极差是最大值与最小值之差。

    For the data set 4, 7, 2, 9, 7, 3: Mean = (4 + 7 + 2 + 9 + 7 + 3) ÷ 6 = 32 ÷ 6 = 5.33 to 2 d.p. Ordered data: 2, 3, 4, 7, 7, 9. Median = (4 + 7) ÷ 2 = 5.5. Mode = 7. Range = 9 − 2 = 7.

    对于数据集 4、7、2、9、7、3：平均数 = (4 + 7 + 2 + 9 + 7 + 3) ÷ 6 = 32 ÷ 6 = 5.33（保留两位小数）。排序后：2、3、4、7、7、9。中位数 = (4 + 7) ÷ 2 = 5.5。众数 = 7。极差 = 9 − 2 = 7。

    Published by TutorHao | Maths Revision Series | aleveler.com

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  • Year 9 Core Answers: Your Compressed Revision Companion | 九年级核心答案精编：你的高效复习指南

    📚 Year 9 Core Answers: Your Compressed Revision Companion | 九年级核心答案精编：你的高效复习指南

    In Year 9, students often face a jump in expectation: answers that were once accepted for effort now need to show precise subject terminology, clear working, and exam-ready structure. This guide compresses the most important Year 9 core answers into a practical revision companion, covering mathematics, English and science.

    九年级学生常会感到要求明显提高：以前凭努力就能拿分的答案，现在需要准确使用学科术语、写清解题步骤，并采用符合考试要求的答题结构。本指南将九年级最核心的答案压缩成一份实用复习手册，覆盖数学、英语和科学。

    1. What Are Core Answers and Why They Matter | 什么是核心答案及其重要性

    A core answer is the minimum complete response that hits all parts of a question: it states the correct result, shows the method, and uses subject vocabulary. Marks in Year 9 assessments are usually awarded for both process and final answer, so a compressed but complete core answer protects easy marks.

    核心答案是指能覆盖题目所有要求的、最精简的完整回答：它写出正确结果、展示解题方法，并使用学科术语。九年级测评通常既给过程分也给结果分，因此一份压缩但完整的核心答案能保住容易得到的分数。

    Core answers are not one-word guesses. For example, in science, writing ‘gas’ may only earn one mark, while ‘carbon dioxide gas is produced’ earns the full mark because it names the substance and its state.

    核心答案不是只写一个词的猜测。例如在科学中，只写 ‘气体’ 可能只得一分，而写 ‘产生二氧化碳气体’ 能得满分，因为它写明了物质名称和状态。

    • English: Show the method, not just the answer. | 中文：展示过程，而不仅仅是答案。
    • English: Use precise terms from the specification. | 中文：使用考纲中的准确术语。
    • English: Match the number of marks to the number of points. | 中文：答案要点数量要与分值匹配。

    2. Reading the Mark Scheme Before You Write | 动笔前先读评分标准

    Before answering any practice question, check the mark scheme or model answer. In revision, mark schemes teach you what examiners actually reward: for a 3-mark question, there are usually three distinct points or steps.

    在回答任何练习题之前，先查看评分标准或标准答案。复习时，评分标准能告诉你考官真正给分的地方：一道 3 分题通常有三个不同的得分点或步骤。

    Write your answer, then compare it line by line with the model answer. Highlight missing keywords and add them to a personal glossary. This turns a simple mark into a reusable core answer.

    写下你的答案，然后逐行与标准答案对照。标出缺失的关键词，并把它们加入个人词汇表。这样就能把一个分数转化为可反复使用的核心答案。

    3. Command Words: The Key to Unlocking Marks | 指令词：解锁分数的关键

    Command words tell you how to answer. In Year 9, common command words include ‘state’, ‘describe’, ‘explain’, ‘calculate’ and ‘compare’. Each requires a different depth of response.

    指令词告诉你答题方式。九年级常见的指令词包括 ‘state’（陈述）、’describe’（描述）、’explain’（解释）、’calculate’（计算）和 ‘compare’（比较）。每个指令词要求的答题深度不同。

    For ‘state’, give a fact without explanation. For ‘describe’, say what happens or what you observe. For ‘explain’, give a reason or cause using ‘because’. For ‘compare’, write both similarities and differences.

    遇到 ‘state’，只给事实不解释；遇到 ‘describe’，说明发生了什么或观察到什么；遇到 ‘explain’，用 ‘因为’ 给出原因或理由；遇到 ‘compare’，要写出相同点和不同点。

    • English: State: Give a fact or name only. | 中文：陈述：只给出事实或名称。
    • English: Describe: Say what happens or what you observe. | 中文：描述：说明发生了什么或观察到什么。
    • English: Explain: Give a reason using ‘because’. | 中文：解释：用 ‘因为’ 给出原因。
    • English: Compare: Write both similarities and differences. | 中文：比较：写出相同点和不同点。

    4. Mathematics: Solving Linear Equations in Compact Steps | 数学：用紧凑步骤解线性方程

    A core answer for solving an equation must show the inverse operation and the final value. For example, solve 3x + 4 = 19.

    解方程的核心答案必须展示逆运算和最终值。例如，解 3x + 4 = 19。

    3x + 4 = 19 → 3x = 15 → x = 5

    Do not skip the middle step. Writing ‘x = 5’ alone may lose method marks, even if the answer is correct. Show subtracting 4 from both sides, then dividing by 3.

    不要跳过中间步骤。只写 ‘x = 5’ 即使答案正确，也可能丢掉过程分。要展示两边同时减 4，再除以 3。

    Always substitute your value back into the original equation to check: 3(5) + 4 = 19, which is true.

    始终把所得值代回原方程检验：3(5) + 4 = 19，等式成立。

    5. Mathematics: Fractions, Decimals and Percentages | 数学：分数、小数和百分比

    Year 9 tests often ask you to convert between fractions, decimals and percentages. A core answer must simplify fractions and show equivalent working.

    九年级测试常要求分数、小数和百分比之间的转换。核心答案必须化简分数并展示等价变形过程。

    3/4 = 0.75 = 75%

    For calculations such as 2/3 + 1/4, write the common denominator 12: 8/12 + 3/12 = 11/12. Do not add numerators and denominators separately.

    计算 2/3 + 1/4 时，先通分为分母 12：8/12 + 3/12 = 11/12。不要分子分母分别相加。

    Percentage increase or decrease needs the formula (change ÷ original) × 100%. If a price rises from £20 to £25, the increase is 25% because 5/20 × 100% = 25%.

    百分比增减要用公式（变化量 ÷ 原始量）× 100%。如果价格从 20 英镑涨到 25 英镑，涨幅为 25%，因为 5/20 × 100% = 25%。

    6. English: Analytical Paragraphs Using PEEL | 英语：使用 PEEL 结构写分析段落

    In English, a core analytical paragraph follows PEEL: Point, Evidence, Explanation, Link. This structure makes your answer readable and directly tied to the question.

    在英语科目中，核心分析段落遵循 PEEL 结构：观点、证据、解释、联系。这种结构使你的答案清晰易读，并紧扣题目。

    Point: state the writer’s technique. Evidence: quote a short phrase. Explanation: say how the technique creates an effect. Link: connect back to the question or to the next point.

    观点：说明作者使用的技巧；证据：引用简短语句；解释：说明该技巧如何产生效果；联系：回扣题目或连接下一点。

    Example: In ‘The old house groaned in the wind’, the writer personifies the house. The verb ‘groaned’ suggests the house is alive and suffering, creating an eerie, unsettling atmosphere.

    示例：在 ‘老房子在风中呻吟’ 一句中，作者使用了拟人手法。动词 ‘呻吟’ 暗示房子像活人一样在受苦，营造出阴森不安的氛围。

    7. English: Persuasive Writing with Rhetorical Devices | 英语：运用修辞手法的说服性写作

    For persuasive writing, a core answer should include a clear opinion, at least two persuasive devices, and a call to action. Devices include rhetorical questions, triples, emotive language and direct address.

    对于说服性写作，核心答案应包含明确观点、至少两种说服技巧和一个行动呼吁。常用技巧包括反问句、三连排比、情感化语言和直接称呼。

    Example: ‘Do we really want our parks buried under concrete? We need clean air, safe play spaces, and a future worth living. Write to your local council today.’

    示例：’难道我们真的想让公园被水泥掩埋吗？我们需要清洁的空气、安全的游戏空间和值得期待的未来。今天就写信给你所在的地方议会。’

    In a short answer, use a rhetorical question to open and end with an imperative. This compressed structure earns content

    Published by TutorHao | Year 9 Revision Series | aleveler.com

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