📚 Essential Maths 9C Homework Answers: Key Concepts | KS3 数学：Essential Maths 9C 作业答案 核心知识点
This article covers the essential mathematical ideas behind typical Year 9 homework tasks from the Essential Maths 9C series. Each section presents a core topic with clear explanations, relevant formulas, and step‑by‑step logic that you can use to check your answers and understand the methods. Whether you are revising for a test or completing homework, these notes will strengthen your grasp of KS3 mathematics.
本文围绕 Essential Maths 9C 教材中常见的家庭作业题目，梳理核心数学知识点。每一节都给出关键概念、公式和解题思路，帮助你核对自己的答案并真正理解背后的方法。无论是准备考试还是完成作业，这些内容都能帮你打牢 KS3 数学基础。
1. Number Operations and BIDMAS | 数的运算与运算顺序
When a calculation involves more than one operation, the BIDMAS rule tells us the correct order: Brackets first, then Indices (powers and roots), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).
当一个算式包含多种运算时，BIDMAS 规则规定了正确的计算顺序：先算括号，再算乘方与根号，接着算乘除（从左到右），最后算加减（从左到右）。
For example, to evaluate 3 + 4 × (5 − 2)², we start inside the brackets: 5 − 2 = 3. Next we apply the index: 3² = 9. Then multiply: 4 × 9 = 36. Finally add: 3 + 36 = 39.
例如计算 3 + 4 × (5 − 2)²，先算括号里的 5 − 2 = 3，然后计算指数 3² = 9，再乘法 4 × 9 = 36，最后加法 3 + 36 = 39。
With division and multiplication having equal priority, you must work from left to right. So 24 ÷ 6 × 2 gives (24 ÷ 6) × 2 = 4 × 2 = 8, not 24 ÷ (6 × 2).
除法和乘法优先级相同，必须从左到右计算。所以 24 ÷ 6 × 2 等于 (24 ÷ 6) × 2 = 4 × 2 = 8，而不是 24 ÷ (6 × 2)。
2. Fractions, Decimals, and Percentages | 分数、小数和百分比
These three are different ways of writing the same value. A fraction represents a part of a whole; a decimal uses a point to show tenths, hundredths, etc.; a percentage is a number out of 100.
这三种形式可以相互转换：分数表示整体的一部分，小数用小数点表示十分位、百分位等，百分比则是一个以 100 为分母的数。
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375. To change a decimal to a percentage, multiply by 100: 0.375 × 100 = 37.5%.
分数化小数只需用分子除以分母，如 3/8 = 3 ÷ 8 = 0.375。小数化百分比乘以 100，即 0.375 × 100 = 37.5%。
Recurring decimals like 1/3 = 0.333… can be written with a dot over the repeating digit. Fractions can be simplified by dividing top and bottom by their highest common factor (HCF).
循环小数如 1/3 = 0.333… 可在循环数字上加一个点。分数可以通过分子分母同时除以它们的最大公因数（HCF）化简。
Always be careful when calculating with mixed numbers; convert them to improper fractions first. For instance, 2½ = 5/2.
处理带分数时要先化成假分数，例如 2½ = 5/2。
3. Ratio and Proportion | 比率与比例
A ratio compares two or more quantities, showing how much of one there is compared to another. Ratios can be simplified by dividing each part by the same number, just like fractions.
比率用于比较两个或以上的数量。可以将比的每一项同时除以同一个数来化简，就像化简分数一样。
If a recipe uses 200 g of flour and 120 g of sugar, the ratio flour : sugar = 200 : 120. Divide both numbers by 40 (the HCF) to get 5 : 3.
例如一个配方用 200 g 面粉和 120 g 糖，面粉与糖的比为 200 : 120，两边同除以 40 得到 5 : 3。
Proportion problems often involve sharing an amount in a given ratio. To share £72 in the ratio 3 : 5, first find the total number of parts (3 + 5 = 8). One part is £72 ÷ 8 = £9. So the shares are £27 and £45.
按比例分配时，先计算总份数。比如按 3 : 5 分配 72 英镑，总份数 8，每份 72 ÷ 8 = 9，最终两份分别为 3×9 = 27 和 5×9 = 45。
The unitary method finds the value of one unit first, using it to scale up or down. Direct proportion means that as one quantity doubles, the other also doubles.
单位法就是先求出一个单位的量，再按比例放大或缩小。正比例意味着一个量翻倍时，另一个量也跟着翻倍。
4. Algebraic Expressions and Simplification | 代数表达式与化简
In algebra, letters stand for unknown numbers. Like terms contain the same letter(s) raised to the same power and can be combined by adding or subtracting their coefficients.
代数中，字母代表未知数。同类项是指含有相同字母且对应指数也相同的项，可以将其系数相加或相减。
For example, 3a + 5a simplifies to 8a, while 4x² + 2x cannot be combined because the powers of x are different. You can multiply terms in any order: 2p × 3q = 6pq.
例如 3a + 5a 化简为 8a，而 4x² + 2x 不能合并，因为 x 的指数不同。乘法可以按任意顺序：2p × 3q = 6pq。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. To expand 3(x + 4), multiply 3 by both x and 4 to get 3x + 12. Double brackets like (x + 2)(x + 5) can be expanded with the FOIL method: x² + 5x + 2x + 10 = x² + 7x + 10.
去括号使用分配律：a(b + c) = ab + ac。如 3(x + 4) = 3x + 12。两个括号相乘如 (x + 2)(x + 5) 可用 FOIL 方法得到 x² + 5x + 2x + 10 = x² + 7x + 10。
Factorising is the reverse of expanding. Look for a common factor and place it outside the bracket: 4x + 12 = 4(x + 3).
因式分解是去括号的逆运算。找出公因式并提到括号外：4x + 12 = 4(x + 3)。
5. Solving Linear Equations | 解一元一次方程
A linear equation contains an unknown, usually represented by a letter, and the highest power of the unknown is 1. The goal is to find the value that makes the equation true, using inverse operations to isolate the unknown.
一元一次方程含有一个未知数（通常用字母表示），未知数的最高次数为 1。目标是利用逆运算将未知数单独放在等号一边，求出它的值。
To solve 2x + 7 = 15, subtract 7 from both sides: 2x = 8, then divide by 2: x = 4. Always do the same operation to both sides to keep the equation balanced.
解 2x + 7 = 15，两边先减 7：2x = 8，再除以 2：x = 4。一定要对等号两边进行相同的运算以保持平衡。
When the unknown appears on both sides, collect like terms first. For 5x − 3 = 2x + 9, subtract 2x from both sides to get 3x − 3 = 9, then add 3 to both sides: 3x = 12, so x = 4.
当未知数出现在等号两边时，先合并同类项。如 5x − 3 = 2x + 9，两边同时减去 2x 得 3x − 3 = 9，再加 3：3x = 12，x = 4。
Equations with brackets should be expanded first. Solve 3(2y − 1) = 15 → 6y − 3 = 15 → 6y = 18 → y = 3.
含有括号的方程应先去括号。解 3(2y − 1) = 15，先展开得 6y − 3 = 15，6y = 18，y = 3。
6. Linear Graphs and Coordinates | 线性图像与坐标
A linear graph is a straight line. Its equation is usually written as y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).
线性图像是一条直线，其方程通常写成 y = mx + c，其中 m 是斜率，c 是直线在 y 轴上的截距。
To draw the graph of y = 2x + 1, create a table of values. Choose x‑values like 0, 1, 2, substitute them in, find the y‑values, and plot the coordinate pairs (x, y).
画 y = 2x + 1 的图像时，先列一个数值表。选取 x 值如 0, 1, 2，代入方程算出 y 值，然后在坐标系中描出对应的 (x, y) 点。
The gradient m can be found using two points on the line: m = change in y / change in x. A positive gradient slopes upwards, a negative gradient slopes downwards.
直线上两个点可以计算斜率：m = y 的变化量 / x 的变化量。正斜率向右上方倾斜，负斜率向右下方倾斜。
Horizontal lines have the equation y = a constant, and their gradient is 0. Vertical lines are x = a constant, but their gradient is undefined.
水平线的方程为 y = 常数，斜率为 0。竖直线的方程为 x = 常数，斜率不存在。
7. Area and Perimeter of 2D Shapes | 平面图形的面积与周长
Perimeter is the total distance around the outside of a shape. Area is the amount of space inside a flat shape, measured in square units.
周长是指围绕图形外边缘的总长度。面积是平面图形内部区域的大小，用平方单位表示。
Key formulas you need to know:
重点需要掌握的公式：
- Square: Perimeter = 4 × side length; Area = side².
- 正方形：周长 = 4 × 边长；面积 = 边长²。
- Rectangle: Perimeter = 2(l + w); Area = length × width.
- 长方形：周长 = 2(长 + 宽)；面积 = 长 × 宽。
- Triangle: Area = ½ × base × height.
- 三角形：面积 = ½ × 底 × 高。
- Parallelogram: Area = base × perpendicular height.
- 平行四边形：面积 = 底 × 垂直高。
- Trapezium: Area = ½ × (a + b) × h, where a and b are the parallel sides.
- 梯形：面积 = ½ × (上底 + 下底) × 高。
For compound shapes, divide them into simpler shapes, find individual areas, and add them together. For circles, the circumference is 2πr or πd, and the area is πr², where r is the radius.
对于组合图形，可以分割成若干个简单图形，分别求面积再相加。圆形的周长使用 2πr 或 πd，面积使用 πr²，其中 r 为半径。
8. Volume and Surface Area of Prisms | 棱柱的体积与表面积
A prism is a 3D shape with the same cross‑section running all the way through. The volume of any prism is found by multiplying the area of the cross‑section by the length.
棱柱是一种横截面处处相同的三维立体。任何棱柱的体积都可以用横截面面积乘以长度计算。
For a cuboid (rectangular prism), Volume = length × width × height. Surface area is the total area of all six faces. For a cube of side s, surface area = 6s².
长方体的体积 = 长 × 宽 × 高。表面积是所有六个面的面积之和。边长为 s 的立方体，表面积 = 6s²。
For a cylinder (a prism with circular cross‑section), Volume = πr²h, where r is the radius and h is the height. Its surface area is 2πr² + 2πrh (the two circles plus the curved surface).
圆柱是一种横截面为圆的棱柱，体积 = πr²h，r 是半径，h 是高。表面积 = 2πr² + 2πrh（两个圆面积加曲面面积）。
Remember to use consistent units and convert where needed. When calculating volume, the answer will be in cubic units (cm³, m³, etc.).
注意单位要统一，必要时要换算。计算体积时，结果用立方单位表示，如 cm³, m³ 等。
9. Transformations: Translation, Rotation, Reflection, Enlargement | 变换：平移、旋转、反射、放大
A transformation moves or changes a shape. The four main types are translation (sliding), rotation (turning), reflection (flipping), and enlargement (resizing).
几何变换可以移动或改变一个图形。四种主要变换是：平移（滑动）、旋转（转动）、反射（翻折）和放大（改变大小）。
Translation is described by a vector (e.g. 3 right, 2 up is written as (3, 2)). Every point moves the same distance and direction.
平移用向量描述，例如向右 3 格、向上 2 格记作 (3, 2)。图形上每个点都沿相同方向和距离移动。
Rotation needs a centre of rotation, an angle (90°, 180°, 270°), and a direction (clockwise or anticlockwise). The shape does not change size or orientation relative to the centre – it simply turns.
旋转要指定旋转中心、旋转角度（90°, 180°, 270°）和旋转方向（顺时针或逆时针）。图形大小和形状不变，只是绕着中心转动。
Reflection creates a mirror image across a given mirror line. Each point is the same perpendicular distance behind the line as the original is in front.
反射将图形沿给定的一条镜线翻折，形成镜像。每个点到镜线的垂直距离保持不变。
Enlargement changes the size by a scale factor from a centre of enlargement. A scale factor of 2 doubles all side lengths. If the scale factor is a fraction, the shape gets smaller.
放大（或缩小）根据中心点和放大系数改变图形大小。放大系数为 2 时所有边长翻倍；系数为分数时图形缩小。
10. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies only to right‑angled triangles. It states that the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它表明：斜边（最长边，正对直角）的平方等于另外两条直角边的平方和。
If the legs are a and b and the hypotenuse is c, then a² + b² = c². To find the hypotenuse, use c = √(a² + b²). To find a shorter side, say a, use a = √(c² − b²).
设两直角边为 a 和 b，斜边为 c，则 a² + b² = c²。求斜边可用 c = √(a² + b²)，求直角边可用 a = √(c² − b²)。
Example: In a triangle with legs 5 cm and 12 cm, the hypotenuse is √(5² + 12²) = √(25 + 144) = √169 = 13 cm.
例：一直角三角形两直角边分别为 5 cm 和 12 cm，则斜边长为 √(5² + 12²) = √(25 + 144) = √169 = 13 cm。
Always label the sides clearly before you start, and check that the triangle really does contain a right angle. Pythagoras is used in many real‑world problems involving distances, ladders, and navigation.
解题时先标出各边，并确认三角形确有直角。勾股定理常用于涉及距离、梯子、导航等实际问题的计算。
11. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It is always a number between 0 (impossible) and 1 (certain), often expressed as a fraction, decimal, or percentage.
概率表示一个事件发生的可能性大小。它的值总是在 0（不可能）和 1（必然）之间，常用分数、小数或百分比表示。
The probability of an event = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
事件概率 = 有利结果的数量 / 所有等可能结果的总数。
For a fair six‑sided die, the probability of rolling a prime number (2, 3, 5) is 3/6 = 1/2. The probability of an event not happening is 1 − P(event).
投掷一个均匀的六面骰子，得到质数（2, 3, 5）的概率是 3/6 = 1/2。某事件不发生的概率等于 1 减去该事件发生的概率。
Two events are mutually exclusive if they cannot happen at the same time. The sum of the probabilities of all mutually exclusive outcomes is 1.
如果两个事件不能同时发生，则它们互斥。所有互斥结果的概率之和为 1。
Sample space diagrams and tree diagrams help list all possible outcomes for combinations of events. The expected frequency of an event is probability × number of trials.
样本空间图和树状图能帮助列出事件组合的所有可能结果。一件事的期望频数等于概率乘以试验次数。
12. Averages and Range | 平均数与极差
Three common averages summarise a set of data: the mean (average), median (middle value), and mode (most frequent value). The range measures how spread out the data are.
表示数据集典型值的三种常用平均数是：平均数（均值）、中位数（中间值）和众数（出现次数最多的值）。极差用来度量数据的分散程度。
Mean = sum of all data values ÷ number of values. For example, the mean of 4, 7, 9, 10, 15 is (4+7+9+10+15)/5 = 45/5 = 9.
平均数 = 所有数据之和除以数据个数。如 4, 7, 9, 10, 15 的平均数为 (4+7+9+10+15)/5 = 45/5 = 9。
Median is the middle number when the data are arranged in order. If there is an even number of values, take the mean of the two middle numbers.
中位数是将数据按顺序排列后位于最中间的数。若数据个数为偶数，则取中间两个数的平均数。
Mode is the most common value. There can be more than one mode, or none if all values are equally frequent. Range = largest value − smallest value.
众数是出现次数最多的数值。可以不止一个众数，如果所有数值都只出现一次则没有众数。极差 = 最大值 − 最小值。
Choosing the most suitable average depends on the data. The mean is affected by outliers, while the median is not. The mode works well for non‑numerical data.
选择哪一种平均数取决于数据特点。平均值容易受极端值影响，中位数则不受影响。众数也适用于非数值数据。
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