Tag: KS3

📚 KS3 Advanced Maths: Typical Example Questions Explained in Detail | KS3 进阶数学：典型例题详解

In Key Stage 3 Maths, students are expected to move beyond basic calculations and tackle more challenging problems involving algebra, geometry, and data handling. This article provides a collection of typical advanced-level questions with step-by-step explanations, designed to strengthen problem-solving skills and deepen understanding.

在 KS3 数学中，学生需超越基础计算，解决更具挑战性的代数、几何和数据处理问题。本文精选了一组进阶难度的典型例题，并配有分步详解，旨在提升解题能力，加深理解。

---

1. Solving Linear Equations with Unknowns on Both Sides | 解两边带未知数的线性方程

Equations with variables on both sides appear frequently in advanced KS3. The key is to collect all variable terms on one side and constant terms on the other.

两边带有未知数的方程在进阶 KS3 中很常见。关键是把所有含未知数的项移到等式一边，常数项移到另一边。

Example: Solve 5x + 2 = 3x + 10.

例题： 解方程 5x + 2 = 3x + 10。

Subtract 3x from both sides: 5x – 3x + 2 = 3x – 3x + 10 → 2x + 2 = 10.

两边同时减去 3x：5x – 3x + 2 = 3x – 3x + 10 → 2x + 2 = 10。

Subtract 2 from both sides: 2x = 8 → x = 4.

两边同时减去 2：2x = 8 → x = 4。

Always check: 5(4) + 2 = 22 and 3(4) + 10 = 22, so the solution is correct.

务必检验：5(4) + 2 = 22，3(4) + 10 = 22，因此解正确。

---

2. Applying Pythagoras’ Theorem in 2D | 在二维图形中应用勾股定理

Pythagoras’ theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b².

勾股定理指出，在直角三角形中，斜边的平方等于两直角边的平方和：c² = a² + b²。

Example: Find the length of the diagonal of a rectangle with sides 6 cm and 8 cm.

例题： 一个矩形边长为 6 cm 和 8 cm，求对角线的长度。

The diagonal is the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm. Using c² = 6² + 8² = 36 + 64 = 100.

对角线即为直角三角形的斜边，两直角边分别为 6 cm 和 8 cm。根据 c² = 6² + 8² = 36 + 64 = 100。

Thus c = √100 = 10 cm. The diagonal is 10 cm long.

因此 c = √100 = 10 cm。对角线长度为 10 cm。

---

3. Working with Standard Form (Scientific Notation) | 科学记数法的运算

Standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Operations with standard form require careful handling of the powers.

科学记数法写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。进行运算时需要谨慎处理幂次。

Example: Calculate (3 × 10⁵) × (2 × 10³).

例题： 计算 (3 × 10⁵) × (2 × 10³)。

Multiply the coefficients: 3 × 2 = 6. Add the exponents: 5 + 3 = 8. The result is 6 × 10⁸.

系数相乘：3 × 2 = 6。指数相加：5 + 3 = 8。结果为 6 × 10⁸。

Example: Divide (8 × 10⁷) ÷ (4 × 10²).

例题： 计算 (8 × 10⁷) ÷ (4 × 10²)。

Divide coefficients: 8 ÷ 4 = 2. Subtract exponents: 7 − 2 = 5. Answer: 2 × 10⁵.

系数相除：8 ÷ 4 = 2。指数相减：7 − 2 = 5。答案：2 × 10⁵。

---

4. Finding the nth Term of Quadratic Sequences | 求二次数列的第 n 项

For a quadratic sequence, the second difference is constant. The nth term has the form an² + bn + c, where a is half the second difference.

对于二次数列，二阶差是常数。第 n 项形式为 an² + bn + c，其中 a 是二阶差的一半。

Example: Find the nth term of the sequence 3, 6, 11, 18, 27, …

例题： 求数列 3, 6, 11, 18, 27, … 的第 n 项。

First differences: 3, 5, 7, 9. Second differences: 2, 2, 2 → constant 2, so a = 1. The rule contains n².

一阶差：3, 5, 7, 9。二阶差：2, 2, 2 → 常数为 2，因此 a = 1。通项包含 n²。

Subtract n² from each term: 3−1=2, 6−4=2, 11−9=2, 18−16=2, 27−25=2. This gives linear sequence 2,2,2,2,2, which is the constant 2. So nth term = n² + 2.

从每一项减去 n²：3−1=2, 6−4=2, 11−9=2, 18−16=2, 27−25=2，得到常数序列 2,2,2,2,2。因此第 n 项为 n² + 2。

---

5. Interior and Exterior Angles of Polygons | 多边形的内角与外角

The sum of exterior angles of any convex polygon is 360°. Each interior angle = 180° − exterior angle. For a regular polygon with n sides, interior angle = (n−2)×180°/n.

任何凸多边形的外角和都是 360°。每个内角 = 180° − 外角。对于有 n 条边的正多边形，内角 = (n−2)×180°/n。

Example: A regular polygon has an interior angle of 156°. How many sides does it have?

例题： 一个正多边形的内角为 156°，它有多少条边？

Exterior angle = 180° − 156° = 24°. Number of sides n = 360° ÷ exterior angle = 360 ÷ 24 = 15.

外角 = 180° − 156° = 24°。边数 n = 360° ÷ 外角 = 360 ÷ 24 = 15。

The polygon has 15 sides.

该多边形有 15 条边。

---

6. Probability Space Diagrams and Two-Way Tables | 概率空间图与双向表

Probability space diagrams help list all possible outcomes when two events occur. The probability of an event is the number of favourable outcomes divided by total outcomes.

概率空间图用于列出两个事件发生的所有可能结果。某事件的概率 = 有利结果数 ÷ 总结果数。

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

例题： 掷两个公平的六面骰子，求点数之和为 7 的概率。

Total outcomes = 6 × 6 = 36. Favourable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes. Probability = 6/36 = 1/6.

总结果数 = 6 × 6 = 36。有利组合：(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 个结果。概率 = 6/36 = 1/6。

---

7. Direct and Inverse Proportion | 正比例与反比例

In direct proportion, y = kx for some constant k. In inverse proportion, y = k/x. Finding k using given values allows calculation for new situations.

正比例关系为 y = kx，k 为常数。反比例关系为 y = k/x。利用已知值求出 k，即可计算其他情况。

Example: y is directly proportional to x. When x = 4, y = 10. Find y when x = 7.

例题： y 与 x 成正比。当 x = 4 时 y = 10，求 x = 7 时 y 的值。

y = kx → 10 = k × 4 → k = 10/4 = 2.5. Then y = 2.5 × 7 = 17.5.

y = kx → 10 = k × 4 → k = 10/4 = 2.5。因此 y = 2.5 × 7 = 17.5。

Example: y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.

例题： y 与 x 成反比。当 x = 3 时 y = 8，求 x = 6 时 y 的值。

y = k/x → 8 = k/3 → k = 24. Then y = 24/6 = 4.

y = k/x → 8 = k/3 → k = 24。因此 y = 24/6 = 4。

---

8. Surface Area and Volume of Prisms | 棱柱的表面积和体积

Volume of a prism = area of cross-section × length. Surface area is the total area of all faces. For a triangular prism, decompose into rectangles and triangles.

棱柱体积 = 横截面积 × 长度。表面积是所有面的面积之和。对于三棱柱，需分解为矩形和三角形来计算。

Example: A triangular prism has a right-angled triangle base with legs 3 cm and 4 cm, and length 10 cm. Find its volume and surface area.

例题： 一个三棱柱的底面为直角三角形，直角边 3 cm 和 4 cm，棱柱长 10 cm。求它的体积和表面积。

Cross-sectional area = (1/2)×3×4 = 6 cm². Volume = 6 × 10 = 60 cm³.

横截面积 = (1/2)×3×4 = 6 cm²。体积 = 6 × 10 = 60 cm³。

Surface area: Hypotenuse of triangle = √(3²+4²) = 5 cm. Area of two triangular faces = 2 × 6 = 12 cm². Lateral area = perimeter of triangle × length = (3+4+5)×10 = 12×10 = 120 cm². Total surface area = 12 + 120 = 132 cm².

表面积：三角形斜边 = √(3²+4²) = 5 cm。两个三角形面面积 = 2 × 6 = 12 cm²。侧面积 = 三角形周长 × 长度 = (3+4+5)×10 = 12×10 = 120 cm²。总表面积 = 12 + 120 = 132 cm²。

---

9. Linear Graphs: Finding the Equation of a Line | 线性图：求直线方程

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Given two points, you can calculate m and then find c.

直线方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。已知两点，可先求 m，再求 c。

Example: Find the equation of the line passing through (2, 5) and (4, 9).

例题： 求经过点 (2, 5) 和 (4, 9) 的直线方程。

Gradient m = (9 − 5) / (4 − 2) = 4/2 = 2. So y = 2x + c. Substitute (2,5): 5 = 2(2) + c → 5 = 4 + c → c = 1. Equation: y = 2x + 1.

斜率 m = (9 − 5) / (4 − 2) = 4/2 = 2。因此 y = 2x + c。代入 (2,5)：5 = 2(2) + c → 5 = 4 + c → c = 1。直线方程为 y = 2x + 1。

Check with the other point: when x=4, y=2(4)+1=9, correct.

用另一点检验：当 x=4，y=2(4)+1=9，无误。

---

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 KS3 Maths: Essential Topics from Book 8H | KS3 数学：8H 核心知识点精讲

This article covers the key knowledge points typically found in Essential Maths Book 8H, a popular resource for Key Stage 3 students. We break down core topics including number operations, algebra, geometry, and statistics, providing clear explanations and worked examples to boost understanding and exam confidence.

本文涵盖《Essential Maths Book 8H》中的核心知识点，这是 KS3 阶段广泛使用的学习资料。我们将深入讲解数系运算、代数、几何与统计等关键主题，通过清晰的讲解和例题，帮助巩固理解并提升应考信心。

1. Numbers and Operations | 数及其运算

Understanding place value and the four operations (addition, subtraction, multiplication, division) with integers and decimals is fundamental. Students must confidently apply the order of operations, often remembered by BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).

理解位值以及整数与小数的四则运算（加、减、乘、除）是基础。学生必须熟练运用运算顺序，通常用 BIDMAS（括号、指数、除法/乘法、加法/减法）来记忆。

Example: Calculate 3 + 4 × (5 – 2)². First brackets: 5 – 2 = 3. Then indices: 3² = 9. Then multiplication: 4 × 9 = 36. Finally addition: 3 + 36 = 39.

示例：计算 3 + 4 × (5 – 2)²。先括号：5 – 2 = 3。再指数：3² = 9。然后乘法：4 × 9 = 36。最后加法：3 + 36 = 39。

Negative numbers also appear frequently. Key rules: (-3) + (-5) = -8; (-3) × (-4) = 12; 6 ÷ (-2) = -3. A number line helps visualise addition and subtraction of negatives.

负数也经常出现。关键规则：(-3) + (-5) = -8；(-3) × (-4) = 12；6 ÷ (-2) = -3。数轴有助于想象负数的加减法。

---

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

Interconverting between fractions, decimals and percentages is a vital skill. For instance, 3/5 = 0.6 = 60%. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.

分数、小数和百分比之间的互化是一项关键技能。例如，3/5 = 0.6 = 60%。将分数化为小数，用分子除以分母；将小数化为百分比，乘以 100。

Operations with fractions: adding/subtracting requires a common denominator. For 1/3 + 1/4, use 12 as denominator: 4/12 + 3/12 = 7/12. Multiplication: multiply numerators and denominators; division: multiply by the reciprocal.

分数运算：加减需通分。例如 1/3 + 1/4，以 12 为公分母：4/12 + 3/12 = 7/12。乘法：分子乘分子，分母乘分母；除法：乘倒数。

Percentages: finding a percentage of an amount and percentage increase/decrease. To find 15% of £80, calculate 10% = £8, 5% = £4, so 15% = £12. A 20% decrease on £50 gives a new price of £40.

百分比：求一个数的百分比以及百分比的增减。求 £80 的 15%，先算 10% = £8，5% = £4，所以 15% = £12。在 £50 基础上减少 20%，新价格为 £40。

---

3. Ratio and Proportion | 比率与比例

Ratio compares quantities. Simplifying ratios is similar to simplifying fractions. The ratio 8:12 can be divided by 4 to get 2:3. Sharing in a given ratio: divide £45 in the ratio 2:3 → total parts = 5, each part = £9, so amounts are £18 and £27.

比率用于比较数量。化简比率与化简分数相似。比率 8:12 可以除以 4 得到 2:3。按比例分配：将 £45 按 2:3 分配 → 总份数 = 5，每份 = £9，因此得到 £18 和 £27。

Direct proportion: as one quantity increases, the other increases at the same rate. If 5 pens cost £3.50, then 8 pens cost (8 ÷ 5) × £3.50 = £5.60. The unitary method is very useful here.

正比例：一个量增加，另一个量也以相同速率增加。若 5 支笔 £3.50，则 8 支笔花费 (8 ÷ 5) × £3.50 = £5.60。单位法在这里非常有效。

Scale drawings and maps use ratio. A scale of 1:50 000 means 1 cm on the map represents 50 000 cm (0.5 km) in real life.

比例尺图和地图用比率表示。比例尺 1:50 000 表示地图上 1 cm 代表实际 50 000 cm（0.5 公里）。

---

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

Algebra uses letters to represent unknown numbers. Simplifying expressions: 3a + 2b – a + 4b = 2a + 6b. Expanding brackets: 3(x + 4) = 3x + 12; and factorising: 6x + 9 = 3(2x + 3).

代数用字母表示未知数。化简表达式：3a + 2b – a + 4b = 2a + 6b。展开括号：3(x + 4) = 3x + 12；分解因式：6x + 9 = 3(2x + 3)。

Solving linear equations: aim to isolate the variable. For 2x + 5 = 13, subtract 5 from both sides: 2x = 8, then divide by 2: x = 4. Always check your answer by substitution.

解一次方程：目标是分离变量。对于 2x + 5 = 13，两边减 5：2x = 8，然后除以 2：x = 4。务必用代入法检查答案。

Equations can involve brackets or unknowns on both sides. Solve 3(2y – 1) = 5y + 4 → expand: 6y – 3 = 5y + 4 → subtract 5y: y – 3 = 4 → y = 7.

方程可能含有括号或两边都有未知数。解 3(2y – 1) = 5y + 4 → 展开：6y – 3 = 5y + 4 → 两边减 5y：y – 3 = 4 → y = 7。

---

5. Sequences and Patterns | 数列与规律

A sequence is an ordered list of numbers following a rule. Arithmetic sequences have a common difference. Find the nth term of 5, 8, 11, 14, … The difference is +3, so nth term = 3n + 2. Check: for n=1, 3(1)+2=5; n=2, 8. Correct.

数列是按一定规则排列的数。等差数列有公差。求数列 5, 8, 11, 14, … 的第 n 项。差为 +3，因此第 n 项 = 3n + 2。检验：n=1，3(1)+2=5；n=2，8。正确。

Other sequences include square numbers (1, 4, 9, 16…), triangular numbers, and Fibonacci where each term is the sum of the two preceding ones. Recognising patterns visually and in numbers is tested.

其他数列包括平方数（1, 4, 9, 16…）、三角形数以及斐波那契数列（每一项是前两项之和）。考试会考察从图形和数字中识别规律。

Generating terms: for nth term = 4n – 3, the first three terms are 1, 5, 9. This is essential for understanding linear patterns.

生成项：对于第 n 项 = 4n – 3，前三项为 1, 5, 9。这对理解线性规律至关重要。

---

6. Geometry: Angles and Shapes | 几何：角度与图形

Angle facts: angles on a straight line sum to 180°, around a point sum to 360°. Vertically opposite angles are equal. In a triangle, the sum of interior angles is 180°.

角度基本事实：直线上的角之和为 180°，围绕一点的角度之和为 360°。对顶角相等。三角形内角和为 180°。

Parallel lines: alternate angles (Z-shape) are equal, corresponding angles (F-shape) are equal, and co-interior angles (C-shape) sum to 180°. These help find missing angles in diagrams.

平行线：内错角（Z 形）相等，同位角（F 形）相等，同旁内角（C 形）之和为 180°。这些规律有助于求图中未知角度。

Properties of quadrilaterals: square (4 equal sides, 4 right angles), rectangle (opposite sides equal, 4 right angles), parallelogram (opposite sides parallel, opposite angles equal), rhombus, trapezium. Sum of interior angles in any quadrilateral is 360°.

四边形的性质：正方形（四边等长，四个直角），矩形（对边等长，四个直角），平行四边形（对边平行，对角相等），菱形，梯形。任何四边形的内角和均为 360°。

---

7. Measures: Area and Perimeter | 测量：面积与周长

Perimeter is the distance around a shape. For a rectangle, P = 2(length + width). Area of rectangle = length × width. Area of triangle = ½ × base × height. Compound shapes require splitting into basic figures.

周长是图形边界总长。矩形周长 P = 2(长 + 宽)。矩形面积 = 长 × 宽。三角形面积 = ½ × 底 × 高。组合图形需要分解为基本图形。

Area of a parallelogram = base × perpendicular height. Area of a trapezium = ½ × (a + b) × h, where a and b are parallel sides and h is the perpendicular height.

平行四边形面积 = 底 × 垂直高度。梯形面积 = ½ × (a + b) × h，其中 a 和 b 为平行边，h 为垂直高度。

Units for area: mm², cm², m². Converting: 1 m² = 10 000 cm² because 1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm². Volume units: cm³, m³, and capacity: 1 litre = 1000 cm³.

面积单位：mm²、cm²、m²。换算：1 m² = 10 000 cm²，因为 1 m = 100 cm，所以 1 m² = 100 × 100 = 10 000 cm²。体积单位：cm³、m³，以及容积：1 升 = 1000 cm³。

---

8. Statistics and Probability | 统计与概率

Averages: mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent; range = highest minus lowest. Choosing the best average is important for interpreting data.

平均数：平均数 = 总和 ÷ 数量；中位数 = 排序后的中间值；众数 = 出现次数最多的值；极差 = 最大值减最小值。选择合适的平均数对数据解读很重要。

Probability scale from 0 (impossible) to 1 (certain). Probability of an event = number of favourable outcomes ÷ total number of outcomes. For a fair six-sided die, P(rolling an even number) = 3/6 = ½.

概率尺度从 0（不可能）到 1（必然发生）。事件概率 = 有利结果个数 ÷ 总结果个数。掷一个公平的六面骰子，掷出偶数的概率 = 3/6 = ½。

Tree diagrams and sample space diagrams can list outcomes for two or more events. Relative frequency from an experiment estimates probability.

树状图和样本空间图可用于列出两个或以上事件的结果。实验得出的相对频数可以估计概率。

Presenting data: bar charts, pie charts, line graphs, and scatter graphs. Scatter graphs show correlation: positive, negative, or none. A line of best fit can be drawn to predict values.

数据展示：条形图、饼图、折线图和散点图。散点图显示相关性：正相关、负相关或无相关。可以画出最佳拟合线来预测数值。

---

9. Graphs and Coordinates | 图与坐标

Coordinates are written (x, y) in the Cartesian plane. Plotting points, drawing straight lines from linear equations, and reading graphs are core skills. The equation of a straight line is often y = mx + c, where m is the gradient and c the y-intercept.

坐标在笛卡尔平面中用 (x, y) 表示。描点、根据线性方程画直线以及读图是核心技能。直线方程通常为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。

To plot y = 2x + 1, choose x-values like -2, -1, 0, 1, 2, calculate y-values, plot and join. The gradient m = 2 means for every 1 unit right, go up 2. The line crosses y-axis at (0,1).

画 y = 2x + 1，选 x 值如 -2, -1, 0, 1, 2，计算 y 值，描点连线。斜率 m = 2 表示每向右 1 单位，向上 2 单位。该直线与 y 轴交于 (0,1)。

Real-life graphs: distance-time graphs have speed as gradient. A horizontal line means stationary. Conversion graphs help change units, e.g., miles to kilometres.

实际应用图：距离-时间图中斜率为速度。水平线表示静止。转换图可用于换算单位，例如英里与公里的转换。

---

10. Problem Solving and Reasoning | 问题解决与推理

Word problems require translating English into maths. Read carefully, identify what is asked, pick out numbers and keywords (more than, less, total, each). Use a step-by-step approach and check units.

应用题需要将文字转化为数学表达。仔细阅读，明确问题，找出数字和关键词（多、少、总共、每个）。采用分步解决法并检查单位。

Reasoning tasks involve justifying statements: ‘Is the sum of two consecutive odd numbers always even?’ Yes, because (2n+1) + (2n+3) = 4n+4 = 2(2n+2), which is a multiple of 2.

推理任务要求论证陈述：“两个连续奇数的和总是偶数吗？” 是的，因为 (2n+1) + (2n+3) = 4n+4 = 2(2n+2)，是 2 的倍数。

Multi-step problems combine topics: find the cost of painting a wall given dimensions, paint coverage and price per litre. Use area, division, and money calculations systematically.

多步骤问题会综合多个主题：根据墙壁尺寸、油漆覆盖率和每升价格计算粉刷成本。需要系统使用面积、除法及货币计算。

Developing logical thinking and checking for reasonableness helps avoid common mistakes and builds mathematical confidence at KS3.

培养逻辑思维并检查答案合理性，有助于避免常见错误，并在 KS3 阶段建立数学信心。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 KS3 Maths: Kinematics Key Points | KS3 数学：运动学考点精讲

Kinematics is the branch of mathematics that describes motion. At KS3 level, you learn to calculate speed, distance, and time using simple formulas, interpret distance‑time graphs, and convert between common units of speed. Mastering these fundamentals will give you confidence when you later encounter more advanced topics like acceleration in GCSE Physics and Maths.

运动学是描述运动的数学分支。在 KS3 阶段，你会学习如何使用简单的公式计算速度、距离和时间，解读距离‑时间图，以及转换常见的速度单位。掌握这些基础知识会让你以后在 GCSE 物理和数学中遇到加速度等更高级主题时充满信心。

---

1. What is Kinematics? | 什么是运动学？

Kinematics focuses on how objects move, without worrying about the forces that cause the motion. You will work with quantities such as distance, time, speed, and later velocity and acceleration. For KS3 maths, the emphasis is on uniform (steady) motion and understanding the relationship between distance, speed, and time.

运动学关注物体如何运动，而不考虑导致运动的力。你会接触路程、时间、速度等量，将来还会学习速率和加速度。在 KS3 数学中，重点在于匀速运动以及理解距离、速度和时间之间的关系。

---

2. Speed, Distance, and Time Basics | 速度、距离和时间基础

Speed tells you how fast an object is moving. Distance is how far the object travels. Time is the duration of the journey. These three are linked by a very important equation. If you know any two of them, you can work out the third.

速度告诉你物体运动有多快。距离是物体移动的路程。时间是行程的持续时长。这三者由一个非常重要的公式联系起来。如果你知道其中任意两个量，就可以求出第三个。

---

3. The Speed Formula | 速度公式

The formula that connects speed, distance, and time is:

Speed = Distance ÷ Time

Some people remember this as the triangle: put D on top, and S and T on the bottom. Cover the quantity you want to leave the correct calculation.

连接速度、距离和时间的公式为：

速度 = 距离 ÷ 时间

有些人用三角形记忆法：把 D 放在顶部，S 和 T 放在底部。盖住你要求的量，剩下的就是正确的计算公式。

---

4. Rearranging the Formula | 公式变形

You must be able to rearrange the speed formula. To find distance, use:

Distance = Speed × Time

To find time, use:

Time = Distance ÷ Speed

Practise switching between these forms until it becomes automatic.

你必须能够对速度公式进行变形。要求距离，用：

距离 = 速度 × 时间

要求时间，用：

时间 = 距离 ÷ 速度

练习在这些形式之间切换，直至能够自动反应。

---

5. Units of Speed | 速度的单位

Speed can be measured in metres per second (m/s) or kilometres per hour (km/h). In KS3 maths, you will often see both. Always check the units given in a problem and make sure your answer uses the correct ones. Sometimes you may need to convert distances or times first (e.g. minutes to hours).

速度可以用米每秒 (m/s) 或千米每小时 (km/h) 来衡量。在 KS3 数学中，你经常会见到这两种单位。始终检查题目给出的单位，确保你的答案使用正确的单位。有时你需要先转换距离或时间（例如把分钟转化为小时）。

---

6. Converting Units: m/s to km/h | 单位转换：米每秒与千米每小时

To change m/s to km/h, multiply by 3.6. This is because 1 m/s means 3600 metres per hour, which is 3.6 km/h. To change km/h to m/s, divide by 3.6.

将米每秒转化为千米每小时，乘以 3.6。这是因为 1 m/s 意味着每小时 3600 米，也就是 3.6 km/h。将千米每小时转化为米每秒，则除以 3.6。

From m/s to km/h	× 3.6
From km/h to m/s	÷ 3.6

---

7. Average Speed | 平均速度

In real life, objects don’t always move at a constant speed. Average speed is the total distance travelled divided by the total time taken. Even if a car speeds up and slows down, the average speed tells you the overall rate of the whole journey.

在现实生活中，物体并不总是以恒定速度运动。平均速度是总路程除以总时间。即使一辆汽车忽快忽慢，平均速度也能告诉你整个行程的整体快慢程度。

Example: A cyclist covers 15 km in 0.75 hours. Average speed = 15 ÷ 0.75 = 20 km/h.

例子：一位骑行者 0.75 小时行驶了 15 千米。平均速度 = 15 ÷ 0.75 = 20 km/h。

---

8. Distance-Time Graphs | 距离‑时间图

A distance‑time graph shows how distance from a starting point changes over time. Time is on the horizontal axis (x‑axis) and distance is on the vertical axis (y‑axis). These graphs are very useful for visualising motion.

距离‑时间图显示从起点出发的距离如何随时间变化。时间在横轴（x 轴）上，距离在纵轴（y 轴）上。这些图对于直观显示运动非常有用。

• A straight line sloping upwards means constant speed.
• A horizontal line means the object is at rest (stationary).
• A steeper line means a greater speed.

• 一条向上倾斜的直线表示匀速运动。
• 一条水平线表示物体静止。
• 线条越陡，速度越大。

---

9. Interpreting Graphs: Steady Speed and Rest | 图线解读：匀速与静止

When the line is straight and diagonal, the gradient (slope) equals the speed. To calculate speed from a distance‑time graph, pick two points on the straight line and find the change in distance (vertical) divided by the change in time (horizontal).

当图线是一条笔直的斜线时，其梯度（斜率）等于速度。要从距离‑时间图计算速度，在直线上取两个点，用距离的变化（纵向）除以时间的变化（横向）。

Speed = (distance₂ – distance₁) ÷ (time₂ – time₁)

速度 = (距离₂ – 距离₁) ÷ (时间₂ – 时间₁)

A curved line indicates changing speed (acceleration or deceleration), but at KS3 you will often focus on straight‑line segments.

曲线表示速度在变化（加速或减速），但在 KS3 阶段，你通常只需要处理直线段。

---

10. Solving Problems with Distance‑Time Graphs | 利用距离‑时间图解题

You may be asked to draw a distance‑time graph from a description of a journey, or to describe a journey from a given graph. Practise turning worded problems into a table of time and distance values, plot the points, and join them with straight lines where appropriate.

你可能会被要求根据行程描述绘制距离‑时间图，或者根据给定的图描述行程。请练习将文字问题转化为时间和距离的数值表格，描点，并在适当位置用直线连接。

Example: ‘A car travels at a steady speed for 2 hours covering 100 km, then stops for a 1‑hour rest, then continues for another 3 hours covering 150 km.’ Draw the graph as three line segments: one sloping up, one horizontal, then another sloping up.

例子：“一辆车以恒定速度行驶了 2 小时，行程 100 千米，然后停下休息 1 小时，再继续行驶 3 小时，行程 150 千米。” 将图画为三条线段：一段向上倾斜，一段水平，然后另一段向上倾斜。

---

11. Common Mistakes | 常见错误

• Confusing minutes with hours: always convert time into hours if speed is in km/h, or into seconds if speed is in m/s.
• Forgetting to use average speed when the journey has different parts.
• Mixing up the axes on a distance‑time graph – remember, time always goes on the x‑axis.
• Using the total distance for a line segment’s speed instead of the distance covered in that segment.

• 混淆分钟与小时：如果速度单位是 km/h，总要把时间转换为小时；如果是 m/s 则转换为秒。
• 忘记了当行程包含不同阶段时应该使用平均速度。
• 在距离‑时间图上混淆坐标轴——记住，时间始终在 x 轴上。
• 用总距离代替某一段线段的速度计算时应使用该段内经过的距离。

---

12. Practice Tips | 练习建议

To become confident with kinematics at KS3, work through plenty of examples that combine unit conversion, speed calculations, and graph interpretation. Set yourself challenges like drawing a distance‑time graph of your journey to school.

要在 KS3 阶段掌握运动学，请大量练习包含单位转换、速度计算和图线解读的例题。给自己一点挑战，比如画一张你上学路程的距离‑时间图。

Also, use the triangle method to check your rearranged formulas quickly, and always write down your working step by step, including units at each stage.

此外，用三角形方法快速核验你的公式变形，并始终一步步写下你的解题过程，每一步都要带上单位。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 KS3 Maths: Clearing Up Common Confusions | KS3 数学：常见混淆概念解析

In Key Stage 3 Mathematics, students encounter many fundamental concepts that can easily be confused. Understanding the subtle differences between similar mathematical ideas is essential for building a strong foundation. This article explores common areas of confusion, such as expressions versus equations, area versus perimeter, and factors versus multiples, providing clear explanations and examples to help students master these distinctions.

在KS3数学学习中，学生会遇到许多基本概念，这些概念很容易混淆。准确理解相似数学概念之间的细微差别，是夯实数学基础的关键。本文探讨了常见的易混淆点，如表达式与方程、面积与周长、因数与倍数等，通过清晰的解释和示例，帮助学生掌握这些区别。

---

1. Expressions vs Equations | 表达式与方程

An expression is a combination of numbers, variables and operators (like +, -, ×, ÷) but contains no equality sign. For example, 3x + 5 is an expression.

表达式是由数字、变量和运算符（如 +、-、×、÷）组合而成的式子，不含等号。例如 3x + 5 就是一个表达式。

An equation, however, states that two expressions are equal by using an equals sign. You can solve an equation to find the value of the unknown. For instance, 3x + 5 = 11 is an equation.

方程则是用等号表示两个表达式相等。你可以解方程来求出未知数的值。例如 3x + 5 = 11 就是一个方程。

In summary, expressions are simplified or evaluated, while equations are solved.

总而言之，表达式需要化简或求值，而方程则需要求解。

---

2. Area vs Perimeter | 面积与周长

Perimeter is the total distance around the outside of a 2D shape. It is measured in units of length (e.g. cm, m).

周长是二维图形外边一周的总长度，用长度单位（如厘米、米）来衡量。

Area is the amount of space inside the shape. It is measured in square units (e.g. cm², m²).

面积是图形内部空间的大小，用平方单位（如 cm²、m²）来衡量。

For a rectangle, perimeter = 2(l + w), area = l × w. Confusing the two often leads to using wrong units or formulas.

对于矩形，周长 = 2(长 + 宽)，面积 = 长 × 宽。混淆这两个概念常导致使用错误的单位或公式。

The following table summarises the differences:

下表总结了区别：

Perimeter	Distance around	cm, m	Rectangle: 2(l + w)
Area	Space inside	cm², m²	Rectangle: l × w

---

3. Mean, Median and Mode | 平均数、中位数与众数

These are all measures of central tendency but calculated differently. The mean (average) is the sum of all values divided by the number of values.

这三者都是集中趋势的度量，但计算方法不同。平均数（平均值）是所有数值之和除以数值的个数。

The median is the middle value when the data is ordered. If there are two middle numbers, the median is their mean.

中位数是将数据按大小排序后处于中间的值。如果有两个中间数，则中位数是这两个数的平均数。

The mode is the value that appears most frequently. A data set can have one mode, more than one mode, or no mode at all.

众数是出现次数最多的数值。一组数据可以有一个众数、多个众数或没有众数。

Students often confuse which measure is affected by extreme values: the mean is affected, while median and mode are more resistant.

学生常混淆哪种度量受极端值影响：平均数受影响，而中位数和众数更具抗干扰性。

---

4. Factors vs Multiples | 因数与倍数

A factor of a number divides exactly into that number with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.

一个数的因数是能整除该数的数，余数为零。例如 3 是 12 的因数，因为 12 ÷ 3 = 4。

A multiple of a number is the product of that number and an integer. So 12 is a multiple of 3 because 3 × 4 = 12.

一个数的倍数是该数与一个整数的乘积。因此 12 是 3 的倍数，因为 3 × 4 = 12。

Remember: factors are smaller or equal to the number, while multiples are larger or equal. The number itself is both a factor and a multiple.

请记住：因数小于或等于原数，而倍数大于或等于原数。该数本身既是因数也是倍数。

---

5. Prime vs Composite Numbers | 质数与合数

A prime number has exactly two distinct factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7.

质数恰好有两个不同的因数：1 和它本身。例如 7 是质数，因为只有 1 和 7 两个因数。

A composite number has more than two factors. For example, 8 has factors 1, 2, 4, 8, so it is composite.

合数有超过两个因数。例如 8 的因数有 1、2、4、8，所以它是合数。

Note that 1 is neither prime nor composite. Also, 2 is the only even prime number.

注意，1 既不是质数也不是合数。另外，2 是唯一的偶质数。

---

6. Direct vs Inverse Proportion | 正比例与反比例

Two quantities are in direct proportion if they increase or decrease together at the same rate. The ratio between them remains constant, so y = kx.

两个量如果以相同的速率同时增加或减少，则成正比例。它们之间的比值保持恒定，所以 y = kx。

Inverse proportion means that as one quantity increases, the other decreases proportionally. Their product is constant: xy = k.

反比例是指一个量增加时，另一个量按比例减少。它们的乘积为常数：xy = k。

A common mistake is mixing up the equations: direct proportion is y/x = k, inverse is xy = k. Check whether multiplying or dividing gives a constant.

常见错误是混淆公式：正比例满足 y/x = k，反比例满足 xy = k。检查相乘或相除哪个得到常数。

---

7. Line Symmetry vs Rotational Symmetry | 轴对称与旋转对称

A shape has line symmetry (reflection symmetry) if it can be folded along a line (the mirror line) so that one half fits exactly onto the other.

如果一个图形可以沿一条直线（对称轴）对折，使得两部分完全重合，则该图形具有轴对称（反射对称）。

Rotational symmetry occurs when a shape can be rotated about a central point and still look the same in less than a full turn. The order of rotational symmetry tells you how many times it matches within 360°.

旋转对称是指图形绕中心点旋转一定角度（小于一整圈）后能与原图重合。旋转对称的阶数表示在 360° 内重合的次数。

For example, a square has 4 lines of symmetry and rotational symmetry of order 4. A rectangle has 2 lines of symmetry and rotational symmetry of order 2.

例如，正方形有 4 条对称轴，旋转对称阶数为 4。长方形有 2 条对称轴，旋转对称阶数为 2。

---

8. Discrete vs Continuous Data | 离散数据与连续数据

Discrete data can only take specific, separate values. These are often counted, like the number of students in a class.

离散数据只能取特定的、分开的数值，通常是计数得到的，比如一个班级的学生人数。

Continuous data can take any value within a range. Measurements like height, weight, or time are continuous because they can include fractions and decimals.

连续数据可以在一定范围内取任意数值。像身高、体重、时间这样的测量值是连续的，因为它们可以包含分数和小数。

In graphs, discrete data is shown with points that are not joined, whereas continuous data points are often connected by a line.

在图表中，离散数据用不相连的点表示，而连续数据点通常用线连接。

---

9. Theoretical vs Experimental Probability | 理论概率与实验概率

Theoretical probability is what we expect to happen based on equally likely outcomes. For a fair coin, P(head) = ½.

理论概率是基于等可能结果我们预期发生的数值。对于一枚均匀硬币，P(正面) = ½。

Experimental (relative frequency) is based on actual trials: number of times the event occurs divided by total trials. It may differ from theoretical probability, especially with few trials.

实验概率（相对频率）是基于实际试验的：事件发生的次数除以总试验次数。它可能与理论概率有差异，尤其是在试验次数较少时。

As the number of trials increases, experimental probability tends to get closer to the theoretical value (law of large numbers).

随着试验次数的增加，实验概率会趋近于理论值（大数定律）。

---

10. Simplifying vs Expanding | 化简与展开

Simplifying an expression means writing it in its most compact form by collecting like terms. e.g. 2x + 3x simplifies to 5x.

化简表达式是指通过合并同类项将其写成最简洁的形式。例如 2x + 3x 化简为 5x。

Expanding means removing brackets by multiplying each term inside by the factor outside. e.g. 3(x + 2) expands to 3x + 6.

展开是指通过将括号内的每一项乘以外面的因式来去掉括号。例如 3(x + 2) 展开为 3x + 6。

These are opposite operations. Expanding converts a product into a sum, while simplifying often does the reverse by grouping.

这两个运算是互逆的。展开将乘积转化为和，而化简常常通过分组做相反的转换。

Published by TutorHao | KS3 Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）