📚 Year 8 CAIE Maths: Summer Preview and Bridging Course | Year 8 CAIE 数学：暑期预习与衔接课程
This summer bridging course is designed to help you transition smoothly from Year 7 to Year 8 Mathematics in the Cambridge Lower Secondary programme. You will revisit key foundations and get a head start on the new topics you will meet in Stage 8, such as scientific notation, linear graphs, inequalities and probability. Regular practice will build your confidence and problem‑solving skills before the new school year begins.
这门暑期衔接课程旨在帮助你从七年级数学平稳过渡到剑桥初中课程的八年级。你将重温关键基础，并提前接触 Stage 8 的新课题，如科学记数法、一次函数图像、不等式和概率。通过规律练习，你将在新学年开始前建立信心，提升解题能力。
1. Number Sense and Place Value | 数字感与位值
en as 4.5 × 10⁴ and 0.00032 as 3.2 × 10⁻⁴. Understanding powers of ten and the role of negative exponents is essential for working with measurements in science and geography.
在八年级，你将把位值概念扩展到极大和极小的数，包括使用科学记数法。例如，45 000 可以写成 4.5 × 10⁴，0.00032 可以写成 3.2 × 10⁻⁴。理解十的幂以及负指数的作用，对科学和地理学科中的测量计算至关重要。
Negative numbers appear frequently in real‑ integers. The rules: −7 + 3 = −4, (−5) × (−6) = 30, and 12 ÷ (−4) = −3 will all be used throughout the year.
负数经常出现在温度和银行账户余额等实际情境中。你必须熟练掌握负整数的排序和四则运算。以下规则将贯穿全年使用：−7 + 3 = −4，(−5) × (−6) = 30，12 ÷ (−4) = −3。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
will continue to convert freely between fractions, decimals and percentages. For example, ⅗ = 0.6 = 60% umbers, is a core skill. When adding ⅓ and ¼, you will rewrite them as ⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂.
你将继续在分数、小数和百分数之间自由转换。例如 ⅗ = 0.6 = 60%。对分数（包括带分数）进行四则运算是一项核心技能。计算 ⅓ + ¼ 时，你会将它们通分为 ⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂。
Percentage increase and decrease are widely used in financial contexts. To increase £240 by 15%, you can find 15% of £240 = £36 and add, giving £276, or use a multiplier: 1.15 × £240 = £276. Problems often involve finding the original amount after a percentage change — reverse percentages will be introduced.
百分比的增减在财务场景中应用广泛。要将 £240 增加 15%，可以先计算 15% × £240 = £36 然后相加，得到 £276，或使用乘数：1.15 × £240 = £276。题目常涉及已知百分比变化后的量求原值——倒推百分比也将被引入。
3. Ratio and Proportion | 比与比例
by their highest common factor. Sharing a quantity in a given ratio, for example dividing £120 in the ratio 2 : 3, means calculating the value of one share (2 + 3 = 5 shares, £120 ÷ 5 = £24 per share) so the parts are 2 × £24 = £48 and 3 × £24 = £72.
你应当能将 24 : 16 这类比化简为 3 : 2，方法是除以它们的最大公因数。按给定比例分配数量，例如将 £120 按 2 : 3 分配，意味着先算出一份的值（2 + 3 = 5 份，£120 ÷ 5 = £24/份），于是两部分分别为 2 × £24 = £48 和 3 × £24 = £72。
proportion means that two quantities increase or decrease at the same rate. If 5 pens cost £3.50, then 8 pens cost (8/5) × £3.50 = £5.60. In Year 8 you will also explore the idea of inverse proportion through real‑life examples, such as the time taken to complete a job with more workers.
正比例意味着两个量以相同的率增加或减少。如果 5 支笔花费 £3.50，那么 8 支笔花费 (8/5) × £3.50 = £5.60。在八年级，你还会通过生活实例探索反比例的概念，例如增加工人人数后完成一项工作所需的时间。
4. Algebraic Expressions and Sequences | 代数表达式与数列
) = 8x − 12. Factorising simple expressions by taking out a common factor, like 6y + 10 = 2(3y + 5), is also introduced.
八年级的代数超越简单的代入。你将合并同类项，例如 3a + 5b − a + 2b = 2a + 7b，并展开括号如 4(2x − 3) = 8x − 12。提取公因式进行因式分解，如 6y + 10 = 2(3y + 5)，也将被引入。
Sequences become more formal: you will generate terms of linear sequences from a term‑to‑term rule or from the position‑to‑term (nth term) rule. For the sequence 5, 8, 11, 14, … the nth term is 3n + 2. You will use this to find, say, the 50th term without writing out all terms.
数列的学习将更加系统化：你将根据项与项之间的规则，或根据位置与项（第 n 项）的规则来生成线性数列的项。对于数列 5, 8, 11, 14, …，第 n 项为 3n + 2。你将利用它来求第 50 项，而不必列出所有项。
5. Solving Linear Equations | 解一次方程
e tackled by performing the same operation to both sides to isolate the variable. Always check your solution by substituting back into the original equation.
你将求解需要两到三步的方程，如 2x + 7 = 19，得到 x = 6。未知数在等号两边的方程，如 5x − 3 = 2x + 9，可通过在等式两边进行相同操作来分离变量。一定要将解代回原方程进行检验。
Forming equations from word problems is a key skill. For example: "I think of a number, multiply it by 4 and add 5. The answer is 29." This translates to 4x + 5 = 29. Setting up and solving such equations helps you model real‑life situations.
根据文字题列方程是一项关键技能。例如：“我想一个数，把它乘 4 再加 5，结果是 29。”这可转化为方程 4x + 5 = 29。建立并求解这类方程有助于你模拟现实生活中的情境。
6. Inequalities | 不等式
n circle at 3 and an arrow to the right. The inequality y ≤ −2 uses a closed circle to include −2.
不等式使用符号 <、>、≤ 和 ≥ 来表示一个范围的值。例如，x > 3 表示所有大于 3 的数，在数轴上用 3 处的空心圆和向右的箭头表示。不等式 y ≤ −2 则用实心圆表示包含 −2。
e inequality sign. For −2x < 8, dividing by −2 gives x > −4. You will also write integer solutions that satisfy an inequality, such as x ≥ −1 and x < 3 giving −1, 0, 1, 2.
解一次不等式与解方程非常相似，但切记：两边同乘或同除以一个负数时，不等号方向要改变。对 −2x < 8，两边除以 −2 得 x > −4。你还将写出满足某个不等式的整数解，如 x ≥ −1 且 x < 3 给出 −1, 0, 1, 2。
7. Coordinates and Linear Graphs | 坐标与一次函数图像
alues for x (e.g. −2, −1, 0, 1, 2), calculate the corresponding y values, plot the points and draw a straight line through them.
在四个象限中描点已是必备知识。八年级你将绘制一次函数的图像。要画出 y = 2x + 1，先构建 x 的取值表（如 −2, −1, 0, 1, 2），计算相应的 y 值，描点，然后过这些点作一条直线。
The equation y = mx + c is central. Here m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑ an equation allows you to sketch graphs quickly and to see the effect of changing m or c.
方程 y = mx + c 是核心。其中 m 是斜率（倾斜度），c 是 y 轴截距（直线与 y 轴的交点）。直接从方程中识别斜率和截距能让你快速画出草图，并理解改变 m 或 c 带来的影响。
8. Angles and Shapes | 角与图形
You will work with angles on parallel lines: alternate angles are equal, corresponding angles are equal and interior angles add up to 180°. These facts are used to find missing angles in diagrams without measuring. Labelling angles with three letters, such as ∠ABC, is standard practice.
你将运用平行线上的角：内错角相等，同位角相等，同旁内角和为 180°。这些性质可用来求出图形中的未知角而不必测量。用三个字母标记角，如 ∠ABC，是标准做法。
Polygon angle rules are extended. The sum of interior angles of an n‑ 6 − 2) × 180° ÷ 6 = 120°. The sum of exterior angles of any convex polygon is always 360°.
多边形内角和的规律得以拓展。n 边形的内角和为 (n − 2) × 180°。对正六边形，每个内角为 (6 − 2) × 180° ÷ 6 = 120°。任何凸多边形的外角和总是 360°。
9. Area, Perimeter and Volume | 面积、周长与体积
a + b) × h). For compound shapes, split them into simpler parts, calculate individual areas and sum them.
你的公式工具箱将扩展到三角形面积（½ × 底 × 高）、平行四边形面积（底 × 高）和梯形面积（½ × (a + b) × h）。对于组合图形，将其分割成简单部分，分别计算面积再求和。
. Converting between units of volume, such as 1 m³ = 1 000 000 cm³, will become second nature.
长方体的体积为长 × 宽 × 高。如果所有尺寸都用厘米，体积单位为 cm³。表面积可通过将所有六个面的面积相加求得。体积单位的换算，如 1 m³ = 1 000 000 cm³，将成为你的第二天性。
10. Statistics and Probability | 统计与概率
Statistical literacy deepens: you will calculate the mean, median, mode and range for sets of data, and decide which average best represents a dataset. Displaying data in bar charts, pie charts and stem‑and‑leaf diagrams helps you to spot patterns and outliers.
统计素养得到深化：你将计算一组数据的平均数、中位数、众数和极差，并判断哪个平均数最能代表该数据集。用条形图、饼图和茎叶图展示数据，有助于你发现规律和异常值。
f an event is (number of favourable outcomes) ÷ (total number of outcomes). For a fair six‑ P(even) = 3/6 = ½. You will learn to list all outcomes systematically using sample spaces.
概率的引入伴随着从 0（不可能）到 1（必然）的量尺。一个事件的概率为（有利结果数）÷（总结果数）。对一枚均匀的六面骰子，P(偶数) = 3/6 = ½。你将学会用样本空间系统地列出所有可能结果。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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