Year 8 Cambridge Mathematics: Core Knowledge Points | Year 8 剑桥数学：核心知识点梳理

📚 Year 8 Cambridge Mathematics: Core Knowledge Points | Year 8 剑桥数学：核心知识点梳理

Welcome to the comprehensive guide for Year 8 Cambridge Mathematics. This article walks you through the essential topics you will encounter, from number operations to algebraic thinking, geometry, and data handling. Each section unpacks key ideas and skills that form the foundation for IGCSE and further study. Whether you are revising, catching up, or preparing ahead, these notes will help you build confidence and fluency.

欢迎阅读 Year 8 剑桥数学全面指南。本文将带你梳理所有核心主题，从数的运算到代数思维、几何以及数据处理。每个小节都拆解了关键概念和技能，为 IGCSE 及更高阶的学习打下基础。无论你是在复习、补漏还是提前预习，这些笔记都将帮助你建立信心，提升解题流畅度。

1. Number Systems and Place Value | 数系与位值

In Year 8, you deepen your understanding of our number system. You work confidently with integers, decimals, and negative numbers. Place value up to millions and down to thousandths is essential for reading, writing, and comparing numbers. You learn to round to a given number of decimal places or significant figures and to estimate answers before calculating.

在 Year 8，你将对数系有更深入的理解。你需要熟练处理整数、小数和负数。掌握从百万位到千分位的位值对于读、写和比较数字至关重要。你将学习按要求保留指定位数的小数或有效数字进行四舍五入，并在计算前进行估算。

Key skills include ordering negative numbers on a number line, using inequality symbols correctly (<, >, ≤, ≥), and applying the four operations with integers. You also explore factors, multiples, prime factors, and index notation. Expressing a number as a product of prime factors using a factor tree builds the basis for later work with HCF and LCM.

核心技能包括在数轴上排列负数、正确使用不等号 (<、>、≤、≥)，以及进行整数的四则运算。你还会探究因数、倍数、质因数和指数记法。利用因子树将一个数表示为质因数的乘积为后续求最大公因数和最小公倍数打下基础。

Standard form (scientific notation) for large numbers is introduced, e.g., 5.6 × 10⁴ = 56 000. | 大数的标准形式（科学记数法）被引入，例如 5.6 × 10⁴ = 56 000。
Squares, cubes, square roots and cube roots are extended to include negative bases and estimation. | 平方、立方、平方根和立方根扩展到负底数及估算。

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

Fluency in converting between fractions, decimals, and percentages is a central goal. You learn to recognise common equivalents such as 1/2 = 0.5 = 50% and to use division to find a fraction as a decimal. Ordering mixed sets of fractions, decimals, and percentages by converting them into the same form becomes a regular exercise.

熟练地在分数、小数和百分数之间进行转换是核心目标。你需要识别常见的等值关系，如 1/2 = 0.5 = 50%，并利用除法将分数化为小数。把混合的分数、小数和百分数转化为同一种形式进行排序是常见的练习。

Calculating with fractions includes adding, subtracting, multiplying, and dividing proper, improper, and mixed numbers. You practice finding a common denominator for addition and subtraction, and using ‘keep-change-flip’ for division. Percentage calculations involve finding a percentage of a quantity, percentage increase and decrease, and expressing one quantity as a percentage of another.

分数的运算包括真分数、假分数和带分数的加、减、乘、除。你需要练习为加减法寻找公分母，以及使用“变乘号并取倒数”的方法进行分数除法。百分数计算涉及求一个量的百分之几、百分数增减，以及将一个量表示为另一个量的百分数。

Ordering fractions on a number line: 2/5, 0.45, 43% → 2/5 = 0.4, so 2/5 < 43% < 0.45. | 在数轴上排列分数：2/5、0.45、43% → 2/5 = 0.4，因此 2/5 < 43% < 0.45。
Percentage change: a T-shirt reduced from $20 to $15 is a 25% decrease. | 百分数变化：T恤从20元降至15元，降幅为25%。

3. Ratio and Proportion | 比与比例

Ratio compares part-to-part or part-to-whole and is written using a colon, e.g., 3:2. You simplify ratios by dividing both sides by common factors, just like simplifying a fraction. Sharing an amount in a given ratio involves finding the total number of parts and multiplying each share accordingly.

比用于比较部分与部分或者部分与整体，并用冒号表示，如 3:2。你通过两边除以公因数来化简比，就像约分一样。按给定比例分配金额需要先求出总份数，再相应计算每一份。

Proportion problems often use the unitary method: finding the value of one unit first. Direct proportion means that as one quantity doubles, the other also doubles. You solve real-life problems such as scaling recipes, map scales, and converting currencies. Understanding that the ratio and the corresponding fractions are different representations of the same situation is crucial.

比例问题常使用单位法：先求出一个单位的值。正比例意味着一个量翻倍时，另一个量也翻倍。你解决现实生活中的问题，如缩放食谱、地图比例尺和货币兑换。理解比与相应的分数是同一情境的不同表示形式至关重要。

Recipe for 4 people needs 300 g of flour. For 10 people, use the unitary method: 1 person → 75 g, so 10 people → 750 g. | 四人份食谱需要300克面粉。十人份用单位法：一人→75克，因此十人→750克。
Maps: a scale of 1:25 000 means 1 cm on the map represents 25 000 cm (250 m) in reality. | 地图：比例尺1:25 000表示图上1厘米代表实际25 000厘米（250米）。

4. Introduction to Algebra | 代数入门

Algebra is generalised arithmetic. You learn to use letters to represent unknown numbers or variables. Key concepts include writing expressions, combining like terms, and understanding the difference between an expression, an equation, and a formula. Substituting values into expressions and formulas builds a bridge to functions.

代数是推广的算术。你学习用字母表示未知数或变量。核心概念包括书写表达式、合并同类项，以及理解表达式、方程和公式的区别。将数值代入表达式和公式为学习函数搭建了桥梁。

You expand brackets using the distributive law, e.g., 3(2x + 4) = 6x + 12, and begin to factorise simple expressions by finding common factors, such as 6x + 9 = 3(2x + 3). Using index notation in algebra is formalised: x × x = x², and you apply the rules of indices for multiplication and division of powers with the same base.

你运用分配律展开括号，如 3(2x + 4) = 6x + 12，并开始通过寻找公因式分解简单的表达式，如 6x + 9 = 3(2x + 3)。代数中的指数记法被形式化：x × x = x²，并且你应用同底数幂的乘除指数法则。

Collect like terms: 5a + 2b – 3a + 7b = 2a + 9b. | 合并同类项：5a + 2b – 3a + 7b = 2a + 9b。
Simplify: (x³ × x²) ÷ x⁴ = x⁽³⁺²⁻⁴⁾ = x¹ = x. | 化简：(x³ × x²) ÷ x⁴ = x⁽³⁺²⁻⁴⁾ = x¹ = x。

5. Solving Linear Equations | 解一次方程

Solving equations means finding the value of the unknown that makes the statement true. You use inverse operations and keep the balance by doing the same to both sides. One-step and two-step equations are mastered before moving on to equations with unknowns on both sides and those involving brackets.

解方程就是求出使等式成立的未知数的值。你使用逆运算，并通过在等式两边同时进行相同操作来保持平衡。在掌握一步和两步方程后，再学习含未知数在两边以及含括号的方程。

Typical steps: simplify both sides if needed, collect variable terms on one side and constants on the other, then solve. You check your solution by substituting it back into the original equation. Word problems require translating a written scenario into an algebraic equation.

典型步骤：需要时先化简两边，将含变量项移到一边，常数移到另一边，然后求解。你将解代回原方程进行检验。应用问题需要将文字场景转化为代数方程。

Solve: 2(x + 3) = 10 → 2x + 6 = 10 → 2x = 4 → x = 2. | 解方程：2(x + 3) = 10 → 2x + 6 = 10 → 2x = 4 → x = 2。
Unknowns on both sides: 5x – 3 = 2x + 9 → 3x = 12 → x = 4. | 两边都有未知数：5x – 3 = 2x + 9 → 3x = 12 → x = 4。

6. Sequences and Patterns | 数列与规律

You explore number sequences and learn to describe the term-to-term rule. Linear sequences have a constant difference between consecutive terms. You generate terms of a sequence given the first term and the term-to-term rule, and also find the rule from a given sequence.

你探究数字数列并学习描述项间变化规则。线性数列的相邻项之间具有固定的差值。给定首项和递推规则，你生成数列的各项；同时也能从已知数列中找出规则。

A major leap forward is finding the nth term of an arithmetic sequence. You learn that for a sequence with common difference d, the nth term is given by an + b, where a = d and b is the zeroth term. Using the nth term formula, you can find any term in the sequence and determine whether a particular number belongs to it.

一个重大进步是找出等差数列的第 n 项公式。你学习到对于一个公差为 d 的数列，第 n 项可用 an + b 表示，其中 a = d，b 是第零项。利用通项公式，你可以求出数列中的任意一项，也能判断某个数是否属于该数列。

Sequence: 3, 7, 11, 15… → difference +4, so nth term = 4n – 1. | 数列：3, 7, 11, 15…→ 公差 +4，因此第 n 项 = 4n – 1。
Use the formula: 10th term = 4(10) – 1 = 39. | 使用公式：第 10 项 = 4(10) – 1 = 39。

7. Geometry: Angles and Shapes | 几何：角与图形

Angle properties form a large part of Year 8 geometry. You revise angles on a straight line (sum 180°), angles around a point (360°), and vertically opposite angles (equal). Parallel lines introduce alternate angles, corresponding angles, and co-interior angles, all of which are used to find missing angles and prove lines are parallel.

角的性质是 Year 8 几何的重要组成部分。你复习直线上的角（和为 180°）、点周角（360°）以及对顶角（相等）。平行线引入了内错角、同位角和同旁内角，这些性质都被用来求未知角度和证明两直线平行。

Properties of triangles include angle sum (180°), the exterior angle theorem, and the special cases of equilateral, isosceles, and right-angled triangles. For quadrilaterals, you learn to classify squares, rectangles, parallelograms, rhombuses, trapeziums, and kites based on side and angle properties. Simple constructions using a ruler, protractor, and compasses are also practised.

三角形的性质包括内角和 (180°)、外角定理，以及等边三角形、等腰三角形和直角三角形的特例。对于四边形，你学习根据边和角的性质分类：正方形、矩形、平行四边形、菱形、梯形和风筝形。还练习使用直尺、量角器和圆规进行简单作图。

Isosceles triangle: base angles are equal. If the vertex angle is 50°, each base angle is (180° – 50°) ÷ 2 = 65°. | 等腰三角形：底角相等。若顶角为 50°，每个底角为 (180° – 50°) ÷ 2 = 65°。
Angles in parallel lines: if a transversal crosses two parallel lines, then corresponding angles are equal. | 平行线中的角：若一直线截两条平行线，则同位角相等。

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

Moving beyond simple rectangles, you calculate the perimeter and area of compound shapes, parallelograms, and trapeziums. The formula for the area of a trapezium is ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height. You learn to estimate areas of irregular shapes by counting squares and parts of squares on a grid.

在简单矩形的基础上，你进一步计算复合图形、平行四边形和梯形的周长与面积。梯形的面积公式是 ½(a + b)h，其中 a 和 b 为平行的两边，h 为垂直高度。你学习通过在网格上数方格和部分方格来估算不规则图形的面积。

Volume and surface area are extended to cubes, cuboids, and right prisms. The volume of a prism is area of cross-section × length. You convert between metric units for length, area, and volume, including 1 cm³ = 1 ml, and 1 m³ = 1000 litres. Working with compound measures such as speed (distance/time) and density (mass/volume) also appears.

体积和表面积扩展到立方体、长方体和直棱柱。棱柱的体积 = 横截面面积 × 长度。你进行长度、面积和体积的公制单位换算，包括 1 cm³ = 1 ml，1 m³ = 1000 升。还会涉及如速度（距离/时间）和密度（质量/体积）等复合量。

Parallelogram area = base × perpendicular height, not the slant height. | 平行四边形面积 = 底 × 垂直高，而不是斜高。
Cuboid volume = length × width × height; surface area = 2(lw + lh + wh). | 长方体体积 = 长 × 宽 × 高；表面积 = 2(长×宽 + 长×高 + 宽×高)。

9. Coordinates and Graphs | 坐标与图像

You work in all four quadrants of the Cartesian plane, plotting points and reading coordinates accurately. The concept of a linear equation is linked to a straight-line graph. You learn that the graph of y = mx + c has gradient m and y-intercept c, and you practice drawing lines from equations and finding equations from given lines.

你在笛卡尔平面的全部四个象限内工作，准确描点和读取坐标。一次方程的概念与直线图像联系起来。你学习到 y = mx + c 的图像具有斜率 m 和 y-截距 c，并练习根据方程画直线以及根据给定的直线确定方程。

Real-life graphs include distance-time graphs of travel, simple conversion graphs, and temperature-time graphs. You learn to interpret the meaning of gradient (speed on distance-time graphs, rate in general) and horizontal sections (stationary). Midpoint and distance between two points on a coordinate grid are also covered.

现实生活中的图像包括旅程的距离-时间图、简单的换算图以及温度-时间图。你学习解读斜率的意义（在距离-时间图中代表速度，一般代表变化率）和水平段（代表静止）。还会涉及坐标网格上两点间的中点和距离。

Gradient = (change in y) ÷ (change in x). For y = 3x – 2, gradient = 3, y-intercept = –2. | 斜率 = (y 的变化) ÷ (x 的变化)。对于 y = 3x – 2，斜率为 3，y-截距为 –2。
Midpoint of (2, 5) and (8, 9): ((2+8)/2, (5+9)/2) = (5, 7). | (2, 5) 和 (8, 9) 的中点：((2+8)/2, (5+9)/2) = (5, 7)。

10. Collecting and Presenting Data | 收集与展示数据

Statistical work in Year 8 strengthens your ability to design surveys, collect data using tally charts, and organise raw data into frequency tables. You choose appropriate diagrams to display data: bar charts for categorical data, pie charts for proportions, and line graphs for time series. Understanding when to use each type is as important as drawing them correctly.

Year 8 的统计学习强化了你设计调查、使用记数表收集数据以及将原始数据整理为频数表的能力。你选择合适的图表展示数据：条形图用于分类数据，饼图用于比例，折线图用于时间序列。懂得何时使用每种图表并正确绘制同样重要。

You learn to calculate and interpret mean, median, mode, and range. The mean is the average, the median is the middle value, and the mode is the most frequent. The range describes how spread out the data are. Comparing two data sets using these averages and the range draws out real-life conclusions.

你学习计算和解读平均数、中位数、众数和极差。平均数是均值，中位数是中间值，众数是出现最频繁的值。极差描述数据的离散程度。利用这些平均数和极差来比较两个数据集可以得出有现实意义的结论。

Pie chart: each sector angle = (category frequency ÷ total frequency) × 360°. | 饼图：每个扇形的角度 = (类别频数 ÷ 总频数) × 360°。
Data: 3, 7, 2, 9, 2 → mean = 4.6, median = 3, mode = 2, range = 7. | 数据：3, 7, 2, 9, 2 → 平均数 = 4.6，中位数 = 3，众数 = 2，极差 = 7。

11. Probability | 概率

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). You express probabilities as fractions, decimals, or percentages. The probability of an event not happening is 1 minus the probability that it does happen. Sample space diagrams and simple two-way tables are used to list all possible outcomes of two combined events.

概率衡量事件发生的可能性，范围从 0（不可能）到 1（确定）。你将概率表示为分数、小数或百分数。一个事件不发生的概率等于 1 减去它发生的概率。样本空间图和简单的双向表用于列出两个组合事件的所有可能结果。

You calculate the probability of mutually exclusive outcomes and use the fact that the sum of probabilities of all outcomes is 1. In games of chance, you compare theoretical probability with experimental probability from simple experiments. Tree diagrams may be introduced for multiple independent events, multiplying probabilities along branches.

你计算互斥结果的概率，并运用所有结果的概率之和为 1 的事实。在机会游戏中，你通过简单实验比较理论概率和实验概率。对于多个独立事件，可能会引入树状图，并沿着分支将概率相乘。

A fair 6-sided die: P(rolling a 4) = 1/6. P(not a 4) = 5/6. | 一个公平的六面骰子：掷出 4 点的概率为 1/6；不掷出 4 点的概率为 5/6。
Two coins: sample space {HH, HT, TH, TT}. P(at least one head) = 3/4. | 两枚硬币：样本空间 {HH, HT, TH, TT}。至少一个正面的概率为 3/4。

12. Problem Solving and Reasoning | 解题与推理

Mathematical thinking is about more than performing calculations; it involves investigating patterns, making conjectures, and justifying answers. Year 8 problems often require multi-step reasoning, selecting appropriate strategies, and communicating solutions clearly. You are encouraged to break down multi-step problems into smaller, manageable parts and to check your work at each stage.

数学思维不仅仅是执行计算，它还包括探究规律、提出猜想和论证答案。Year 8 的题目往往要求多步推理、选择合适的策略并清晰地表达解题过程。鼓励你将多步问题拆解为更小、更可控的部分，并在每个阶段检查自己的答案。

Common problem-solving strategies include working backwards, drawing a diagram, looking for a pattern, making an organised list or table, and solving a simpler problem first. You also learn to use algebra to generalise and prove simple number patterns, such as why the sum of three consecutive integers is always a multiple of 3.

常见的解题策略包括倒推法、画图、寻找规律、列出有序表格或清单，以及先解决一个更简单的问题。你还会用代数概括和证明简单的数字规律，例如为什么三个连续整数的和总是 3 的倍数。

Prove: n + (n+1) + (n+2) = 3n + 3 = 3(n+1), which is divisible by 3. | 证明：n + (n+1) + (n+2) = 3n + 3 = 3(n+1)，可被 3 整除。
Strategy: if a puzzle asks to find the original amount after several percentage changes, try working backwards step by step. | 策略：如果题目问到经过若干次百分数变化后的原始量，尝试一步步倒推。

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Year 8 Cambridge Mathematics: Core Knowledge Points | Year 8 剑桥数学：核心知识点梳理