Year 8 Cambridge Advanced Mathematics: Core Topic Review | Year 8 剑桥进阶数学核心知识点梳理

📚 Year 8 Cambridge Advanced Mathematics: Core Topic Review | Year 8 剑桥进阶数学核心知识点梳理

Year 8 Advanced Mathematics under the Cambridge curriculum builds on the foundational skills learned in Year 7 and introduces more abstract concepts that are vital for success in IGCSE. This review summarises the core knowledge areas, from algebraic manipulation and equations to probability and geometry, ensuring that students are well-prepared for the challenges ahead. Each topic is explained with clear examples, so you can consolidate your understanding and tackle exam questions confidently.

剑桥 Year 8 进阶数学在七年级的基础上引入了更抽象的概念，为 IGCSE 成功打下坚实基础。本文梳理了代数运算、方程、概率、几何等核心知识点，配合清晰示例，帮助你巩固理解，自信应对各类考题。

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

Algebra is the language of mathematics. In Year 8, you will work with letters (variables) and numbers, combining like terms and using the distributive law to expand brackets. Remember: like terms have exactly the same variable part; for instance, 3x and 5x can be added, but 2x² and 3x cannot.

代数是数学的语言。在八年级，你将处理字母（变量）和数字，合并同类项并运用分配律展开括号。记住：同类项必须变量部分完全相同，比如 3x 和 5x 可以相加，但 2x² 和 3x 不能合并。

ample, simplify 4a + 3b − 2a + 5b: first group like terms to get (4a − 2a) + (3b + 5b) = 2a + 8b.

化简时务必遵循运算顺序：括号、乘方、乘除、加减。例如化简 4a + 3b − 2a + 5b：先合并同类项得 (4a − 2a) + (3b + 5b) = 2a + 8b。

Expanding brackets involves multiplying each term inside the bracket by the term outside. For 3(x + 4), multiply 3 by x and 3 by 4 to get 3x + 12. For more complex cases like 2(3x − 5) + 4(x + 1), expand first: 6x − 10 + 4x + 4, then simplify to 10x − 6.

展开括号就是用外面的项乘以括号内的每一项。如 3(x + 4) 得 3x + 12。对于 2(3x − 5) + 4(x + 1)，先展开：6x − 10 + 4x + 4，再化简为 10x − 6。

You will also learn to factorise simple expressions, which is the reverse of expanding. For example, 6x + 9 can be factorised by taking out the common factor 3 to give 3(2x + 3).

你还将学习因式分解，它是展开的逆过程。例如 6x + 9 可以通过提取公因数 3 分解为 3(2x + 3)。

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2. Solving Linear Equations | 解一元一次方程

Solving equations means finding the value of the variable that makes the statement true. In Year 8, you focus on linear equations with one unknown, often needing two or three steps. The golden rule is: whatever you do to one side, you must do to the other to keep the equation balanced.

解方程就是找到使等式成立的变量值。八年级主要学习解一元一次方程，通常需要两步或三步。黄金法则是：对方程一边做的任何操作，另一边也必须做相同操作，以保持等式平衡。

h sides to get 3x − 3 = 9. Then add 3 to both sides: 3x = 12. Finally, divide by 3: x = 4.

例如解方程 5x − 3 = 2x + 9。首先把含变量的项移到一边：两边减去 2x 得 3x − 3 = 9。然后两边加 3：3x = 12。最后两边除以 3：x = 4。

= 12, x = 2. Always check your answer by substituting it back into the original equation.

有时方程会含有括号，如 2(3x + 1) = 14。展开得到 6x + 2 = 14，再求解：6x = 12，x = 2。始终要把答案代回原方程检验。

roduced: for x/3 + 2 = 5, subtract 2 to get x/3 = 3, then multiply both sides by 3: x = 9.

还会引入含分数的方程：如 x/3 + 2 = 5，先减去 2 得 x/3 = 3，然后两边乘以 3：x = 9。

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3. Inequalities | 不等式

tions. Solving an inequality follows the same rules as solving an equation, with one critical exception: if you multiply or divide by a negative number, you must reverse the inequality sign.

不等式与方程类似，但用 <、>、≤、≥ 代替等号，表示一个解的范围。解不等式的规则和方程相同，但有一个关键例外：乘以或除以负数时，必须改变不等号方向。

Solve 3x + 4 < 13: subtract 4 to get 3x < 9, then divide by 3: x < 3. This means any number less than 3 is a solution. You can represent this on a number line with an open circle at 3 and an arrow pointing left.

解不等式 3x + 4 < 13：减 4 得 3x < 9，除以 3 得 x < 3。这意味着任何小于 3 的数都是解。可以在数轴上用 3 处的空心圆和向左的箭头表示。

by −2 and reverse the sign: x ≥ −4. The closed circle on −4 indicates that −4 is included.

对于 −2x ≤ 8，两边除以 −2 并改变不等号方向：x ≥ −4。数轴上 −4 处用实心圆表示 −4 包含在解集中。

You will also solve double inequalities like 5 < 2x + 1 < 11. Break it into two parts or solve simultaneously: subtract 1 from all three sides: 4 < 2x < 10, then divide by 2: 2 < x < 5.

还将学习双不等式如 5 < 2x + 1 < 11。可以拆分成两部分或同时求解：三边减 1 得 4 < 2x < 10，再除以 2 得 2 < x < 5。

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4. Sequences and nth Term | 数列与第 n 项

A sequence is a set of numbers following a pattern. Year 8 focuses on arithmetic sequences where the difference between consecutive terms is constant. This constant is called the common difference, d.

数列是按某种模式排列的一组数。八年级重点学习等差数列，即相邻两项的差为常数 d，称为公差。

is: uₙ = a + (n − 1)d, where a is the first term. For example, the sequence 4, 7, 10, 13, … h term is uₙ = 4 + (n − 1)×3 = 3n + 1. You can check: n=1 gives 4, n=2 gives 7, and so on.

等差数列的第 n 项公式：uₙ = a + (n − 1)d，其中 a 为首项。例如数列 4, 7, 10, 13, …，a = 4，d = 3，所以通项为 uₙ = 4 + (n − 1)×3 = 3n + 1。验证：n=1 得 4，n=2 得 7，等等。

You can use the nth term to find any term, e.g., the 50th term of 5, 8, 11, … 9×3 = 152. You may also be asked to find whether a given number is a term in the sequence.

利用通项可求任意项，如数列 5, 8, 11, … 的第 50 项：a = 5, d = 3，u₅₀ = 5 + 49×3 = 152。也可能要求判断某个数是否在数列中出现。

Other sequences include geometric progressions and patterns with squares or cubes. Recognising a sequence from its terms and describing the rule in words is also key.

还会涉及等比数列或平方、立方数列等。通过项来识别数列并用语言描述规则同样重要。

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5. Ratio and Proportion | 比与比例

Ratio compares two or more quantities, showing their relative sizes. It is written as a : b and must be simplified by dividing by the highest common factor. For example, a ratio of 12 : 8 simplifies to 3 : 2 (divide both by 4).

比用来比较两个或多个量，写成 a : b，并需除以最大公因数化简。如 12 : 8 化简为 3 : 2（同除以 4）。

You can share a quantity in a given ratio. To divide £ etween two people in the ratio 3 : 7, add the parts (3+7=10), then one person gets (3/10)×£60 = £18 and the other gets £42.

可按给定比例分配数量。例如将 60 英镑按 3 : 7 分给两人，总份数为 3+7=10，一人得 (3/10)×60=18 英镑，另一人得 42 英镑。

itten as a : b = c : d. Cross-multiplication can solve proportion problems: if 3 pens cost £1.20, how much for 5 pens? 3/1.20 = 5/x gives 3x = 6, x = £2.00.

比例表示两个比相等，写成 a : b = c : d。交叉相乘可解比例问题：若 3 支笔 1.20 英镑，5 支笔多少钱？3/1.20 = 5/x，得 3x = 6，x = 2.00 英镑。

Direct proportion means as one quantity increases, the other increases at the same rate; inverse proportion means one increases while the other decreases. Year 8 introduces basic direct proportion graphs, which are straight lines through the origin.

正比例指一个量增加，另一个量以相同速率增加；反比例则相反。八年级引入正比例图像，是一条过原点的直线。

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6. Percentages | 百分比

convert between fractions, decimals, and percentages fluently. For example, 3/5 = 0.6 = 60%.

百分比是分母为 100 的分数。你应该能在分数、小数和百分数之间灵活转换。如 3/5 = 0.6 = 60%。

Calculating a percentage of a quantity: to find 15% of 80, multiply 0.15 × 80 = 12. You can also use fractions: 15% = 15/100, so (15/100)×80 = 12.

求一个量的百分之几：求 80 的 15%，计算 0.15 × 80 = 12。也可用分数：15% = 15/100，所以 (15/100)×80 = 12。

Percentage increase and decrease are core skills. An increase of 20% on £40 means new value = 40 + 0.2×40 = £48, or multiply by 1.2. A decrease of 30% means multiply by 0.7.

百分数增减是核心技能。在 40 英镑基础上增加 20%，新值 = 40 + 0.2×40 = 48 英镑，或乘以 1.2。减少 30% 则乘以 0.7。

Reverse percentages: if a salary of £33,000 is after a 10% pay rise, the original was 33,000 ÷ 1.1 = £30,000. Always identify whether you are working forwards or backwards.

逆向百分数：若加薪 10% 后工资为 33,000 英镑，原工资 = 33,000 ÷ 1.1 = 30,000 英镑。做题时先明确是正向还是逆向计算。

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7. Geometry of Straight Lines and Angles | 直线与角的几何

Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal. These rules are used to find missing angles without a protractor.

直线上的角之和为 180°。绕一点一周的角之和为 360°。对顶角相等。这些规则用于不用量角器求未知角。

Parallel lines create special angle relationships: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°. Identifying the angle pairs 'F' (corresponding), 'Z' (alternate), and 'C' (co-interior) shapes is a helpful visual trick.

平行线产生特殊的角关系：同位角相等，内错角相等，同旁内角之和为 180°。通过 'F' 型（同位）、'Z' 型（内错）、'C' 型（同旁内）来识别角对，是非常有用的视觉技巧。

In triangles, the interior angles sum to 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles. You can use these facts to solve problems involving isosceles or right-angled triangles.

三角形内角和为 180°。三角形的外角等于不相邻的两内角之和。可利用这些事实解决等腰三角形或直角三角形的问题。

Bearings are measured clockwise from north, always expressed as three figures (e.g., 045°, 230°). They are used in navigation and require angle reasoning.

方位角以正北为基准顺时针测量，总写成三位数（如 045°、230°）。方位角用于导航，需要角度推理。

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8. Area and Perimeter of 2D Shapes | 二维图形的面积和周长

. For triangles, it is the sum of all sides. Be careful with compound shapes: break them into simpler parts.

周长是围绕图形的总长度。长方形周长 P = 2(l + w)。三角形周长是三边之和。处理组合图形时要小心，可将其拆分为简单图形。

erpendicular height, and for a trapezium, A = ½(a + b)h, where a and b are the parallel sides.

长方形面积 = 长 × 宽。三角形面积 = ½ × 底 × 高，高必须垂直于底。平行四边形面积 = 底 × 垂直高，梯形面积 = ½(a + b)h，其中 a 和 b 为平行边。

Here is a summary of common area formulas:

以下是常见面积公式总结：

Shape / 图形	Area formula / 面积公式
Rectangle / 长方形	A = l × w
Triangle / 三角形	A = ½ b h
Parallelogram / 平行四边形	A = b h
Trapezium / 梯形	A = ½(a + b)h

Remember that area is expressed in square units (cm², m²) and perimeter in linear units (cm, m). Always write the units in your answer.

注意面积单位为平方单位（cm²、m²），周长单位为长度单位（cm、m）。答案中务必写出单位。

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9. Volume and Surface Area of 3D Shapes | 三维图形的体积和表面积

cuboid with length l, width w, height h: V = l × w × h; surface area SA = 2(lw + wh + lh).

体积衡量立体内部空间，单位为立方单位。表面积是所有面的总面积。对于长 l、宽 w、高 h 的长方体：V = l × w × h；表面积 SA = 2(lw + wh + lh)。

m cross-section. For a cylinder, V = πr²h, but at Year 8 you may work with simpler prisms.

棱柱体积 = 横截面积 × 长度。这适用于三棱柱、圆柱和任何等截面立体。圆柱体积 V = πr²h，但八年级可能先接触较简单的棱柱。

all the faces. For a cylinder, the curved surface area = 2πrh, and total SA = 2πr² + 2πrh.

棱柱表面积：求出每个面的面积再相加。借助展开图可帮助想象所有面。圆柱的侧面积 = 2πrh，总表面积 = 2πr² + 2πrh。

You may also explore the volume of composite solids by splitting them into known prisms and adding volumes. Never forget to cube the units for volume (e.g., cm³).

还可能学习组合体的体积，通过拆分成已知棱柱并相加。体积单位必须使用立方（如 cm³）。

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10. Statistics: Averages and Range | 统计：平均数与极差

Three averages are used to describe data sets: the mean (add all values and divide by the number of values), the median (middle value when ordered), and the mode (most frequent value). Each measure gives a different insight.

描述数据集有三种平均数：平均数（所有值之和除以数量）、中位数（排序后中间的值）和众数（出现次数最多的值）。每种度量提供不同信息。

: order as 2,3,5,7,7. Median = 5, mode = 7, mean = (2+3+5+7+7)/5 = 4.8, range = 7 − 2 = 5.

极差是最大值与最小值的差，一个简单的离散度量。对于数据 3, 7, 7, 2, 5：排序为 2,3,5,7,7。中位数 = 5，众数 = 7，平均数 = (2+3+5+7+7)/5 = 4.8，极差 = 7 − 2 = 5。

Year 8 also covers reading and interpreting bar charts, line graphs, pie charts, and scatter graphs. You should be able to find statistics from a frequency table.

八年级还要求阅读并解释条形图、折线图、饼图和散点图。你应当能从频率表中获取统计数据。

For grouped data, you estimate the mean using midpoints of the class intervals. Outliers are extreme values that affect the mean more than the median.

对于分组数据，使用组中值估算平均数。极端值（离群值）对平均数的影响比中位数更大。

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11. Probability | 概率

ten as a fraction, decimal, or percentage. For a fair six-sided die, P(rolling a 3) = 1/6.

概率是事件发生的可能性，用 0（不可能）到 1（必然）之间的数值衡量。可写成分数、小数或百分数。对一个正常的六面骰子，掷出 3 的概率 P(3) = 1/6。

tcomes / total number of outcomes. If a bag contains 3 red and 5 blue balls, P(red) = 3/8.

对等可能结果，使用 P(事件) = 有利结果数 / 总结果数。若袋中有 3 个红球和 5 个蓝球，P(红) = 3/8。

The probability of an event not happening is 1 – P(event). So P(not red) = 1 − 3/8 = 5/8. All probabilities in a sample space sum to 1.

事件不发生的概率是 1 − P(事件)。因此 P(非红) = 1 − 3/8 = 5/8。样本空间中所有概率之和为 1。

r two dice, a 6x6 grid shows 36 outcomes, and the probability of a sum of 7 is 6/36 = 1/6.

样本空间图用于列出两个事件的所有可能结果。掷两粒骰子，6×6 网格显示 36 种结果，和为 7 的概率是 6/36 = 1/6。

fact that P(A or B) = P(A) + P(B). Experimental probability is based on data from trials.

你还将学习互斥事件（不能同时发生），并使用 P(A 或 B) = P(A) + P(B)。实验概率基于试验数据。

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12. Transformations | 变换

Transformations move or change shapes on a coordinate grid. The four types are: translation (slides), reflection (flips), rotation (turns), and enlargement (scales). In Year 8, you work on describing and carrying out single transformations.

变换是在坐标网格上移动或改变图形。四种类型：平移（滑动）、反射（翻转）、旋转（转动）和放大（缩放）。八年级要求描述并执行单一变换。

Translation is described by a column vector (x/y), e.g., (3/−2) means shift 3 right, 2 down. The shape does not change size or orientation.

平移用列向量 (x/y) 描述，如 (3/−2) 表示向右移 3，向下移 2。图形大小和方向不变。

d its image are equidistant from the mirror line, and the shape is reversed but congruent.

反射需要一条镜像线（如 x = 1、y = 0、y = x）。每个点与其像到镜像线的距离相等，图形翻转但全等。

Rotation needs a centre, angle, and direction (clockwise or anticlockwise). The centre of rotation is a point; every other point moves on an arc around it. A 180° rotation of point (x, y) about the origin becomes (−x, −y).

旋转需要旋转中心、角度和方向（顺时针或逆时针）。旋转中心是一个点，其余各点绕其做弧线运动。点 (x, y) 绕原点旋转 180° 成为 (−x, −y)。

Enlargement is specified by a centre and a positive scale factor. If the scale factor is 2, all sides double and distances from the centre double. A scale factor less than 1 makes the shape smaller. The centre of enlargement is the fixed point.

放大通过中心和正比例因子描述。若比例为 2，所有边长加倍，各点距中心的距离也加倍。比例因子小于 1 会使图形缩小。放大中心是固定点。

Year 8 also covers combining transformations and using a coordinate grid to plot images precisely.

八年级也涉及组合变换，并在网格上精确绘制变换后的图形。

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