📚 Grade 8 Mathematics Student Book: Core Concepts and Exam Skills | 八年级数学学生用书：核心概念与应试技能
This student book covers the essential Grade 8 mathematics topics, including number sense, algebra, geometry, functions, statistics, and probability. Each section presents key ideas with clear explanations and worked examples to help students build confidence and succeed in exams.
本学生用书涵盖八年级数学的核心主题，包括数感、代数、几何、函数、统计和概率。每个部分都以清晰的解释和例题呈现关键概念，帮助学生建立信心并在考试中取得好成绩。
1. Number Sense and Operations | 数感与运算
In Grade 8, students extend number sense to include irrational numbers such as √2 and π. A rational number can be written as a fraction a/b where b ≠ 0, while an irrational number cannot be expressed as a simple fraction.
八年级学生将数感扩展到无理数，如 √2 和 π。有理数可以写成 a/b 的分数形式且 b ≠ 0，而无理数不能表示为简单分数。
inates or repeats, for example 3/8 = 0.375 and 2/3 = 0.666... . Irrational numbers such as √2 and π = 3.14159265... never terminate and never repeat, so they can only be approximated.
每个有理数的小数形式要么有限、要么循环，例如 3/8 = 0.375，2/3 = 0.666…。而 √2 = 1.41421356…、π = 3.14159265… 这类无理数既不终止也不循环，只能用近似值表示。
a number line. A useful strategy is to compare squares: since 1.4² = 1.96 and 1.5² = 2.25, √2 must lie between 1.4 and 1.5, closer to 1.4.
学生应能通过估算在数轴上的位置来比较和排列一组混合数字。一个实用策略是比较平方值：因为 1.4² = 1.96、1.5² = 2.25，所以 √2 必在 1.4 与 1.5 之间，且更接近 1.4。
Key vocabulary to master:
- Natural numbers: 1, 2, 3, ... / 自然数：1, 2, 3, …
- Integers: ..., −2, −1, 0, 1, 2, ... / 整数：…, −2, −1, 0, 1, 2, …
- Rationals: a/b with b ≠ 0 / 有理数：a/b 且 b ≠ 0
- Irrationals: non-repeating, non-terminating decimals / 无理数：不循环、不终止的小数
Real numbers are the union of rational and irrational numbers, and every point on a number line corresponds to exactly one real number. This idea is the foundation for later work with functions and coordinates.
实数是有理数与无理数的并集，数轴上的每一个点都恰好对应一个实数。这一观念是后续学习函数与坐标系的基础。
2. Scientific Notation, Roots, and Order of Operations | 科学计数法、方根与运算顺序
an integer. This is useful for comparing sizes and performing multiplication or division efficiently.
科学计数法用于表示非常大或非常小的数，形式为 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数。这有助于比较大小并高效地进行乘除运算。
ultiply the coefficients and add the exponents; when dividing, divide the coefficients and subtract the exponents.
例如，地球到太阳的距离约为 1.5 × 10⁸ km，氢原子的直径约为 1.06 × 10⁻¹⁰ m。相乘时系数相乘、指数相加；相除时系数相除、指数相减。
Evaluating square roots and cube roots is a key skill. For example, √64 = 8 because 8² = 64, and ∛ n-perfect squares and identify the perfect squares up to 225 and perfect cubes up to 1000.
计算平方根与立方根是一项关键技能。例如，√64 = 8，因为 8² = 64；∛27 = 3，因为 3³ = 27。学生还应估算非完全平方数的平方根，并熟记 225 以内的完全平方数与 1000 以内的完全立方数。
e applied consistently, especially with negative numbers. For example, −3² = −9, but (−3)² = +9.
运算顺序（括号、指数、从左到右的乘除、从左到右的加减）必须一致应用，尤其在处理负数时。例如 −3² = −9，而 (−3)² = +9。
PEMDAS: ( ) → xⁿ → × ÷ → + −
A reliable exam habit is to rewrite each step on a new line, keeping the equals signs aligned. This reduces sign errors and makes it easier for markers to award method marks.
一个可靠的应试习惯是每步另起一行书写，并让等号对齐。这样能减少符号错误，也便于阅卷老师给出过程分。
3. Ratios, Proportions, and Percent | 比、比例与百分数
A ratio compares two quantities, and a proportion states that two ratios are equal. Unit rates, such as price per item or speed in km/h, allow direct comparisons.
比用于比较两个量，比例则表示两个比相等。单位率（如每件商品的价格或速度 km/h）可用于直接比较。
t 12 yuan, then 7 pens cost 12 ÷ 3 × 7 = 28 yuan, because the unit rate is 4 yuan per pen.
解比例可用交叉相乘。若 a/b = c/d，则 a × d = b × c。例如 3 支笔 12 元，则 7 支笔为 12 ÷ 3 × 7 = 28 元，因为单位率为每支 4 元。
given a part and a percent. Percent change is calculated as (new value − original value) ÷ original value × 100%.
百分数问题有三种标准题型：求一个数的百分之几、求一个数是另一个数的百分之几、已知部分与百分数求整体。百分变化率的计算为（新值 − 原值）÷ 原值 × 100%。
, where P is the principal, r is the annual rate as a decimal, and t is the time in years.
实际应用包括折扣、加价、税费、小费、单利和比例尺图。单利公式为 I = P × r × t，其中 P 为本金，r 为以小数表示的年利率，t 为年数。
Common exam trap: a 20% decrease followed by a 20% increase does not return to the original value. Starting from 100, a 20% decrease gives 80, and a 20% increase on 80 gives only 96.
常见考试陷阱：先降 20% 再升 20% 并不会回到原值。从 100 出发，降 20% 得 80，再在 80 上升 20% 只得到 96。
4. Exponents and Algebraic Expressions | 指数与代数式
× aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Also a⁰ = 1 for any a ≠ 0, and a⁻ⁿ = 1 ÷ aⁿ.
指数运算律可以快速化简表达式。同底数时：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。此外，对任意 a ≠ 0 有 a⁰ = 1，且 a⁻ⁿ = 1 ÷ aⁿ。
implified by adding exponents, because exponents only combine under multiplication and division.
例如 2³ × 2⁴ = 2⁷ = 128，(3²)³ = 3⁶ = 729。注意 aᵐ + aⁿ 不能通过指数相加来化简，因为指数只在乘除运算中合并。
+ 3x = 8x, but 5x + 3y stays as it is. The distributive property a(b + c) = ab + ac is used constantly to expand brackets.
代数式包含项、系数、变量和常数。同类项具有完全相同的字母部分，可以合并：5x + 3x = 8x，但 5x + 3y 保持不变。分配律 a(b + c) = ab + ac 在去括号时被频繁使用。
Expanding and factoring are inverse processes. Expanding 3(x + 4) gives 3x + 12, while factoring 3x + 12 gives 3(x + 4). Factoring out the greatest common factor is the fastest way to check whether an answer is fully simplified.
展开与因式分解是互逆过程。展开 3(x + 4) 得 3x + 12，而分解 3x + 12 得 3(x + 4)。提取最大公因式是检验答案是否完全化简的最快方法。
5. Linear Equations and Inequalities | 一元一次方程与不等式
solate the variable using inverse operations, keeping the equation balanced at every step.
一元一次方程可写成 ax + b = c 的形式。目标是用逆运算把变量单独留在一边，并保证每一步方程两边保持平衡。
= 16 and x = 8. Always substitute the answer back into the original equation to verify it.
解 5x − 7 = 3x + 9 时先把变量集中到一边：5x − 3x = 9 + 7，得 2x = 16，x = 8。务必把答案代回原方程检验。
o solution; if it gives a true statement such as 4 = 4, every value of the variable works.
方程可能无解或有无穷多解。若化简后得到 4 = 7 这样的假命题，则方程无解；若得到 4 = 4 这样的真命题，则变量的任意取值都成立。
Inequalities are solved in almost the same way, but the inequality sign reverses when both sides are multiplied or divided by a negative number. For example, −2x > 6 gives x < −3, not x > −3.
不等式的解法几乎相同，但当两边同时乘以或除以负数时，不等号方向要反转。例如 −2x > 6 得 x < −3，而不是 x > −3。
id values. Compound inequalities such as −1 < x ≤ 4 are written as a single interval on the line.
不等式的解用数轴表示：< 或 > 用空心圆，≤ 或 ≥ 用实心圆，箭头指向所有有效值的方向。像 −1 < x ≤ 4 这样的复合不等式在数轴上表示为一段区间。
6. Linear Functions and Slope | 一次函数与斜率
pt. Proportional relationships are the special case y = kx that passes through the origin.
一次函数具有恒定的变化率，可写成斜截式 y = mx + b，其中 m 为斜率，b 为 y 轴截距。正比例关系 y = kx 是经过原点的特殊情形。
m = (y₂ − y₁) ÷ (x₂ − x₁)
ontal, and an undefined slope belongs to a vertical line. A slope of 3 means that y increases by 3 units for every 1 unit increase in x.
斜率同时表示方向和陡峭程度：m 为正时从左到右上升，m 为负时下降，m = 0 为水平线，斜率不存在则对应竖直线。斜率为 3 表示 x 每增加 1，y 增加 3。
Functions can be represented as equations, tables, graphs or verbal descriptions, and exam questions often ask you to compare two functions given in different forms. Compare slopes to compare rates of change, and compare y-intercepts to compare starting values.
函数可用方程、表格、图象或文字描述来呈现，考试常要求比较用不同形式给出的两个函数。比较斜率即比较变化率，比较 y 轴截距即比较初始值。
r kilometre, then y = 2x + 5, where 2 is the cost per kilometre and 5 is the fixed charge.
在情境中解释斜率与截距是高价值技能。若出租车起步价 5 元、每公里 2 元，则 y = 2x + 5，其中 2 是每公里费用，5 是固定费用。
7. Systems of Linear Equations | 二元一次方程组
A system of two linear equations can be solved by graphing, substitution or elimination. The solution is the ordered pair (x, y) that satisfies both equations at the same time, and on a graph it is the point where the two lines intersect.
由两个一次方程组成的方程组可用图象法、代入法或加减消元法求解。解是同时满足两个方程的有序数对 (x, y)，在图象上就是两条直线的交点。
x = 14, so x = 7 and y = 3. Substitution works well when one variable is already isolated.
当系数比较接近时消元法更方便：解 x + y = 10 与 x − y = 4，两式相加得 2x = 14，所以 x = 7、y = 3。当某个变量已被单独表示时，代入法更方便。
The number of solutions depends on the lines. If the slopes differ, there is exactly one solution; if the slopes are equal and the intercepts differ, the lines are parallel and there is no solution; if both are equal, the lines coincide and there are infinitely many solutions.
解的个数取决于两条直线。若斜率不同，则恰有一个解；若斜率相同而截距不同，两线平行，无解；若两者都相同，两线重合，有无穷多解。
Word problems usually involve two unknown quantities, such as the number of adult and child tickets, or the number of hours worked at two different pay rates. Define the variables clearly in words before writing the equations.
应用题通常涉及两个未知量，例如成人票与儿童票的数量，或按两种不同时薪工作的小时数。写方程前先用文字清楚定义变量。
8. Geometry: Angles, Similarity, and the Pythagorean Theorem | 几何：角、相似与勾股定理
When a transversal crosses two parallel lines, several angle pairs are formed: corresponding angles are equal, alternate interior angles are equal, and co-interior angles add to 180°. The sum of the interior angles of any triangle is 180°.
当一条截线与两条平行线相交时，会形成多组角：同位角相等，内错角相等，同旁内角之和为 180°。任意三角形的内角和为 180°。
An exterior angle of a triangle equals the sum of the two opposite interior angles. In a right triangle, the two acute angles are complementary, so they add to 90°.
三角形的外角等于与它不相邻的两个内角之和。在直角三角形中，两个锐角互余，即和为 90°。
Two triangles are similar if two pairs of corresponding angles are equal (AA), or if corresponding sides are proportional with an included angle equal (SAS). Similar figures have the same shape but not necessarily the same size, and the ratio of corresponding sides is the scale factor.
若两对对应角相等（AA），或对应边成比例且夹角相等（SAS），则两个三角形相似。相似图形形状相同但大小不一定相同，对应边的比即相似比。
a² + b² = c²
then the triangle must be right-angled. Common triples to memorise are 3-4-5, 5-12-13, 8-15-17 and their multiples.
勾股定理适用于直角三角形，其中 c 为斜边。其逆定理同样重要：若三边满足 a² + b² = c²，则该三角形必为直角三角形。需记住的常用勾股数有 3-4-5、5-12-13、8-15-17 及其倍数。
₁)² + (y₂ − y₁)²). This links geometry to algebra and appears frequently in later courses.
该定理还可求坐标平面上两点间的距离：d = √((x₂ − x₁)² + (y₂ − y₁)²)。这把几何与代数联系起来，在后续课程中频繁出现。
9. Transformations and Volume | 图形变换与体积
Grade 8 geometry includes four transformations: translation (slide), reflection (flip), rotation (turn) and dilation (resize). The first three preserve size and shape, producing congruent figures, while dilation produces a similar figure.
八年级几何包含四种变换：平移（滑动）、反射（翻转）、旋转（转动）和位似（缩放）。前三种保持大小与形状不变，得到全等图形；位似则得到相似图形。
ction in the x-axis, (x, y) → (x, −y); under a reflection in the y-axis, (x, y) → (−x, y).
平移下每一点沿同一方向移动相同距离，记作 (x, y) → (x + a, y + b)。关于 x 轴反射时 (x, y) → (x, −y)；关于 y 轴反射时 (x, y) → (−x, y)。
(−x, −y). A dilation with centre at the origin and scale factor k maps (x, y) → (kx, ky).
绕原点逆时针旋转 90° 将 (x, y) 映射为 (−y, x)；旋转 180° 映射为 (−x, −y)。以原点为中心、相似比为 k 的位似将 (x, y) 映射为 (kx, ky)。
Volume formulas for three-dimensional solids are exam favourites:
- Cylinder: V = πr²h / 圆柱：V = πr²h
- Cone: V = (1/3)πr²h / 圆锥：V = (1/3)πr²h
- Sphere: V = (4/3)πr³ / 球：V = (4/3)πr³
When shapes are similar with scale factor k, lengths scale by k, areas scale by k², and volumes scale by k³. This single relationship solves many comparison questions without recalculating everything.
当图形相似且相似比为 k 时，长度按 k 缩放，面积按 k² 缩放，体积按 k³ 缩放。这一关系可解决许多比较类问题，无需重新计算全部数据。
10. Statistics: Scatter Plots and Data Analysis | 统计：散点图与数据分析
A scatter plot displays paired numerical data and reveals whether two variables are associated. Patterns to describe include positive or negative association, linear or non-linear form, clusters, gaps and outliers.
散点图展示成对的数值数据，可揭示两个变量是否相关。需要描述的特征包括正相关或负相关、线性或非线性、聚集、空缺和离群点。
hange between the variables, and its y-intercept describes the predicted value when x = 0.
最佳拟合直线应尽量靠近尽可能多的点，且上下两侧的点数大致相等。其斜率描述变量间的变化率，y 轴截距描述 x = 0 时的预测值。
The strength of an association is judged by how closely the points follow the line: tight clustering indicates a strong association, while widely scattered points indicate a weak one. An outlier is a point that does not follow the overall pattern and is often explained by a measurement error or a special case.
相关强度取决于点沿直线的紧密程度：点越紧密，相关性越强；点越分散，相关性越弱。离群点是不遵循整体模式的点，通常由测量误差或特殊情况造成。
Two-way tables summarise categorical data by counting how many individuals belong to each combination of categories. Relative frequencies are computed by dividing each cell by the grand total, which makes groups of different sizes comparable.
双向表通过统计属于各类别组合的个体数量来汇总分类数据。相对频数用每个单元格除以总计得出，使不同规模的组别可以比较。
Mean absolute deviation measures how spread out data are around the mean: find the mean, subtract it from each value, take absolute values, then average those distances. A larger deviation means greater variability in the data set.
平均绝对偏差衡量数据围绕平均值的分散程度：先求平均值，再用每个值减去平均值并取绝对值，最后求这些距离的平均。偏差越大，数据集的离散程度越大。
11. Probability | 概率
ally likely outcomes, P(event) = number of favourable outcomes ÷ total number of outcomes.
概率衡量事件发生的可能性，取值范围从 0（不可能）到 1（必然）。当结果等可能时，P(事件) = 有利结果数 ÷ 结果总数。
in tomorrow is 0.3, then the probability of no rain is 0.7. Probabilities of all possible outcomes in a sample space must sum to 1.
对立事件法则指出 P(非 A) = 1 − P(A)。若明天下雨的概率为 0.3，则不下雨的概率为 0.7。样本空间中所有可能结果的概率之和必为 1。
twice in a row is ½ × ½ = ¼. For mutually exclusive events, add: P(A or B) = P(A) + P(B).
对于两个独立事件，用乘法：P(A 且 B) = P(A) × P(B)。例如连续两次掷得正面的概率为 ½ × ½ = ¼。对于互斥事件，用加法：P(A 或 B) = P(A) + P(B)。
Tree diagrams and organised
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