📚 Year 8 SQA Advanced Mathematics: Core Topics Summary | Year 8 SQA 进阶数学：核心知识点梳理
Building a strong foundation in Year 8 Advanced Mathematics means mastering a set of interconnected topics, from algebra and geometry to statistics and probability. This revision guide walks you through the essential ideas you will encounter in the SQA curriculum, with clear explanations and key formulas to support your learning.
要在 Year 8 进阶数学中打下坚实基础，需要掌握一系列相互关联的主题，从代数、几何到统计与概率。这本复习指南带你梳理 SQA 课程中的核心概念，用清晰的解释和关键公式为你的学习提供支持。
1. Integers and Indices | 整数与指数
Integers include positive and negative whole numbers, and working with them confidently is essential for all later topics. The rules for adding, subtracting, multiplying and dividing negative numbers must become second nature.
整数包括正整数和负整数，熟练地进行整数运算是后续所有主题的基础。掌握负数加减乘除的规则应当成为一种本能。
The index laws allow us to simplify expressions involving powers. When multiplying powers of the same base, we add the indices: am × an = am+n. For division, we subtract the indices: am ÷ an = am-n.
指数法则使我们能够简化含有幂的表达式。同底数幂相乘，指数相加：am × an = am+n。同底数幂相除，指数相减：am ÷ an = am-n。
We also need to know that any non-zero number raised to the power zero is 1 (a0 = 1), and that a negative exponent indicates a reciprocal: a-n = 1 / an. A power of a power multiplies the indices: (am)n = amn.
我们还需要知道任何非零数的零次幂等于 1（a0 = 1），而负指数表示倒数：a-n = 1 / an。幂的乘方将指数相乘：(am)n = amn。
2. Algebraic Expressions | 代数表达式
Algebraic expressions use letters to represent unknown numbers. We simplify them by collecting like terms – terms that have exactly the same variable part. For example, 3x + 5y – 2x + y simplifies to x + 6y.
代数表达式用字母表示未知数。通过合并同类项（变量部分完全相同的项）可以化简表达式。例如，3x + 5y – 2x + y 化简为 x + 6y。
Expanding brackets involves multiplying each term inside the bracket by the term outside. For instance, 4(2x – 3) becomes 8x – 12. When we have a double bracket like (x + 2)(x + 5), we multiply every term in the first bracket by every term in the second, often using the FOIL method, giving x2 + 7x + 10.
展开括号要用括号外的项乘以括号内的每一项。例如，4(2x – 3) 展开为 8x – 12。当遇到 (x + 2)(x + 5) 这样的双括号时，需要将第一个括号内的每一项乘以第二个括号内的每一项，通常使用 FOIL 方法，得到 x2 + 7x + 10。
Factorising is the reverse of expanding. We look for the highest common factor and take it outside the bracket: 6x + 9 = 3(2x + 3). Quadratic expressions like x2 + 5x + 6 can be factorised into two binomials (x + 2)(x + 3) by finding numbers that multiply to give 6 and add to give 5.
因式分解是展开的逆运算。需要找出最大公因数并将其提到括号外：6x + 9 = 3(2x + 3)。像 x2 + 5x + 6 这样的二次表达式可以通过找到两个相乘得 6、相加得 5 的数，分解为两个二项式 (x + 2)(x + 3)。
3. Solving Linear Equations | 解线性方程
Solving an equation means finding the value of the unknown that makes the statement true. We keep the equation balanced by performing the same operation on both sides. For a simple equation like 2x + 5 = 13, we first subtract 5 from both sides (2x = 8) and then divide by 2 (x = 4).
解方程意味着找到使等式成立的未知数值。我们通过对等式两边进行相同操作来保持平衡。对于像 2x + 5 = 13 这样的简单方程，先将两边减去 5（2x = 8），再除以 2（x = 4）。
When equations involve brackets, we expand first: 3(2x – 1) = 15 becomes 6x – 3 = 15. Adding 3 to both sides gives 6x = 18, so x = 3. If there are unknowns on both sides, we collect variable terms on one side and constants on the other. For example, 5x + 2 = 3x + 10 leads to 2x = 8, hence x = 4.
当方程含有括号时，先展开：3(2x – 1) = 15 变为 6x – 3 = 15。两边加 3 得 6x = 18，所以 x = 3。如果未知数在等式两边，则将含变量的项移到一边，常数项移到另一边。例如，5x + 2 = 3x + 10 推导出 2x = 8，因此 x = 4。
Equations with fractions are cleared by multiplying every term by the lowest common denominator. Always check your solution by substituting it back into the original equation.
含有分数的方程可以通过每一项乘以最小公分母来去分母。解出答案后，务必代入原方程检验。
4. Inequalities | 不等式
Inequalities compare expressions using symbols: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). They are solved similarly to equations, but with one critical rule: when multiplying or dividing by a negative number, the inequality sign must be reversed.
不等式使用符号比较表达式：<（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）。解不等式与解方程类似，但有一条关键规则：当乘以或除以一个负数时，不等号方向必须改变。
For instance, to solve -2x ≤ 8, we divide both sides by -2, which reverses the sign to give x ≥ -4. The solution set can be represented on a number line with a closed circle for ≤ or ≥ and an open circle for < or >.
例如，解 -2x ≤ 8，两边除以 -2，不等号反向，得到 x ≥ -4。解集可以在数轴上表示，≤ 或 ≥ 用实心圆点，< 或 > 用空心圆点。
Double inequalities like -3 < 2x + 1 ≤ 7 are solved by isolating x in the middle. Subtract 1 throughout: -4 < 2x ≤ 6, then divide by 2: -2 < x ≤ 3. This means x is greater than -2 and at most 3.
像 -3 < 2x + 1 ≤ 7 这样的双向不等式可通过将中间的 x 分离来求解。整体减 1：-4 < 2x ≤ 6，再除以 2：-2 < x ≤ 3。这意味着 x 大于 -2 且不大于 3。
5. Sequences and the nth Term | 数列与第 n 项
A sequence is an ordered list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant. For example, in 5, 9, 13, 17, …, the common difference is 4.
数列是按一定规则排列的一列数。在等差数列中，相邻两项的差是常数。例如，在 5, 9, 13, 17, … 中，公差为 4。
The nth term formula allows us to find any term in the sequence without listing all previous ones. For the sequence above, the nth term is 4n + 1. We can generate terms: when n=1, term = 5; n=2, term = 9, and so on. Checking the 10th term gives 4(10) + 1 = 41.
第 n 项公式使我们无需列出前面所有项就能找到数列中的任意一项。对于上面的数列，第 n 项为 4n + 1。可以生成各项：当 n=1，项 = 5；n=2，项 = 9，以此类推。求第 10 项：4(10) + 1 = 41。
We can also find a term given its position or find n when a term is given. If the nth term is 3n – 2 and a term equals 25, we set 3n – 2 = 25, so n = 9. Sequences help us recognise patterns and form the basis for functions.
我们还可以根据位置求项，或已知某项求 n。若第 n 项为 3n – 2，且某项等于 25，则设 3n – 2 = 25，解得 n = 9。数列有助于识别模式，并为函数奠定基础。
6. Coordinates and Linear Graphs | 坐标与线性图
Points on a flat surface are located using coordinates (x, y). The equation of a straight line is often written as y = mx + c, where m represents the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
平面上的点用坐标 (x, y) 定位。直线的方程通常写成 y = mx + c，其中 m 表示斜率（坡度），c 是 y 轴截距（直线与 y 轴的交点）。
To draw a graph, we construct a table of values, choose at least three x-values, calculate the corresponding y-values, plot the points and join them with a straight line. For y = 2x – 1, when x = 0, y = -1; when x = 1, y = 1; when x = 2, y = 3.
要绘制图形，我们构建数值表，选取至少三个 x 值，计算相应的 y 值，描点并用直线连接。对于 y = 2x – 1，当 x = 0 时 y = -1；x = 1 时 y = 1；x = 2 时 y = 3。
The gradient m can be found by choosing two points on the line and using the formula: m = (change in y) / (change in x). Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to give -1.
斜率 m 可以通过选取直线上两点，使用公式 m = (y 的变化量) / (x 的变化量) 求得。平行线斜率相等，垂直线斜率之积为 -1。
7. Ratio, Proportion and Rates | 比率、比例和速率
A ratio compares parts of a whole. The ratio 3:5 means that for every 3 parts of one quantity, there are 5 parts of another. To divide £48 in the ratio 3:5, we find the total parts (8), so one part is £48 ÷ 8 = £6, giving £18 and £30.
比率比较整体中的各个部分。比率 3:5 意味着每 3 份某一量对应 5 份另一量。将 £48 按 3:5 分配，总份数为 8，每份为 £48 ÷ 8 = £6，分别得到 £18 和 £30。
Proportion describes when two quantities change at the same rate. If 4 pens cost £2.60, then 10 pens cost £6.50 by first finding the cost of 1 pen (£0.65) and multiplying. This is direct proportion.
比例描述两个量以相同速率变化的关系。如果 4 支笔花费 £2.60，则先求出 1 支笔的价钱 (£0.65)，再相乘可得 10 支笔花费 £6.50。这就是正比例。
Rates involve compound measures like speed, density and pressure. Speed = distance ÷ time. If a car travels 150 miles in 2.5 hours, its average speed is 60 mph. We often need to convert units, for example metres per second to kilometres per hour.
速率涉及复合量度，如速度、密度和压强。速度 = 距离 ÷ 时间。如果一辆汽车 2.5 小时行驶 150 英里，平均速度即为 60 mph。我们经常需要进行单位换算，比如米每秒换算为千米每小时。
8. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, and angles around a point sum to 360°. Vertically opposite angles are equal. When parallel lines are crossed by a transversal, corresponding angles are equal and alternate angles are equal.
直线上的角之和为 180°，一点周角之和为 360°。对顶角相等。当平行线被一条横截线所截时，同位角相等，内错角相等。
The sum of interior angles in a triangle is always 180°. In any quadrilateral, the sum is 360°. For a regular polygon with n sides, the sum of interior angles is (n – 2) × 180°, and each interior angle is that total divided by n.
三角形的内角和总是 180°。任意四边形的内角和为 360°。对于 n 条边的正多边形，内角和为 (n – 2) × 180°，每个内角为内角和除以 n。
Exterior angles of any polygon, one at each vertex, always add up to 360°. In a regular polygon, each exterior angle is 360° / n. This property can be used to find the number of sides if the exterior or interior angle is known.
任何多边形在每个顶点处的外角之和始终为 360°。在正多边形中，每个外角为 360° / n。这一性质可用于已知外角或内角时求边数。
9. Perimeter, Area and Volume | 周长、面积和体积
Perimeter is the total distance around the outside of a shape. For rectangles, P = 2(l + w). The circumference of a circle is given by C = πd or C = 2πr, where r is the radius and d is the diameter. Use the π button on your calculator for accuracy.
周长是围绕图形外部的总距离。对于矩形，P = 2(l + w)。圆的周长由 C = πd 或 C = 2πr 给出，其中 r 是半径，d 是直径。使用计算器上的 π 键以获得精确值。
Area measures the surface inside a shape. Key formulas include: rectangle A = lw, triangle A = ½bh, parallelogram A = bh, trapezium A = ½(a + b)h, and circle A = πr2. Remember that area is always expressed in square units.
面积度量图形内部的表面。关键公式包括：矩形 A = lw，三角形 A = ½bh，平行四边形 A = bh，梯形 A = ½(a + b)h，以及圆 A = πr2。记住面积始终以平方单位表示。
Volume is the space occupied by a 3D shape. For a cuboid, V = l × w × h. The volume of a prism is found by multiplying the area of its cross-section by its length. For a cylinder, V = πr2h. Volume is measured in cubic units.
体积是三维形状占据的空间。对于长方体，V = l × w × h。棱柱的体积通过将其截面面积乘以长度求得。对于圆柱，V = πr2h。体积以立方单位度量。
10. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2 + b2 = c2, where c is the hypotenuse.
勾股定理适用于直角三角形。它指出斜边（直角所对的边）的平方等于另外两条边的平方和：a2 + b2 = c2，其中 c 为斜边。
To find the length of the hypotenuse when the legs are 3 cm and 4 cm, we calculate c = √(32 + 42) = √(9 + 16) = √25 = 5 cm. When finding a shorter side, say b, rearrange to b2 = c2 – a2 and then take the square root.
当两直角边分别为 3 cm 和 4 cm 时，求斜边长：c = √(32 + 42) = √(9 + 16) = √25 = 5 cm。当求较短的直角边时，例如 b，移项得 b2 = c2 – a2，再开平方根。
The theorem can also be used to check whether a triangle is right-angled. If the sides satisfy a2 + b2 = c2, it is a right-angled triangle. Real-life applications include finding heights, distances and navigation problems.
该定理还可用于检验三角形是否为直角三角形。如果三边满足 a2 + b2 = c2，则为直角三角形。实际应用包括求高度、距离和导航问题。
11. Statistics: Averages and Range | 统计：平均数与极差
Statistics involves collecting, analysing and interpreting data. The three main averages are the mean, median and mode. The mean is the sum of all values divided by the number of values. For the data set 4, 7, 7, 9, 13, the mean is (4+7+7+9+13) ÷ 5 = 8.
统计学涉及数据的收集、分析和解读。三个主要的平均数是平均数、中位数和众数。平均数等于所有数值之和除以数值个数。对于数据集 4, 7, 7, 9, 13，平均数为 (4+7+7+9+13) ÷ 5 = 8。
The median is the middle value when data are arranged in order. For the same set, the median is 7. If there is an even number of values, the median is the mean of the two middle numbers. The mode is the value that appears most frequently – here it is 7.
中位数是将数据按顺序排列后的中间值。对于同一组数据，中位数为 7。若数据个数为偶数，中位数是中间两个数的平均数。众数是出现频率最高的值，这里为 7。
The range measures spread: it is the difference between the largest and smallest values. In our example, the range is 13 – 4 = 9. A small range indicates consistent data, while a large range shows variation. These measures help us summarise and compare data sets.
极差度量数据的分散程度，它是最大值与最小值的差。在我们的例子中，极差为 13 – 4 = 9。极差小表明数据集中，极差大则显示变异。这些度量有助于概括和比较数据集。
12. Probability | 概率
Probability measures how likely an event is to happen, expressed as a number between 0 (impossible) and 1 (certain). The theoretical probability of an event is P(event) = number of favourable outcomes / total number of possible outcomes, assuming all outcomes are equally likely.
概率度量事件发生的可能性，用 0（不可能）到 1（必然）之间的数字表示。事件的理论概率为 P(事件) = 有利结果的数量 / 所有可能结果的总数，前提是每种结果等可能出现。
When rolling a fair six-sided die, the probability of rolling a 3 is 1/6, and the probability of rolling an even number is 3/6 = 1/2. Probabilities can also be written as fractions, decimals or percentages.
抛掷一个均匀的六面骰子时，掷得 3 的概率为 1/6，掷得偶数的概率为 3/6 = 1/2。概率也可用分数、小数或百分数表示。
We can estimate probability from experimental data using relative frequency: relative frequency = number of times the event occurs / total number of trials. The more trials we carry out, the closer this estimate tends to get to the theoretical probability. For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities.
我们可以通过实验数据用相对频率估计概率：相对频率 = 事件发生的次数 / 试验总次数。进行的试验次数越多，估计值往往越接近理论概率。对于互斥事件，任一事件发生的概率等于各自概率之和。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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