📚 Year 9 SQA Advanced Mathematics: Core Knowledge Review | Year 9 SQA 进阶数学：核心知识点梳理
This comprehensive guide covers the core topics in Year 9 SQA Advanced Mathematics, designed to help you consolidate key skills and prepare for progress into National 5 and beyond. Each section presents clear explanations, essential formulae, and worked examples using proper mathematical notation without any LaTeX – just clean Unicode symbols.
这篇全面指南涵盖了 Year 9 SQA 进阶数学的核心主题，旨在帮助你巩固关键技能，为升入 National 5 及更高阶段做好准备。每个部分都提供了清晰的解释、基本公式和示例，使用整齐的 Unicode 符号，没有采用 LaTeX。
1. Algebraic Manipulation | 代数运算
In advanced algebra, fluency in expanding, factorising, and simplifying expressions with brackets and powers is essential. These skills form the foundation for solving equations and working with functions.
在进阶代数中，熟练展开、因式分解以及简化带有括号和幂的表达式至关重要。这些技能是解方程和研究函数的基础。
= x² + (a + b)x + ab. When the coefficients are not 1, use the distributive law carefully.
两个二项式的基本展开公式是 (x + a)(x + b) = x² + (a + b)x + ab。当系数不为 1 时，需要仔细使用分配律。
finding two numbers that multiply to 12 and add to 7. Thus x² + 7x + 12 = (x + 3)(x + 4).
对二次三项式如 x² + 7x + 12 进行因式分解，就是找出两个乘积为 12、和为 7 的数。因此 x² + 7x + 12 = (x + 3)(x + 4)。
a common factor first. For 2x² + 8x + 6, factor out 2 to get 2(x² + 4x + 3) = 2(x+1)(x+3).
永远要先寻找公因子。对于 2x² + 8x + 6，先提取 2 得到 2(x² + 4x + 3) = 2(x+1)(x+3)。
2. Linear and Quadratic Equations | 线性与二次方程
Solving equations moves from simple linear cases to quadratics that may have two, one, or no real solutions. Mastery of the quadratic formula and discriminant is expected.
解方程从简单的线性方程发展到可能有两个、一个或没有实数解的二次方程。要求掌握二次公式和判别式。
For a linear equation like 3x – 7 = 2x + 5, isolate x to find x = 12. Check your solution by substituting back.
对于像 3x – 7 = 2x + 5 这样的线性方程，通过移项解出 x = 12。代入原方程检验你的解。
To solve a quadratic, first try factorising. If x² – 5x + 6 = 0, then (x – 2)(x – 3) = 0 gives x = 2 or x = 3.
解二次方程时，首先尝试因式分解。若 x² – 5x + 6 = 0，则 (x – 2)(x – 3) = 0，得到 x = 2 或 x = 3。
factorising is not straightforward, use the quadratic formula: for ax² + bx + c = 0, x = [ –b ± √b² – 4ac ] / (2a).
当因式分解不易时，使用二次公式：对于 ax² + bx + c = 0，x = [ –b ± √b² – 4ac ] / (2a)。
The discriminant Δ = b² – Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
判别式 Δ = b² – 4ac 可判断根的性质。若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根。
3. Functions and Graphs | 函数与图像
Understanding the relationship between algebraic equations and their graphs is crucial. You will work with linear functions, quadratic parabolas, and transformations of graphs.
理解代数方程与其图像之间的关系至关重要。你将学习线性函数、二次抛物线和图像的变换。
, where m is the gradient and c is the y-intercept. Parallel lines have the same gradient.
直线的斜截式是 y = mx + c，其中 m 是梯度，c 是 y 轴截距。平行的直线具有相同的梯度。
transformations: y = (x + a)² shifts the graph horizontally, y = x² + k shifts vertically.
二次函数 y = x² 的图像是一条 U 形抛物线。它的变换：y = (x + a)² 将图像水平移动，y = x² + k 垂直移动。
Completing the square helps to find the turning point. Write y = x² – 4x + 1 as y = (x – 2)² – 3; the vertex is (2, –3).
配方法有助于找到转折点。将 y = x² – 4x + 1 写成 y = (x – 2)² – 3，顶点为 (2, –3)。
urning point. The axis of symmetry can be read from the completed square: x = h for y = (x – h)² + k.
描绘图像时总要标出截距和转折点。对称轴可由配方法后的形式读出：对于 y = (x – h)² + k，对称轴为 x = h。
4. Trigonometry: From Right Triangles to General Angles | 三角学：从直角三角形到一般角
Trigonometry begins with right-angled triangles and extends to solving any triangle using the sine and cosine rules. Exact values for special angles are also required.
三角学从直角三角形开始，并扩展到使用正弦定理和余弦定理解任意三角形。还要求记住特殊角的精确值。
osite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use SOH CAH TOA.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。记住口诀 SOH CAH TOA。
Exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
精确值：sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
C, used for non-right triangles when you know two angles and a side, or two sides and a non-included angle.
正弦定理：a/sin A = b/sin B = c/sin C，适用于已知两角一边或两边及其一边对角的非直角三角形。
The cosine rule: a² = b² + c² – 2bc cos A. Use it when you know two sides and the included angle, or all three sides.
余弦定理：a² = b² + c² – 2bc cos A。当已知两边及其夹角，或已知三边时使用。
5. Circle Geometry and Angle Properties | 圆几何与角度性质
Circle theorems describe relationships among angles, chords, tangents, and arcs. Memorising these theorems helps to find unknown angles quickly.
圆定理描述了角、弦、切线和弧之间的关系。熟记这些定理有助于快速求出未知角度。
The angle at the centre is twice the angle at the circumference subtended by the same arc. If the arc subtends an angle of 80° at the centre, the circumference angle is 40°.
圆心角是同一弧所对圆周角的两倍。如果一条弧所对的圆心角为 80°，则圆周角为 40°。
Angles in the same segment are equal. A tangent and radius meet at right angles; two tangents from an external point are equal in length.
同弧上的圆周角相等。切线与半径相交成直角；从圆外一点引出的两条切线长度相等。
A cyclic quadrilateral has opposite angles summing to 180°. If one angle is 110°, the opposite angle is 70°.
圆内接四边形的对角之和为 180°。若一个角为 110°，则其对角为 70°。
Always mark known angles on the diagram and identify which theorem applies before calculating.
在计算前，始终在图上标出已知角，并判断适用哪个定理。
6. Vectors and Their Operations | 向量及其运算
Vectors represent magnitude and direction. In two dimensions, they are expressed as column vectors or in component form, and can be added, subtracted, and multiplied by scalars.
向量表示大小和方向。在二维中，它们用列向量或分量形式表示，并可进行加减法和标量乘法。
AB→. In components, a vector u = (3, 4) means 3 units right, 4 units up. Its magnitude is |u| = √3² + 4² = 5.
从 A 到 B 的向量记作 AB→。用分量表示，向量 u = (3, 4) 表示向右 3 个单位，向上 4 个单位。其模长为 |u| = √3² + 4² = 5。
lar k: ku = (ku₁, ku₂). Two vectors are parallel if one is a scalar multiple of the other.
向量加法：u + v = (u₁+v₁, u₂+v₂)。乘以标量 k：ku = (ku₁, ku₂)。若一个向量是另一个向量的标量倍数，则两向量平行。
To find a resultant vector, draw them tip-to-tail or use algebra. The zero vector (0, 0) has no length and no specific direction.
求合向量时，可用三角形法则或代数计算。零向量 (0, 0) 没有长度，也没有特定方向。
duct: u · v = u₁v₁ + u₂v₂. It helps to find the angle between vectors: cos θ = (u · v) / (|u| |v|).
标量积（点积）：u · v = u₁v₁ + u₂v₂。它可用来求两向量夹角：cos θ = (u · v) / (|u| |v|)。
7. Probability Using Tree Diagrams | 使用树形图的概率
Probability trees are powerful tools for representing sequences of independent and conditional events. Always multiply along branches and add across different outcomes.
概率树形图是表示一系列独立事件和条件事件的强大工具。总是沿着分支相乘，并将不同结果的分支概率相加。
). Draw two sets of branches: first event possibilities, then second event from each node.
对于两个独立事件 A 和 B，P(A 且 B) = P(A) × P(B)。画出两组分支：第一组表示第一个事件的可能结果，然后从每个节点出发画出第二个事件的分支。
When probabilities depend on earlier outcomes, use conditional probability. For example, a bag with 3 red and 5 blue marbles: remove one without replacement; the second draw's probabilities change.
当概率依赖于之前的结果时，使用条件概率。例如，一个袋子里有 3 颗红色和 5 颗蓝色弹珠：不放回地取出一颗；第二次抽取的概率发生变化。
Final probabilities for combined outcomes are found by multiplying along the path. Check that all final probabilities sum to 1.
组合事件的最终概率通过沿路径相乘得到。检查所有最终概率之和是否为 1。
Always label branches with the event and its probability. Simplify fractions at the end but keep accuracy during multiplication.
始终在分支上标注事件及其概率。最后可化简分数，但乘法过程中要保持精确。
8. Statistical Diagrams and Measures | 统计图表与度量
Data handling in SQA Advanced Mathematics involves cumulative frequency graphs, box plots, and measures of spread such as interquartile range and standard deviation.
SQA 进阶数学中的数据处理涉及累积频数图、箱线图以及诸如四分位距和标准差等离散程度的度量。
A cumulative frequency table adds frequencies as you go up the class intervals. Plot the upper class boundary against cumulative frequency and join with a smooth curve.
累积频数表沿区间依次累加频数。将上限值与累积频数作图，并用光滑曲线连接。
mate the median, lower quartile (Q1) and upper quartile (Q3). Interquartile range IQR = Q3 – Q1.
从图中可以估计中位数、下四分位数 (Q1) 和上四分位数 (Q3)。四分位距 IQR = Q3 – Q1。
A box plot displays the five-number summary: minimum, Q1, median, Q3, maximum. It shows the spread and any skewness in the data.
箱线图显示五数概括：最小值、Q1、中位数、Q3、最大值。它展示数据的分布和偏态。
Standard deviation measures average distance from the mean. For a population, σ = √[ Σ(x – μ)² / n ]. A larger standard deviation means more spread.
标准差衡量数据与均值的平均距离。对于总体，σ = √[ Σ(x – μ)² / n ]。标准差越大，数据越分散。
9. Surds and Rationalising Denominators | 根式与分母有理化
Surds are irrational roots like √2, √3. Working with surds means simplifying, adding, multiplying, and rationalising denominators so that no surd remains in the bottom of a fraction.
根式是指像 √2、√3 这样的无理根式。处理根式包括化简、加减、乘法以及分母有理化，使分数的分母中不含根式。
Simplify a surd by extracting square factors: √72 = √36×2 = 6√2. Multiplication: √a × √b = √ab. Addition: only like surds can be combined, e.g. 2√3 + 5√3 = 7√3.
通过提取平方因子化简根式：√72 = √36×2 = 6√2。乘法：√a × √b = √ab。加法：只有同类根式才能合并，例如 2√3 + 5√3 = 7√3。
Rationalising a denominator like 1/√2: multiply top and bottom by √2 to get √2/2. For a+b√c in the denominator, use the conjugate.
对分母如 1/√2 进行有理化：上下同乘 √2 得到 √2/2。如果分母是 a+b√c 的形式，则使用其共轭式。
Example: (3)/(1+√2) multiply by (1–√2)/(1–√2) = 3(1–√2)/(1–2) = 3(1–√2)/(–1) = –3(1–√2) = –3 + 3√2.
例子：(3)/(1+√2)，上下同乘 (1–√2)，得到 3(1–√2)/(1–2) = 3(1–√2)/(–1) = –3(1–√2) = –3 + 3√2。
10. Sequences: Finding the nth Term | 数列：求第 n 项
Recognising patterns and expressing the general term of a sequence is vital. You will deal with linear, quadratic, and perhaps simple geometric sequences.
识别模式并写出数列的通项至关重要。你将处理线性数列、二次数列，可能还有简单的几何数列。
A linear sequence has a constant difference d. The nth term is given by an = dn + (a – d), where a is the first term. For 5, 8, 11, 14, …, d = 3 so an = 3n + 2.
线性数列的公差 d 是常数。第 n 项公式为 an = dn + (a – d)，其中 a 是首项。对于 5, 8, 11, 14, …，d = 3，因此 an = 3n + 2。
nstant second difference. Find the nth term by matching to an² + bn + c. For 3, 6, 10, 15, …, second difference is 1, so a = ½, then solve for b and c.
二次数列的二级差分是常数。通过匹配 an² + bn + c 来求第 n 项。对于数列 3, 6, 10, 15, …，二级差为 1，因此 a = ½，然后求解 b 和 c。
The nth term for that sequence is ½n(n+3), or equivalently ½n² + (3/2)n + 1 after checking. Write it in the simplest polynomial form.
该数列的第 n 项为 ½n(n+3)，或经验算后写为 ½n² + (3/2)n + 1。用最简多项式形式表示。
each term is multiplied by a common ratio r. The nth term is arⁿ⁻¹. Example: 2, 6, 18, 54,… has r = 3, an = 2 × 3ⁿ⁻¹.
在几何数列中，每一项乘以公比 r。第 n 项为 arⁿ⁻¹。例如：2, 6, 18, 54,…，r = 3，an = 2 × 3ⁿ⁻¹。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图