📚 SQA National 5 Mathematics: Core Topic Review for Year 10 | 苏格兰国家5级数学：Year 10 核心知识点梳理
National 5 Mathematics is the key qualification for Scottish students in Year 10 (S4). This article brings together the most important concepts, techniques, and problem types across algebra, geometry, trigonometry, statistics, and more. Use this structured guide to consolidate your understanding, spot gaps in revision, and build confidence for the SQA exam.
国家5级数学是苏格兰 Year 10（S4）学生升学的关键学科。这篇文章梳理了代数、几何、三角、统计等模块中最重要的概念、题型和解题方法。你可以借助这个结构化指南来巩固理解、查漏补缺，并建立应对 SQA 考试的信心。
1. Algebraic Expressions & Simplification | 代数表达式与化简
a(b + c) = ab + ac, and (x + 3)(x + 2) = x² + 5x + 6. Always watch out for negative signs: –2(x – 4) = –2x + 8.
你必须熟练掌握展开括号与合并同类项。例如 a(b + c) = ab + ac，以及 (x + 3)(x + 2) = x² + 5x + 6。还要注意负号的处理：–2(x – 4) = –2x + 8。
Simplifying algebraic fractions is another core skill. Cancel common factors only after factoring numerators and denominators. For instance, (x² – 9)/(x + 3) simplifies to x – 3, provided x ≠ –3.
化简代数分式也是核心技能。只有先将分子和分母因式分解后，才能约去公因式。例如 (x² – 9)/(x + 3) 可化简为 x – 3，其中 x ≠ –3。
aᵐⁿ, and a⁻ⁿ = 1/aⁿ. Indices often appear in surds and standard form calculations as well.
处理指数时牢记基本法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ。根式和标准形式计算中也会频繁使用指数律。
2. Factorising | 因式分解
For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开括号的逆运算。首先检查是否有公因式：6x² + 9x = 3x(2x + 3)。对于形如 x² + bx + c 的二次式，找到两个数，乘积为 c，和为 b。
The difference of two squares is a vital pattern: a² – b² = (a + b)(a – b). You will see this in contexts like x² – 16 = (x + 4)(x – 4) or 9y² – 25 = (3y + 5)(3y – 5).
平方差公式至关重要：a² – b² = (a + b)(a – b)。常见形式有 x² – 16 = (x + 4)(x – 4) 或 9y² – 25 = (3y + 5)(3y – 5)。
For harder quadratics where the coefficient of x² is not 1, you can use the 'ac method' or systematic trial. For example, 2x² + 7x + 3 = (2x + 1)(x + 3).
对于二次项系数不为 1 的二次三项式，可使用 "ac 方法" 或系统性试凑。例如 2x² + 7x + 3 = (2x + 1)(x + 3)。
3. Quadratic Functions & Equations | 二次函数与方程
The sign of a determines whether the parabola opens up (a > 0) or down (a < 0). The turning point and axis of symmetry are essential for sketching.
二次函数具有形式 f(x) = ax² + bx + c，其图像是抛物线。系数 a 的符号决定开口向上 (a > 0) 或向下 (a < 0)。顶点和对称轴对画草图非常关键。
To solve quadratic equations, factorising is the quickest method. If a quadratic cannot be factorised, use the quadratic formula:
解二次方程最快的方法是因式分解。若不能分解，则使用求根公式：
x = (–b ± √b² – 4ac) / 2a
The discriminant Δ = b² – two real and distinct roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² – 4ac 揭示根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
4. Straight Line | 直线方程
he y-intercept. To find the gradient between two points (x₁, y₁) and (x₂, y₂), use m = (y₂ – y₁)/(x₂ – x₁).
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。计算过两点 (x₁, y₁) 和 (x₂, y₂) 的斜率，公式为 m = (y₂ – y₁)/(x₂ – x₁)。
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to –1: m₁ × m₂ = – ind equations of lines parallel or perpendicular to a given line through a specific point.
平行直线斜率相等。垂直直线的斜率乘积为 –1：m₁ × m₂ = –1。要会求过某定点且平行或垂直于已知直线的直线方程。
You should also be able to use the alternative forms y – b = m(x – al form ax + by + c = 0. Diagrams often require interpreting intercepts and intersections.
你还应该会用点斜式 y – b = m(x – a) 以及一般式 ax + by + c = 0。图形题常常涉及截距和交点的解读。
5. Right-Angled Triangle Trigonometry | 直角三角形三角比
In a right-angled triangle, remember SOH CAH TOA:
在直角三角形中，牢记 SOH CAH TOA：
- sin θ = Opposite / Hypotenuse
- sin θ = 对边 / 斜边
- cos θ = Adjacent / Hypotenuse
- cos θ = 邻边 / 斜边
- tan θ = Opposite / Adjacent
- tan θ = 对边 / 邻边
You can use these to find missing sides or angles. When finding an angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. Always check your calculator is in degree mode.
你可以用这些比来求未知边或角。求角时使用反三角函数 sin⁻¹、cos⁻¹、tan⁻¹。务必检查计算器处于角度模式。
Problems often involve 'angles of elevation and depression', which are measured from the horizontal. Draw a clear diagram and label the sides relative to the given angle.
应用题中常见 "仰角与俯角"，它们从水平线开始测量。画清晰的示意图，并根据已知角标记相关边。
6. Non-Right-Angled Triangles: Sine & Cosine Rules | 非直角三角形：正弦与余弦定理
When triangles are not right-angled, you need the sine rule or cosine rule. The sine rule states:
当三角形不是直角三角形时，需要用正弦定理或余弦定理。正弦定理为：
a / sin A = b / sin B = c / sin C
Use the sine rule when you know two angles and one side, or two sides and a non-included angle. The cosine rule is used for two sides and the included angle, or all three sides:
已知两角一边，或两边及一个非夹角时用正弦定理。余弦定理用于已知两边及其夹角，或已知三边：
a² = b² + c² – 2bc cos A
= ½ ab sin C. This is particularly useful when you know two sides and the included angle.
非直角三角形的面积公式为：面积 = ½ ab sin C。当知道两边及其夹角时，这个公式特别有用。
7. Similarity & Pythagoras | 相似形与毕达哥拉斯定理
Pythagoras' c is the hypotenuse. Use it to find a missing side or to prove a triangle is right-angled.
毕达哥拉斯定理是基础：任意直角三角形中，a² + b² = c²，其中 c 为斜边。用它求未知边长，或证明三角形是直角三角形。
Similar shapes have the same angles and proportional sides. If the scale factor for lengths is k, then the scale factor for area is k² and for volume is k³. This appears in many composite figure problems.
相似形具有相同角度和对应边成比例。若长度比例因子为 k，则面积比例因子为 k²，体积比例因子为 k³。这种关系常出现在复合图形题中。
You should be able to use similarity to find unknown lengths in overlapping triangles, particularly when parallel lines create proportionate segments.
你要会用相似性求重叠三角形中的未知长度，尤其当平行线截得比例线段时。
8. Statistics: Mean & Standard Deviation | 统计：平均数和标准差
The mean x̄ of a data set is the sum of all values divided by the number of values. For grouped frequency data, use the midpoints to estimate the mean.
一组数据的平均数 x̄ 等于所有数值之和除以数据个数。对于分组频数资料，用组中值来估算平均数。
Standard deviation measures the spread of data. The sample standard deviation formula is:
标准差衡量数据的离散程度。样本标准差的计算公式为：
s = √( Σ(x – x̄)² / (n – 1) )
A small standard deviation indicates data are close to the mean; a large standard deviation shows greater variability. You may be asked to compare two data sets using mean and standard deviation together.
标准差小表示数据集中在均值附近；标准差大表明变异性强。题目可能要求你用平均数和标准差一起比较两组数据。
Quartiles and the interquartile range are also part of the N5 syllabus. The median is Q₂, the lower quartile Q₁, and the upper quartile Q₃.
四分位数和四分位距也在 N5 考纲中。中位数即 Q₂，下四分位数 Q₁，上四分位数 Q₃。
9. Area, Arc Length & Volume | 面积、弧长与体积
For a circle of radius r, circumference = 2πr, area = πr². When dealing with sectors:
对于半径为 r 的圆，周长 = 2πr，面积 = πr²。处理扇形时：
- Arc length = θ/360 × 2πr
- 弧长 = θ/360 × 2πr
- Sector area = θ/360 × πr²
- 扇形面积 = θ/360 × πr²
e (V = ⁴/₃ πr³). You often need to round answers to a given number of significant figures.
三维立体的体积和表面积经常出现：圆柱 (V = πr²h)，圆锥 (V = ⅓πr²h)，球 (V = ⁴/₃ πr³)。解题时常需要将结果四舍五入到指定有效数字。
Composite shapes require you to add or subtract areas and volumes. Don't forget to convert units consistently – e.g. cm to m for volume calculations.
复合图形需要加减面积和体积。不要忘记统一单位 – 例如体积计算时需要把 cm 换算为 m。
10. Percentages & Compound Interest | 百分比与复利
Percentage increase or decrease can be calculated using a decimal multiplier. For an increase of r%, the multiplier is 1 + r/100; for a decrease it is 1 – r/100.
百分比增减可用小数乘数计算。增加 r% 时乘数为 1 + r/100；减少时乘数为 1 – r/100。
Compound interest is an essential application. After n periods, the amount accumulated is:
复利是重要的应用。经过 n 个周期后，本息合计为：
A = P(1 + r/100)n
where P is the principal and r is the interest rate per period. Appreciation and depreciation use the same structure, with a depreciation multiplier being less than 1.
其中 P 为本金，r 为每期利率。升值与折旧使用相同结构，折旧的乘数小于 1。
Reverse percentage problems ask you to find the original amount before a change: divide by the decimal multiplier. Be careful with wording: 'including VAT' implies 120% is the given total if VAT is 20%.
逆向百分比问题要求找出变化前的原始量：除以相应的小数乘数。注意措辞："含增值税" 意味着若增值税率 20%，则给定总量是 120%。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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