📚 Year 11 SQA Mathematics: Core Topics Summary | SQA 数学核心知识点梳理
This guide brings together the essential topics for Year 11 SQA Mathematics (typically National 5). It is designed to help you review key concepts, reinforce your understanding and spot any gaps before the exam. Each section is paired in English and Chinese so you can study comfortably in either language while mastering the mathematical vocabulary.
本指南总结了 Year 11 SQA 数学（通常对应 National 5 课程）的核心知识点。旨在帮助你梳理关键概念、巩固理解，并在考前查漏补缺。每个小节都以英文和中文配对呈现，便于你使用任意语言复习，同时掌握数学术语。
1. Algebraic Operations & Factorising | 代数运算与因式分解
Algebraic manipulation includes expanding brackets, simplifying expressions and factorising. You must be confident using the distributive law to expand single and double brackets, and to factorise by taking out a common factor, by recognising a difference of two squares, or by splitting a trinomial.
代数运算包括去括号、化简表达式以及因式分解。你需要熟练运用分配律展开单层和双层括号，并且能通过提取公因式、识别平方差公式或拆分三项式来进行因式分解。
For example, expanding (x + 3)(2x – 5) gives 2x² – 5x + 6x – 15, which simplifies to 2x² + x – 15. Factorising x² – 9 leads to (x + 3)(x – 3) because it is a difference of squares. A trinomial such as x² + 5x + 6 can be expressed as (x + 2)(x + 3).
例如，展开 (x + 3)(2x – 5) 得到 2x² – 5x + 6x – 15，化简为 2x² + x – 15。将 x² – 9 因式分解为 (x + 3)(x – 3)，因为它属于平方差。三项式 x² + 5x + 6 可表示为 (x + 2)(x + 3)。
Always check your factorisation by expanding back. Common mistakes include missing a negative sign or forgetting to factor out the highest common factor first.
务必通过再次展开来检验因式分解的结果。常见的错误包括遗漏负号，或者忘记先提取最大公因式。
2. Equations and Inequalities | 方程与不等式
Linear equations often involve brackets or fractions. To solve an equation like 3(2x – 4) = 18, expand first to get 6x – 12 = 18, then add 12 to both sides and divide by 6, giving x = 5. When fractions appear, multiply every term by the common denominator.
线性方程通常会包含括号或分数。解方程 3(2x – 4) = 18 时，先展开得 6x – 12 = 18，然后两边加 12 再除以 6，得到 x = 5。出现分数时，每项都乘以公分母。
Inequalities are solved similarly, but remember to reverse the inequality sign when multiplying or dividing by a negative number. For example, from -2y ≥ 8, dividing by -2 gives y ≤ -4.
解不等式的方法类似，但要记住当乘以或除以负数时，不等号方向要改变。例如，由 –2y ≥ 8，两边除以 –2 得到 y ≤ –4。
Solutions to linear inequalities are often shown on a number line. An open circle indicates strict inequality (< or >), while a closed circle means the value is included (≤ or ≥). Compound inequalities like 1 < 2x + 3 ≤ 7 are solved by subtracting 3 throughout to get -2 < 2x ≤ 4, dividing by 2 to obtain -1 < x ≤ 2.
一元一次不等式的解集通常用数轴表示。空心圆圈表示严格不等关系（< 或 >），实心圆圈表示包含该点（≤ 或 ≥）。复合不等式如 1 < 2x + 3 ≤ 7 可通过整体减 3 得到 –2 < 2x ≤ 4，再除以 2 得到 –1 < x ≤ 2。
3. Functions and Graphs | 函数与图像
A function is a rule that connects an input (x) to exactly one output (f(x)). For the function f(x) = 3x – 2, evaluating f(4) gives 3(4) – 2 = 10. The domain is the set of allowed input values, and the range is the set of possible output values.
函数是连接输入值 x 与唯一输出值 f(x) 的规则。对于函数 f(x) = 3x – 2，计算 f(4) 等于 3(4) – 2 = 10。定义域是允许的输入值的集合，值域是所有可能输出值的集合。
Graphs of linear functions are straight lines. Their steepness is given by the gradient, and where they cross the y-axis gives the y-intercept. Recognising the shape of y = x² (a parabola), y = 2ˣ (exponential growth) and y = 1/x (hyperbola) is essential for interpreting graphs.
一次函数的图像是直线。直线的倾斜度由斜率决定，与 y 轴的交点即为 y 轴截距。识别 y = x² 的图像形状（抛物线）、y = 2ˣ（指数增长）和 y = 1/x（双曲线）对于解读图像至关重要。
To find the equation of a straight line from its graph, pick two points and use the gradient formula m = (y₂ – y₁) / (x₂ – x₁). Then substitute one point into y = mx + c to find c.
若要根据图像求直线方程，可选取两点，用斜率公式 m = (y₂ – y₁) / (x₂ – x₁) 计算斜率，再将其中一个点代入 y = mx + c 求出截距 c。
4. Quadratic Graphs & Their Features | 二次函数图像及其特征
The graph of a quadratic function y = ax² + bx + c is a smooth, symmetrical curve called a parabola. If a > 0, the parabola opens upwards with a minimum turning point; if a < 0, it opens downwards with a maximum turning point.
二次函数 y = ax² + bx + c 的图像是一条光滑、对称的曲线，称为抛物线。若 a > 0，抛物线开口向上，并有一个最低点（极小值）；若 a < 0，则开口向下，有一个最高点（极大值）。
The axis of symmetry runs vertically through the turning point. The x-coordinate of the turning point can be found using x = -b / (2a). The y-coordinate is then found by substituting this x-value into the function. The roots or x-intercepts are the solutions of ax² + bx + c = 0.
对称轴是一条通过转折点的铅垂线。转折点的 x 坐标可用公式 x = –b / (2a) 求得，然后将此 x 值代回原函数求出对应的 y 坐标。方程的根或图像与 x 轴的交点就是方程 ax² + bx + c = 0 的解。
The discriminant Δ = b² – 4ac tells you how many real roots the quadratic has. If Δ > 0, there are two distinct real roots; if Δ = 0, exactly one real root (the graph touches the x-axis); if Δ < 0, there are no real roots (the graph does not cross the x-axis).
判别式 Δ = b² – 4ac 可判断二次方程实数根的个数。若 Δ > 0，有两个相异实根；若 Δ = 0，只有一个实根（图像与 x 轴相切）；若 Δ < 0，没有实根（图像不与 x 轴相交）。
x = ( -b ± √(b² – 4ac) ) / 2a
5. Trigonometry | 三角学
In right-angled triangles, the three basic trigonometric ratios are defined as: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. A common memory aid is ‘SOH CAH TOA’.
在直角三角形中，三个基本三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。常用的记忆口诀是 “SOH CAH TOA”。
To find a missing side, choose the ratio that connects the known angle, the known side and the unknown side. To find an angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. Always ensure your calculator is in degree mode.
要求未知边长时，应选择能将已知角、已知边和未知边联系起来的三角比。要求角度时，则使用反函数 sin⁻¹, cos⁻¹, tan⁻¹。确保计算器处于角度模式。
For non-right-angled triangles, the sine rule and cosine rule extend your problem-solving toolkit. The sine rule states: a / sin A = b / sin B = c / sin C. Use it when you know two angles and a side, or two sides and a non-included angle. The cosine rule a² = b² + c² – 2bc cos A is used when you know three sides or two sides and the included angle.
对于非直角三角形，正弦定理和余弦定理拓展了解题工具。正弦定理为：a / sin A = b / sin B = c / sin C。当已知两个角和一个边，或已知两边及一个非夹角时使用。余弦定理 a² = b² + c² – 2bc cos A 则适用于已知三边或两边及其夹角。
The area of a triangle using trigonometry is given by Area = ½ab sin C. This is useful when you know two sides and the angle between them.
利用三角学计算三角形面积的公式为 面积 = ½ab sin C。当已知两边及其夹角时可使用此公式。
6. Vectors | 向量
A vector represents a quantity with both magnitude and direction. In 2D, a vector can be written as a column [x, y] or using component form xi + yj. Adding vectors is done by adding corresponding components: [a, b] + [c, d] = [a + c, b + d].
向量表示既有大小又有方向的量。在二维空间中，向量可表示为列向量 [x, y] 或分量形式 xi + yj。向量加法只需将对应分量相加：[a, b] + [c, d] = [a + c, b + d]。
Subtraction follows a similar pattern. Multiplying a vector by a scalar k changes its magnitude but not its direction (unless k is negative, which reverses the direction). The scalar multiple kv multiplies each component by k.
减法同理。标量 k 乘向量只改变其大小而不改变方向（若 k 为负，则方向相反）。标量乘法 kv 即将每个分量乘以 k。
The magnitude (length) of a vector v = [x, y] is given by |v| = √(x² + y²). Vectors are particularly useful for describing journeys and paths; for example, from A to B is AB = OB – OA.
向量 v = [x, y] 的模（长度）由 |v| = √(x² + y²) 给出。向量在描述路径和行程时特别有用；例如，从 A 到 B 的向量为 AB = OB – OA。
Collinearity means three points lie on a straight line. This can be proved by showing that the direction vectors are scalar multiples of each other.
共线意味着三点在同一直线上，可通过证明方向向量互为标量倍数来验证。
7. Statistics | 统计
Descriptive statistics summarise a data set using measures of central tendency (mean, median, mode) and measures of spread (range, interquartile range, standard deviation). The mean is the sum of all values divided by the count, while the median is the middle value when data are ordered.
描述统计通过集中趋势量数（平均数、中位数、众数）和离散程度量数（极差、四分位距、标准差）来概括数据集。平均数是所有数据之和除以数据个数，中位数则是排序后处于中间位置的数值。
The interquartile range (IQR = Q₃ – Q₁) measures the spread of the middle 50% of data. A box plot (or box-and-whisker diagram) visually displays the minimum, Q₁, median, Q₃ and maximum, making it easy to compare distributions.
四分位距 (IQR = Q₃ – Q₁) 度量了中间 50% 数据的分散程度。箱线图（箱须图）能够直观地展示最小值、第一四分位数、中位数、第三四分位数和最大值，便于比较分布。
Standard deviation quantifies how much the data values deviate from the mean on average. In SQA National 5, you are expected to calculate the standard deviation of a small data set using the formula s = √[Σ(x – x̄)² / (n – 1)] and interpret what it means: a smaller standard deviation indicates data are clustered closely around the mean.
标准差量化了数据值平均偏离平均数的程度。在 SQA National 5 考试中，你需要使用公式 s = √[Σ(x – x̄)² / (n – 1)] 计算小数据集的标准差，并解释其含义：标准差越小，数据越紧密地聚集在平均数周围。
8. Probability | 概率
Probability measures the chance of an event occurring. It is calculated as (number of favourable outcomes) / (total number of possible outcomes). Probabilities always lie between 0 and 1, inclusive.
概率衡量事件发生的可能性大小，计算公式为（有利结果数目）/（所有可能结果的总数）。概率值始终介于 0 到 1 之间，包括 0 和 1。
Expected frequency is found by multiplying the probability of an event by the number of trials. If a fair coin is tossed 200 times, the expected number of heads is 0.5 × 200 = 100.
期望频数等于事件发生的概率乘以试验次数。如果抛一枚均匀硬币 200 次，期望出现正面的次数为 0.5 × 200 = 100。
Tree diagrams help to list the outcomes of multistage experiments systematically. Multiply probabilities along a branch to find the probability of a sequence of events. For independent events, P(A and B) = P(A) × P(B).
树状图有助于系统地列出多阶段试验的所有结果。沿一条分支将概率相乘，即可得到这一系列事件发生的概率。对于独立事件，有 P(A 且 B) = P(A) × P(B)。
Venn diagrams and two-way tables can also be used to organise outcomes and calculate conditional probabilities, although conditional probability is not always assessed at National 5. Always simplify fractions where possible.
韦恩图和双向表也可用于整理结果且计算条件概率，尽管条件概率并不总是在 National 5 考试中出现。计算结果应尽可能化简分数。
9. Geometry & Circle Properties | 几何与圆的性质
Angle facts are fundamental: angles on a straight line add up to 180°, angles around a point sum to 360°, vertically opposite angles are equal. In a triangle, the interior angles sum to 180°; in any quadrilateral, the sum is 360°.
角的基本性质是基础：直线上的邻角之和为 180°，绕一点的周角之和为 360°，对顶角相等。三角形的内角和为 180°，任意四边形的内角和为 360°。
When parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and interior (allied) angles sum to 180°. Recognising these relationships allows you to calculate unknown angles and prove results.
当平行线被一条截线所切割时，同位角相等，内错角相等，同旁内角互补（和为 180°）。识别这些关系有助于计算未知角并完成证明。
Circle geometry includes several key theorems: the angle in a semicircle is a right angle; a tangent to a circle is perpendicular to the radius at the point of contact; angles standing on the same chord in the same segment are equal. These properties help solve problems involving inscribed triangles and tangents.
圆相关的几何定理包括：半圆上的圆周角是直角；圆的切线垂直于经过切点的半径；同弦上的圆周角相等。这些性质有助于解决涉及内接三角形和切线的几何问题。
10. Similar Shapes & Pythagoras | 相似形与勾股定理
Two shapes are similar if they have exactly the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are in proportion. The scale factor k is the ratio of corresponding lengths.
如果两个图形形状完全相同而大小可能不同，则它们相似。对应角相等，对应边成比例。比例因子 k 就是对应边的长度之比。
For similar figures, the relationship between areas is given by the square of the linear scale factor (area factor = k²), and the relationship between volumes is given by the cube of the linear scale factor (volume factor = k³). This is crucial when solving problems involving enlargement or reduction.
对于相似图形，面积之间的关系由线性比例因子的平方给出（面积比 = k²），体积之间的关系由线性比例因子的立方给出（体积比 = k³）。这在进行放大或缩小的计算时尤为重要。
Pythagoras’ theorem applies exclusively to right-angled triangles: a² + b² = c², where c is the hypotenuse. It can be used to find a missing side in a right-angled triangle or to test whether a triangle contains a right angle by checking if the sides satisfy the relation.
勾股定理仅适用于直角三角形：a² + b² = c²，其中 c 为斜边。此定理可用来求直角三角形的未知边长，或通过验证三边是否满足该关系来判断三角形是否含有直角。
When applying Pythagoras’ theorem in 3D problems, such as finding the diagonal of a cuboid, break the problem into two right-angled triangles. The space diagonal d of a box with sides a, b, c is given by d = √(a² + b² + c²).
在三维问题中应用勾股定理时（如求长方体对角线），可将问题分解为两个直角三角形。边长为 a、b、c 的长方体的空间对角线长为 d = √(a² + b² + c²)。
11. Indices & Surds | 指数与根式
The rules of indices (exponents) must be applied confidently: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. A zero exponent gives a⁰ = 1 (for a ≠ 0), and negative exponents mean the reciprocal: a⁻ⁿ = 1 / aⁿ.
必须熟练掌握指数运算规则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。零指数满足 a⁰ = 1（a ≠ 0），负指数则表示倒数：a⁻ⁿ = 1 / aⁿ。
Fractional indices connect indices with roots. The notation a^(1/ⁿ) means the nth root of a, so a^(½) = √a. In general, a^(m/ⁿ) = (ⁿ√a)ᵐ = ⁿ√(aᵐ). This is used to simplify expressions like 16^(3/4), which equals (⁴√16)³ = 2³ = 8.
分数指数将指数与根式联系起来。符号 a^(1/ⁿ) 表示 a 的 n 次方根，因此 a^(½) = √a。一般地，a^(m/ⁿ) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)。这可用于简化诸如 16^(3/4) 的表达式，它等于 (⁴√16)³ = 2³ = 8。
Surds are irrational roots like √2 or ³√5. They can be simplified by looking for square factors: √48 = √(16 × 3) = 4√3. When adding or subtracting surds, only like terms can be combined, just as with algebraic terms.
根式是指像 √2 或 ³√5 这样的无理数根号。可以通过寻找平方因子来化简根式：√48 = √(16 × 3) = 4√3。加减根式时，只有同类项才能合并，与代数项的处理方式相同。
Rationalising the denominator means rewriting a fraction so that no surd appears in the denominator. For a simple surd denominator like 1/√5, multiply numerator and denominator by √5 to get √5/5. When the denominator is of the form a + √b, multiply by its conjugate a – √b.
分母有理化是指将分数改写成分母中不含根式的形式。对于如 1/√5 这样的简单根式分母，分子分母同乘 √5 得到 √5/5。若分母形如 a + √b，则同时乘以其共轭表达式 a – √b。
12. Straight Line & Linear Relationships | 直线与线性关系
Every non-vertical straight line can be expressed in the slope-intercept form y = mx + c, where m is the gradient and c is the y-intercept. The gradient is calculated as m = (y₂ – y₁) / (x₂ – x₁). If m is positive, the line rises from left to right; if negative, it falls.
任意一条非竖直的直线都可以表示为斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率计算公式为 m = (y₂ – y₁) / (x₂ – x₁)。斜率为正时，直线从左向右上升；为负时则下降。
To find the equation of a line given two points, first compute m, then substitute one point’s coordinates into y = mx + c to solve for c. Once the equation is written, you can use it to find missing coordinates or to test whether a point lies on the line.
给定两点求直线方程，先计算斜率 m，然后将其中一点坐标代入 y = mx + c 求出 c。得到方程后，可据此求缺失的坐标，或检验某点是否在该直线上。
Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines satisfy m₁ × m₂ = -1, provided neither line is vertical. These facts allow you to find equations of lines parallel or perpendicular to a given line through a given point.
平行直线的斜率相等（m₁ = m₂）。互相垂直的直线满足 m₁ × m₂ = –1（假设两条直线都不垂直于 x 轴）。借助这一性质可求出过某定点且与已知直线平行或垂直的直线方程。
Linear models also describe direct proportion: variables x and y are in direct proportion if y = kx. The graph is a straight line through the origin. Understanding linear relationships is fundamental for interpreting real-world data and modelling.
线性模型同样描述正比例关系：若 y = kx，则变量 x 和 y 成正比，图像是一条通过原点的直线。理解线性关系是解读实际数据和建立数学模型的基础。
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