📚 Prior Learning for IB Mathematics: SL and HL | IB数学预备知识：标准水平与高级水平
Before embarking on the IB Mathematics: Analysis and Approaches or Applications and Interpretation courses at either Standard Level or Higher Level, students are expected to have a solid grasp of foundational mathematical concepts. This prior learning encompasses topics typically covered in secondary school mathematics, including number systems, algebra, coordinate geometry, trigonometry, functions, statistics, and basic sequences. A confident command of these areas ensures a smooth transition into the more abstract and rigorous thinking demanded by the IB Diploma Programme. The following review highlights the essential knowledge and skills, presented with concise explanations and examples, to help you identify any gaps and consolidate your understanding before the course begins.
在开始学习IB数学：分析与方法或应用与解释课程之前，无论是标准水平还是高级水平，学生都应扎实掌握基础的数学概念。这些预备知识通常涵盖中学数学的内容，包括数系、代数、坐标几何、三角学、函数、统计以及基本数列。对这些领域的熟练掌握将确保你顺利过渡到IB文凭课程所要求的更抽象和严谨的思维中。下面的复习内容提炼了核心知识与技能，并配以简洁的解释和例子，帮助你在课程开始前找出薄弱环节并巩固理解。
1. Number Systems and Operations | 数系与运算
The real number system is the foundation of all quantitative reasoning. It includes natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), and irrational numbers, which together form the real numbers (ℝ). Rational numbers can be expressed as a fraction a/b where a and b are integers and b ≠ 0, while irrational numbers cannot be written as a simple fraction, e.g., √2 : 2² = 4, then multiplication: 2 × 4 = 8, and finally addition: 3 + 8 = 11. Squares and square roots reverse each other: if x² = 25, then x = ±5. Cube roots and higher roots extend this idea.
实数系是所有定量推理的基石。它包括自然数(ℕ)、整数(ℤ)、有理数(ℚ)和无理数，它们共同构成了实数集(ℝ)。有理数可以表示为分数a/b（a、b为整数且b≠0），而无理数不能写成简单的分数，例如√2或π。掌握运算顺序（通常记为PEMDAS或BIDMAS：括号、指数/指标、乘除、加减）至关重要。例如，在3 + 2 × (8 ÷ 4)²中，先算括号内：8 ÷ 4 = 2，再算指数：2² = 4，然后乘法：2 × 4 = 8，最后加法：3 + 8 = 11。平方与平方根互为逆运算：若x² = 25，则x = ±5。立方根和更高次根的概念依此类推。
2. Algebraic Expressions and Identities | 代数表达式与恒等式
distributive property or FOIL method, and factoring common factors. Three fundamental identities are worth memorising: (a + b)² = a² + 2ab + b², (a - b)² = a² - 2ab + b², and (a + b)(a - b) = a² - b². These help in both expanding and factoring. For example, to expand (2x - 3)², use (a - b)² with a = 2x, b = 3: (2x)² - 2(2x)(3) + 3² = 4x² - 12x + 9. Substitution means replacing variables with numerical values, e.g., if x = -1 and y = 4, then 2x² - y = 2(-1)² - 4 = 2 - 4 = -2.
代数是我们概括算术关系的语言。代数表达式通过运算将数字和变量组合起来；例如，3x² - 2x + 5。基本技能包括合并同类项（如4a + 3b - 2a + b = 2a + 4b），使用分配律或FOIL法则展开二项式乘积，以及提取公因子。三个基本恒等式值得牢记：(a + b)² = a² + 2ab + b²，(a - b)² = a² - 2ab + b²，和(a + b)(a - b) = a² - b²。它们对展开和分解都很有帮助。例如，展开(2x - 3)²，使用(a - b)²，代入a = 2x, b = 3：得(2x)² - 2(2x)(3) + 3² = 4x² - 12x + 9。代入法是指用数值替换变量，例如，若x = -1且y = 4，那么2x² - y = 2(-1)² - 4 = 2 - 4 = -2。
3. Linear Equations and Inequalities | 线性方程与不等式
ominator to clear them. Solving inequalities is similar, but remember that multiplying or dividing by a negative number reverses the inequality sign. For instance, -2x > 6 becomes x < -3. Systems of two linear equations in two unknowns can be solved by substitution, elimination, or graphically as the intersection point of two lines. The solution to y = 2x + 1 and y = -x + 4 is x = 1, y = 3. Always check your solution in both original equations.
一元线性方程可以写成 ax + b = 0 的形式。通过逆运算分离变量即可求解。例如，解 3x - 7 = 5：两边加7得 3x = 12，再除以3得 x = 4。若方程含有分数，可乘以最小公分母来去分母。解不等式与之类似，但要记住：当乘或除以一个负数时，不等号方向要改变。例如，-2x > 6 变为 x < -3。两个二元一次方程组可通过代入法、消元法或用作图法（两条直线的交点）来求解。方程组 y = 2x + 1 和 y = -x + 4 的解为 x = 1, y = 3。解出后务必代回原方程进行检验。
4. Coordinate Geometry | 坐标几何
d x₁ ≠ x₂. A horizontal line has gradient 0; a vertical line has undefined gradient. The equation of a line can be written in gradient-intercept form y = mx + c, where c is the y-intercept, or in point-gradient form y - y₁ = m(x - x₁). Parallel lines have equal gradients, while perpendicular lines have gradients that are negative reciprocals (m₁ × m₂ = -1). For example, the line perpendicular to y = 3x + 2 would have gradient -1/3.
笛卡尔平面让我们能用代数方法描述几何图形。两点 (x₁, y₁) 和 (x₂, y₂) 间的距离公式为 d = √[(x₂ - x₁)² + (y₂ - y₁)²]。中点为 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。通过这两点的直线斜率（坡度）为 m = (y₂ - y₁)/(x₂ - x₁)，前提是 x₁ ≠ x₂。水平线斜率为0；竖直线斜率无定义。直线方程可写成斜截式 y = mx + c（c是y轴截距），或点斜式 y - y₁ = m(x - x₁)。平行线斜率相等，而垂直线的斜率互为负倒数（m₁ × m₂ = -1）。例如，与 y = 3x + 2 垂直的直线斜率为 -1/3。
5. Right-Angled Trigonometry | 直角三角学
ou can use the mnemonic SOH CAH TOA. The Pythagorean theorem, a² + b² = c², is essential for finding missing sides. For example, in a triangle with opposite = 3, adjacent = 4, the hypotenuse is √3² + 4² . Angles of elevation and depression are measured from the horizontal. Inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) find the angle when two sides are known.
直角三角形中的三角学将角度与边之比联系起来。三个基本三角比是正弦 (sin)、余弦 (cos) 和正切 (tan)。对于直角三角形中的锐角 θ：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。你可使用助记符 SOH CAH TOA。勾股定理 a² + b² = c² 对于求未知边至关重要。例如，在一个对边为3、邻边为4的三角形中，斜边为 √3² + 4² = 5，因此 sin θ = 3/5，cos θ = 4/5，tan θ = 3/4。仰角和俯角均从水平线量起。已知两边求角时，使用反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）。
6. Functions and Graphs | 函数与图像
izontal axis and the dependent variable y = f(x) on the vertical axis. Key features include intercepts (where the graph crosses the axes), maximum and minimum points, and intervals where the function is increasing or decreasing. The graph of a quadratic function y = ax² + bx + c is a parabola; if a > 0 it opens upward and has a minimum, if a < 0 it opens downward and has a maximum. The vertex can be found by completing the square or using x = -b/(2a).
函数是一种给每个输入值分配唯一输出值的规则。通常记为 f(x)，例如 f(x) = 2x + 3。定义域是所有可能输入值的集合，值域是所有可能输出值的集合。绘制函数图像时，我们在横轴上标注自变量 x，在纵轴上标注因变量 y = f(x)。关键特征包括截距（图像与坐标轴的交点）、最大值和最小值点，以及函数的增减区间。二次函数 y = ax² + bx + c 的图像是一条抛物线；若 a > 0 则开口向上且有最小值，若 a < 0 则开口向下且有最大值。顶点可通过配方法或用公式 x = -b/(2a) 求得。
7. Statistics and Probability | 统计与概率
value. For grouped data, we estimate the mean using class midpoints. Probability quantifies the likelihood of an event, ranging from 0 (impossible) to 1 (certain). The probability of an event A is P(A) = number of favorable outcomes / total number of outcomes, assuming equally likely outcomes. For combined events, "A or B" uses addition, while "A and B" events uses multiplication: P(A and B) = P(A) × P(B). A tree diagram helps visualise multi‑stage experiments.
描述性统计利用集中趋势量数（平均数、中位数、众数）和离散量数（极差、四分位距、标准差）来汇总数据。平均数即算术平均。中位数是将数据排序后的中间值。众数是出现频率最高的值。对于分组数据，我们使用组中点来估算平均数。概率用于量化事件的可能性，范围从0（不可能）到1（必然）。事件A的概率 P(A) = 有利结果数 / 总结果数，前提是等可能结果。对于复合事件，“A或B”用加法法则，而独立事件的“A且B”用乘法法则：P(A且B) = P(A) × P(B)。树状图有助于展现多阶段试验。
8. Exponents and Radicals | 指数与根式
0), (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), and a⁻ⁿ = 1/aⁿ. Rational exponents connect to radicals: a1/n = ⁿ√a and am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ. For example, 82/3 = (³√8 a < 10 and n is an integer. Simplifying radicals involves removing perfect square factors: √48 = √16 × 3 = 4√3. When rationalising a denominator like 1/√2, multiply numerator and denominator by √2 to get √2/2.
指数表示连乘：aⁿ = a × a × ... × a （n次）。指数的运算法则是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ （a ≠ 0），(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1 （a ≠ 0），以及 a⁻ⁿ = 1/aⁿ。有理指数与根式相关：a1/n = ⁿ√a，且 am/n = (ⁿ√a)ᵐ = ⁿ√aᵐ。例如，82/3 = (³√8)² = 2² = 4。科学记数法将非常大或非常小的数表示为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。简化根式包括去除完全平方因子：√48 = √16 × 3 = 4√3。当要将如 1/√2 的分母有理化时，将分子和分母同乘 √2 得 √2/2。
9. Polynomials and Factoring | 多项式与因式分解
iables and coefficients combined using addition, subtraction, and multiplication, with non‑ to c and add to b; difference of squares: a² - b² = (a + b)(a - b); and factoring by grouping. For x² - 5x + 6, numbers -2 and -3 give the factored form (x - 2)(x - 3). To factor 2x² + 7x + 3, multiply a×c = 2×3 = 6; numbers 6 and 1 work since 6+1=7, so rewrite as 2x² + 6x + x + 3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
多项式是由变量和系数通过加法、减法和乘法组合而成的表达式，且指数为非负整数。次数是最高指数。因式分解是展开的逆过程。常用技巧包括提取最大公因子，如 6x³ + 9x² = 3x²(2x + 3)；对于 x² + bx + c 形式的二次式，找出两个数使其乘积为c且和为b；利用平方差公式 a² - b² = (a + b)(a - b)；以及分组分解法。对于 x² - 5x + 6，数字 -2 和 -3 使得分解结果为 (x - 2)(x - 3)。要分解 2x² + 7x + 3，先算 a×c = 2×3 = 6；数字6和1符合6+1=7，因此改写为 2x² + 6x + x + 3，再分组：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
10. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers following a pattern. Two common types are arithmetic and geometric sequences. In an arithmetic sequence, the difference d between consecutive terms is constant: un + 19×3 = 62. In a geometric sequence, the ratio r between consecutive terms is constant: un = u₁ × rn-1 the sum of the terms of a sequence. The sum of the first n terms of an arithmetic series is Sn = n/2 (u₁ + un) or Sn = n/2 [2u₁ + (n - 1)d]. For a geometric series with r ≠ 1, Sn - r). These formulae are vital for solving problems involving savings, growth, and decay.
数列是按照一定规律排列的一列数。最常见的是等差数列和等比数列。在等差数列中，相邻项的差 d 是常数：un = u₁ + (n - 1)d。例如，5, 8, 11, 14,... 中 d = 3；第20项为 5 + 19×3 = 62。在等比数列中，相邻项的比 r 是常数：un = u₁ × rn-1。例如，3, 6, 12, 24,... 中 r = 2；第8项为 3 × 2⁷ = 384。级数是数列各项的和。等差数列前n项和为 Sn = n/2 (u₁ + un) 或 Sn = n/2 [2u₁ + (n - 1)d]。对于 r ≠ 1 的等比级数，Sn = u₁(1 - rⁿ)/(1 - r)。这些公式对于解决储蓄、增长和衰减等问题至关重要。
11. Perimeter, Area, and Volume | 周长、面积与体积
: volume of a prism = area of cross-section × length; volume of a cylinder = πr²h; volume of a sphere = ⁴⁄₃πr³; surface area of a sphere = 4πr²; volume of a pyramid or cone = ⅓ × base area × height. When the object is a composite solid, break it into simpler parts. Units must be consistent; converting between linear, square, and cubic units requires careful attention (e.g., 1 m² = 10,000 cm²).
几何测量公式需要熟记。对于平面图形：圆的周长 = 2πr；圆的面积 = πr²；三角形的面积 = ½ × 底 × 高；梯形的面积 = ½(a + b)h，其中a和b是平行边；平行四边形的面积 = 底 × 高。对于立体：棱柱的体积 = 截面面积 × 长度；圆柱的体积 = πr²h；球的体积 = ⁴⁄₃πr³；球的表面积 = 4πr²；棱锥或圆锥的体积 = ⅓ × 底面积 × 高。当物体为组合体时，应将其拆分为简单部分。单位必须一致；线性、平方和立方单位之间的换算需仔细处理（例如，1 m² = 10,000 cm²）。
12. Sets and Venn Diagrams | 集合与文氏图
-defined collection of objects, often denoted by capital letters. Basic notation includes: ∈ (element of), ∉ (not an element of), ∪ (union), ∩ (intersection), and A' (complement of A). The empty set is ∅ a visual way to represent sets and their relationships. For two sets A and B, the union A ∪ B contains elements in A or B or both; the intersection A ∩ B contains elements common to both. n(A ∪ B) = n(A) + n(B) - n(A ∩ g with three sets, a careful shading and labelling approach helps solve problems involving "at least one" or "exactly two" conditions. These concepts underpin probability calculations with combined events.
集合是由明确定义的对象组成的整体，常用大写字母表示。基本符号包括：∈（属于）、∉（不属于）、∪（并集）、∩（交集）和 A'（A的补集）。空集记为 ∅。n(A) 表示集合A中元素的个数。文氏图提供了直观表示集合及其关系的方式。对于两个集合 A 和 B，并集 A ∪ B 包含属于 A 或 B 或两者兼有的元素；交集 A ∩ B 包含同时属于两者的元素。关键公式为 n(A ∪ B) = n(A) + n(B) - n(A ∩ B)。在处理三个集合时，仔细的填色和标注方法有助于解决包含“至少一个”或“恰好两个”条件的问题。这些概念是计算复合事件概率的基础。
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