📚 Year 7 Cambridge Advanced Mathematics: Core Knowledge Review | 剑桥七年级进阶数学：核心知识点梳理
Welcome to this comprehensive review of the core topics in Year 7 Cambridge Advanced Mathematics. This guide covers essential concepts that extend beyond the basic curriculum, helping students build a solid foundation for IGCSE and beyond. We will explore number theory, algebra, geometry, data handling, and more, with clear explanations and examples.
欢迎来到这篇关于剑桥七年级进阶数学核心知识点的综合梳理。本指南涵盖了超越基础课程的重要概念，帮助学生为IGCSE及更高阶段的学习奠定坚实基础。我们将深入探讨数论、代数、几何、数据处理等内容，并提供清晰的解释与示例。
1. Number Theory and Properties | 数论与性质
Number theory deals with the properties and relationships of integers. You must be able to identify factors, multiples, prime numbers, and composite numbers. A prime number has exactly two distinct factors (1 and itself), while a composite number has more than two factors.
数论研究整数的性质与关系。你需要能够识别因数、倍数、素数与合数。素数只有两个不同的因数（1和自身），而合数有两个以上的因数。
To find the highest common factor (HCF) of two numbers, list all factors of each and choose the largest common one. The lowest common multiple (LCM) can be found by listing multiples or using prime factorisation with Venn diagrams.
求两个数的最大公因数(HCF)时，列出每个数的所有因数并选取最大的共同因数。最小公倍数(LCM)可以通过列出倍数或利用维恩图进行质因数分解求得。
Prime factorisation expresses a number as a product of prime numbers, e.g., 60 = 2² × 3 × 5. This method helps in simplifying fractions and finding square roots. You should also understand square numbers and cube numbers.
质因数分解将一个数表示为素数的乘积，例如 60 = 2² × 3 × 5。该方法有助于化简分数和求平方根。你还应理解平方数和立方数。
2. Fractions, Decimals, and Percentages | 分数、小数和百分数
Conversions between fractions, decimals, and percentages are essential. A fraction like 3/4 can be written as 0.75 or 75%. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%.
分数、小数和百分数之间的转换至关重要。例如 3/4 可写作 0.75 或 75%。将分数化为小数时，用分子除以分母。将小数化为百分数时，乘以 100%。
Adding and subtracting fractions require a common denominator. Multiply, divide and simplify fractions by cancelling common factors. Recurring decimals can be represented with dot notation, e.g., 0.3̇ for 0.333… .
分数加减需要公分母。乘除分数时可通过约分进行化简。循环小数可用点记号表示，例如 0.3̇ 表示 0.333… 。
Percentage increase and decrease problems involve adding or subtracting a percentage of the original amount. Understanding the relationship between ratios, fractions, and percentages helps solve real-world problems such as discounts and interest.
百分数增减问题涉及在原数值上加上或减去某个百分比。理解比率、分数和百分数之间的关系有助于解决折扣、利息等实际问题。
3. Integers and Operations | 整数与运算
Working with negative numbers is a key skill. Rules for adding, subtracting, multiplying, and dividing integers: same signs give a positive product, different signs give negative. Remember that subtracting a negative number is equivalent to addition: a − (−b) = a + b.
处理负数是关键技能。整数加减乘除的法则：同号得正，异号得负。切记减去一个负数等同于加法：a − (−b) = a + b。
Order of operations (BIDMAS/BODMAS) must be followed: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right). For example, −3² means −(3²) = −9, whereas (−3)² = 9.
必须遵循运算顺序（BIDMAS/BODMAS）：括号、指数、除法/乘法（从左至右）、加法/减法（从左至右）。例如 −3² 表示 −(3²) = −9，而 (−3)² = 9。
Using a number line can help visualise addition and subtraction of integers. You should also be confident with long multiplication and division involving decimals and whole numbers.
使用数轴有助于直观理解整数的加减。你还应熟练掌握涉及小数和整数的长乘法和长除法。
4. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra uses letters to represent unknown values. Terms like 3a, −5b, and a² are combined by collecting like terms: e.g., 3a + 2a = 5a. You cannot combine terms with different variables or powers.
代数用字母表示未知值。诸如 3a、−5b 和 a² 这样的项通过合并同类项组合：例如 3a + 2a = 5a。不能合并变量或指数不同的项。
Expanding brackets involves multiplying the term outside by each term inside: a(b + c) = ab + ac. Be careful with negative signs: −2(x − 3) = −2x + 6. Factorising is the reverse process, writing an expression as a product, e.g., 6x + 9 = 3(2x + 3).
展开括号需要将括号外的项乘以括号内的每一项：a(b + c) = ab + ac。注意负号：−2(x − 3) = −2x + 6。因式分解是逆过程，将表达式写成乘积形式，例如 6x + 9 = 3(2x + 3)。
You will also encounter substitution, where you replace variables with given numbers to evaluate an expression. Always follow BIDMAS when substituting.
你还会遇到代入求值，即用给定的数值替换变量来计算表达式。代入时始终遵循运算顺序。
5. Solving Linear Equations | 解线性方程
Linear equations can be solved by performing inverse operations to isolate the variable. For example, 2x + 3 = 11 → subtract 3 → 2x = 8 → divide by 2 → x = 4. Always check your solution by substituting back.
解线性方程可以通过逆运算将变量分离。例如 2x + 3 = 11 → 减3 → 2x = 8 → 除以2 → x = 4。务必通过回代检验解。
Equations with brackets should be expanded first. Equations with variables on both sides require moving terms: e.g., 3x − 5 = x + 7 → 3x − x = 7 + 5 → 2x = 12 → x = 6.
含括号的方程应先展开。变量在等号两边的方程需要移项：例如 3x − 5 = x + 7 → 3x − x = 7 + 5 → 2x = 12 → x = 6。
Word problems leading to equations are common. Translate statements into algebraic forms: ‘three more than a number’ → x + 3; ‘twice a number decreased by 4’ → 2x − 4.
常见源自文字问题的方程。将语句翻译成代数形式：“比一个数多3” → x + 3；“一个数的两倍减4” → 2x − 4。
6. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. Recognising patterns helps find the next term and the nth term formula. Arithmetic sequences have a constant difference: e.g., 5, 8, 11, 14, … has a common difference of +3.
数列是按规则排列的一列数。识别规律有助于找出下一项和第n项公式。算术数列具有固定的差：例如 5, 8, 11, 14, … 公差为 +3。
The nth term of an arithmetic sequence can be written as a + (n−1)d, where a is the first term and d is the common difference. For the sequence above, the nth term is 5 + (n−1)×3 = 3n + 2.
算术数列的第n项可写作 a + (n−1)d，其中 a 为首项，d 为公差。上述数列的第n项为 5 + (n−1)×3 = 3n + 2。
Geometric sequences multiply by a constant ratio. Special sequences like square numbers (1, 4, 9, 16…) and Fibonacci-type sequences are also explored. You should be able to continue a sequence and generate terms from a given rule.
等比数列乘以一个固定的公比。还会探索平方数（1, 4, 9, 16…）和斐波那契式数列等特殊数列。你应能接续数列并根据给定规则生成项。
7. Geometry: Angles and Lines | 几何：角与线
Angles are measured in degrees. Types of angles: acute (less than 90°), right (90°), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°). Angles on a straight line add up to 180°; angles around a point sum to 360°.
角度以度为单位测量。角的类型：锐角（小于90°）、直角（90°）、钝角（90°到180°之间）、平角（180°）、优角（180°到360°之间）。平角等于180°；周角为360°。
When two lines intersect, vertically opposite angles are equal. Parallel lines create angle relationships: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. Use these to find unknown angles.
两直线相交时，对顶角相等。平行线产生角的关系：同位角相等，内错角相等，同旁内角互补（和为180°）。利用这些性质求未知角。
Calculate angles in triangles: the sum of interior angles is always 180°. Exterior angle of a triangle equals the sum of the two opposite interior angles. In polygons, the sum of interior angles = (n−2)×180°.
计算三角形内角：内角和恒为180°。三角形的外角等于两个不相邻的内角之和。多边形的内角和 = (n−2)×180°。
8. Perimeter, Area, and Volume | 周长、面积与体积
Perimeter is the total distance around a 2D shape. For rectangles, P = 2(l + w); for circles, circumference C = 2πr or πd. Area measures the space inside: rectangle A = lw, triangle A = ½bh, circle A = πr².
周长是二维图形一周的长度。矩形周长 P = 2(l + w)；圆的周长 C = 2πr 或 πd。面积测量内部空间：矩形 A = lw，三角形 A = ½bh，圆 A = πr²。
Area of compound shapes can be found by splitting into known figures. Use π ≈ 3.14 or the π button. For parallelograms, A = base × perpendicular height; for trapeziums, A = ½(a + b)h.
复合形状的面积可通过拆分成已知图形求得。使用 π ≈ 3.14 或 π 键。平行四边形面积 A = 底 × 高；梯形面积 A = ½(a + b)h。
Volume of 3D solids: cuboid V = l × w × h, prism V = area of cross-section × length. Surface area is the total area of all faces. Year 7 may introduce cylinders: volume = πr²h, surface area = 2πr² + 2πrh.
三维立体的体积：长方体 V = l × w × h，棱柱 V = 横截面积 × 长。表面积是所有面的总面积。七年级可能接触圆柱体：体积 = πr²h，表面积 = 2πr² + 2πrh。
9. Ratios and Proportions | 比和比例
Ratios compare quantities. A ratio a:b can be simplified by dividing both parts by their HCF. Equivalent ratios are obtained by multiplying
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