📚 Year 7 CAIE Mathematics: Core Topic Summary | Year 7 CAIE 数学：核心知识点梳理
Year 7 CAIE Mathematics consolidates and extends the skills learned in primary school, introducing more formal reasoning in arithmetic, algebra, geometry, and statistics. This revision guide provides a clear overview of the essential topics, complete with bilingual explanations and worked examples to support understanding.
Year 7 CAIE 数学巩固并拓展了小学阶段所学技能，在算术、代数、几何和统计方面引入了更规范的推理方法。本复习指南提供了核心知识点的清晰概述，并配有双语解释和示例，以帮助理解。
1. Whole Numbers and Operations | 整数与运算
The foundation of Year 7 mathematics rests on a confident handling of whole numbers. We read and write large numbers using place value up to millions. Each position – millions, hundred thousands, ten thousands, thousands, hundreds, tens, and ones – tells us the value of a digit.
Year 7 数学的基础在于对整数的熟练掌握。我们使用百万以内的数位来读写大数。每个数位——百万位、十万位、万位、千位、百位、十位和个位——都表明了数字的值。
Addition and subtraction with whole numbers require careful alignment of place values. We use carrying when a column sum exceeds 9 and borrowing when we need to subtract a larger digit from a smaller one. For example, to add 4567 + 892, we carry from the ones to the tens and later to the hundreds.
整数的加法和减法需要仔细对齐数位。当某一列之和超过 9 时，我们需要进位；当从一个较小的数字减去较大的数字时，则需要借位。例如，计算 4567 + 892 时，我们需要从个位向十位进位，然后再向百位进位。
Multiplication extends these ideas. Long multiplication breaks the multiplier into place values, while multiplying by multiples of 10, 100, or 1000 simply shifts digits to the left. Division is introduced as both sharing and grouping, with methods such as short division and interpreting remainders.
乘法拓展了这些概念。长乘法将乘数按数位拆分，而乘以 10、100 或 1000 的倍数时，只需将数字向左移动相应数位。除法则以“平均分配”和“分组”两种形式引入，具体方法包括短除法以及余数的处理。
Order of operations is essential for multi-step calculations. We use the BODMAS rule: Brackets first, then Orders (powers and roots), then Division and Multiplication from left to right, and finally Addition and Subtraction from left to right.
运算顺序对于多步计算至关重要。我们使用 BODMAS 规则：先算括号，再算阶（乘方和方根），然后从左到右进行乘除运算，最后从左到右进行加减运算。
BODMAS: 2 + 3 × 4 = 2 + 12 = 14
Word problems help apply these operations to everyday contexts, such as calculating totals, differences, and sharing amounts.
应用题帮助将这些运算应用到日常生活中，例如计算总数、差异和分配金额。
2. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a whole number that divides another number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple is the result of multiplying a number by an integer, so the multiples of 3 are 3, 6, 9, 12, and so on.
因数是指能整除另一个整数且没有余数的整数。例如，12 的因数有 1、2、3、4、6 和 12。倍数是指一个数乘以某个整数得到的结果，因此 3 的倍数有 3、6、9、12 等等。
Prime numbers have exactly two distinct factors, 1 and the number itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Composite numbers have more than two factors. The number 1 is neither prime nor composite.
质数恰好有两个不同的因数：1 和它本身。最初的几个质数有 2、3、5、7、11、13、17 和 19。合数有两个以上的因数。数字 1 既不是质数也不是合数。
We can express any composite number as a product of prime factors by drawing a factor tree. This prime factorisation helps in finding the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two or more numbers.
我们可以通过画出因子树，将任何一个合数表示为质因数的乘积。这种质因数分解有助于求出两个或更多个数的最大公因数 (HCF) 和最小公倍数 (LCM)。
Prime factorisation of 24 = 2³ × 3
The HCF is the largest factor shared by the numbers, while the LCM is the smallest multiple they share. Listing factors and multiples, or using Venn diagrams with prime factors, are common methods.
HCF 是几个数共有的最大因数，而 LCM 是它们共有的最小倍数。列举因数和倍数，或使用带有质因数的文氏图，都是常见的方法。
3. Negative Numbers | 负数
Negative numbers are numbers less than zero. They are written with a minus sign, such as −3, and are found on the left side of zero on a number line. Everyday examples include temperatures below freezing, floors below ground level, or debts.
负数是小于零的数。它们用减号书写，如 −3，位于数轴上零的左侧。常见的例子有零度以下的温度、地下的楼层或负债。
Comparing negative numbers can be tricky: a smaller number means one further to the left on the number line, so −7 is less than −2. To order a mix of positive and negative numbers, we place them on a number line in ascending order from left to right.
比较负数时容易混淆：较小的数在数轴上更靠左，因此 −7 小于 −2。要对包含正数和负数的数列进行排序，我们可以将它们放在数轴上，从左到右按升序排列。
Adding and subtracting with negatives follows clear rules. Adding a positive moves right on the number line; adding a negative moves left. Subtracting a negative is the same as adding its opposite. For example, 5 − (−3) = 5 + 3 = 8.
负数的加减法遵循明确的规则。加上一个正数沿数轴向右移动；加上一个负数向左移动。减去一个负数等同于加上它的相反数。例如，5 − (−3) = 5 + 3 = 8。
Multiplication and division with negatives also follows a simple sign rule: if the signs are the same, the result is positive; if the signs are different, the result is negative.
负数的乘除法也遵循简单的符号规则：同号得正，异号得负。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions represent parts of a whole. Equivalent fractions look different but have the same value, such as ½ and 2/4. We simplify fractions by dividing the numerator and denominator by their HCF. Improper fractions (where the numerator is larger) can be converted to mixed numbers, like 7/3 = 2⅓.
分数表示整体的一部分。等值分数形式不同但数值相同，如 ½ 和 2/4。我们通过将分子和分母除以它们的最大公因数来化简分数。假分数（分子大于分母）可以转换为带分数，如 7/3 = 2⅓。
Adding and subtracting fractions require a common denominator. Once the fractions have the same denominator, we add or subtract the numerators and keep the denominator unchanged. For example, ¼ + ⅓ = 3/12 + 4/12 = 7/12.
分数的加减法要求分母相同。一旦分数有了相同的分母，我们只需将分子相加或相减，分母保持不变。例如，¼ + ⅓ = 3/12 + 4/12 = 7/12。
Decimals are another way to write fractions with denominators 10, 100, 1000, and so on. We compare decimals by looking at each place value from left to right. To convert a fraction to a decimal, divide the numerator by the denominator.
小数是以 10、100、1000 等为分母的分数的另一种书写形式。我们通过从左到右比较每个数位来比较小数的大小。要将分数转换为小数，用分子除以分母即可。
Percentages mean ‘out of 100’. To convert a decimal to a percentage, multiply by 100; to convert a percentage to a decimal, divide by 100. Finding a percentage of a quantity is a key skill, often done by finding 1% or 10% first.
百分数表示“每一百份”。要将小数转换为百分数，乘以 100；将百分数转换为小数，除以 100。求一个数的百分之几是一项关键技能，通常先求出 1% 或 10% 的值再推算。
5. Introduction to Algebra | 代数入门
Algebra uses letters to stand for unknown numbers or variables. An algebraic expression contains numbers, letters, and operation symbols, such as 3a + 2. We call 3a a term; 3 is the coefficient, and ‘a’ is the variable.
代数使用字母来代表未知数或变量。代数表达式包含数字、字母和运算符号，如 3a + 2。我们称 3a 为一个项，其中 3 是系数，’a’ 是变量。
We can simplify expressions by collecting like terms – terms that have exactly the same variable part. For example, 2a + 3a simplifies to 5a, while 2a + 3b cannot be combined further.
我们可以通过合并同类项来化简表达式——同类项指变量部分完全相同的项。例如，2a + 3a 可化简为 5a，而 2a + 3b 则不能进一步合并。
Writing expressions from word problems is an important skill. The phrase ‘5 more than a number’ can be written as n + 5, and ‘twice a number’ as 2n. Substitution means replacing the letter with a given number and working out the value of the expression.
根据文字描述写出表达式是一项重要技能。“一个数加上 5”可写成 n + 5，“一个数的两倍”写成 2n。代入求值是指用给定的数替换字母，并计算出表达式的值。
If n = 4, then 3n + 2 = 3 × 4 + 2 = 14
6. Solving Simple Equations | 简单方程求解
An equation shows that two expressions are equal, using an equals sign. Solving an equation means finding the value of the unknown that makes the statement true. We use inverse operations to undo what has been done to the unknown.
方程通过等号表示两个表达式相等。解方程就是求出使等式成立的未知数的值。我们使用逆运算来抵消对未知数所做的运算。
For a one-step equation like x + 5 = 12, we subtract 5 from both sides to find x = 7. For 3x = 18, we divide both sides by 3 to get x = 6. The balance method reminds us that whatever we do to one side, we must do to the other.
对于像 x + 5 = 12 这样的一步方程，我们将两边同时减去 5，得到 x = 7。对于 3x = 18，两边同时除以 3，得到 x = 6。天秤法提醒我们，对等式一边所做的任何操作，都必须同样地施加在另一边。
Two-step equations involve two operations. For example, to solve 2x + 3 = 11, we first subtract 3 and then divide by 2, giving x = 4. Setting out the work clearly, one step per line, helps avoid mistakes.
两步方程包含两种运算。例如，要解 2x + 3 = 11，我们先减去 3，然后再除以 2，得到 x = 4。书写清晰，每行只写一步，有助于避免错误。
7. Sequences and Patterns | 数列与规律
A sequence is a list of numbers that follow a rule. The term-to-term rule tells us how to move from one term to the next. For example, in the sequence 5, 9, 13, 17, … , the rule is ‘add 4’.
数列是按一定规则排列的一列数。项到项规则告诉了我们如何从一项得到下一项。例如，在数列 5, 9, 13, 17, … 中，规则是“加 4”。
We also look at patterns in shapes, such as number of matchsticks used to build a pattern of squares. Describing the pattern first in words helps us later to write an algebraic rule, often called the nth term rule. For Year 7, we focus on recognising and continuing linear sequences.
我们还会观察图形中的规律，例如搭建一排正方形所需的火柴棍数量。先用语言描述规律有助于我们后续写出代数规则，这通常被称为第 n 项规则。在 Year 7 阶段，我们重点在于识别并延续线性数列。
Generating a sequence from a given rule, such as ‘start at 3 and add 5 each time’, produces 3, 8, 13, 18, … . We also learn to find missing terms by working backwards using the inverse of the term-to-term rule.
根据给定的规则生成数列，例如“从 3 开始，每次加 5”，得到的数列为 3, 8, 13, 18, … 。我们还学习利用项到项规则的逆运算，通过反向推导找出缺失的项。
8. Angles and Lines | 角与线
An angle measures the amount of turn between two lines that meet at a vertex. We measure angles in degrees (°). A full turn is 360°, a straight line is 180°, and a right angle is exactly 90°.
角度衡量的是两条线在顶点相交处的旋转量。我们以度 (°) 为单位来测量角度。一个完整的周角是 360°，一条直线是 180°，一个直角正好是 90°。
Angles are classified by their size: acute (less than 90°), right angle (exactly 90°), obtuse (between 90° and 180°), and reflex (greater than 180°). We use a protractor to draw and measure angles accurately.
根据大小，角可以分为几类：锐角（小于 90°）、直角（恰好 90°）、钝角（大于 90° 且小于 180°）和优角（大于 180°）。我们使用量角器来精确地画角和测角。
| Angle type | Size in degrees | Example |
|---|---|---|
| Acute | < 90° | 45° |
| Right | = 90° | corner of a square |
| Obtuse | > 90° and < 180° | 120° |
| Reflex | > 180° | 210° |
Important angle facts include: angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal. These facts help us to find missing angles without measuring.
重要的角度事实包括：直线上的角之和为 180°，围绕一点的周角之和为 360°，对顶角相等。这些事实能帮助我们不用测量就能求出未知角的度数。
9. Shapes, Symmetry and Triangles | 图形、对称与三角形
Two-dimensional shapes are classified by their properties. Triangles are named by their sides: equilateral (3 equal sides and 3 equal 60° angles), isosceles (2 equal sides and 2 equal base angles), and scalene (no equal sides). Triangles can also be classified by their largest angle: acute-angled, right-angled, or obtuse-angled.
二维图形依据其性质进行分类。三角形根据边长命名：等边三角形（三条边相等，三个角均为 60°）、等腰三角形（两条边相等，两个底角相等）和不等边三角形（三边均不相等）。三角形也可根据其最大角来分类：锐角三角形、直角三角形或钝角三角形。
Quadrilaterals are four-sided polygons. A square has four equal sides and four right angles. A rectangle has opposite sides equal and four right angles. A rhombus has four equal sides but angles that are not necessarily 90°. A parallelogram has both pairs of opposite sides parallel. A trapezium has exactly one pair of parallel sides.
四边形是有四条边的多边形。正方形有四条相等的边和四个直角。长方形对边相等且四个角均为直角。菱形有四条相等的边，但角不一定为 90°。平行四边形有两组对边分别平行。梯形只有一对边平行。
Symmetry is an important geometric property. A shape has line symmetry if it can be folded along a line so that one half fits exactly onto the other. The number of lines of symmetry can be 0, 1, 2, 3, or more. Rotational symmetry tells us how many times a shape looks exactly the same during a full turn – the order of rotational symmetry.
对称是一项重要的几何性质。如果一个图形沿某条线对折后，一半能完全与另一半重合，则该图形具有线对称。对称轴的数量可以是 0、1、2、3 或更多。旋转对称则描述了在一个完整的旋转过程中，图形与其自身重合的次数，即旋转对称的阶。
10. Perimeter and Area | 周长与面积
Perimeter is the total distance around the outside of a shape. For a rectangle, the perimeter is found by adding the lengths of all four sides, or by using the formula P = 2(l + w) where l is length and w is width. Compound shapes made of rectangles can be tackled by carefully adding all outer side lengths.
周长是图形外部一周的总长度。对于长方形，周长可以是将四条边长相加，或者使用公式 P = 2(长 + 宽)，其中 l 为长，w 为宽。由长方形组成的复合图形，只需仔细将所有的外边长度相加即可。
Area measures the surface space inside a shape, recorded in square units such as cm² or m². The area of a rectangle is length × width: A = l × w. To find the area of compound shapes, we often split them into smaller rectangles, calculate each area, and add them together.
面积度量的是图形内部的表面空间，单位为平方厘米 (cm²) 或平方米 (m²) 等。长方形的面积等于长乘宽：A = l × w。要计算复合图形的面积，我们经常将其分割成
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