Year 7 CCEA Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 CCEA 数学：公式定理速查手册

📚 Year 7 CCEA Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 CCEA 数学：公式定理速查手册

This handbook brings together the essential formulas, rules and theorems you will encounter in Year 7 CCEA Mathematics. Use it as a quick reference to reinforce your understanding, check your working and build confidence with number, algebra, shape, data and measurement topics. Each section gives the key idea in plain English, followed by the same explanation in Chinese, so you can study in the language that suits you best.

本手册汇集了 Year 7 CCEA 数学课程中将会遇到的基本公式、法则和定理。你可以把它当作快速参考，用来巩固理解、检查解题过程，并在数字、代数、图形、数据和测量等主题上建立信心。每个部分先用简单的英文给出核心概念，然后提供相应的中文解释，方便你以最适合自己的语言进行学习。

1. Number Operations | 数字运算

Addition, subtraction, multiplication and division are the four building blocks of arithmetic. The sum of a and b is a + b; the difference is a − b. Multiplication can be written as a × b, a · b or simply ab, while division is written as a ÷ b or a/b. The result of a multiplication is called the product, and the result of a division is called the quotient.

加法、减法、乘法和除法是算术的四大基石。a 与 b 的和为 a + b；差为 a − b。乘法可以写作 a × b、a · b 或简写为 ab，除法则写作 a ÷ b 或 a/b。乘法的结果叫做积，除法的结果叫做商。

When adding or subtracting negative numbers, remember the sign rules: a + (−b) = a − b, and a − (−b) = a + b. On a number line, adding a negative means moving left; subtracting a negative means moving right.

在对负数进行加减运算时，牢记符号法则：a + (−b) = a − b，而 a − (−b) = a + b。在数轴上，加一个负数意味着向左移动；减一个负数意味着向右移动。

2. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)

To avoid confusion in calculations with more than one operation, follow the BIDMAS rule: Brackets, Indices (powers), Division and Multiplication (working from left to right), Addition and Subtraction (left to right). The order ensures that everyone interprets an expression in the same way.

为避免含有多个运算的算式出现歧义，请遵循 BIDMAS 法则：先算括号（Brackets），再算指数（Indices，即幂），然后从左到右计算乘除（Division and Multiplication），最后从左到右计算加减（Addition and Subtraction）。该顺序确保所有人对同一个表达式有相同的理解。

Example: 3 + 4 × 2 = 3 + 8 = 11, not 7 × 2 = 14. With brackets: (3 + 4) × 2 = 7 × 2 = 14.

例如：3 + 4 × 2 = 3 + 8 = 11，而不是 7 × 2 = 14。若加上括号：(3 + 4) × 2 = 7 × 2 = 14。

3. Factors, Multiples and Primes | 因数、倍数与质数

A factor of a number divides it exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple of a number is the product of that number and any whole number: the multiples of 5 are 5, 10, 15, 20, …

一个数的因数是指能整除该数而没有余数的数。例如，12 的因数有 1、2、3、4、6 和 12。一个数的倍数是指该数与任意整数的乘积：5 的倍数有 5、10、15、20……

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are the first few primes. The only even prime is 2. Every whole number greater than 1 can be written as a unique product of prime factors – this is the Fundamental Theorem of Arithmetic.

质数是大于 1 且只有两个因数（1 和它自身）的整数。2、3、5、7、11 和 13 是最前面的几个质数。唯一的偶质数是 2。每个大于 1 的整数都可以唯一地写成质因数的乘积——这就是算术基本定理。

The highest common factor (HCF) of two numbers is the largest factor they share. The lowest common multiple (LCM) is the smallest multiple they share. These are used in fraction arithmetic.

两个数的最大公因数（HCF）是它们共有的最大因数。最小公倍数（LCM）是它们共有的最小倍数。这在分数运算中会用到。

4. Fractions, Decimals and Percentages | 分数、小数与百分比

A fraction a/b represents a ÷ b. The top number a is the numerator, the bottom b is the denominator. Equivalent fractions are found by multiplying or dividing numerator and denominator by the same number: 1/2 = 2/4 = 3/6.

分数 a/b 表示 a ÷ b。上面的数字 a 是分子，下面的 b 是分母。将分子和分母同时乘或除以同一个非零数，可得到等值分数：1/2 = 2/4 = 3/6。

To add or subtract fractions, first find a common denominator. For 1/3 + 1/4, the LCM of 3 and 4 is 12, so rewrite as 4/12 + 3/12 = 7/12. To multiply fractions, multiply numerators and denominators: 2/3 × 3/5 = 6/15 = 2/5. To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.

加减分数时，首先要找到公分母。计算 1/3 + 1/4，3 和 4 的最小公倍数是 12，因此改写为 4/12 + 3/12 = 7/12。分数相乘，分子乘分子、分母乘分母：2/3 × 3/5 = 6/15 = 2/5。除以一个分数，等于乘它的倒数：a/b ÷ c/d = a/b × d/c。

A decimal is another way of writing a fraction whose denominator is a power of 10. The percentage sign % means ‘out of 100’. To convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply by 100%: 3/5 = 60%.

小数是分母为 10 的幂的另一种分数写法。百分号 % 表示“每百分之一”。将分数转化为百分比，可以找出分母为 100 的等值分数，或乘以 100%：3/5 = 60%。

5. Ratio and Proportion | 比和比例

A ratio compares the sizes of two or more quantities. It is written as a : b and read as ‘a to b’. Ratios can be simplified by dividing both sides by the same number, just like fractions.

比用来比较两个或多个量的大小，写作 a : b，读作“a 比 b”。比可以通过两边同除以一个数来化简，就像分数一样。

When a ratio is written in the form 1 : n or n : 1, it is in its simplest unit form. The quantities in a ratio must be measured in the same units.

当比写成 1 : n 或 n : 1 的形式时，就是最简单的单位比。比中的各个量必须用相同的单位来衡量。

Direct proportion means that as one quantity increases, the other increases at the same rate. If y is directly proportional to x, then y = kx, where k is the constant of proportionality.

正比例意味着当一个量增加时，另一个量以相同的比率增加。如果 y 与 x 成正比，那么 y = kx，其中 k 是比例常数。

6. Algebra: Expressions and Equations | 代数：表达式与方程

In algebra, letters stand for unknown numbers. An expression such as 3x + 2 combines numbers and letters using operations. Like terms contain exactly the same letters and powers: 2x and 5x are like terms, whereas 3x and 3x² are not.

在代数中，字母代表未知数。像 3x + 2 这样的代数式通过运算把数字和字母组合在一起。同类项包含完全相同的字母和指数：2x 和 5x 是同类项，而 3x 和 3x² 则不是。

To simplify, collect like terms: 4a + 3b − 2a + b = 2a + 4b. Expanding brackets uses the distributive law: a(b + c) = ab + ac. Factorising is the reverse: taking a common factor out, e.g. 6x + 9 = 3(2x + 3).

化简代数式时要合并同类项：4a + 3b − 2a + b = 2a + 4b。展开括号用到分配律：a(b + c) = ab + ac。因式分解是其逆过程：把公因式提取出来，如 6x + 9 = 3(2x + 3)。

An equation states that two expressions are equal. To solve an equation, perform the same operation on both sides to isolate the variable. For 2x + 5 = 13, subtract 5 from both sides: 2x = 8, then divide by 2: x = 4.

方程表示两个代数式相等。解方程时，需要在等式两边进行相同的运算以分离出变量。对于 2x + 5 = 13，两边都减去 5 得 2x = 8，再除以 2 得 x = 4。

7. Geometry: Angles and Lines | 几何：角与线

An angle measures the turn between two rays that meet at a vertex. Angles are measured in degrees (°). A full turn is 360°, a straight line is 180°, and a right angle is 90°.

角用来度量从同一个顶点引出的两条射线之间的转动程度，单位是度（°）。一整圈是 360°，一条直线是 180°，直角是 90°。

Angles on a straight line add up to 180°. / 直线上的角之和为 180°。
Angles around a point add up to 360°. / 围绕一个点的角之和为 360°。
Vertically opposite angles are equal. / 对顶角相等。
Alternate angles on parallel lines are equal; corresponding angles are equal; allied (co‑interior) angles add up to 180°. / 平行线中的内错角相等；同位角相等；同旁内角之和为 180°。

In a triangle, the three interior angles always sum to 180°. An equilateral triangle has three equal 60° angles; an isosceles triangle has two equal base angles.

在三角形中，三个内角之和总为 180°。等边三角形的三个角均等于 60°；等腰三角形的两个底角相等。

8. Perimeter, Area and Volume | 周长、面积与体积

Perimeter is the total distance around the outside of a shape. For a rectangle with length l and width w, P = 2l + 2w. For a regular polygon with n sides of length s, P = n × s.

周长是图形外边界的总长度。对长为 l、宽为 w 的矩形，P = 2l + 2w。对边长为 s 的正 n 边形，P = n × s。

Area measures the surface inside a shape. The area of a rectangle is A = l × w. The area of a triangle is A = ½ × base × height. The area of a parallelogram is A = base × height, and the area of a trapezium is A = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.

面积度量图形内部的表面大小。矩形面积 A = l × w。三角形面积 A = ½ × 底 × 高。平行四边形面积 A = 底 × 高；梯形面积 A = ½ × (a + b) × h，其中 a 和 b 是两条平行边，h 是它们之间的垂直距离。

Volume is the amount of space a solid object occupies. The volume of a cuboid is V = l × w × h. Volume is measured in cubic units such as cm³ and m³.

体积是立体物体所占空间的大小。长方体的体积 V = l × w × h。体积的单位是立方单位，如 cm³ 和 m³。

9. Coordinates and Graphs | 坐标与图表

A coordinate pair (x, y) gives the position of a point on a grid. The x‑coordinate comes first and tells you how far to move horizontally from the origin (0,0); the y‑coordinate tells you how far to move vertically.

坐标对 (x, y) 给出了点在网格上的位置。x 坐标在前，表示从原点 (0,0) 沿水平方向移动的距离；y 坐标表示沿垂直方向移动的距离。

The midpoint of two points (x₁, y₁) and (x₂, y₂) is found by averaging the coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

两点 (x₁, y₁) 和 (x₂, y₂) 的中点通过计算坐标的平均值得出：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

A graph of a linear relationship is a straight line. The equation of a straight line is often written as y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).

线性关系的图像是一条直线。直线方程通常写作 y = mx + c，其中 m 是斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴交点的 y 坐标）。

10. Averages and Data | 平均数与数据

The mean of a set of numbers is found by adding all the values and dividing by the number of values: mean = sum of values ÷ number of values. The median is the middle value when the numbers are arranged in order. The mode is the value that appears most often. The range is the difference between the largest and smallest values: range = maximum − minimum.

一组数据的平均数（均值）等于所有数值之和除以数值的个数。将数据按大小顺序排列后，位于中间位置的值叫做中位数。众数是出现次数最多的数值。极差是最大值与最小值之差：极差 = 最大值 − 最小值。

Data can be displayed in bar charts, pictograms, pie charts and line graphs. A pie chart uses a circle divided into sectors where the angle of each sector represents the proportion of that category: sector angle = (category frequency ÷ total frequency) × 360°.

数据可以用条形图、象形图、饼图和折线图来呈现。饼图是将一个圆划分成若干扇形，每个扇形的圆心角代表该类别所占的比例：扇形角 = (类别频数 ÷ 总数) × 360°。

11. Transformations | 图形变换

A reflection flips a shape over a mirror line to produce a mirror image. The shape remains congruent (same size and shape). A rotation turns a shape around a centre by a given angle and direction (clockwise or anticlockwise). A translation slides a shape by a vector: moving every point the same distance in the same direction. An enlargement changes the size of a shape by a scale factor from a centre of enlargement; if the scale factor is greater than 1 the shape gets bigger, if between 0 and 1 it gets smaller.

反射是指图形沿着一条镜面线翻转，得到镜像图形。反射后的图形与原图全等（大小形状相同）。旋转是指图形绕一个旋转中心按给定的角度和方向（顺时针或逆时针）转动。平移是指图形通过一个平移向量移动：所有点都沿相同的方向和距离滑动。放缩是指从一个放缩中心出发，按一个比例因子改变图形的大小；比例因子大于 1 时图形变大，介于 0 和 1 之间时图形变小。

12. Units of Measurement | 测量单位

Metric units for length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. For mass: 1 tonne = 1000 kg, 1 kg = 1000 g. For capacity: 1 litre = 1000 ml, 1 cm³ = 1 ml.

公制长度单位：1 km = 1000 m，1 m = 100 cm，1 cm = 10 mm。质量单位：1 吨 = 1000 kg，1 kg = 1000 g。容量单位：1 升 = 1000 ml，1 cm³ = 1 ml。

Time units: 1 hour = 60 minutes, 1 minute = 60 seconds. When converting, multiply to change from larger to smaller units, and divide to change from smaller to larger units.

时间单位：1 小时 = 60 分钟，1 分钟 = 60 秒。换算时，从较大单位转换为较小单位用乘法，从较小单位转换为较大单位用除法。

Perimeter and area units: length in m gives perimeter in m and area in m². When finding the area of a composite shape, split it into rectangles and triangles, find each area separately, then add or subtract as needed.

周长与面积单位：若长度单位是米，周长的单位就是米，面积的单位就是平方米。计算组合图形的面积时，可将其分割为矩形和三角形，分别求出面积后再按需要进行加减。

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Year 7 CCEA Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 CCEA 数学：公式定理速查手册