📚 Year 7 SQA Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 SQA 数学：公式定理速查手册
This handbook provides a concise summary of the essential formulas and theorems you need in Year 7 SQA Mathematics. Use it for quick revision, homework help, or test preparation. Each topic is explained in clear English and Chinese so that you can fully grasp the concepts and apply them with confidence.
本手册浓缩了 Year 7 SQA 数学必备的公式与定理，可用于快速复习、辅助作业或备考。每个主题都以清晰的英文和中文阐释，帮助你透彻理解概念并自信运用。
1. Number Types and Place Value | 数的类型与位值
A digit’s value depends on its position in a number. For example, in 3,467, the digit 3 stands for 3 thousands, 4 for 4 hundreds, 6 for 6 tens, and 7 for 7 ones. Understanding place value helps when comparing and ordering numbers.
数字的大小取决于它在一个数中所处的位置。例如在数字 3,467 中，3 表示 3 个千，4 表示 4 个百，6 表示 6 个十，7 表示 7 个一。掌握位值有助于比较和排序数字。
Integers are whole numbers that can be positive, negative, or zero. A negative integer is always less than a positive one. Zero is neither positive nor negative. We use the symbols < and > to compare numbers.
整数是正整数、负整数和零的总称。负数永远小于正数，零既不是正数也不是负数。我们用 < 和 > 符号来比较数字的大小。
2. Operations with Whole Numbers | 整数的四则运算
Addition and subtraction are inverse operations. When adding, align digits by place value and carry over if a column sum reaches 10 or more. For subtraction, borrow from the next column if needed.
加法和减法互为逆运算。做加法时，将相同数位对齐，满十进一；做减法时，如果不够减就从前一位借一当十。
Multiplication can be seen as repeated addition. For instance, 6 × 4 means adding 6 four times. The standard algorithm multiplies each digit and adds the partial products, paying attention to place value shifts.
乘法可以看作重复加法，例如 6 × 4 表示 4 个 6 相加。笔算乘法时逐位相乘，将部分积对齐相加，并注意数位的变化。
Division is sharing into equal groups. The relationship is: dividend = divisor × quotient + remainder, where the remainder must be smaller than the divisor. For example, 38 ÷ 5 gives quotient 7 and remainder 3.
除法是平均分配的过程。基本关系为：被除数 = 除数 × 商 + 余数，且余数必须小于除数。例如 38 ÷ 5 得到商 7 余 3。
3. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent fractions represent the same amount. You can create equivalent fractions by multiplying or dividing the numerator and denominator by the same number. Simplifying a fraction means writing it in its lowest terms.
等值分数表示相同的量。将分子和分母同时乘或除以同一个非零数，可以得到等值分数。化简分数就是把它写成最简形式。
To add or subtract fractions, they must have the same denominator. Find the lowest common denominator (LCD), adjust the numerators, then add or subtract and simplify if possible.
异分母分数相加减，需要先通分，找到最小公分母，再调整分子，最后相加减并化简。
A decimal point separates the whole part from the fractional part. To convert a fraction to a decimal, divide the numerator by the denominator. Common decimals: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75.
小数点把整数部分和小数部分分开。分数化小数，用分子除以分母即可。常用转换：1/2 = 0.5，1/4 = 0.25，3/4 = 0.75。
‘Percent’ means ‘per hundred’. To change a fraction or decimal to a percentage, multiply by 100%. For example, 0.6 = 60% and 3/5 = 60%.
“百分数”表示”每一百中的几”。将小数或分数化为百分数，乘以 100% 即可。例如 0.6 = 60%，3/5 = 60%。
4. Order of Operations (BODMAS/BIDMAS) | 运算法则（BODMAS / BIDMAS）
When a calculation involves more than one operation, we must follow a specific order. BODMAS tells us the sequence: Brackets first, then Orders (powers or roots), then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
当一道算式中包含多种运算时，必须按照规定的顺序计算。BODMAS 法则说明了计算顺序：先算括号，接着是指数或根号，再算乘除（从左到右），最后算加减（从左到右）。
For example, in 3 + 2 × (8 – 5), you solve inside the brackets first: 8 – 5 = 3. Then multiply: 2 × 3 = 6. Finally add: 3 + 6 = 9.
例如算式 3 + 2 × (8 – 5)，先算括号内：8 – 5 = 3；再算乘法：2 × 3 = 6；最后算加法：3 + 6 = 9。
5. Algebraic Expressions and Simplifying | 代数表达式与化简
In algebra, letters (variables) stand for unknown numbers. A term is a number, a variable, or the product of numbers and variables. Like terms have the same variable raised to the same power; they can be combined by adding or subtracting their coefficients.
在代数中，字母（变量）表示未知数。项指数、字母或数与字母的乘积。同类项指所含字母及其指数完全相同的项，可以通过加减它们的系数来合并。
For example, 3a + 5a simplifies to 8a, but 3a + 5b cannot be simplified further. Always follow the rules: only like terms combine.
例如，3a + 5a 可以化简为 8a，但 3a + 5b 无法合并。记住：只有同类项才能相加减。
When multiplying terms, multiply the coefficients and then multiply the variables, e.g., 4x × 2y = 8xy. Brackets are expanded using the distributive law: a(b + c) = ab + ac.
项与项相乘时，系数相乘，变量相乘，例如 4x × 2y = 8xy。去括号使用分配律：a(b + c) = ab + ac。
6. Solving Simple Equations | 解简单方程
An equation shows that two expressions are equal. To solve an equation means to find the unknown value that makes the statement true. Use inverse operations to isolate the variable.
方程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。使用逆运算把变量孤立出来。
Example: x + 7 = 12. Subtract 7 from both sides: x = 5. Always check your answer by substituting it back into the original equation.
例：x + 7 = 12，两边同时减去 7 得到 x = 5。记得将答案代回原方程检验。
For equations like 3y = 18, divide both sides by 3 to get y = 6. If the variable appears on both sides, collect like terms first.
像 3y = 18 这样的方程，两边同时除以 3 得到 y = 6。如果未知数出现在两边，先合并同类项。
7. Angles and Their Properties | 角及其性质
An angle measures the turn between two lines meeting at a vertex. We measure angles in degrees (°). Acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and straight angles are exactly 180°.
角度量两条射线在顶点处的张开程度，单位为度 (°)。锐角小于 90°，直角等于 90°，钝角介于 90° 和 180° 之间，平角等于 180°。
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles (when two lines cross) are equal.
直线上的角之和为 180°。绕一点的所有角之和为 360°。对顶角（两条直线相交时）相等。
Complementary angles add to 90°, supplementary angles add to 180°. For example, 30° and 60° are complementary; 110° and 70° are supplementary.
互余角之和为 90°，互补角之和为 180°。例如 30° 和 60° 互余，110° 和 70° 互补。
8. Triangles and Their Properties | 三角形及其性质
The sum of the interior angles of any triangle is always 180°. You can use this fact to find a missing angle when the other two are known.
任何三角形的内角和恒为 180°。利用这一性质，已知两个角就可以求出第三个角。
Triangles are classified by their angles: an acute triangle has all angles less than 90°, a right triangle has one 90° angle, and an obtuse triangle has one angle greater than 90°. They can also be classified by sides: equilateral (all sides equal), isosceles (two sides equal), and scalene (all sides different).
三角形按角分类：锐角三角形三个角都小于 90°，直角三角形有一个 90° 角，钝角三角形有一个大于 90° 的角。按边分类：等边三角形三边相等，等腰三角形有两边相等，不等边三角形三边都不等。
In an isosceles triangle, the base angles are equal. In an equilateral triangle, all angles are 60°.
等腰三角形中，两底角相等；等边三角形中，每个角都是 60°。
9. Perimeter and Area of Common Shapes | 常见图形的周长与面积
Perimeter is the total distance around a shape. For a rectangle, perimeter P = 2(l + w), where l is length and w is width. For a square, P = 4s, where s is the side length.
周长是一个图形所有边的总长。长方形周长公式为 P = 2(l + w)，其中 l 为长，w 为宽。正方形周长为 P = 4s，s 为边长。
Area measures the surface inside a shape, in square units. Rectangle area A = l × w. Square area A = s². Triangle area A = ½ × base × height.
面积衡量图形内部的大小，单位为平方单位。长方形面积 A = l × w，正方形面积 A = s²，三角形面积 A = ½ × 底 × 高。
For a parallelogram, area = base × perpendicular height. Remember to use the vertical height, not the slant height.
平行四边形的面积 = 底 × 垂直高。注意要用垂直高，而不是斜边长。
10. Symmetry and Transformations | 对称与变换
A shape has line symmetry if it can be folded along a line so that the two halves match exactly. That fold line is called a line of symmetry. A square has four lines of symmetry, a rectangle has two, an equilateral triangle has three.
如果一个图形能沿某条直线对折后完全重合，就说它有线对称性。那条折痕叫做对称轴。正方形有 4 条对称轴，长方形有 2 条，等边三角形有 3 条。
Reflection flips a shape over a mirror line. All points stay the same distance from the mirror line. Rotation turns a shape around a fixed point, called the centre of rotation, through a given angle. Translation slides a shape without turning or flipping it.
反射是把图形沿镜线翻折，所有点到镜线的距离保持不变。旋转是让图形绕一个固定点（旋转中心）转动给定角度。平移是只移动图形的位置，方向与大小都不变。
11. Data Handling and Averages | 数据处理与平均数
The mean (average) is calculated by adding all values and dividing by the number of values. The median is the middle value when data are ordered; if there are two middle numbers, take their mean. The mode is the value that occurs most often. The range is the difference between the largest and smallest values, a measure of spread.
平均数（均值）的计算为：所有数据之和 ÷ 数据个数。中位数是将数据排序后位于中间的数，如果中间有两个数，则取它们的平均值。众数是出现频率最高的数。极差是最大值与最小值的差，用于衡量数据的离散程度。
Example: Data set 3, 7, 2, 9, 7. Mean = (3+7+2+9+7)÷5 = 28÷5 = 5.6. Order: 2, 3, 7, 7, 9; median = 7; mode = 7; range = 9 – 2 = 7.
例如数据集 3, 7, 2, 9, 7。平均数：(3+7+2+9+7)÷5 = 28÷5 = 5.6。排序后为 2, 3, 7, 7, 9，中位数 = 7，众数 = 7，极差 = 9 – 2 = 7。
12. Interpreting Graphs and Charts | 图表解读
Bar charts use bars to show frequencies for categories. The height of each bar represents the value. Always read the scale carefully. Pictograms use symbols to represent quantities; a key tells you how many each symbol represents.
条形图用条柱的高度来表示各类别的频数。阅读时务必注意纵轴的刻度。象形图用图形符号表示数量，图例会标出一个符号代表多少个单位。
Line graphs show how data changes over time. Points are plotted and connected with straight lines. Pie charts display proportions: the whole circle (360°) represents 100%, and each sector’s angle is calculated as (category ÷ total) × 360°.
折线图用于展示数据随时间变化的趋势，将数据点用直线连接起来。饼图用于显示各项所占比例：整个圆（360°）代表 100%，每个扇形的角度 = (类别数值 ÷ 总数) × 360°。
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