📚 Year 7 SQA Advanced Mathematics: Quick-Reference Formula and Theorem Handbook | Year 7 SQA 进阶数学：公式定理速查手册
This quick-reference handbook gathers the essential formulae, theorems, and key facts that every Year 7 student needs for SQA Advanced Mathematics. Use it to revise topics from number operations and algebra through geometry and data handling. Each section pairs an English explanation with its Chinese translation, ensuring clarity for bilingual learners.
这本速查手册汇集了每位 Year 7 学生在 SQA 进阶数学中需要掌握的基本公式、定理和关键知识点。从数的运算、代数到几何与数据处理，均可用于复习。每个小节都将英文解释与中文翻译配对，确保双语学习者都能清晰理解。
1. Order of Operations (BIDMAS) | 运算顺序
BIDMAS tells us the order in which to calculate parts of a mathematical expression: Brackets, Indices (powers and roots), Division, Multiplication, Addition, Subtraction.
BIDMAS 告诉我们计算数学表达式各部分时应遵循的顺序：括号、指数（幂和根）、除法、乘法、加法、减法。
Always begin by simplifying anything inside brackets. Next, evaluate indices, then perform division and multiplication as they appear from left to right. Finally, carry out addition and subtraction from left to right.
始终从化简括号内的内容开始。接着计算指数，然后按从左到右的顺序进行除法和乘法。最后按从左到右的顺序进行加法和减法。
Division and multiplication share the same priority; the same is true for addition and subtraction. A common mistake is to perform addition before subtraction simply because it appears first in the acronym.
除法和乘法具有相同的优先级；加法和减法也是如此。一个常见错误是仅仅因为缩写词中加法出现在减法之前就先做加法。
Example: (3 + 2²) × 5 ÷ 2 = (3 + 4) × 5 ÷ 2 = 7 × 5 ÷ 2 = 35 ÷ 2 = 17.5
2. Fractions, Decimals, and Percentages | 分数、小数与百分比
A fraction represents a part of a whole. The top number is the numerator, and the bottom number is the denominator. Equivalent fractions have the same value but different numerators and denominators.
分数表示整体的一部分。分子在上，分母在下。等值分数数值相同，但分子和分母不同。
To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100 and add the % sign. Reversing the process: a percentage becomes a decimal by dividing by 100, and a decimal becomes a fraction by writing it over a power of ten and simplifying.
要将分数转换为小数，用分子除以分母。将小数转换为百分比，乘以 100 并加上百分号。反向转换：百分比除以 100 得到小数，小数写成分母为 10 的幂的分数再化简就得到分数形式。
Fraction to decimal: a/b = a ÷ b
Decimal to percentage: 0.75 × 100 = 75%
Percentage to fraction: 40% = 40/100 = 2/5
When adding or subtracting fractions, find a common denominator first. For multiplication, multiply the numerators together and the denominators together. For division, multiply by the reciprocal of the second fraction.
分数加减时，先找到公分母。乘法：分子相乘，分母相乘。除法：乘第二个分数的倒数。
3. Algebraic Notation and Simplification | 代数符号与化简
In algebra, letters represent unknown numbers or variables. Terms are separated by + or − signs. A coefficient is the number multiplying a variable, such as 3 in 3x. A constant term has no variable.
在代数中，字母表示未知数或变量。项由加号或减号分隔。系数是乘以变量的数字，如 3x 中的 3。常数项不含变量。
Like terms contain exactly the same variable(s) raised to the same power. Only like terms can be combined by adding or subtracting their coefficients. For example, 5a and 2a can be added to give 7a, but 5a and 2b cannot be combined.
同类项包含完全相同的变量且对应指数相同。只有同类项才能通过系数相加或相减来合并。例如，5a 和 2a 相加得到 7a，但 5a 和 2b 不能合并。
Simplification: 4x + 3y − 2x + y = (4x − 2x) + (3y + y) = 2x + 4y
When multiplying variables, write them together, e.g., a × b = ab. Powers are used for repeated multiplication: a × a = a². When expanding brackets, multiply each term inside the bracket by the term outside, paying attention to signs.
变量相乘时写在一起，如 a × b = ab。重复相乘用幂表示：a × a = a²。去括号时，用括号外的项乘以括号内的每一项，注意符号。
Expanding: 3(2x − 5) = 3 × 2x + 3 × (−5) = 6x − 15
4. Solving Linear Equations | 解一元一次方程
A linear equation in one variable can be written in the form ax + b = c, where a, b, and c are numbers and a ≠ 0. The goal is to isolate the variable on one side of the equation using inverse operations.
一元一次方程可以写成 ax + b = c 的形式，其中 a、b、c 是数字且 a ≠ 0。目的是通过逆运算将变量单独移到方程的一边。
Always perform the same operation on both sides of the equation to maintain equality. Start by undoing addition or subtraction, then undo multiplication or division. Check your answer by substituting it back into the original equation.
始终在方程两边进行相同的运算以维持等式。先通过加减消去常数项，再通过乘除消去系数。将答案代回原方程进行检验。
General solution: If ax + b = c, then x = (c − b) / a
Example: Solve 2x + 7 = 19 → 2x = 12 → x = 6
Equations with variables on both sides should be rearranged so that all variable terms end up on one side and all constants on the other. Always simplify each side first if possible.
变量在等号两边的方程，应重新整理，使所有含变量项移到一边，所有常数项移到另一边。如果可能，先化简每一边。
5. Coordinates and the Midpoint Formula | 坐标与中点公式
The position of a point on a coordinate grid is given by an ordered pair (x, y). The x-coordinate tells you how far to move horizontally from the origin; the y-coordinate tells you how far to move vertically.
点在坐标网格上的位置用有序对 (x, y) 表示。x 坐标表示从原点水平移动的距离，y 坐标表示垂直移动的距离。
The midpoint of a line segment joining two points (x₁, y₁) and (x₂, y₂) is found by taking the average of the x-coordinates and the average of the y-coordinates. This gives the point exactly halfway between them.
连接两点 (x₁, y₁) 和 (x₂, y₂) 的线段中点，通过取 x 坐标的平均值和 y 坐标的平均值得到。该点恰好在两点正中间。
Midpoint formula: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
If you are given the midpoint and one endpoint, you can work backwards to find the missing endpoint. Double the midpoint’s coordinates and subtract the known endpoint’s coordinates.
如果已知中点和一个端点，可以反向求出缺失的端点：将中点坐标加倍，再减去已知端点的坐标。
Coordinates are also essential for plotting straight lines, recognising symmetry, and solving geometric problems on a grid.
坐标对于绘制直线图像、识别对称性以及解决网格上的几何问题也至关重要。
6. Angles and Parallel Lines | 角与平行线
Angles are measured in degrees (°). A right angle is exactly 90°, an acute angle is less than 90°, an obtuse angle lies between 90° and 180°, and a straight line forms 180°.
角以度 (°) 为单位。直角恰好是 90°，锐角小于 90°，钝角在 90° 到 180° 之间，平角是 180°。
When two straight lines intersect, vertically opposite angles are equal. Angles around a single point sum to 360°. Angles on a straight line always add up to 180°.
两条直线相交时，对顶角相等。围绕一个点的所有角之和为 360°。直线上的角之和始终为 180°。
Vertically opposite angles: a = b and c = d
Angles on a straight line: p + q = 180°
When a transversal crosses a pair of parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°. Recognising these relationships helps to find unknown angles without measuring.
当一条截线与一对平行线相交时，同位角相等，内错角相等，同旁内角（同侧内角）之和为 180°。识别这些关系有助于在不需要度量的情况下求出未知角。
In a triangle: interior angles sum to 180°
7. Perimeter, Area, and Volume Formulae | 周长、面积与体积公式
The perimeter is the total distance around the outside of a 2D shape. It is found by adding together the lengths of all its sides. For regular polygons, you can multiply the side length by the number of sides.
周长是二维图形外部边界的总距离，通过将所有边长相加得到。对于正多边形，可以用边长乘以边数计算。
Perimeter of a rectangle: P = 2(l + w)
The area of a shape measures the space inside it. Common area formulae are essential for quick calculation.
面积衡量图形内部的空间大小。熟记常见的面积公式对快速计算至关重要。
Area of rectangle: A = l × w
Area of triangle: A = ½ × base × height
Area of parallelogram: A = base × perpendicular height
Area of trapezium: A = ½(a + b)h, where a and b are the parallel sides
Volume measures the space occupied by a 3D object. For a cuboid, multiply length, width, and height. For prisms, multiply the area of the cross-section by the length.
体积衡量三维物体所占空间的大小。对于长方体，体积等于长 × 宽 × 高。对于棱柱，体积等于横截面积乘以长。
Volume of cuboid: V = l × w × h
8. Mean, Median, Mode, and Range | 平均数、中位数、众数与极差
The mean is the average of a set of numbers. It is calculated by adding all the values together and then dividing by how many values there are.
平均数是一组数据的平均值，计算方法是将所有数值相加，再除以数据的个数。
Mean = (sum of all values) ÷ (number of values)
The median is the middle value when the numbers are arranged in order. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据按顺序排列后位于中间的数。如果有偶数个数据，中位数是中间两个数的平均数。
The mode is the value that appears most often. A data set can have one mode, more than one mode, or no mode at all if all values occur equally often.
众数是出现次数最多的数值。一组数据可能有一个众数、多个众数，或者如果所有数值出现次数相同则没有众数。
The range measures the spread of the data. It is found by subtracting the smallest value from the largest value.
极差衡量数据的分散程度，计算方法是用最大值减去最小值。
Range = largest value − smallest value
These four measures together give a quick summary of a data set, helping you compare distributions and draw conclusions.
这四个度量指标一起可以快速总结一组数据，帮助你比较分布并得出结论。
9. Ratio and Proportion | 比与比例
A ratio compares the sizes of two or more quantities. Ratios can be written in the form a : b or as a fraction a/b. They have no units and should be simplified by dividing both sides by common factors, just like fractions.
比用于比较两个或多个量的大小。比可以写成 a : b 或分数 a/b 的形式。比没有单位，应像分数一样通过同时除以公因数来化简。
To share a quantity in a given ratio, first add the parts of the ratio to find the total number of equal shares. Divide the quantity by that total, then multiply by each part of the ratio.
要按给定比例分配一个数量，先将比的各项相加得到总份数。用总数除以总份数，再分别乘以比的每一项。
Divide 50 in the ratio 2 : 3 → total parts = 5, each part = 10, so shares are 20 and 30
Proportion describes the relationship between quantities. Two quantities are in direct proportion if they increase or decrease at the same rate. The ratio between them stays constant.
比例描述量之间的关系。如果两个量以相同的速率增加或减少，它们就成正比，两者之间的比值保持不变。
If y is directly proportional to x, then y = kx, where k is the constant of proportionality
Scaling recipes, maps, and scale drawings all rely on ratio and proportion. The scale factor is the number you multiply dimensions by to enlarge or reduce a shape accurately.
调整食谱、地图和比例图都依赖于比和比例。比例因子是你乘以尺寸以准确放大或缩小图形所用的数。
10. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
勾股定理适用于直角三角形。它表明，斜边（直角所对的边）长度的平方等于另外两边长度的平方之和。
a² + b² = c², where c is the hypotenuse and a, b are the legs
If you know the lengths of any two sides, you can use the theorem to calculate the length of the third side. To find the hypotenuse, add the squares and then take the square root. To find a shorter side, subtract one square from the square of the hypotenuse and then take the square root.
如果已知任意两边的长度，你可以使用该定理计算第三边的长度。求斜边时，将两边的平方相加再开平方。求直角边时，用斜边的平方减去已知直角边的平方再开平方。
Find c: c = √(a² + b²)
Find a: a = √(c² − b²)
Always check whether the triangle is right-angled before applying the theorem. The converse is also true: if for a triangle a² + b² = c², then the triangle is right-angled.
在应用定理前，务必检查三角形是否为直角三角形。其逆定理也成立：如果一个三角形满足 a² + b² = c²，则该三角形是直角三角形。
11. Sequences and the nth Term | 数列与第n项
A number sequence is a list of numbers written in a particular order according to a rule. Each number in the sequence is called a term. The rule allows you to find the next term or any term in the sequence.
数列是按特定规则排列的一列数。数列中的每一个数称为项。规则可以帮助你找到下一项或任意一项。
In a linear (arithmetic) sequence, the difference between consecutive terms is always the same. This fixed difference is called the common difference, d.
在线性（等差）数列中，相邻两项的差始终相同，这个固定的差称为公差，记作 d。
Sequence: 3, 7, 11, 15, … has common difference d = 4
The nth term of a linear sequence can be expressed using a formula. To find it, start with the common difference multiplied by n, then adjust by adding or subtracting a constant so that the formula matches the first term.
线性数列的第 n 项可以用公式表示。首先用公差乘以 n，然后通过加减一个常数使公式与首项一致。
nth term formula: Tₙ = dn + (a − d), where a is the first term
For example, in the sequence 5, 9, 13, 17, …, d = 4 and a = 5, so the nth term is 4n + 1. You can use this formula to find the 100th term without listing all the terms.
例如，数列 5, 9, 13, 17, … 的公差 d = 4，首项 a = 5，因此第 n 项为 4n + 1。你可以用这个公式求出第 100 项而无需列出所有项。
Understanding the nth term also helps you recognise whether a given number belongs to the sequence, by setting the formula equal to that number and solving for n; n must be a positive integer.
理解第 n 项还能帮助你判断一个给定的数是否属于该数列：令公式等于该数并解出 n，n 必须是正整数。
12. Introduction to Probability | 概率入门
Probability measures how likely an event is to occur. It is given as a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. Probabilities can also be written as fractions, decimals, or percentages.
概率衡量一个事件发生的可能性，用一个 0 到 1 之间的数表示，0 表示不可能，1 表示必然发生。概率也可以写成分数、小数或百分比。
Probability of an event = (number of favourable outcomes) ÷ (total number of possible outcomes)
All outcomes must be equally likely for this basic formula to apply directly. For a fair six-sided die, the probability of rolling a 3 is 1/6. The sum of probabilities of all possible mutually exclusive outcomes of an experiment is always 1.
只有当所有结果等可能时，这个基本公式才能直接应用。对于一枚公平的六面骰子，掷出 3 的概率是 1/6。一项实验中所有可能的互斥结果的概率之和总是 1。
The complement of an event is the event that it does not happen. The probability of the complement is 1 minus the probability of the event.
一个事件的补事件是该事件不发生的事件。补事件的概率等于 1 减去该事件的概率。
P(not A) = 1 − P(A)
In more advanced work, probability trees and sample space diagrams help you list all possible outcomes systematically, making it easier to calculate probabilities of combined events.
在更进阶的学习中，概率树和样本空间图能帮助你系统地列出所有可能的结果，从而更容易计算复合事件的概率。
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