📚 Year 8 SQA Mathematics: Quick Reference Formula & Theorems Handbook | Year 8 SQA 数学：公式定理速查手册
This quick reference handbook covers the essential formulas and theorems every SQA Year 8 student needs to master. Use it to revise key concepts in number, algebra, geometry, measurement, and statistics — all clearly explained with examples and paired bilingual notes.
这份速查手册涵盖了 SQA Year 8 学生必须掌握的核心公式和定理。你可以用它来复习数字、代数、几何、测量和统计方面的关键概念，每个知识点都附有清晰的示例和中英双语对照讲解。
1. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS stands for Brackets, Indices, Division/Multiplication, Addition/Subtraction. You must follow this order when evaluating any mathematical expression.
BIDMAS 代表括号、指数、除法/乘法、加法/减法。在计算任何数学表达式时，都必须遵循这一顺序。
First, evaluate expressions inside brackets (or parentheses).
首先，计算括号内的表达式。
Next, simplify any powers or roots (indices). For example, 3² or √49.
然后，化简任何乘方或开方（指数）。例如 3² 或 √49。
Then, carry out divisions and multiplications as they appear from left to right.
接着，从左到右依次进行除法和乘法运算。
Finally, perform additions and subtractions from left to right.
最后，从左到右依次进行加法和减法运算。
Example: 2 + 3 × 4 = 2 + 12 = 14
示例：2 + 3 × 4 = 2 + 12 = 14
2. Fractions, Decimals & Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of representing parts of a whole. They can be converted into each other using simple rules.
分数、小数和百分数是表示整体部分的三种方式，可以通过简单规则相互转换。
To change a fraction to a decimal, divide the numerator (top) by the denominator (bottom).
将分数化为小数，用分子（上方的数）除以分母（下方的数）。
To change a decimal to a percentage, multiply by 100 and add the % symbol.
将小数化为百分数，乘以 100 并加上 % 符号。
To find a percentage of an amount, use the formula: part = (percentage ÷ 100) × whole.
求一个数的百分之几，使用公式：部分 = (百分数 ÷ 100) × 整体。
Fraction to decimal: 3/4 = 3 ÷ 4 = 0.75
分数化小数：3/4 = 3 ÷ 4 = 0.75
Decimal to percentage: 0.75 × 100 = 75%
小数化百分数：0.75 × 100 = 75%
When adding or subtracting fractions, find a common denominator first. To multiply fractions, multiply the numerators and denominators straight across. To divide by a fraction, multiply by its reciprocal (flip the second fraction).
分数加减法要先通分。乘法直接将分子相乘、分母相乘。除以一个分数，等于乘以其倒数（将第二个分数上下颠倒）。
3. Simplifying Algebraic Expressions | 代数表达式化简
Algebraic expressions contain numbers, variables (letters) and operation signs. Simplifying them makes them easier to work with.
代数表达式包含数字、变量（字母）和运算符号。化简后使用起来更方便。
Combine like terms (terms with exactly the same variable raised to the same power).
合并同类项（变量和指数完全相同的项）。
Example: 2x + 3y + 5x - y = 7x + 2y
示例：2x + 3y + 5x - y = 7x + 2y
Use the distributive law to expand brackets: a(b + c) = ab + ac.
使用分配律展开括号：a(b + c) = ab + ac。
Distributive law: 3(x + 4) = 3x + 12
分配律：3(x + 4) = 3x + 12
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. Be careful — this only works if the base is the same.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。注意，只有当底数相同时才能这样做。
4. Solving Linear Equations | 解一元一次方程
A linear equation is an equation where the unknown variable is raised to the power 1. The goal is to find the value of the variable that makes the equation true.
一元一次方程是指未知数的次数为 1 的方程。目标是求出使等式成立的未知数的值。
Use the balance method: whatever you do to one side of the equation, you must do to the other side.
运用等式的平衡原理：对等式一边所做的任何操作，都必须在另一边进行同样的操作。
To isolate the variable, perform inverse operations in reverse BIDMAS order — first add or subtract, then multiply or divide.
为了将变量单独分离出来，需要按照逆 BIDMAS 顺序进行逆运算——先加减，再乘除。
Solve: 2x + 3 = 7
解方程：2x + 3 = 7
Step 1: Subtract 3 from both sides → 2x = 4
步骤 1：两边减去 3 → 2x = 4
Step 2: Divide both sides by 2 → x = 2
步骤 2：两边除以 2 → x = 2
Always check your solution by substituting it back into the original equation.
始终将所得的解代回原方程进行检验。
5. Sequences & Patterns | 序列与规律
A sequence is an ordered list of numbers following a rule. Finding the nth term allows you to calculate any term in the sequence without listing all previous terms.
序列是一组按规则排列的数。找到第 n 项公式后，无需列出前面各项就能直接计算任意项。
(linear) sequence with constant difference d, the nth term formula is: aₙ = a₁ + (n - 1)d.
对于公差为常数 d 的等差数列，第 n 项公式为：aₙ = a₁ + (n - 1)d。
Here a₁ is the first term, n is the term number, and d is the common difference (can be positive or negative).
其中 a₁ 是首项，n 是项数，d 是公差（可为正或负）。
Example: In 4, 7, 10, 13, ..., a₁ = 4, d = 3.
示例：在 4, 7, 10, 13, ... 中，a₁ = 4，d = 3。
nth term: aₙ = 4 + (n - 1) × 3 = 3n + 1
第 n 项：aₙ = 4 + (n - 1) × 3 = 3n + 1
6. Angles & Lines | 角与线
Angles are measured in degrees (°). Knowing angle facts helps you find missing angles without a protractor.
角以度 (°) 为单位进行测量。掌握角度基本定理后，无需量角器就能求出未知角度。
On a straight line, angles sum to 180° (supplementary). Angles around a point sum to 360°.
平角为 180°（补角）；周角（绕一点一周）为 360°。
Vertically opposite angles (formed when two lines cross) are equal.
对顶角（两直线相交形成的对角）相等。
When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.
当一条截线与两条平行线相交时，同位角相等，内错角相等，同旁内角互补（和为 180°）。
Angle sum of a triangle: a + b + c = 180°
三角形内角和：a + b + c = 180°
7. Triangles & Quadrilaterals | 三角形与四边形
Triangles can be classified by their sides (equilateral, isosceles, scalene) or by their angles (acute, right-angled, obtuse).
三角形可按边分为等边三角形、等腰三角形和不规则三角形，也可按角分为锐角三角形、直角三角形和钝角三角形。
The sum of interior angles in a triangle is always 180°. In an isosceles triangle, base angles are equal.
三角形内角和恒为 180°。在等腰三角形中，两个底角相等。
Area of a triangle = ½ × base × vertical height
三角形面积 = ½ × 底 × 垂直高度
A quadrilateral is a four-sided polygon. The sum of interior angles of any quadrilateral is 360°.
四边形是有四条边的多边形。任意四边形的内角和均为 360°。
= base × perpendicular height. Area of a trapezium = ½ × (sum of parallel sides) × height.
平行四边形面积 = 底 × 垂直高度。梯形面积 = ½ × (上底 + 下底) × 高。
8. Perimeter, Area & Volume | 周长、面积与体积
Perimeter is the distance around a 2D shape. Area is the amount of space inside a 2D shape. Volume is the space occupied by a 3D solid.
周长是二维图形一周的长度；面积是二维图形内部所占区域的大小；体积是三维立体所占空间的大小。
- Rectangle perimeter: P = 2(l + w)
- Rectangle area: A = lw
- 矩形周长：P = 2(l + w)
- 矩形面积：A = lw
circles, use the constant π (pi ≈ 3.14 or 22/7). Circumference C = 2πr = πd. Area A = πr².
对于圆，使用常数 π（约等于 3.14 或 22/7）。周长 C = 2πr = πd。面积 A = πr²。
uboid: V = length × width × height. Volume of a prism: V = area of cross-section × length.
立方体/长方体体积：V = 长 × 宽 × 高。柱体体积：V = 截面积 × 长。
Right triangular prism volume: V = (½ bh) × L
直角三棱柱体积：V = (½ bh) × L
9. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem applies only to right-angled triangles. It relates the lengths of the three sides.
勾股定理仅适用于直角三角形，它揭示了三边长度之间的关系。
If c is the hypotenuse (the side opposite the right angle), and a and b are the other two legs, then:
如果 c 为斜边（直角所对的边），a 和 b 为另外两条直角边，则有：
a² + b² = c²
a² + b² = c²
You can use this theorem to find a missing side in a right-angled triangle. If the hypotenuse is unknown, square and add the legs, then take the square root. If a leg is missing, rearrange the formula to subtract.
你可以用这一定理求直角三角形中未知边的长度。如果斜边未知，则将两直角边分别平方相加后开平方。如果一直角边未知，则移项后用斜边平方减去另一直角边平方，再开方。
Example: a = 3, b = 4 → c² = 9 + 16 = 25 → c = √25 = 5
示例：a = 3，b = 4 → c² = 9 + 16 = 25 → c = √25 = 5
10. Statistics & Probability | 统计与概率
Statistics involves collecting, organizing and interpreting data. Three key measures of central tendency are the mean, median and mode.
统计学涉及数据的收集、整理和解读。三种重要的集中趋势量数分别是平均数、中位数和众数。
values. Median is the middle value when data is ordered. Mode is the most frequent value.
平均数 = 所有数值之和 ÷ 数值的个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。
Range = maximum value - minimum value, showing the spread.
极差 = 最大值 - 最小值，用来表示数据的分散程度。
Probability measures how likely an event is to happen. It is always between 0 (impossible) and 1 (certain), often written as a fraction, decimal or percentage.
概率衡量某一事件发生的可能性大小，取值总是在 0（不可能）和 1（必然）之间，常以分数、小数或百分数表示。
Probability of an event = Number of favourable outcomes ÷ Total number of outcomes
事件发生的概率 = 有利结果的数量 ÷ 所有可能结果的总数
For equally likely outcomes: P(heads) = ½ when flipping a fair coin.
对于等可能结果：抛一枚均匀硬币时，P(正面) = ½。
11. Ratio & Proportion | 比例与比率
A ratio compares the sizes of two or more quantities in the same unit. Ratios can be simplified by dividing all parts by their highest common factor.
比率用来比较两个或多个同单位量的大小。将比率中的每一项都除以它们的最大公约数，即可化简比率。
Simplify 8:12 → divide by 4 → 2:3
化简 8:12 → 除以 4 → 2:3
To share a quantity in a given ratio, first add the total number of parts, then divide the quantity by that total, and multiply by each part.
按给定比率分配一个量时，先将各部分份数相加，用总量除以总份数，再分别乘以每一份的份数。
Direct proportion means that as one quantity increases, the other increases at the same constant rate. The ratio stays the same.
正比例关系是指一个量增加时，另一个量以相同的恒定速率增加，两者的比率保持不变。
If y is directly proportional to x, we write y ∝ x and y = kx, where k is the constant of proportionality.
如果 y 与 x 成正比例，记作 y ∝ x，且 y = kx，其中 k 为比例常数。
12. Coordinates & Straight Line Graphs | 坐标与直线图
The coordinate plane has a horizontal x-axis and a vertical y-axis. Points are written as ordered pairs (x, y).
坐标平面有一条水平的 x 轴和一条垂直的 y 轴。点的位置用有序数对 (x, y) 表示。
The midpoint between two points (x₁, y₁) and (x₂, y₂) is given by:
两点 (x₁, y₁) 和 (x₂, y₂) 的中点公式为：
Midpoint = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
中点 = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
re m is the gradient (slope) and c is the y-intercept (where the line crosses the y-axis).
直线的图像遵循方程 y = mx + c。其中 m 为斜率（坡度），c 为 y-截距（直线与 y 轴交点的纵坐标）。
e in x) = Δy / Δx. A positive gradient slopes upward, a negative gradient slopes downward.
斜率 m = (y 的变化量) / (x 的变化量) = Δy / Δx。斜率为正时直线向上倾斜，斜率为负时直线向下倾斜。
To draw a straight line graph, find at least three points that satisfy the equation, plot them, and join with a ruler.
画直线图像时，至少找出三个满足方程的点，描点后用直尺连接即可。
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