📚 Year 8 SQA Maths: Quick Reference Handbook of Formulas and Theorems | Year 8 SQA 数学：公式定理速查手册
This quick reference handbook brings together all the essential formulas and theorems you need for Year 8 SQA Mathematics. Use it to check key facts, practise recall, and build confidence for tests and assessments. Each topic is presented with a short rule or formula, followed by a clear explanation in both English and Chinese, so you can learn bilingually and master the terms.
这本速查手册汇集了 Year 8 SQA 数学所需的所有重要公式和定理。你可以用它核对关键知识，练习回忆，为考试和评估建立信心。每个主题都先用简明的规则或公式呈现，再用中英双语给出清晰解释，帮助你双语学习、牢固掌握术语。
1. Number and Place Value | 数字与位值
Use place value to compare and order decimals by looking at the digits from left to right. The first digit that differs decides which number is larger.
通过从左向右看各位数字，利用位值来比较和排列小数。第一个不同的数位决定哪个数更大。
Round numbers to a given place value: if the next digit is 5 or more, round up; if it is 4 or less, round down.
将数字舍入到指定位值：如果下一位数字是 5 或更大，则进位；如果是 4 或更小，则舍去。
Follow the order of operations with BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
遵循运算顺序 BIDMAS：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
When adding a negative number, move left on the number line: a + (⁻b) = a ⁻ b. Subtracting a negative is the same as adding: a ⁻ (⁻b) = a + b.
加上一个负数时，在数轴上向左移：a + (⁻b) = a ⁻ b。减去一个负数等同于加上其相反数：a ⁻ (⁻b) = a + b。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
To add or subtract fractions, find a common denominator first, then add or subtract the numerators. Keep the denominator the same.
加减分数时，先找到公分母，然后将分子相加或相减，分母不变。
Multiply fractions by multiplying the numerators together and the denominators together: a/b × c/d = (a×c)/(b×d).
分数相乘时，分子乘分子，分母乘分母：a/b × c/d = (a×c)/(b×d)。
Divide by a fraction by multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.
除以一个分数等于乘它的倒数：a/b ÷ c/d = a/b × d/c。
Convert between fractions, decimals and percentages: to change a fraction to a decimal, divide the numerator by the denominator. Multiply the decimal by 100 to get the percentage.
分数、小数和百分数互相转换：把分数转为小数，用分子除以分母；小数乘以 100 得到百分数。
To find a percentage of an amount, multiply the amount by the percentage as a decimal (e.g. 15% of 200 = 0.15 × 200).
求一个数的百分之几，用这个数乘以百分数对应的小数（例如 200 的 15% = 0.15 × 200）。
Compare fractions by converting them to equivalent fractions with a common denominator or to decimals.
通过化为同分母的等值分数或化为小数来比较分数的大小。
3. Ratio and Proportion | 比和比例
Write ratios in their simplest form by dividing both sides by their highest common factor. The order matters.
把比写成最简形式，两边同时除以它们的最大公因数。顺序很重要。
Share a quantity in a given ratio: add the parts of the ratio to find the total number of shares, then divide the quantity by that total and multiply by each part.
按给定比例分配一个量：将比的各份相加得到总份数，然后用总量除以总份数，再乘以各份数。
Use proportion to scale recipes or maps: if 3 apples cost £1.50, then 1 apple costs £0.50 and 5 apples cost £2.50.
利用比例来调整食谱或地图的尺度：如果 3 个苹果 1.50 英镑，那么一个苹果 0.50 英镑，5 个苹果 2.50 英镑。
Direct proportion means two quantities increase or decrease at the same rate; their ratio stays constant.
正比例意味着两个量以相同速率增加或减少；它们的比值保持不变。
4. Algebraic Expressions | 代数表达式
Simplify expressions by collecting like terms: 3a + 2b + 5a = 8a + 2b. Only terms with exactly the same variable part can be combined.
通过合并同类项来简化表达式：3a + 2b + 5a = 8a + 2b。只有变量部分完全相同的项才能合并。
Multiply out brackets using the distributive law: a(b + c) = a×b + a×c = ab + ac.
使用分配律去括号：a(b + c) = a×b + a×c = ab + ac。
Factorise an expression by taking out the highest common factor: 6x + 9 = 3(2x + 3).
通过提取最大公因数来因式分解：6x + 9 = 3(2x + 3)。
The product of a variable and a number is written with the number first: write 4 × y as 4y, not y4.
数字与变量的乘积，数字写在前面：4 × y 写作 4y，而不是 y4。
5. Linear Equations | 一次方程
Solve an equation by doing the same operation to both sides to keep it balanced. Always aim to isolate the variable.
解方程时，两边同时进行相同运算，保持等式平衡。始终目标是孤立未知数。
For the equation 2x + 3 = 11, subtract 3 from both sides: 2x = 8, then divide by 2: x = 4.
对于方程 2x + 3 = 11，两边同时减 3 得 2x = 8，再除以 2 得 x = 4。
Check your solution by substituting it back into the original equation. The left-hand side must equal the right-hand side.
将解代入原方程检验，左边必须等于右边。
When the variable appears on both sides, collect variable terms on one side and constant terms on the other first.
当未知数在等式两边都出现时，先将含未知数的项移到一边，常数项移到另一边。
6. Sequences | 数列
An arithmetic sequence has a constant difference between consecutive terms. The nth term formula is:
等差数列的相邻项之差为常数。第 n 项公式为：
aₙ = a₁ + (n – 1)d
where a₁ is the first term, d is the common difference, and n is the term position.
其中 a₁ 是首项，d 是公差，n 是项的位置编号。
For example, in the sequence 5, 8, 11, 14, …, a₁ = 5 and d = 3. The nth term is aₙ = 5 + (n – 1) × 3 = 3n + 2.
例如，数列 5, 8, 11, 14, … 中，a₁ = 5，d = 3。第 n 项为 aₙ = 5 + (n – 1) × 3 = 3n + 2。
Use the nth term to find any term: put the position number into the formula. Term 10 is 3×10 + 2 = 32.
用第 n 项公式求任意一项：把位置编号代入公式。第 10 项为 3×10 + 2 = 32。
A geometric sequence multiplies by a constant ratio each time, but arithmetic sequences are the main focus in Year 8.
等比数列每次乘以一个固定比例，但 Year 8 主要关注等差数列。
7. Angles and Lines | 角与直线
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°.
绕一点一周的角之和为 360°。
Vertically opposite angles are equal when two lines intersect.
两直线相交时，对顶角相等。
When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles add up to 180°.
当一条截线穿过两条平行线时：同位角相等，内错角相等，同旁内角互补（和为 180°）。
In a triangle, interior angles sum to 180°. An exterior angle equals the sum of the two opposite interior angles.
三角形内角和为 180°。一个外角等于与它不相邻的两个内角之和。
8. 2D Shapes: Perimeter and Area | 二维图形：周长与面积
Perimeter is the total distance around a shape. Add all the side lengths.
周长是图形一周的总长度，把所有边长相加。
Area of a rectangle: A = l × w
矩形面积：A = l × w
Area of a triangle: A = ½ × b × h
三角形面积：A = ½ × b × h
Area of a parallelogram: A = b × h, where h is the perpendicular height.
平行四边形面积：A = b × h，其中 h 是垂直高度。
Area of a trapezium: A = ½ × (a + b) × h, where a and b are the parallel sides.
梯形面积：A = ½ × (a + b) × h，其中 a 和 b 是两条平行边。
Circumference of a circle: C = π × d or C = 2 × π × r
圆的周长：C = π × d 或 C = 2 × π × r
Area of a circle: A = π × r²
圆的面积：A = π × r²
Compound area: split the shape into simpler rectangles or triangles, find each area, then add or subtract them.
组合图形的面积：将图形分割成简单的矩形或三角形，分别求面积，然后相加或相减。
9. 3D Shapes: Volume and Surface Area | 三维图形：体积与表面积
Volume of a cuboid: V = l × w × h
长方体体积：V = l × w × h
Volume of a prism: V = area of cross-section × length. The cross‑section is the same shape all through.
棱柱的体积：V = 横截面积 × 长。横截面沿全长保持不变。
Surface area of a cuboid: sum of the areas of all six rectangular faces. A net can help: SA = 2lw + 2lh + 2wh.
长方体的表面积：所有六个矩形面的面积之和。展开图有助于：SA = 2lw + 2lh + 2wh。
Volume of a cylinder: V = π × r² × h, where the cross‑section is a circle.
圆柱的体积：V = π × r² × h，其横截面为圆。
Capacity is the liquid a 3D shape can hold, often measured in litres (1 litre = 1000 cm³).
容量是三维形状可盛放的液体量，常用升（1 升 = 1000 立方厘米）计量。
10. Pythagoras’ Theorem | 勾股定理
In a right‑angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
在直角三角形中，斜边（直角对边）的平方等于另外两条边的平方和。
a² + b² = c²
where c is the hypotenuse, and a and b are the legs.
其中 c 是斜边长，a 和 b 是直角边长。
To find the hypotenuse: c = √(a² + b²)
求斜边：c = √(a² + b²)
To find a shorter side: a = √(c² ⁻ b²)
求直角边：a = √(c² ⁻ b²)
Check if a triangle is right‑angled by testing whether a² + b² = c² holds for the longest side.
要检验一个三角形是否为直角三角形，检验最长边是否满足 a² + b² = c²。
11. Statistics: Averages and Range | 统计：平均数和极差
Mean: add all the values together and divide by the number of values. Mean = sum of data / number of data items.
平均数：将所有数值相加，再除以数值的个数。 平均数 = 数据总和 / 数据个数。
Median: the middle value when the data is ordered. For an even number of values, the median is the mean of the two middle numbers.
中位数：将数据按大小排列后位于中间的数。若数据个数为偶数，则中位数是中间两个数的平均数。
Mode: the value that appears most often. A data set can have one mode, more than one mode, or no mode at all.
众数：出现次数最多的数值。一组数据可能有一个众数、多个众数，或者没有众数。
Range: the difference between the largest and smallest values. Range = maximum – minimum.
极差：最大值与最小值的差。 极差 = 最大值 – 最小值。
A bar chart uses bars to show frequencies; the height of each bar represents the count. Pie charts show proportions as angles: each category’s angle = (category frequency / total frequency) × 360°.
条形图用直条表示频数；每个直条的高度代表频数。饼图用角度表示比例：每个类别的角度 = (类别频数 / 总频数) × 360°。
12. Probability | 概率
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
概率衡量事件发生的可能性，取值范围在 0（不可能）到 1（必然）之间。
Probability = number of favourable outcomes / total number of possible outcomes
概率 = 有利结果的数量 / 所有可能结果的总数
For a fair six‑sided die, the probability of rolling a 3 is 1/6.
掷一枚公平的六面骰子，掷出 3 的概率为 1/6。
The probabilities of all possible outcomes add up to 1. The probability of an event not happening is 1 minus the probability of it happening.
所有可能结果的概率之和为 1。事件不发生的概率等于 1 减去该事件发生的概率。
Use tree diagrams or sample space diagrams to list all possible outcomes for two or more events, then count favourable ones.
对两个或多个事件，利用树形图或样本空间图列出所有可能结果，再数出有利结果的个数。
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