📚 Year 8 SQA Advanced Mathematics: Formula & Theorem Quick Reference | SQA Year 8 进阶数学：公式定理速查手册
This quick reference handbook brings together every essential formula, theorem and key fact you need for Year 8 SQA Advanced Mathematics. Keep it handy while revising, working through exercises or checking your answers — clear definitions, worked examples and paired bilingual notes make it easy to review.
这本速查手册汇总了 SQA Year 8 进阶数学所需的所有核心公式、定理和关键知识点。复习、做题或核对答案时随时翻阅——清晰的定义、示例和双语注解让回顾更轻松。
1. Algebraic Expressions & Simplifying | 代数表达式与化简
A term is a single number, a variable or the product of numbers and variables. Like terms contain exactly the same variable(s) raised to the same power and can be added or subtracted.
项是指一个单独的数、变量或数与变量的乘积。同类项含有完全相同的变量且变量的指数相同，可以相加或相减。
3x = 8x. The variable part stays unchanged. Constants are also like terms with each other.
合并同类项时，只有系数发生变化：5x + 3x = 8x。变量部分保持不变。常数项之间也互为同类项。
erm inside the bracket must be multiplied by the factor outside, watching signs carefully.
分配律用于展开括号：a(b + c) = ab + ac。括号内的每一项都要乘以外面的因数，并注意正负号。
ghest common factor (HCF) of all terms and write it outside a bracket: 6x + 9 = 3(2x + 3).
分解因式时，找出各项的最高公因式（HCF）并将其写在括号外：6x + 9 = 3(2x + 3)。
2. Linear Equations & Inequalities | 线性方程与不等式
he variable by performing inverse operations on both sides, keeping the equation balanced.
线性方程可写成 ax + b = 0 的形式。目标是通过在等式两边进行逆运算来分离变量，始终保持方程平衡。
Check your solution by substituting the value back into the original equation. If both sides give the same result, the solution is correct.
通过将解代回原方程来检验。若两边得到相同的值，则解正确。
), treat them exactly like equations except: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
解不等式（<, >, ≤, ≥）时，操作与方程一致，但有一条例外：如果两边同时乘以或除以一个负数，不等号方向必须反转。
The solution to an inequality is typically a range of values and is shown on a number line with an open or closed circle, or written in set notation such as x > 2.
不等式的解通常是一个取值范围，在数轴上用空心或实心圆点表示，也可写作集合形式，如 x > 2。
3. Indices (Exponents) & Surds | 指数与根式
you how many times the base is multiplied by itself. For example, 2⁴ = 2 × 2 × 2 × 2 = 16.
指数表示底数自乘的次数。例如，2⁴ = 2 × 2 × 2 × 2 = 16。
Key index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
核心指数律：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1（a ≠ 0）, a⁻ⁿ = 1/aⁿ。
A surd is a root that cannot be simplified to a whole or fractional number, such as √2 or ∛5. Surds can be simplified using √ab = √a × √b and rationalised in denominators.
根式是不可化简为整数或分数的根，如 √2 或 ∛5。可利用 √ab = √a × √b 化简根式，并对分母进行有理化。
| Rule | Example |
| aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x⁵ = x⁷ |
| (aᵐ)ⁿ = aᵐⁿ | (y³)² = y⁶ |
| a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
公式速查：
4. Sequences & the nth Term | 数列与第 n 项
A number sequence is a list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant — this is the common difference, d.
数列是按某一规则排列的一串数字。在等差数列中，相邻两项的差是常数，称为公差 d。
re a is the first term. This allows you to find any term directly without writing out the whole list.
等差数列的第 n 项公式为：第 n 项 = a + (n − 1)d，其中 a 为首项。利用该公式可直接求出任意项，无需列出全部项。
To find the rule for a sequence from a pattern of shapes or sticks, look for how the count changes each time. That change becomes the coefficient of n, then adjust with a constant to match the first term.
从图形或火柴棍规律中找数列通项时，先观察每次数量的变化，该变化即为 n 的系数，再通过常数调整使首项吻合。
A quadratic sequence has a changing difference; its nth term involves n². In Year 8, the focus is recognising and continuing simple quadratic patterns.
二次序列的差是变化的，其通项包含 n²。Year 8 阶段重点在于识别并延续简单的二次模式。
5. Straight Line Graphs & Coordinates | 直线图像与坐标
Coordinates are written as (x, y). The x-coordinate gives the horizontal position, the y-coordinate the vertical position. Plot points carefully, then connect them with a ruler for straight line graphs.
坐标写作 (x, y)。x 坐标表示水平位置，y 坐标表示垂直位置。精确描点，然后用直尺连成直线图像。
is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
直线的方程常写作 y = mx + c，其中 m 为斜率（倾斜程度），c 为 y 轴截距（直线与 y 轴的交点）。
₁) / (x₂ − x₁). A positive gradient means the line slopes upward, a negative gradient slopes downward.
计算两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率：m = (y₂ − y₁) / (x₂ − x₁)。正斜率表示直线向上倾斜，负斜率表示向下倾斜。
and vertical lines have x = a constant. Their gradients are 0 and undefined respectively.
水平线方程为 y = 常数，垂直线方程为 x = 常数。它们的斜率分别为 0 和无定义。
6. Angles, Polygons & Parallel Lines | 角、多边形与平行线
Angles on a straight line add up to 180°. Angles around a point total 360°. Vertically opposite angles are equal.
直线上的角之和为 180°。围绕一个点的角之和为 360°。对顶角相等。
In triangles, the interior angles always sum to 180°. An isosceles triangle has two equal sides and two equal base angles. An equilateral triangle has three 60° angles.
三角形的内角和恒为 180°。等腰三角形有两条边相等且两个底角相等。等边三角形的每个角均为 60°。
With parallel lines: alternate angles are equal, corresponding angles are equal, and allied (interior) angles add up to 180°. Recognising these ‘Z’, ‘F’ and ‘C’ patterns helps solve problems quickly.
平行线性质：内错角相等，同位角相等，同旁内角互补（和为 180°）。识别这些“Z”、“F”和“C”形模式有助于快速解题。
polygon = (n − 2) × 180°. The sum of exterior angles of any convex polygon is always 360°.
n 边形内角和 = (n − 2) × 180°。任意凸多边形的外角和恒为 360°。
7. Perimeter, Area & Volume | 周长、面积与体积
of a 2D shape. For a rectangle, P = 2(l + w). For a triangle, add the three side lengths.
周长是二维图形外边界的总长度。矩形周长 P = 2(长 + 宽)。三角形周长将三边相加。
Area measures the surface inside a shape. Key formulas:
面积测量图形内部的表面大小。关键公式如下：
Rectangle: A = l × w
矩形：A = 长 × 宽
Triangle: A = ½ × base × height
三角形：A = ½ × 底 × 高
Parallelogram: A = base × vertical height
平行四边形：A = 底 × 垂直高
Trapezium: A = ½ (a + b) × h
梯形：A = ½ (上底 + 下底) × 高
cross-section × length. For a cuboid, V = l × w × h. Units for volume are cubic, e.g. cm³.
棱柱体积 = 横截面积 × 长。长方体体积 V = 长 × 宽 × 高。体积单位为立方，如 cm³。
8. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right‑angled triangles and states: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
勾股定理适用于直角三角形，它指出：斜边的平方等于另外两条边长的平方和。
a² + b² = c²
where c is the hypotenuse (longest side).
其中 c 为斜边（最长边）。
l the sides carefully before substituting and remember to take the square root at the end.
求直角边时变形为：a² = c² − b²。始终先明确标注各边再代入，并在最后开平方根。
Pythagoras also helps determine whether a triangle is right‑angled: if a² + b² = c² holds true for the three sides, the triangle is right‑angled.
勾股定理也可用于判断三角形是否为直角三角形：若三边满足 a² + b² = c²，则该三角形为直角三角形。
9. Trigonometry in Right Triangles | 直角三角形中的三角学
The three trigonometric ratios link angles to side lengths in right‑angled triangles:
三个三角比在直角三角形中将角度与边长联系起来：
sin θ = opposite / hypotenuse
sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
tan θ = 对边 / 邻边
Identify the sides relative to the angle you are working with: opposite is the side across from the angle, adjacent is the side next to it (not the hypotenuse). Use SOH CAH TOA to remember.
识别与所用角度相关的边：对边是该角对面的边，邻边是相邻的边（非斜边）。熟记 SOH CAH TOA 帮助记忆。
unknown side, then solve. To find an angle, use the inverse function: θ = sin⁻¹(opp/hyp).
求未知边时，选择包含已知边和未知边的比，然后求解。求角度时使用反函数：θ = sin⁻¹(对/斜)。
Always set your calculator to degree mode when working with angles in degrees. Show steps clearly and round final answers to one decimal place unless told otherwise.
处理以度为单位的角度时，确保计算器处于度数模式。清晰写出步骤，除特殊说明外，最终答案保留一位小数。
10. Statistics: Averages & Charts | 统计：平均数与图表
Three common measures of central tendency: the mean (sum of values divided by number of values), the median (middle value when ordered) and the mode (most frequent value).
三种常见的集中趋势度量：平均数（总和除以个数）、中位数（排序后中间的值）和众数（出现次数最多的值）。
The range, calculated as maximum − minimum, describes the spread of data. A small range means the data are closely grouped; a large range indicates wider spread.
极差（最大值 − 最小值）描述数据的离散程度。极差小意味着数据聚集紧密，极差大表示分布较广。
Data can be displayed in bar charts, line graphs, pie charts or scatter graphs. Choose the type that best shows patterns, comparisons or trends.
数据可用条形图、折线图、饼图或散点图呈现。选择最能展示模式、对比或趋势的图表类型。
sector angle is proportional to the frequency: sector angle = (frequency ÷ total) × 360°.
阅读饼图时，记住整个圆表示总数，每个扇形的圆心角与频数成比例：扇形角 = (频数 ÷ 总数) × 360°。
11. Probability Basics | 概率基础
Probability is a measure of how likely an event is to happen, always between 0 (impossible) and 1 (certain). It can be written as a fraction, decimal or percentage.
概率衡量事件发生可能性的大小，总是在 0（不可能）到 1（必定）之间。可表示为分数、小数或百分比。
omes) / (total number of possible outcomes). The sample space lists all possible outcomes.
对于等可能结果：P(事件) = （有利结果的数量）/（可能结果的总数）。样本空间列出所有可能的结果。
of all mutually exclusive outcomes in a sample space is 1. Therefore, P(not A) = 1 − P(A).
样本空间中所有互斥结果的概率之和为 1。因此，P(非 A) = 1 − P(A)。
= P(A) × P(B). Use tree diagrams to map out sequential events and multiply along branches.
对于两个独立事件，P(A 且 B) = P(A) × P(B)。可用树状图展示连续事件，并沿分支相乘。
12. Units & Conversions | 单位与换算
mm, 1 m = 100 cm, 1 km = 1000 m. For mass, 1 kg = 1000 g. For capacity, 1 litre = 1000 ml.
公制换算必不可少：1 厘米 = 10 毫米，1 米 = 100 厘米，1 千米 = 1000 米。质量：1 千克 = 1000 克。容量：1 升 = 1000 毫升。
ar conversion factor: 1 m² = 10000 cm². For volume, cube the factor: 1 m³ = 1 000 000 cm³.
换算面积单位时，需将长度换算系数平方：1 平方米 = 10000 平方厘米。换算体积时则立方：1 立方米 = 1 000 000 立方厘米。
ween hours and minutes, multiply or divide by 60. Use decimal fraction for parts of an hour.
时间换算：1 分 = 60 秒，1 小时 = 60 分。小时和分钟互化时乘以或除以 60。小时的小数部分用小数表示。
e smallest unit first. Compound measures like speed use formulas: Speed = Distance ÷ Time.
加减混合单位时，应对齐数位（米与米、厘米与厘米）或先全部转为最小单位。复合单位如速度使用公式：速度 = 路程 ÷ 时间。
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