📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference guide brings together the essential formulas, rules and theorems you will meet in Year 7 CCEA Further Mathematics. Keep it handy when working through practice questions, and use it to build confidence with the facts that underpin more advanced problem-solving.
这本速查手册汇总了你在 CCEA 进阶数学（Year 7）中将遇到的关键公式、法则和定理。在做练习时把它放在手边，利用它来巩固基础知识，帮你更有信心地应对更高阶的题目。
1. Order of Operations (BIDMAS) | 运算顺序
When an expression contains more than one operation, always follow the BIDMAS rule: Brackets first, then Indices (powers and roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right).
当一个算式中含有多个运算时，始终遵循 BIDMAS 法则：先算括号，再算指数（乘方和开方），然后算除法和乘法（从左到右），最后算加法和减法（从左到右）。
Example: 3 + 4 × (2 + 1)² = 3 + 4 × 3² = 3 + 4 × 9 = 3 + 36 = 39.
示例：3 + 4 × (2 + 1)² = 3 + 4 × 3² = 3 + 4 × 9 = 3 + 36 = 39。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
To convert a fraction to a decimal, divide the numerator (top) by the denominator (bottom). To change a decimal to a percentage, multiply by 100 and add the % sign.
将分数转化为小数，用分子除以分母。将小数转化为百分数，先乘以 100，再加上百分号。
Key equivalences to memorise: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333… = 33.3%, 1/10 = 0.1 = 10%.
需要记住的常见等值关系：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/3 ≈ 0.333… = 33.3%，1/10 = 0.1 = 10%。
To find a percentage of an amount, write the percentage as a fraction over 100 and multiply, or change it to a decimal first. For example, 20% of 60 = 0.20 × 60 = 12.
求一个数量的百分比，可以把百分数写成分母为 100 的分数然后相乘，或者先化为小数。例如，60 的 20% = 0.20 × 60 = 12。
3. Algebraic Expressions and Simplification | 代数表达式与化简
In algebra, letters (variables) stand for unknown numbers. Terms that have exactly the same variable parts can be collected together. This is called ‘collecting like terms’.
在代数中，字母（变量）代表未知数。含有完全相同的变量部分的项可以合并，这叫做“合并同类项”。
Example: 3a + 5b – 2a + 4b = (3a – 2a) + (5b + 4b) = a + 9b.
示例：3a + 5b – 2a + 4b = (3a – 2a) + (5b + 4b) = a + 9b。
When multiplying, multiply the numbers (coefficients) and then the letters. Remember: a × a = a², and b × b × b = b³.
做乘法时，先把数字（系数）相乘，再处理字母部分。记住：a × a = a²，b × b × b = b³。
4. Solving Linear Equations | 解一元一次方程
A linear equation has the unknown (usually x) raised to the power 1. The golden rule is: always do the same operation to both sides of the equation to keep it balanced.
一元一次方程中的未知数（通常是 x）的指数为 1。黄金法则是：始终在方程两边进行相同的运算，以保持等式的平衡。
To solve 3x + 5 = 20, first subtract 5 from both sides: 3x = 15. Then divide both sides by 3: x = 5.
解方程 3x + 5 = 20，首先两边同时减去 5：3x = 15，然后两边同时除以 3：x = 5。
Always check your answer by substituting it back into the original equation.
务必将答案代回原方程进行检验。
5. Sequences and the nth Term | 数列与第 n 项
A number sequence follows a rule. An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, d.
数字序列按照一定规则排列。等差数列的相邻两项之差保持不变，这个差叫做公差，记作 d。
The nth term of an arithmetic sequence can be written as n-th term = a + (n – 1)d, where a is the first term. In Year 7 you will often see simpler expressions like 2n + 1.
等差数列的第 n 项可以写成 第 n 项 = a + (n – 1)d，其中 a 为首项。在 Year 7 你更多看到像 2n + 1 这样的简单表达式。
To find the term-to-term rule, work out what is added or subtracted each time. For example, in 5, 8, 11, 14,…, the rule is ‘add 3’. The nth term is 3n + 2.
要找逐项变化规律，先看每次加或减了多少。例如在 5, 8, 11, 14,… 中，规律是“每次加 3”。第 n 项公式是 3n + 2。
6. Perimeter and Area | 周长与面积
Perimeter is the total distance around a 2D shape. For a rectangle with length l and width w, perimeter P = 2(l + w). For a square with side s, P = 4s.
周长是围绕一个二维图形一周的总长度。对于长为 l、宽为 w 的长方形，周长 P = 2(l + w)。对于边长为 s 的正方形，P = 4s。
Area measures the surface inside the shape. Rectangle: A = l × w. Square: A = s². Triangle: A = ½ × base × height, where the height is perpendicular to the base.
面积衡量图形内部的表面大小。长方形：A = l × w。正方形：A = s²。三角形：A = ½ × 底 × 高，这里的高与底边垂直。
For a compound shape, split it into simpler shapes, find each area separately, then add or subtract as needed.
对于组合图形，将其拆分成几个简单图形，分别求出各自的面积，再按需要相加或相减。
7. Properties of Angles | 角的性质
Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal.
平角（直线上的角）之和为 180°。周角（一个点周围的角）之和为 360°。对顶角相等。
In a triangle, the three interior angles always add up to 180°. In a quadrilateral, the interior angles add up to 360°.
三角形三个内角之和总是 180°。四边形内角之和为 360°。
Parallel lines create special angle relationships: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles add up to 180°.
平行线产生特殊的角度关系：内错角相等，同位角相等，同旁内角互补（和为 180°）。
8. Volume and Surface Area of 3D Shapes | 立体图形的体积与表面积
Volume measures the space inside a 3D object. For a cuboid with length l, width w and height h: Volume V = l × w × h. Surface area = 2(lw + lh + wh).
体积衡量一个三维物体内部的空间大小。对于长为 l、宽为 w、高为 h 的长方体：体积 V = l × w × h。表面积 = 2(lw + lh + wh)。
For a cube of side s: V = s³, surface area = 6s². Remember to give units: cm³ for volume, cm² for surface area.
对于边长为 s 的立方体：V = s³，表面积 = 6s²。记得写上单位：体积用 cm³，表面积用 cm²。
When using volume formulas, make sure all lengths are in the same unit first.
使用体积公式时，要确保所有长度单位一致。
9. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is written as a² + b² = c², where c is the hypotenuse.
在直角三角形中，斜边（直角所对的边）的平方等于两条直角边的平方和。此定理写作 a² + b² = c²，其中 c 为斜边。
You can use Pythagoras’ theorem to find the length of a missing side in right-angled triangles, and to check whether a triangle is right-angled.
你可以用勾股定理求直角三角形中未知边的长度，也可以用它检验一个三角形是否为直角三角形。
10. Averages and Range | 平均数与极差
The three common averages are the mean, median and mode. The mean is found by adding all values and dividing by the number of values. The median is the middle value when the data is in order. The mode is the most frequent value.
三种常用的平均数是均值、中位数和众数。均值是把所有数值相加再除以数据的个数。中位数是将数据按从小到大排序后位于中间位置的数。众数是出现次数最多的数。
The range is a measure of spread: range = largest value – smallest value. It tells you how spread out the data are.
极差是衡量数据分散程度的一个量：极差 = 最大值 – 最小值。它告诉你数据的分布有多广。
11. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It is always a number between 0 (impossible) and 1 (certain), and can be written as a fraction, decimal or percentage.
概率衡量一个事件发生的可能性有多大。它总是一个介于 0（不可能）与 1（必然）之间的数，可以用分数、小数或百分数表示。
Probability of an event = number of favourable outcomes ÷ total number of possible outcomes, provided all outcomes are equally likely.
事件概率 = 有利结果的数量 ÷ 所有可能结果的总数，前提是每个结果发生的可能性都相等。
The probabilities of all mutually exclusive outcomes in an experiment add up to 1. Probability of an event not happening = 1 – probability of the event happening.
一个试验中所有互斥结果的概率之和为 1。事件不发生的概率 = 1 – 事件发生的概率。
12. Coordinates and Graphs | 坐标与图像
A point on a grid is located by an ordered pair (x, y). The x-coordinate tells you how far to move horizontally, and the y-coordinate tells you how far to move vertically. The origin is (0,0).
网格上的一个点通过有序数对 (x, y) 来定位。x 坐标表示水平移动的距离，y 坐标表示垂直移动的距离。原点是 (0,0)。
A straight-line graph has an equation of the form y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
一次函数的图像是一条直线，其方程形式为 y = mx + c，其中 m 是斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴的交点）。
To draw a straight-line graph, create a table of values for x, work out the corresponding y values, plot the points and join them with a straight line.
画一次函数图像时，先列出 x 的值表，计算出对应的 y 值，在坐标平面上描点，再用直线将它们连接起来。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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