📚 Year 7 CCEA Further Mathematics Formula & Theorem Quick Reference | 七年级CCEA进阶数学公式定理速查手册
This quick reference handbook provides all the essential formulas and theorems needed for Year 7 CCEA Further Mathematics. It covers number operations, fractions, algebra, geometry, measurement, statistics, and probability. Each section is carefully structured to help you memorise and apply key rules effectively.
本速查手册涵盖了七年级CCEA进阶数学所需的所有基本公式和定理，内容包括数的运算、分数、代数、几何、测量、统计和概率。每个部分都经过精心编排，帮助你有效记忆并运用关键规则。
1. Order of Operations and Number Laws | 运算顺序与运算律
The order of operations is remembered by BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Left-to-right rule applies for division and multiplication, and for addition and subtraction.
运算顺序可用 BIDMAS 记忆：括号、指数、除法、乘法、加法、减法。除法和乘法、加法和减法需按从左到右的顺序计算。
Commutative law: a + b = b + a, a × b = b × a. This does not work for subtraction or division.
交换律：a + b = b + a，a × b = b × a。减法和除法不遵循交换律。
Associative law: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
结合律：(a + b) + c = a + (b + c)，(a × b) × c = a × (b × c)。
Distributive law: a × (b + c) = a × b + a × c. This is very useful for expanding brackets.
分配律：a × (b + c) = a × b + a × c。这在展开括号时非常有用。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
To add or subtract fractions, make the denominators the same using equivalent fractions. Then add or subtract the numerators and keep the denominator.
加减分数时，先用等值分数将分母化为相同，然后分子相加减，分母不变。
To multiply fractions: multiply the numerators together and the denominators together. Simplify where possible.
分数相乘：分子乘分子，分母乘分母，尽可能化简。
To divide fractions: multiply by the reciprocal. That is, a/b ÷ c/d = a/b × d/c.
分数相除：乘以倒数。即 a/b ÷ c/d = a/b × d/c。
Converting between fractions, decimals and percentages: a fraction to a decimal by division; decimal to percentage by multiplying by 100; percentage to fraction by writing over 100 and simplifying.
分数、小数和百分数互化：分数化小数用除法；小数化百分数乘以100；百分数化分数写成百分之几并化简。
Common equivalences to remember: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333… = 33⅓%, 1/10 = 0.1 = 10%.
需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… = 33⅓%，1/10 = 0.1 = 10%。
3. Ratio, Proportion and Scales | 比、比例和比例尺
A ratio compares quantities. It can be simplified by dividing all parts by their greatest common factor.
比用于比较数量。可以通过除以最大公因数来化简比。
To share a quantity in a given ratio, find the total number of parts, divide the quantity by that total, and then multiply by each part of the ratio.
按给定比例分配数量时，先求出总份数，用总量除以总份数，再乘以比的各部分。
Direct proportion: as one quantity increases, the other increases at the same rate. Represented by y = kx, where k is the constant of proportionality.
正比例：一个量增加，另一个以相同速率增加。表示为 y = kx，其中 k 是比例常数。
Scale in maps and models: scale = drawing length ÷ actual length. All measurements must be in the same units.
地图和模型的比例尺：比例尺 = 图上长度 ÷ 实际长度。所有测量单位必须一致。
4. Algebraic Expressions and Simplification | 代数表达式与化简
A term is a single number, a variable, or numbers and variables multiplied together. Like terms contain exactly the same variable part (e.g. 3x and -5x are like terms).
项指一个单独的数、一个变量，或数与变量的乘积。同类项包含完全相同的变量部分（例如 3x 和 -5x 是同类项）。
Collecting like terms: add or subtract the coefficients. For example, 4a + 2a = 6a, 7b – 3b = 4b.
合并同类项：将系数相加或相减。例如 4a + 2a = 6a，7b – 3b = 4b。
Expanding a bracket using the distributive law: a(b + c) = ab + ac. For double brackets: (x + a)(x + b) = x² + (a+b)x + ab.
运用分配律展开括号：a(b + c) = ab + ac。双层括号展开：(x + a)(x + b) = x² + (a+b)x + ab。
Factorising an expression: find the greatest common factor and write the expression as a product. e.g. 6x + 9 = 3(2x + 3).
因式分解：找出最大公因数，将表达式写成乘积形式。例如 6x + 9 = 3(2x + 3)。
5. Solving Linear Equations | 解线性方程
The goal is to isolate the variable on one side of the equation. Always do the same operation to both sides to keep the equation balanced.
目的是将变量单独解在方程的一边。始终对方程两边执行相同运算以保持等式平衡。
Steps: simplify each side if needed; use inverse operations to remove added or subtracted terms, then remove multiplication or division.
步骤：必要时每边先化简；用逆运算消去加减项，再消去乘除项。
Example: 3x + 5 = 17. Subtract 5: 3x = 12. Divide by 3: x = 4.
例如：3x + 5 = 17。减去5：3x = 12。除以3：x = 4。
Checking solution: substitute the value back into the original equation to verify both sides are equal.
检验解：将求出的值代回原方程，验证两边相等。
6. Sequences and the nth Term | 数列与第n项
A sequence is a list of numbers following a rule. In an arithmetic sequence the difference between consecutive terms is constant (common difference).
数列是按规则排列的一列数。在等差数列中，相邻两项的差是常数（公差）。
The nth term of an arithmetic sequence is given by: nth term = a + (n-1)d, where a is the first term and d is the common difference.
nth term = a + (n-1)d
等差数列的第n项公式为：第n项 = a + (n-1)d，其中a为首项，d为公差。
To generate a sequence from the nth term, substitute n = 1, 2, 3, … into the expression.
根据第n项公式生成数列，将 n=1,2,3… 代入表达式即可。
Finding the rule for a sequence: find the common difference and adjust to match the first term.
寻找数列通项规则：先找公差，再调整以匹配首项。
7. Angles and Basic Geometry | 角与基础几何
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
平角（直线上的角）和为180°。绕一点一周的角和为360°。
Vertically opposite angles are equal. When two lines intersect, opposite angles are the same.
对顶角相等。两直线相交时，对顶角相等。
Angles in a triangle add up to 180°. In an equilateral triangle all angles are 60°. In an isosceles triangle the base angles are equal.
三角形内角和为180°。等边三角形每个角都是60°。等腰三角形两底角相等。
Parallel lines: alternate angles are equal, corresponding angles are equal, co-interior (allied) angles sum to 180°.
平行线：内错角相等，同位角相等，同旁内角互补（和为180°）。
8. Perimeter and Area of 2D Shapes | 二维图形的周长与面积
Perimeter is the distance around a shape. For a rectangle: P = 2(l + w), where l is length and w is width.
P = 2(l + w)
周长是围绕图形的距离。矩形的周长：P=2(l+w)，其中 l 为长，w 为宽。
Area of a rectangle: A = l × w.
A = l × w
矩形面积：A = l × w。
Area of a triangle: A = ½ × base × height, or A = ½ bh. The height must be perpendicular to the base.
A = ½ bh
三角形面积：A = ½ × 底 × 高，或 A = ½ bh。高必须垂直于底。
Area of a parallelogram: A = base × perpendicular height.
A = b × h
平行四边形面积：A = 底 × 高（垂直高）。
Area of a trapezium: A = ½ (a + b)h, where a and b are the parallel sides and h is the perpendicular height.
A = ½ (a + b)h
梯形面积：A = ½ (a + b)h，其中 a 和 b 是平行边，h 是高。
Circumference of a circle: C = 2πr or C = πd, where r is radius and d is diameter. Use π ≈ 3.14 or 22/7.
C = 2πr = πd
圆的周长：C = 2πr 或 C = πd，其中 r 为半径，d 为直径。π 常用 3.14 或 22/7。
Area of a circle: A = πr².
A = πr²
圆的面积：A = πr²。
9. Volume and Surface Area of 3D Shapes | 立体图形的体积与表面积
Volume of a cuboid (rectangular prism): V = l × w × h, where l, w, h are length, width, height.
V = lwh
长方体体积：V = 长 × 宽 × 高。
Surface area of a cuboid: SA = 2(lw + lh + wh). It is the sum of the areas of all six rectangular faces.
SA = 2(lw + lh + wh)
长方体表面积：SA = 2(lw + lh + wh)。即所有六个矩形面的面积之和。
Volume of a prism: V = area of cross-section × length. For a triangular prism, the cross-section is a triangle.
V = Ah
棱柱体积：V = 底面积 × 高（长度）。对于三棱柱，底面是三角形。
Units of volume: cm³, m³, mm³. Remember that 1 litre = 1000 cm³ and 1 m³ = 1,000,000 cm³.
体积单位：cm³, m³, mm³。记住 1 升 = 1000 cm³，1 m³ = 1,000,000 cm³。
10. Statistics: Averages and Range | 统计：平均数与极差
The mean is found by adding all data values and dividing by the number of values. Mean = sum of data ÷ number of data points.
Mean = Σx ÷ n
平均数：将所有数据相加，除以数据个数。平均数 = 总和 ÷ 个数。
The median is the middle value when data is ordered. If there is an even number of data points, median is the mean of the two middle numbers.
中位数是将数据排序后位于中间的值。如果数据个数为偶数，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.
众数是出现次数最多的值。一组数据可以有一个众数、多个众数，或者没有众数。
The range is the difference between the largest and smallest values. Range = maximum – minimum.
Range = max − min
极差是最大数据与最小数据的差。极差 = 最大值 − 最小值。
When working with frequency tables, multiply each value by its frequency to find the total sum, then divide by total frequency to find the mean.
使用频数表时，将每个数值乘以它的频数得到总和，再除以总频数求平均数。
11. Probability Basics | 概率基础
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
概率衡量事件发生的可能性，范围从0（不可能）到1（必然）。可用分数、小数或百分数表示。
For equally likely outcomes: P(event) = number of favourable outcomes ÷ total number of outcomes.
P(A) = favourable outcomes / total outcomes
对于等可能结果：P(事件) = 有利结果数 ÷ 所有可能结果总数。
The sum of probabilities of all possible outcomes is 1. The probability of an event not happening = 1 − P(event).
所有可能结果的概率之和为1。事件不发生的概率 = 1 − P(事件发生)。
Expected frequency: expected number of occurrences = probability × number of trials.
Expected = P × n
期望频数：期望的次数 = 概率 × 试验次数。
12. Coordinates and Linear Graphs | 坐标与一次函数图像
Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. The origin is (0,0).
坐标记为 (x, y)，x 是水平位置，y 是垂直位置。坐标原点为 (0,0)。
In a linear graph, the equation is usually given as y = mx + c, where m is the gradient and c is the y-intercept.
y = mx + c
一次函数图像方程常写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
The gradient (m) is calculated by: m = change in y ÷ change in x = (y₂ − y₁) ÷ (x₂ − x₁).
m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)
斜率 (m) 计算公式：m = y 的变化 ÷ x 的变化 = (y₂ − y₁) / (x₂ − x₁)。
Plotting a linear graph: choose at least three x-values, calculate the corresponding y-values using the equation, plot the points, and draw a straight line through them.
绘制一次函数图像：选择至少三个 x 值，代入方程计算对应 y 值，描点，过点画直线。
Coordinates can also be used to reflect shapes across the axes, simply by changing the sign of the appropriate coordinate.
坐标还可用于图形对称：关于坐标轴反射，只需改变相应坐标的正负号。
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