📚 Year 7 WJEC Further Maths: Quick Reference Formula & Theorem Handbook | Year 7 WJEC 进阶数学：公式定理速查手册
Welcome to your handy revision guide for Year 7 WJEC Further Mathematics. This handbook collects the essential formulas, theorems, and key rules you will need throughout the year. Use it to check definitions, refresh your memory before a test, or support your homework. Every topic is broken down into clear statements, so you can learn or revise at your own pace.
欢迎使用这本实用的 Year 7 WJEC 进阶数学复习指南。本手册汇集了整个学年中你需要掌握的基本公式、定理和重要规则。你可以用它来查阅定义、考前突击记忆，或在写作业时寻求支持。每个主题都以清晰的要点形式展开，方便你按照自己的节奏学习或复习。
1. Number Properties | 数的性质
A prime number has exactly two distinct factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19.
质数恰好有两个不同的因数：1 和它本身。最开始的几个质数是 2、3、5、7、11、13、17、19。
The highest common factor (HCF) of two or more numbers is the largest number that divides into all of them without a remainder. The lowest common multiple (LCM) is the smallest number that is a multiple of all the given numbers.
两个或更多个数的最大公因数 (HCF) 是可以整除所有这些数的最大的数。最小公倍数 (LCM) 是所有这些数的倍数中最小的一个数。
iplying an integer by itself twice. For example, 4² = 4 × 4 = 16, and 3³ = 3 × 3 × 3 = 27.
一个整数自乘得到平方数。一个整数连续两次自乘得到立方数。例如，4² = 4 × 4 = 16，而 3³ = 3 × 3 × 3 = 27。
ents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. A power raised to another power multiplies the exponents: (aᵐ)ⁿ = aᵐˣⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐˣⁿ。
Always follow the order of operations: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Remember it as BIDMAS.
始终遵循运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。记住口诀 BIDMAS。
2. Fractions, Decimals & Percentages | 分数、小数与百分数
= 3 ÷ 4 = 0.75. To convert a decimal to a percentage, multiply by 100. So 0.75 × 100 = 75%.
把分数化为小数，用分子除以分母。例如，¾ = 3 ÷ 4 = 0.75。把小数化为百分数，乘以 100。所以 0.75 × 100 = 75%。
To add or subtract fractions, rewrite them with a common denominator. Then add or subtract the numerators and keep the denominator unchanged.
分数的加减，先通分化为同分母，然后分子相加减，分母保持不变。
To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the result if possible.
分数相乘，将分子相乘的积作分子，分母相乘的积作分母，能约分的要化简。
To divide by a fraction, multiply by its reciprocal. For instance, ⅔ ÷ ¼ = ⅔ × ⁴⁄₁ = ⁸⁄₃.
除以一个分数，等于乘以它的倒数。例如，⅔ ÷ ¼ = ⅔ × ⁴⁄₁ = ⁸⁄₃。
A percentage increase or decrease can be found using a multiplier. For a 15% increase, multiply by 1.15. For a 20% decrease, multiply by 0.80.
百分比增减可以用乘数来计算。上涨 15%，乘以 1.15；下降 20%，乘以 0.80。
3. Algebraic Expressions | 代数表达式
An algebraic expression contains numbers, variables, and operation symbols. A term is a single number, a variable, or numbers and variables multiplied together. Like terms have exactly the same variable part and can be combined.
代数表达式包含数字、变量和运算符。项是一个单独的数、一个变量或多个数和变量相乘的结果。同类项具有完全相同的变量部分，可以合并。
sion, collect like terms by adding or subtracting their coefficients. For example, 3x + 5y – x + 2y = 2x + 7y.
化简表达式，通过合并同类项，将它们的系数相加或相减。例如，3x + 5y – x + 2y = 2x + 7y。
+ c) = ab + ac. For two brackets, use the FOIL method: (x + a)(x + b) = x² + (a+b)x + ab.
去括号时，用括号外的项乘以括号内的每一项。单项乘括号：a(b + c) = ab + ac。两个括号相乘，用“先外后内再交叉”的方法：(x + a)(x + b) = x² + (a+b)x + ab。
factor of all terms and write it in front of a bracket. For instance, 6x + 9 = 3(2x + 3).
因式分解是去括号的逆运算。寻找所有项的最高公因式，并写在括号前面。例如，6x + 9 = 3(2x + 3)。
4. Equations & Inequalities | 方程与不等式
An equation shows that two expressions are equal. To solve an equation, perform the same operation on both sides to isolate the variable. Always keep the equation balanced.
方程表示两个表达式相等。解方程时，在等号两边同时进行相同的运算，以把变量单独解出来。始终保持等式平衡。
tract 5 from both sides to get x = 7. For 3x = 18, divide both sides by 3 to obtain x = 6.
对于一步方程，如 x + 5 = 12，两边同时减去 5，得 x = 7。对于 3x = 18，两边同时除以 3，得 x = 6。
uation like 2x + 3 = 11, subtract 3 from both sides (2x = 8) and then divide by 2 (x = 4).
对于两步方程，如 2x + 3 = 11，先从两边减去 3（2x = 8），再除以 2（x = 4）。
An inequality compares two expressions and uses symbols such as <, >, ≤, ≥. To solve an inequality, follow the same steps as for equations, but remember: if you multiply or divide by a negative number, you must reverse the inequality sign.
不等式用来比较两个表达式，使用 <、>、≤、≥ 等符号。解不等式时，按解方程的步骤进行，但要记住：如果乘以或除以一个负数，必须反转不等号的方向。
Represent an inequality on a number line using an open circle for < or > and a closed circle for ≤ or ≥. Shade the region that satisfies the inequality.
在数轴上表示不等式，用空心圆圈表示 < 或 >，用实心圆点表示 ≤ 或 ≥。然后将满足不等式的区域涂上阴影。
5. Sequences & Patterns | 序列与模式
A sequence is an ordered list of numbers following a rule. The term-to-term rule describes how to get from one term to the next. The position-to-term rule (nth term) lets you find any term directly.
数列是根据某种规则排列的一列数。逐项规则描述如何从上一项得到下一项。由位到项的规则（第 n 项）可以直接求出任意一项。
difference is the common difference, d. The nth term can be written as: nth term = dn + (a – d), where a is the first term.
在等差数列里，相邻两项之间的差是常数，这个差叫做公差 d。第 n 项可以写为：第 n 项 = dn + (a – d)，其中 a 是首项。
term. For example, for the sequence 5, 8, 11, 14, ..., d = 3, a = 5, so nth term = 3n + 2.
找第 n 项时，用首项减去公差得到第零项。例如，对数列 5, 8, 11, 14, ...，d = 3，a = 5，那么第 n 项 = 3n + 2。
Other types of sequences include square numbers (1, 4, 9, 16, ..., n²) and cube numbers (1, 8, 27, ..., n³).
其他类型的数列包括平方数数列（1, 4, 9, 16, ...，n²）和立方数数列（1, 8, 27, ...，n³）。
6. Angles & Lines | 角与线
Angles are measured in degrees (°). An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°.
角以度（°）为单位。锐角小于 90°，直角等于 90°，钝角介于 90° 到 180° 之间，优角大于 180°。
Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal.
平角（直线上邻角）之和为 180°。绕一点一周的角之和为 360°。对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.
两条平行线被一条截线所截，同位角相等，内错角相等，同旁内角互补（和为 180°）。
The sum of the interior angles of a triangle is 180°. The sum of the interior angles of a quadrilateral is 360°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为 180°。四边形内角和为 360°。三角形的一个外角等于其两个相对内角之和。
7. Perimeter, Area & Volume | 周长、面积与体积
d the outside of a shape. For a rectangle: P = 2(l + w), where l is length and w is width.
周长是一个图形外边界的总长度。矩形的周长：P = 2(l + w)，其中 l 为长，w 为宽。
Area measures the space inside a 2D shape. Key formulas:
Rectangle: A = l × w
Triangle: A = ½ × b × h
面积衡量二维图形内部的平面空间。重要公式：
矩形：A = l × w
三角形：A = ½ × b × h
Parallelogram: A = b × h
Trapezium: A = ½ (a + b)h
Circle: C = 2πr, A = πr² (π ≈ 3.14)
平行四边形：A = b × h
梯形：A = ½ (a + b)h
圆：C = 2πr, A = πr²（π 约等于 3.14）
id occupies. For a cuboid: V = l × w × h. For a prism: V = area of cross-section × length.
体积是三维立体所占空间的大小。长方体的体积：V = l × w × h。棱柱的体积：V = 底面积 × 高（长）。
8. Coordinates & Graphs | 坐标与图形
Points are plotted on a Cartesian plane using an ordered pair (x, y). The x-coordinate shows the horizontal position, the y-coordinate shows the vertical position.
点在直角坐标系中用有序对 (x, y) 来确定位置。x 坐标表示水平位置，y 坐标表示竖直位置。
+ c, where m is the gradient and c is the y-intercept (where the line crosses the y-axis).
直线方程常写为 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴的交点）。
To find the gradient between two points (x₁, y₁) and (x₂, y₂), use m = (y₂ – y₁) / (x₂ – x₁).
求两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率，用 m = (y₂ – y₁) / (x₂ – x₁)。
₁) and (x₂, y₂) is given by ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between them is √[(x₂ – x₁)² + (y₂ – y₁)²].
连接 (x₁, y₁) 和 (x₂, y₂) 线段的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点之间的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。
9. Statistics & Probability | 统计与概率
The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value. The range is the difference between the maximum and minimum.
平均数（均值）是所有数值的总和除以数值的个数。中位数是将数据排序后位于中间的数。众数是出现次数最多的值。极差是最大值与最小值之差。
Probability measures how likely an event is to happen. It is always between 0 (impossible) and 1 (certain).
P(event) = Number of favourable outcomes / Total number of possible outcomes
概率衡量一个事件发生的可能性大小，始终介于 0（不可能）和 1（必然）之间。
P(事件) = 有利结果数 / 所有可能结果的总数
A and B, P(A or B) = P(A) + P(B). The sum of probabilities of all possible outcomes is 1.
互斥事件不可能同时发生。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。所有可能结果的概率之和为 1。
In experimental probability, the relative frequency of an event is the number of times it occurs divided by the total number of trials. As the number of trials increases, relative frequency approaches the theoretical probability.
在实验概率中，一个事件的相对频率等于它发生的次数除以试验总次数。随着试验次数增加，相对频率会趋近于理论概率。
10. Ratio & Proportion | 比与比例
A ratio compares the sizes of two or more quantities. Simplify a ratio by dividing all parts by their highest common factor. For example, 8:12 simplifies to 2:3.
比用来比较两个或两个以上数量的大小。化简比时，用各项的最大公因数去除各项。例如，8:12 化简为 2:3。
To share a quantity in a given ratio, add the parts of the ratio to find the total number of shares. Divide the quantity by this total and then multiply by each part.
按给定比例分配一个数量时，先把比的各项相加得到总份数，然后用总量除以总份数，再乘以各部分的份数。
ty increases, the other increases at the same rate. Their relationship can be written as y ∝ x, or y = kx, where k is the constant of proportionality.
正比例意味着一个量增加时，另一个量以相同速率增加。它们的关系可写为 y ∝ x，或 y = kx，其中 k 是比例常数。
e quantity increases and the other decreases at a constant product. It can be written as y ∝ 1/x, or y = k/x.
反比例发生在当一个量增加而另一个量减少，且它们的乘积不变时。可写为 y ∝ 1/x，或 y = k/x。
Scale drawings and maps use ratios to represent real distances. A scale of 1:100 means that 1 cm on the drawing represents 100 cm in real life.
比例图和地图使用比来表示实际距离。比例尺 1:100 表示图上 1 cm 代表实际 100 cm。
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