📚 Year 7 CIE Mathematics: Formulas & Theorems Quick Reference | CIE 七年级数学公式定理速查手册
This quick-reference handbook covers all the essential formulas and theorems required for Year 7 CIE Mathematics. It is designed to help students consolidate key concepts in number, algebra, geometry, statistics, and probability. Keep it handy for revision and exam preparation.
本速查手册涵盖了CIE七年级数学所需的所有基本公式和定理，旨在帮助学生巩固数字、代数、几何、统计和概率等关键概念。便于随时复习和备考。
1. Integers and Order of Operations | 整数与运算顺序
Integers include positive whole numbers, zero, and negative whole numbers. When performing calculations, always follow the order of operations known as BIDMAS: Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
整数包括正整数、零和负整数。进行计算时，务必遵循运算顺序BIDMAS：先算括号，再算指数，然后进行除法和乘法（从左到右），最后进行加法和减法（从左到右）。
Example: 3 + 4 × 2 = 3 + 8 = 11, not 14. Without following the order, you would incorrectly add first and then multiply.
例如：3 + 4 × 2 = 3 + 8 = 11，而不是14。如果不按顺序先加后乘，就会错误地得到14。
ices (powers) show repeated multiplication. For any non-zero integer a: a¹ = a and a⁰ = 1.
指数（幂）表示重复乘法。对于任何非零整数a：a¹ = a 且 a⁰ = 1。
a⁰ = 1 (a ≠ 0)
2. Fractions, Decimals, and Percentages | 分数、小数和百分数
A fraction represents part of a whole. Equivalent fractions have the same value but different numerators and denominators. The following conversions are essential.
分数表示整体的一部分。等值分数虽然分子分母不同，但数值相同。以下转换关系非常重要。
| Conversion (English) | 转换（中文） |
|---|---|
| Fraction to decimal: divide numerator by denominator | 分数转小数：分子除以分母 |
| Decimal to percentage: multiply by 100% | 小数转百分数：乘以100% |
| Percentage to decimal: divide by 100 | 百分数转小数：除以100 |
| Percentage of a quantity: (p/100) × total | 求总量的百分之几：(p/100) × 总量 |
When adding or subtracting fractions, first find a common denominator. To multiply fractions, multiply the numerators and the denominators separately. To divide by a fraction, multiply by its reciprocal.
分数加减时，先找到公分母。分数乘法，分子乘分子、分母乘分母。分数除法，乘以除数的倒数。
a/b + c/d = (ad + bc)/bd | a/b × c/d = ac/bd | a/b ÷ c/d = a/b × d/c
3. Algebraic Expressions | 代数表达式
Algebra uses letters (variables) to represent numbers. A term is a number, a variable, or a product of numbers and variables. Like terms have the same variable part and can be combined by adding or subtracting their coefficients.
代数学中用字母（变量）表示数。项是数、变量或数与变量的乘积。同类项具有相同的字母部分，可以通过加减它们的系数来合并。
Key rules for manipulating algebraic expressions:
处理代数表达式的关键法则：
Distributive law: a(b + c) = ab + ac
分配律：a(b + c) = ab + ac
Combining like terms: 2a + 3a = 5a
合并同类项：2a + 3a = 5a
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ
When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power: (aᵐ)ⁿ = aᵐⁿ
幂的乘方：(aᵐ)ⁿ = aᵐⁿ
Always simplify expressions by removing brackets and collecting like terms before solving equations.
在解方程之前，务必去括号、合并同类项，将表达式化简。
4. Linear Equations | 一元一次方程
, where a, b, and c are numbers and a ≠ 0. The goal is to isolate the variable x using inverse operations.
一元一次方程形式为 ax + b = c，其中a、b、c为已知数且a ≠ 0。解方程的目标是利用逆运算将变量x单独留在等号一边。
lve 2x + 3 = 11: subtract 3 from both sides → 2x = 8, then divide both sides by 2 → x = 4.
解方程 2x + 3 = 11：两边同时减3 → 2x = 8，然后两边同时除以2 → x = 4。
If ax + b = c, then x = (c − b) / a
nto the original equation. For the above example, 2(4) + 3 = 8 + 3 = 11, which is correct.
总是将解代入原方程进行检验。如上例，2(4) + 3 = 8 + 3 = 11，正确。
5. Angles and Lines | 角与直线
An angle measures the amount of turn between two lines meeting at a vertex. Angles are measured in degrees (°).
角度量的是两条线段在顶点处的张合程度，单位为度 (°)。
Types of angles:
角的类型：
Acute angle: less than 90°
锐角：小于90°
Right angle: exactly 90°
直角：等于90°
Obtuse angle: between 90° and 180°
钝角：大于90°且小于180°
Straight angle: exactly 180°
平角：等于180°
Reflex angle: between 180° and 360°
优角：大于180°且小于360°
Full turn: 360°
周角：360°
Angle facts on lines and around a point:
关于直线与点周围的角的基本事实：
Angles on a straight line sum to 180°.
直线上的角之和为180°。
Angles around a point sum to 360°.
绕一点一周的角之和为360°。
Vertically opposite angles are equal.
对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and co-interior (allied) angles sum to 180°.
一条横截线与两条平行线相交时，同位角相等，内错角相等，同旁内角之和为180°。
6. Triangles and Quadrilaterals | 三角形与四边形
The sum of the interior angles in any triangle is 180°. The sum of interior angles in any quadrilateral is 360°.
任何三角形的内角和都是180°。任何四边形的内角和都是360°。
Triangle: A + B + C = 180° | Quadrilateral: A + B + C + D = 360°
Special triangles:
特殊三角形：
Equilateral triangle: all three sides equal, each interior angle 60°.
等边三角形：三条边都相等，每个内角为60°。
Isosceles triangle: two sides equal, base angles are equal.
等腰三角形：两条边相等，底角相等。
Right-angled triangle: one angle is 90°. The hypotenuse is the longest side.
直角三角形：有一个角为90°。斜边是最长边。
Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums, and kites. Their properties are based on side lengths, parallel sides, and angle measures.
四边形包括正方形、长方形、平行四边形、菱形、梯形和筝形。它们的性质取决于边长、平行边和角度大小。
7. Perimeter, Area, and Volume | 周长、面积与体积
Perimeter is the distance around a 2D shape. Area measures the surface enclosed. Volume measures the space occupied by a 3D solid.
周长是二维图形边界的总长度。面积测量的是图形内部的表面大小。体积测量的是三维立体所占空间的多少。
| Shape | Perimeter / Area / Volume | 图形 | 周长/面积/体积 |
|---|---|---|---|
| Rectangle | P = 2(l + w), A = lw | 矩形 | 周长 = 2(长 + 宽)，面积 = 长 × 宽 |
| Triangle | A = ½bh | 三角形 | 面积 = ½ × 底 × 高 |
| Parallelogram | A = bh | 平行四边形 | 面积 = 底 × 高 |
| Trapezium | A = ½(a + b)h | 梯形 | 面积 = ½ × (上底 + 下底) × 高 |
| Cuboid | V = lwh | 长方体 | 体积 = 长 × 宽 × 高 |
For composite shapes, break them into simpler shapes, calculate individual areas, and then add or subtract as needed.
对于组合图形，将其拆分成简单图形，分别计算面积，再相加或相减。
8. Ratios and Proportions | 比与比例
A ratio compares two or more quantities. Ratios can be simplified by dividing all parts by their highest common factor.
比用来比较两个或多个量。比可以通过将各部分除以它们的最大公因数进行化简。
Example: 8:12 simplifies to 2:3 (divide both by 4). Ratios do not have units.
例如：8:12化简为2:3（同时除以4）。比没有单位。
ns two ratios are equal. In a proportion a:b = c:d, the cross products are equal: ad = bc.
比例表示两个比相等。在比例式 a:b = c:d 中，交叉乘积相等：ad = bc。
If a:b = c:d, then ad = bc
When sharing a quantity in a given ratio, add the parts, then divide the total by that sum to find the value of one part. Multiply by each ratio number to get the shares.
当按给定的比分配一个量时，先将比的各项相加，用总和除以项数之和得到一个单位量，再乘以比的各项得到各部分。
9. Averages and Range | 平均数与极差
Three types of average are used in statistics: mean, median, and mode. The range measures how spread out the data are.
统计学中用到三种平均数：平均数（均值）、中位数和众数。极差用来衡量数据的分散程度。
Mean: sum of all values divided by the number of values.
平均数（均值）：所有数值之和除以数值的个数。
Median: the middle value when the data are arranged in order. If there are two middle values, take their mean.
中位数：将数据按大小顺序排列后位于中间的数。如果中间有两个数，则取它们的平均数。
Mode: the value that appears most often. There can be more than one mode or no mode at all.
众数：出现次数最多的数值。可以有一个以上的众数，也可能没有众数。
Range: largest value − smallest value.
极差：最大值 − 最小值。
he set 3, 7, 7, 2, 5: mean = (3+7+7+2+5)/5 = 4.8, median = 5, mode = 7, range = 7 − 2 = 5.
例如，在数据组3, 7, 7, 2, 5中：均值 = (3+7+7+2+5)/5 = 4.8，中位数 = 5，众数 = 7，极差 = 7 − 2 = 5。
10. Data Representation | 数据表示
Data can be displayed in various charts and graphs to make patterns and comparisons easier to see.
数据可以用多种图表展示，以便于观察规律和进行比较。
Bar chart: uses bars of equal width to show frequencies. Gaps between bars for discrete data.
条形图：用等宽的条形表示频数。离散数据的条形之间留有空隙。
chart: a circle divided into sectors where each sector angle = (frequency / total) × 360°.
饼图：将一个圆分成扇形，每个扇形的圆心角 = (频数 / 总频数) × 360°。
Line graph: points connected by lines to show trends over time.
折线图：将点用线连接，用于显示随时间变化的趋势。
Frequency table: a table listing data values and the number of times each value occurs.
频数表：列出各个数值及其出现次数的表格。
When reading a chart, always check the scales on the axes and the key or labels to understand what the data represent.
阅读图表时，务必检查坐标轴上的刻度和图例或标签，以理解数据代表什么。
11. Probability Basics | 概率基础
Probability is a measure of how likely an event is to happen. It ranges from 0 (impossible) to 1 (certain).
概率是衡量事件发生可能性大小的量。范围从0（不可能）到1（一定发生）。
robability of an event = Number of favourable outcomes / Total number of possible outcomes
事件的概率 = 有利结果的数量 / 所有可能结果的总数
All possible outcomes are called the sample space. When outcomes are equally likely, the above formula applies. Probability can be written as a fraction, decimal, or percentage.
所有可能的结果构成的集合叫样本空间。当每个结果等可能发生时，可使用上述公式。概率可以用分数、小数或百分数表示。
ling a fair six-sided die, the probability of rolling an even number is 3/6 = ½ = 0.5 = 50%.
例如：掷一个均匀的六面骰子，掷出偶数的概率是3/6 = ½ = 0.5 = 50%。
12. Symmetry and Transformations | 对称与变换
A shape has line symmetry if it can be folded along a line so that the two halves match exactly. This line is called a line of symmetry.
如果一个图形能够沿一条直线对折，两侧完全重合，则称该图形具有线对称。这条直线叫对称轴。
Rotational symmetry occurs when a shape fits onto itself in less than a full turn. The number of times it fits is the order of rotational symmetry.
如果一个图形绕中心旋转小于一周的角度后能与自身重合，则称该图形具有旋转对称。重合的次数叫旋转对称的阶数。
Transformations map a shape onto a new position. The four types are:
变换将一个图形移动到新的位置。四种基本变换是：
Translation: sliding a shape without turning or flipping. All points move the same distance in the same direction.
平移：只滑动不旋转、不翻转。所有点沿同一方向移动相同距离。
Reflection: flipping a shape over a mirror line. The image is the same perpendicular distance from the mirror line as the object.
反射（翻折）：沿镜面翻转，像上的每一点到镜面的垂直距离与原图形对应点到镜面的距离相等。
Rotation: turning a shape around a fixed centre by a given angle (90°, 180°, etc.) clockwise or anticlockwise.
旋转：绕一个固定中心按给定角度（90°，180°等）顺时针或逆时针转动。
Enlargement: changing the size of a shape by a scale factor from a centre of enlargement. If scale factor > 1, the shape gets bigger; if between 0 and 1, it gets smaller.
放大：从放大中心出发，按比例因子改变图形大小。比例因子大于1放大，介于0和1之间则缩小。
In a reflection, the mirror line is the perpendicular bisector of the segment joining corresponding points. In a rotation, the distance from the centre to each point remains the same.
在反射中，镜面是连接对应点线段的垂直平分线。在旋转中，旋转中心到各点的距离保持不变。
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