📚 Year 7 AQA Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 AQA 数学：公式定理速查手册
This quick reference handbook gathers the essential formulas, rules and theorems you will meet in Year 7 AQA Mathematics. Use it alongside your classwork to build confidence, check facts and prepare for assessments. Each topic is presented with a clear statement in English followed immediately by its Chinese equivalent, so you can study comfortably in both languages.
本速查手册汇集了 Year 7 AQA 数学所涉及的核心公式、法则与定理。你可以在学习过程中随时查阅，巩固知识、核对要点，为课堂测验和考试做好准备。每个知识点都先给出英文表述，紧接着提供中文对照，方便双语学习。
1. Number & Place Value | 数字与位值
The value of a digit depends on its place in the number. In 4 372, the digit 3 stands for 300 because it is in the hundreds column.
数字的大小由它所在的数位决定。在 4 372 中，数字 3 表示 300，因为它在百位上。
To round a whole number to the nearest 10, look at the units digit. If it is 5 or more, round up; if it is 4 or less, round down. For example, 36 rounds to 40, and 34 rounds to 30.
将整数四舍五入到最近的 10，看个位数字。如果是 5 或更大，就向上入；如果是 4 或更小，就向下舍。例如 36 入到 40，34 舍到 30。
Negative numbers are all the numbers less than zero. On a number line, moving left means the value decreases. The inequality signs are < (less than) and > (greater than). For instance, -5 < -2.
负数是所有小于零的数。在数轴上，向左移动表示数值减小。不等号有 < (小于) 和 > (大于)。例如 -5 < -2。
2. Addition & Subtraction | 加法与减法
Addition is commutative: a + b = b + a. For example, 7 + 9 = 9 + 7.
加法满足交换律：a + b = b + a。例如 7 + 9 = 9 + 7。
Subtraction is not commutative: a – b and b – a are usually different. Order matters, so always subtract the second number from the first.
减法不满足交换律：a – b 与 b – a 通常不相等。顺序很重要，总是从第一个数里减去第二个数。
When adding or subtracting decimals, align the decimal points and place values before working column by column. Missing places can be filled with zeros.
进行小数加减时，先将小数点对齐，再按数位列竖式计算。缺位数可以用零补足。
The inverse of addition is subtraction. If a + b = c, then c – b = a. This link helps check answers.
加法的逆运算是减法。如果 a + b = c，那么 c – b = a。利用这一关系可以验算。
3. Multiplication & Division | 乘法与除法
Multiplication is commutative: a × b = b × a. For example, 6 × 8 = 8 × 6.
乘法满足交换律：a × b = b × a。例如 6 × 8 = 8 × 6。
Multiplication is also associative: (a × b) × c = a × (b × c). This means that when you multiply three or more numbers, the grouping does not change the product.
乘法也满足结合律：(a × b) × c = a × (b × c)。也就是说三个或更多数相乘时，分组方式不会改变积。
The distributive law links multiplication and addition: a × (b + c) = a × b + a × c. For instance, 4 × (20 + 3) = 4 × 20 + 4 × 3.
分配律将乘法与加法联系起来：a × (b + c) = a × b + a × c。例如 4 × (20 + 3) = 4 × 20 + 4 × 3。
Division is the inverse of multiplication. If a × b = c (and b ≠ 0), then c ÷ b = a. The relation dividend = divisor × quotient + remainder holds for whole-number division.
除法是乘法的逆运算。如果 a × b = c (且 b ≠ 0)，那么 c ÷ b = a。对于整数除法，有 被除数 = 除数 × 商 + 余数。
Multiplying by 10, 100 or 1000 moves all digits to the left by 1, 2 or 3 places. Dividing by these powers of 10 moves digits to the right.
乘以 10、100 或 1000 将使所有数字向左移动 1、2 或 3 位。除以这些 10 的幂次方则使数字向右移动。
4. Fractions, Decimals & Percentages | 分数、小数与百分比
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.
将分数化为小数，用分子除以分母。例如 3/8 = 3 ÷ 8 = 0.375。
To convert a decimal to a percentage, multiply by 100 and add the % sign. 0.47 becomes 47%.
将小数化为百分数，乘以 100 并加上百分号。0.47 就是 47%。
A percentage is a fraction with denominator 100. So 25% = 25/100 = 1/4. To find a percentage of a quantity, multiply the quantity by the fraction or decimal equivalent.
百分数是分母为 100 的分数，因此 25% = 25/100 = 1/4。求一个数量的百分之几，用该数量乘以对应的分数或小数。
To add or subtract fractions with the same denominator, keep the denominator and add or subtract the numerators. a/c + b/c = (a+b)/c.
同分母分数相加减，分母不变，分子相加减。a/c + b/c = (a+b)/c。
To multiply fractions, multiply the numerators and multiply the denominators: a/b × c/d = (a×c)/(b×d). Simplify the result if possible.
分数乘法：分子乘分子，分母乘分母：a/b × c/d = (a×c)/(b×d)。结果能约分就约分。
To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.
除以一个分数，等于乘以它的倒数：a/b ÷ c/d = a/b × d/c。
A fraction can be simplified by dividing the numerator and denominator by their greatest common factor. Equivalent fractions are made by multiplying or dividing both parts by the same non‑zero number.
分数可以通过分子分母同除以它们的最大公因数来约分。等值分数则是将分子分母同乘或同除以同一个非零数得到。
5. Algebraic Expressions | 代数表达式
Letters stand for unknown numbers. A term is a number, a letter or a product; e.g. 5y, 3ab, -2x². Like terms have exactly the same variable part and can be combined: 3a + 2a = 5a.
字母表示未知数。项可以是数字、字母或它们的乘积，如 5y、3ab、-2x²。同类项具有完全相同的字母部分，可以合并：3a + 2a = 5a。
When multiplying terms, multiply the numbers and then the letters, writing letters in alphabetical order. a × b = ab, 3 × p × q = 3pq.
项相乘时，先将数字相乘，再将字母相乘，并按字母表顺序书写。a × b = ab，3 × p × q = 3pq。
Using brackets means everything inside is multiplied by what is outside: 2(x + 5) = 2x + 10. This is expanding.
使用括号表示括号内的每一项都乘以外面的乘数：2(x + 5) = 2x + 10。这个过程叫做展开。
Factoring is the reverse of expanding. Look for the highest common factor and write it outside the bracket: 4x + 12 = 4(x + 3).
因式分解是展开的逆过程。找出各项的最高公因式，并把它写在括号外面：4x + 12 = 4(x + 3)。
Substitution means replacing a letter with a given number. If a = 3, then 2a + 1 = 2 × 3 + 1 = 7.
代入法是指用给定的数值替换字母。若 a = 3，则 2a + 1 = 2 × 3 + 1 = 7。
6. Solving Equations | 解方程
An equation says two expressions are equal. To solve it, find the value of the letter that makes the statement true.
方程表示两个表达式相等。解方程就是求出使等式成立的字母的值。
Keep the equation balanced: whatever you do to one side, do exactly the same to the other side. This is the balance method.
保持方程平衡：对等式一边进行的任何操作，都要同时施加到另一边。这就是平衡法。
To solve x + 7 = 15, subtract 7 from both sides. To solve 3x = 18, divide both sides by 3.
解 x + 7 = 15 时，两边同减 7。解 3x = 18 时，两边同除以 3。
In two‑step equations undo the operations in reverse order. For 2x + 5 = 13, first subtract 5, then divide by 2.
解两步方程时，按照与运算顺序相反的顺序逆推。解 2x + 5 = 13，先减 5，再除以 2。
Always check your solution by substituting it back into the original equation.
务必把求得的解代入原方程进行验算。
7. Perimeter, Area & Volume | 周长、面积与体积
Perimeter is the total distance around a shape. For a rectangle, P = 2(l + w) where l is length and w is width.
周长是围绕一个图形一周的总长度。矩形周长：P = 2(l + w)，其中 l 为长，w 为宽。
Area of a rectangle: A = l × w. Use the same unit for both lengths; the area will be in square units.
矩形面积：A = l × w。长和宽使用相同单位，面积的单位是平方单位。
Area of a triangle: A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex.
三角形面积：A = ½ × 底 × 高，其中高是指从底边到对顶点的垂直距离。
Area of a parallelogram: A = base × perpendicular height. This works just like a rectangle, because a parallelogram can be rearranged into one.
平行四边形面积：A = 底 × 垂直高。与矩形公式类似，因为平行四边形可以重新拼补成矩形。
Volume of a cuboid: V = length × width × height. Count unit cubes if the dimensions are whole numbers.
长方体体积：V = 长 × 宽 × 高。当边长为整数时，可以直接数单位立方体。
Area of compound shapes: split the shape into rectangles, find each area separately and then add or subtract them.
组合图形面积：将图形分割成若干矩形，分别求面积，然后相加或相减。
8. Angles & Lines | 角度与直线
Angles are measured in degrees (°). A full turn is 360°, a right angle is 90°, an acute angle is less than 90°, an obtuse angle is between 90° and 180°.
角度用度 (°) 衡量。一整圈是 360°，直角是 90°，锐角小于 90°，钝角介于 90° 到 180° 之间。
Angles on a straight line add up to 180°. If you know two angles, subtract their sum from 180° to find the third.
直线上的角之和为 180°。如果已知两个角，用 180° 减去它们之和即可得第三个角。
Vertically opposite angles are equal. When two lines cross, the angles opposite each other have the same measure.
对顶角相等。两条直线相交时，相对的两个角大小相等。
Angles around a point sum to 360°. This is useful when several lines meet at a single point.
围绕一个点的所有角之和为 360°。这用来处理多条线交于一点的情况。
Angles in a triangle always add up to 180°. This fact helps find a missing angle if the other two are known.
三角形内角和总是 180°。已知两个角即可求出第三个角。
9. Mean, Median, Mode & Range | 平均数、中位数、众数与极差
The mean (average) of a set of numbers is found by adding all the values and dividing by the number of values. Mean = sum of values ÷ number of values.
平均数（均值）的计算方法是把所有数值加起来，再除以数值的个数。平均数 = 总和 ÷ 个数。
The median is the middle value when the numbers are arranged in order. If there are two middle numbers, the median is their mean.
中位数是将数据按大小顺序排列后，位于中间的那个数。如果有两个中间数，则取它们的平均数作为中位数。
The mode is the value that appears most often. A set of data may have one mode, more than one mode, or no mode at all.
众数是出现次数最多的数值。一组数据可能有一个众数、多个众数，也可能没有众数。
The range is the difference between the largest and the smallest value. Range = maximum – minimum. It shows how spread out the data are.
极差是最大值与最小值的差。极差 = 最大值 − 最小值。它反映数据的分散程度。
10. Metric Units & Conversions | 公制单位与换算
Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. To convert a larger unit to a smaller one, multiply; to go smaller to larger, divide.
长度：1 公里 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米。大单位化小单位乘进率，小单位化大单位除以进率。
Mass (weight): 1 tonne = 1000 kg, 1 kg = 1000 g. When measuring small masses, use grams; for heavier objects, use kilograms.
质量（重量）：1 吨 = 1000 千克，1 千克 = 1000 克。测较轻物体用克，较重物体用千克。
Capacity (volume of liquid): 1 L = 1000 mL. A cubic centimetre (cm³) is equivalent to 1 mL, so 1 litre = 1000 cm³.
容量（液体体积）：1 升 = 1000 毫升。1 立方厘米 (cm³) 等于 1 毫升，因此 1 升 = 1000 cm³。
Area conversions: 1 m² = 10 000 cm² (since 1 m² = 100 cm × 100 cm). Volume conversions: 1 m³ = 1 000 000 cm³.
面积单位换算：1 平方米 = 10 000 平方厘米 (因为 1 m² = 100 cm × 100 cm)。体积单位换算：1 立方米 = 1 000 000 立方厘米。
When working with imperial units for everyday sense, 1 inch ≈ 2.54 cm, 1 foot ≈ 30 cm, 1 mile ≈ 1.6 km. These are approximations.
在涉及英制单位时，常近似换算：1 英寸 ≈ 2.54 厘米，1 英尺 ≈ 30 厘米，1 英里 ≈ 1.6 公里。这些都是近似值。
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