📚 Year 7 AQA Maths: Quick Reference Formula Handbook | 七年级 AQA 数学：公式定理速查手册
This handy handbook covers all the essential formulas and theorems you need for Year 7 AQA Mathematics. Keep it as a quick reference to revise key concepts and build a solid foundation for future maths topics.
本速查手册涵盖七年级 AQA 数学中所有必备的公式和定理。它可作为快速复习关键概念的参考，为今后的数学学习打下坚实的基础。
1. Number Basics | 数字基础
Commutative Law: For any numbers a and b, a + b = b + a and a × b = b × a.
交换律： 对于任意数字 a 和 b，a + b = b + a，a × b = b × a。
Associative Law: (a + b) + c = a + (b + c) and (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.
分配律： a × (b + c) = a × b + a × c。
Factors and Multiples: A factor divides exactly into a number. A multiple is the result of multiplying a number by an integer.
因数与倍数： 因数能整除该数，倍数是一个数与整数相乘的结果。
Prime Numbers: A prime number has exactly two distinct factors: 1 and itself (e.g., 2, 3, 5, 7).
质数： 质数恰好只有两个不同的因数，即 1 和它本身（例如 2、3、5、7）。
Squares and Cubes: 3² = 3 × 3 = 9, 5³ = 5 × 5 × 5 = 125. The square root √9 = 3, and the cube root ∛125 = 5.
平方与立方： 3² = 3 × 3 = 9，5³ = 5 × 5 × 5 = 125。平方根 √9 = 3，立方根 ∛125 = 5。
Negative Number Rules: Adding a negative is like subtracting; subtracting a negative is like adding. Multiplying or dividing two negatives gives a positive.
负数规则： 加负数相当于减法，减负数相当于加法；两个负数相乘或相除结果为正。
Order of Operations (BIDMAS): Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right).
运算顺序（BIDMAS）： 括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
2. Fractions | 分数
Equivalent Fractions: Multiply or divide the numerator and denominator by the same non-zero number to get an equivalent fraction.
等价分数： 将分子和分母同时乘以或除以同一个非零数，即可得到等价分数。
Simplifying Fractions: Divide numerator and denominator by their highest common factor (HCF).
化简分数： 用分子和分母的最大公因数（HCF）分别除分子和分母。
Mixed Numbers and Improper Fractions: ying the whole number by the denominator and adding the numerator. For example, 2 ⅓ = 7/3.
带分数与假分数： 转换时将整数乘以分母再加上分子。例如，2 ⅓ = 7/3。
Adding and Subtracting Fractions: For common denominators, add/subtract numerators. For different denominators, find the least common multiple (LCM) to make equivalent fractions first.
分数加减法： 同分母时，直接对分子进行加减；异分母时，先求最小公倍数（LCM）通分，再加减分子。
Multiplying Fractions: Multiply the numerators together and multiply the denominators together. Simplify if possible.
分数乘法： 分子乘分子，分母乘分母，若可能则化简。
Dividing Fractions: Keep the first fraction, change the sign to multiply, and flip the second fraction (use its reciprocal). Then multiply.
分数除法： 保留第一个分数，把除号变为乘号，并将第二个分数取倒数（翻转分子分母），然后相乘。
3. Decimals | 小数
Place Value: Digits after the decimal point represent tenths, hundredths, thousandths, etc.
位值： 小数点后各位依次代表十分位、百分位、千分位等。
Adding and Subtracting Decimals: Align the decimal points, then add or subtract as with whole numbers, keeping the decimal point in place.
小数加减法： 对齐小数点，然后按整数加减法计算，并保留小数点在同样位置。
Multiplying Decimals: Multiply as whole numbers, then place the decimal point so the product has the total number of decimal places from both factors.
小数乘法： 先按整数相乘，然后根据两个因数的小数位数之和确定积的小数位数，点上小数点。
Dividing Decimals: Multiply both the divisor and dividend by a power of 10 to make the divisor a whole number, then divide.
小数除法： 将被除数和除数同时乘以10的幂，使除数变为整数，然后再相除。
Fraction-Decimal Conversions: Divide the numerator by the denominator to convert a fraction to a decimal. For terminating decimals, write as a fraction over 10, 100, 1000, etc., and simplify.
分数与小数转换： 分子除以分母可将分数化为小数；对于有限小数，可写成分母为 10、100、1000 等的分数并化简。
Rounding: Look at the digit to the right of the place value you are rounding to; if it is 5 or more, round up.
四舍五入： 观察要保留位右面的一位数字，若该数字 ≥ 5，则向前进一。
4. Percentages | 百分数
Meaning of Percentage: Percent means 'out of 100'. To write a percentage as a fraction, divide by 100 and simplify.
百分数的意义： 百分数表示“每一百份中所占的份数”。将百分数化为分数时，除以100并化简。
Common Equivalences: 50% = 1/2, 25% = 1/4, 75% = 3/4, 10% = 1/10, 20% = 1/5.
常见等价关系： 50% = 1/2，25% = 1/4，75% = 3/4，10% = 1/10，20% = 1/5。
Finding a Percentage of an Amount: Convert the percentage to a fraction or decimal and multiply. For example, 30% of 200 = 0.3 × 200 = 60.
求一个数量的百分比： 将百分数转换为分数或小数，再相乘。例如，200 的 30% = 0.3 × 200 = 60。
Percentage Increase/Decrease: Find the percentage of the original amount and add/subtract it. Alternatively, use a multiplier (e.g., 1.15 for a 15% increase).
增减百分比： 先求出原数量的某个百分比，再与原来数量相加或相减。也可使用乘数（例如，增长 15% 则乘 1.15）。
Expressing as a Percentage: Write the part over the whole as a fraction, then convert to a percentage by multiplying by 100.
用百分数表示： 将部分除以整体写成分数，再乘以 100% 即可。
5. Algebraic Expressions | 代数表达式
Terms, Coefficients, Constants: In 3x + 2, '3x' is a term, '3' is the coefficient, 'x' is the variable, and '2' is a constant.
项、系数与常数： 在 3x + 2 中，“3x”是一项，“3”是系数，“x”是变量，“2”是常数。
Collecting Like Terms: y : Only terms with exactly the same variable part can be combined. 5a + 2a = 7a; 3y² + y² = 4y².
合并同类项： 只有变量部分完全相同的项才能合并。5a + 2a = 7a；3y² + y² = 4y²。
Multiplying Algebraic Terms: Multiply the coefficients and multiply the variables. 4x × 3y = 12xy. 2a × 5a = 10a².
代数项相乘： 系数相乘，变量相乘。4x × 3y = 12xy；2a × 5a = 10a²。
Expanding Brackets: ch : Multiply the term outside the bracket by each term inside. a(b + c) = ab + ac. Always check sign when expanding negative multipliers.
展开括号： 用括号外的项乘以括号内的每一项。a(b + c) = ab + ac。乘数为负数时，注意符号变化。
Factorising: e : Find the highest common factor of all terms and write it outside the bracket. 6x + 9 = 3(2x + 3).
因式分解： 找出所有项的最大公因数，并将其写在括号外。6x + 9 = 3(2x + 3)。
Substitution: iables with given numbers and evaluate using BIDMAS. If a = 3, then 2a + 5 = 2×3 + 5 = 11.
代入求值： 将数字代入变量，并按照 BIDMAS 规则计算。若 a = 3，则 2a + 5 = 2×3 + 5 = 11。
6. Linear Equations | 线性方程
Solving One-Step Equations: tions. For x + 4 = 10, subtract 4 from both sides: x = 6. For 3x = 15, divide by 3: x = 5.
解一步方程： 使用逆运算。若 x + 4 = 10，两边同时减 4 得 x = 6；若 3x = 15，两边同时除以 3 得 x = 5。
Solving Two-Step Equations: dition/subtraction first, then undo multiplication/division. 2x + 3 = 11 → 2x = 8 → x = 4.
解两步方程： 先消去加减法，再消去乘除法。2x + 3 = 11 → 2x = 8 → x = 4。
Equations with Unknowns on Both Sides: e : Collect the variable terms on one side and constants on the other. 5x - 2 = 3x + 6 → 2x = 8 → x = 4.
未知数在两边： 将所有含未知数的项移到一边，常数移到另一边。5x - 2 = 3x + 6 → 2x = 8 → x = 4。
Checking Solutions: Substitute your answer back into the original equation to verify both sides are equal.
检验解： 将求得的解代入原方程，检查左右两边是否相等。
Key principle: Whatever you do to one side, do to the other.
核心原理：方程两边必须进行相同的运算。
7. Perimeter and Area | 周长与面积
Perimeter: e : The total distance around the edge of a shape. For a rectangle, P = 2l + 2w or P = 2(l + w).
周长： 图形所有边的总长度。对于矩形，P = 2l + 2w 或 P = 2(l + w)。
Area of Rectangle: A = l × w
矩形面积：A = 长 × 宽
Area of Square: A = s²
正方形面积：A = 边长²
Area of Triangle: A = ½ × base × height
三角形面积：A = ½ × 底 × 高
Area of Parallelogram: A = base × vertical height
平行四边形面积：A = 底 × 垂直高
Area of Trapezium: A = ½ × (a + b) × h
梯形面积：A = ½ × (上底 + 下底) × 高
Circle: Circumference C = π × d or C = 2 × π × r
圆周长：C = π × d 或 C = 2 × π × r
Area of Circle: A = π × r²
圆面积：A = π × r²
Compound Shapes: Split them into rectangles, triangles, etc., find individual areas, then add or subtract as needed.
复合图形： 将其分割为矩形、三角形等简单图形，分别计算面积后再相加或相减。
Units: Perimeter is measured in units (cm, m); area in square units (cm², m²).
单位： 周长用长度单位（cm、m），面积用平方单位（cm²、m²）。
8. Volume and Surface Area | 体积与表面积
Volume of a Cuboid: V = length × width × height (V = lwh)
长方体体积：V = 长 × 宽 × 高
Volume of a Cube: V = s³
立方体体积：V = 边长³
Volume of a Prism: rea of cross-section × length. The cross-section is the shape that runs through the prism.
棱柱体积： V = 横截面积 × 长度。横截面是贯穿棱柱的形状。
Surface Area of a Cuboid: Total area of all six faces: SA = 2lw + 2lh + 2wh = 2(lw + lh + wh).
长方体表面积： 所有六个面的总面积：SA = 2lw + 2lh + 2wh = 2(lw + lh + wh)。
Units for Volume: acity can be measured in millilitres (ml) or litres (l); 1 cm³ = 1 ml, 1000 cm³ = 1 litre.
体积单位： 以立方单位计量，如 cm³、m³。容量可用毫升（ml）或升（l）表示；1 cm³ = 1 ml，1000 cm³ = 1 升。
| Unit Conversion | 单位换算 |
|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² |
| 1 m³ = 1,000,000 cm³ | 1 litre = 1000 cm³ |
9. Angles and Lines | 角与直线
Angle Types: Acute (0°–90°), Right angle (90°), Obtuse (90°–180°), Straight line (180°), Reflex (180°–360°), Full turn (360°).
角的类型： 锐角（0°–90°），直角（90°），钝角（90°–180°），平角（180°），优角（180°–360°），周角（360°）。
Angles on a Straight Line: Sum to 180°. If one angle is known, subtract from 180° to find the other.
直线上的邻角： 和为 180°。若已知其中一个角，用 180° 减去它即得另一个角。
Vertically Opposite Angles: When two lines cross, opposite angles are equal.
对顶角： 两直线相交，对顶角相等。
Angles in a Triangle: Sum is always 180°. For an isosceles triangle, base angles are equal. For an equilateral triangle, each angle is 60°.
三角形内角和： 三角形三内角之和为 180°。等腰三角形两底角相等；等边三角形每个角都是 60°。
Parallel Lines: If two parallel lines are cut by a transversal: corresponding angles are equal, alternate interior angles are equal, co-interior (allied) angles sum to 180°.
平行线： 当一条截线与两条平行线相交时：同位角相等，内错角相等，同旁内角互补（和为 180°）。
Angles in Polygons: Sum of interior angles = (n - 2) × 180°, where n is the number of sides.
多边形内角和： 内角和 = (n - 2) × 180°，其中 n 为边数。
Exterior Angles: Sum of exterior angles of any convex polygon is 360°.
外角和： 任何凸多边形的外角和总是 360°。
10. Transformations and Symmetry | 变换与对称
Reflection: A mirror image across a line (mirror line). Points are the same perpendicular distance from the mirror line.
反射： 沿一条直线（镜像线）做镜像。每个点到镜像线的垂直距离相等。
Rotation: Turn around a fixed centre by a given angle and direction (clockwise/anticlockwise). The centre of rotation is the point that does not move.
旋转： 绕一个固定的中心点按指定角度和方向（顺时针/逆时针）转动。旋转中心是不动的点。
Translation: Sliding a shape by a vector. The vector a b (write as a over b) means move a right (or left if negative) and b up (or down).
平移： 将一个图形按某个向量平移。向量 （a 在上 b 在下）表示向右移 a（负数则向左），向上移 b（负数则向下）。这里我们用 column vector [a, b] 描述。
Enlargement: Changing size by a scale factor from a centre of enlargement. If scale factor > 1, it gets bigger; between 0 and 1, it shrinks.
放大： 以放大中心为基准，按比例因子改变大小。比例因子大于 1 时放大，在 0 和 1 之间
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