📚 Year 7 AQA Maths: Core Topic Summary | Year 7 AQA 数学：核心知识点梳理
Welcome to your Year 7 AQA maths revision guide. This article covers all the essential topics you will meet during the year, from number work and algebra to geometry and statistics. Each section pairs clear English explanations with Chinese translations, making it easy for bilingual learners to grasp and retain the key ideas.
欢迎来到 Year 7 AQA 数学复习指南。本文涵盖了你这一年将会遇到的各个核心主题，从整数运算和代数到几何与统计。每个部分都以清晰的英文解释搭配中文翻译，帮助双语学习者轻松理解和记忆关键概念。
1. Place Value and Ordering | 位值与排序
Place value is the value of a digit depending on its position in a number. For example, in 3,724, the digit 3 has a place value of three thousands, 7 is seven hundreds, 2 is two tens, and 4 is four ones (or four units).
位值是根据数字在数中的位置而具有的值。例如，在 3,724 中，数字 3 的位值是 3 个千，7 是 7 个百，2 是 2 个十，4 是 4 个一（或 4 个位）。
When ordering whole numbers, compare the digits from the leftmost place. If the number of digits differs, the number with more digits is larger.
比较整数大小时，从最左边的数位开始比较。如果数字位数不同，位数多的数更大。
For decimals, align the decimal points and compare digits column by column. 0.45 is less than 0.5 because 4 tenths (0.45) is less than 5 tenths (0.5), even though 0.45 has more digits.
对于小数，对齐小数点后逐列比较。0.45 小于 0.5，因为 0.45 有 4 个十分位，而 0.5 有 5 个十分位，尽管 0.45 的位数更多。
Rounding to the nearest 10, 100 or 1000 uses the digit to the right of the place you are rounding to. If it is 5 or more, round up; otherwise round down.
四舍五入到最近的十、百或千时，观察目标位右边的一位数。若该数大于或等于 5，则进一；否则舍去。
2. Addition and Subtraction | 加法与减法
Use column addition and column subtraction with clear alignment of place values. Always start from the rightmost column (ones) and carry or borrow when needed.
使用竖式加法和竖式减法时，要将位值对齐。始终从最右列（个位）开始，需要时进行进位或借位。
Mental strategies include partitioning: 58 + 36 = (50 + 30) + (8 + 6) = 80 + 14 = 94. For subtraction, 74 − 29 can be thought of as 74 − 30 + 1 = 44 + 1 = 45.
心算策略包括拆分法：58 + 36 = (50 + 30) + (8 + 6) = 80 + 14 = 94。减法方面，74 − 29 可以看作 74 − 30 + 1 = 44 + 1 = 45。
Estimate answers by rounding before calculating, e.g., 198 + 405 ≈ 200 + 400 = 600. This helps check the reasonableness of your answer.
计算前先估算答案，例如 198 + 405 ≈ 200 + 400 = 600。这有助于检查你的答案是否合理。
Inverse operations are useful: addition reverses subtraction, and vice versa. So 125 − 47 = 78 can be checked by 78 + 47 = 125.
逆运算很有用：加法是减法的逆运算，反之亦然。因此 125 − 47 = 78 可以通过 78 + 47 = 125 来验证。
3. Multiplication and Division | 乘法与除法
Times tables up to 12 × 12 are fundamental. Knowing these facts instantly makes larger calculations much faster.
掌握到 12 × 12 的乘法表是基础。熟记这些口诀能让更复杂的计算快得多。
Multiply by 10, 100, 1000 by shifting digits to the left. Divide by these powers of 10 by shifting digits to the right. 4.2 × 10 = 42; 79 ÷ 100 = 0.79.
乘以 10、100、1000 时，将数字向左移动。除以这些 10 的幂时，将数字向右移动。4.2 × 10 = 42；79 ÷ 100 = 0.79。
Long multiplication: for 34 × 26, multiply 34 by 6, then 34 by 20, and add. Division: 345 ÷ 15 can be done by long division (bus stop method) or by finding how many 15s fit.
长乘法：计算 34 × 26 时，先用 34 乘 6，再用 34 乘 20，然后相加。除法：345 ÷ 15 可以用长除法（公车法）或寻找含有多少个 15 来进行。
Order of operations (BIDMAS/BODMAS) tells us to work out Brackets first, then Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). 12 + 3 × 4 = 12 + 12 = 24, not 15 × 4.
运算顺序 (BIDMAS/BODMAS) 告诉我们先算括号，再算指数，接着算除法和乘法（从左到右），最后算加法和减法（从左到右）。12 + 3 × 4 = 12 + 12 = 24，而不是 15 × 4。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
A fraction like 3/8 means 3 out of 8 equal parts. Equivalent fractions can be made by multiplying or dividing the numerator and denominator by the same number, e.g., 3/8 = 6/16.
像 3/8 这样的分数表示 8 等份中的 3 份。将分子和分母同时乘或除以相同的数，可以得到等值分数，例如 3/8 = 6/16。
Simplest form means there is no whole number (except 1) that divides both the numerator and denominator. 12/15 simplifies to 4/5 by dividing by 3.
最简形式是指除了 1 以外，没有其他整数能同时整除分子和分母。12/15 除以 3 后化简为 4/5。
Fractions ↔ Decimals: divide the numerator by the denominator. 3/4 = 0.75. To convert a decimal to a fraction, write the place value as denominator: 0.35 = 35/100 = 7/20.
分数与小数互化：分子除以分母。3/4 = 0.75。将小数化为分数时，以相应的位值作分母：0.35 = 35/100 = 7/20。
Percentages mean ‘per 100’. 43% = 43/100 = 0.43. To find a percentage of a quantity, multiply by the decimal form, e.g., 20% of 60 = 0.20 × 60 = 12.
百分数表示“每一百”。43% = 43/100 = 0.43。求某个数量的百分之几，用小数形式相乘，如 60 的 20% = 0.20 × 60 = 12。
Find a common denominator to compare or add fractions. 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
比较或相加分数时，先通分。1/3 + 1/4 = 4/12 + 3/12 = 7/12。
5. Negative Numbers | 负数
Negative numbers are numbers less than zero. On a number line, they lie to the left of zero. −5 is smaller than −2 because it is further left.
负数是小于零的数。在数轴上，它们位于零的左边。−5 小于 −2，因为它在数轴上更靠左。
Adding a negative is like subtracting: 7 + (−3) = 7 − 3 = 4. Subtracting a negative is like adding: 5 − (−2) = 5 + 2 = 7.
加上一个负数相当于减法：7 + (−3) = 7 − 3 = 4。减去一个负数相当于加法：5 − (−2) = 5 + 2 = 7。
Multiplying or dividing two negatives gives a positive: (−4) × (−6) = 24. A positive and a negative gives a negative: 4 × (−6) = −24.
两个负数相乘或相除结果为正：(−4) × (−6) = 24。一正一负相乘或相除结果为负：4 × (−6) = −24。
In real‑life contexts, temperatures below zero and bank overdrafts use negative numbers. A rise from −3°C by 8 degrees gives 5°C.
在实际生活中，零下的温度和银行透支都用到负数。从 −3 °C 上升 8 度，达到 5 °C。
6. Factors, Multiples and Primes | 因数、倍数与质数
A factor of a number divides exactly into that number with no remainder. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
一个数的因数可以整除该数而没有余数。24 的因数有 1, 2, 3, 4, 6, 8, 12, 24。
A multiple is the product of a number and any integer. The first five multiples of 6 are 6, 12, 18, 24, 30.
倍数是一个数与任意整数的乘积。6 的前五个倍数是 6, 12, 18, 24, 30。
Prime numbers have exactly two distinct factors: 1 and the number itself. 2, 3, 5, 7, 11, 13, 17, 19, 23 are the primes below 25. Note 1 is not prime.
质数只有两个不同的因数：1 和它本身。2, 3, 5, 7, 11, 13, 17, 19, 23 是 25 以下的质数。注意 1 不是质数。
Prime factorisation breaks a number into prime factors. 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
质因数分解是把一个数拆成质数的乘积。60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。
Highest common factor (HCF) and lowest common multiple (LCM) can be found from lists or prime factors. HCF of 24 and 36 is 12; LCM is 72.
最大公因数 (HCF) 和最小公倍数 (LCM) 可通过列举或质因数求得。24 和 36 的 HCF 是 12，LCM 是 72。
7. Algebraic Expressions | 代数表达式
In algebra, letters represent unknown numbers. Called variables, they allow us to write general rules. For example, 3 × a is written as 3a.
在代数中，字母表示未知数，称为变量，它们让我们能写出通用的规则。例如 3 × a 写作 3a。
Like terms can be combined. 2x + 5x = 7x; 4y − y = 3y. Terms with different letters or powers cannot be added, e.g., 3x + 2y stays as it is.
同类项可以合并。2x + 5x = 7x；4y − y = 3y。字母或指数不同的项不能直接相加，如 3x + 2y 保持不变。
Expanding brackets means multiplying each term inside. 3(a + 4) = 3a + 12. Factorising is the reverse: taking a common factor out, e.g., 6b + 9 = 3(2b + 3).
去括号指把括号内的每一项都相乘。3(a + 4) = 3a + 12。因式分解是反过来：提取公因数，如 6b + 9 = 3(2b + 3)。
Substitution means replacing letters with numbers. When a = 5 and b = 2, the expression 3a − 2b = 3(5) − 2(2) = 15 − 4 = 11.
代入是指用数字替换字母。当 a = 5、b = 2 时，表达式 3a − 2b = 3(5) − 2(2) = 15 − 4 = 11。
8. Solving Equations | 解方程
An equation states that two expressions are equal, e.g., x + 3 = 7. Solving means finding the value of the unknown that makes it true.
方程表示两个表达式相等，例如 x + 3 = 7。解方程就是找出使等式成立的未知数的值。
Use inverse operations to isolate the unknown. For x + 3 = 7, subtract 3 from both sides: x = 4. For 2x = 10, divide both sides by 2: x = 5.
运用逆运算分离未知数。对于 x + 3 = 7，两边同时减 3：x = 4。对于 2x = 10，两边同时除以 2：x = 5。
Always keep the balance: whatever you do to one side of the equation, do to the other. 5x − 3 = 2x + 9 becomes 5x − 2x = 9 + 3 → 3x = 12 → x = 4.
始终要保持平衡：等号一边做什么，另一边也要做同样的运算。5x − 3 = 2x + 9 → 5x − 2x = 9 + 3 → 3x = 12 → x = 4。
Check your answer by substituting back into the original equation. If both sides give the same value, the solution is correct.
把解代入原方程检验。如果等号两边的值相等，解就是正确的。
9. Sequences | 数列
A number sequence is a list of numbers that often follows a rule. The rule tells you how to get from one term to the next.
数列是一串按照一定规则排列的数字。规则告诉你如何从一个项得到下一项。
Arithmetic sequences go up or down by a constant amount, called the common difference. Sequence 5, 9, 13, 17, … has a term‑to‑term rule of add 4.
等差数列以恒定的差值递增或递减，这个差叫作公差。数列 5, 9, 13, 17 … 的项到项规则是每次加 4。
You can find the nth term of an arithmetic sequence using the formula:
nth term = first term + (n − 1) × common difference
你可以用公式求等差数列的第 n 项：
第 n 项 = 首项 + (n − 1) × 公差
For the sequence above: first term = 5, difference = 4, so nth term = 5 + (n − 1) × 4 = 4n + 1. Check: for n=1, 4(1)+1=5; n=2, 9, etc.
以上述数列为例：首项 = 5，公差 = 4，所以第 n 项 = 5 + (n − 1) × 4 = 4n + 1。验证：n=1 时，4(1)+1=5；n=2 时为 9，以此类推。
Other sequences may use different rules, such as doubling each time (geometric) or referencing earlier terms (Fibonacci‑type).
其他数列可能使用不同的规则，例如每次翻倍（等比数列）或者利用前几项进行计算（斐波那契型）。
10. Angles | 角
An angle measures the turn between two lines meeting at a point. Angles are measured in degrees (°) using a protractor.
角度量的是从一点出发的两条射线之间的旋转量。角度用度 (°) 作单位，使用量角器测量。
Angle types: acute angle (less than 90°), right angle (exactly 90°), obtuse angle (between 90° and 180°), and reflex angle (greater than 180°).
角的类型：锐角（小于 90°）、直角（恰好 90°）、钝角（介于 90° 和 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 when two lines cross.
直线上的角之和为 180°。一点周围的角之和为 360°。当两条直线相交时，对顶角相等。
In a triangle, the three interior angles always sum to 180°. You can find a missing angle by subtracting the known angles from 180°.
三角形的三个内角之和总是 180°。通过用 180° 减去已知角，就能求出未知的内角。
When a transversal cuts parallel lines, alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles add to 180°. Recognising these helps solve problems.
一条截线与两条平行线相交时，内错角相等，同位角相等，同旁内角（内角）相加等于 180°。识别这些关系有助于解题。
11. Perimeter and Area | 周长与面积
Perimeter is the total distance around the outside of a shape. It is measured in units like cm, m, or km. For a polygon, add all side lengths.
周长是图形外部一周的总长度。它的单位是厘米、米或千米等。对于多边形，将各边长相加即可。
Perimeter of a rectangle can be found by:
P = 2 × (length + width) or P = 2L + 2W
长方形的周长可以用以下公式计算：
周长 = 2 × (长 + 宽) 或 2L + 2W
Area is the amount of space inside a 2D shape. Area of a rectangle = length × width. Compound shapes can be split into rectangles then areas added.
面积是二维图形内部所占的空间。长方形的面积 = 长 × 宽。复合图形可以分割成多个长方形，然后求面积总和。
Area of a triangle = (base × height) ÷ 2. For a triangle with base 8 cm and height 5 cm, area = (8 × 5) ÷ 2 = 20 cm².
三角形的面积 = (底 × 高) ÷ 2。对于一个底边长 8 厘米、高 5 厘米的三角形，面积 = (8 × 5) ÷ 2 = 20 平方厘米。
Area of a parallelogram = base × perpendicular height. The formula works because a parallelogram can be rearranged into a rectangle of equal area.
平行四边形的面积 = 底 × 垂直高。该公式成立是因为平行四边形可以重新拼成一个等面积的长方形。
12. Statistics | 统计
Data is information collected to answer questions. It can be categorical (colours, types) or numerical (heights, scores).
数据是为回答问题而收集的信息。它可以是分类数据（颜色、类型）或数值型数据（身高、得分）。
Bar charts and pictograms show discrete data. In a bar chart, the height of each bar represents frequency. In a pictogram, each picture stands for a number of items.
条形图和象形图用于展示离散数据。在条形图中，每个条的高度代表频数。在象形图中，每个图形代表一定数量的个体。
Line graphs are used to show change over time. The horizontal axis shows time, and the vertical axis shows the quantity measured.
折线图用于显示随时间变化的趋势。横轴表示时间，纵轴表示测量的量。
The mean is the average found by adding all values and dividing by the number of values. The median is the middle value when data is ordered. The mode is the most frequent value. The range is the difference between largest and smallest values.
平均数（均值）是总和除以个数。中位数是排序后位于中间的值。众数是出现频率最高的值。极差是最大值与最小值的差。
For the data set 3, 5, 5, 7, 9: mean = (3+5+5+7+9)÷5 = 5.8; median = 5; mode = 5; range = 9 − 3 = 6.
对于数据集 3, 5, 5, 7, 9：平均数 = (3+5+5+7+9)÷5 = 5.8；中位数 = 5；众数 = 5；极差 = 9 − 3 = 6。
Pie charts show proportions of a whole. The angle for each category is found by (number in category ÷ total) × 360°.
饼图展示各部分在整体中所占的比例。每个类别的角度通过 (类别数量 ÷ 总数) × 360° 计算得出。
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