📚 KS3 Maths: Essential Maths Book 7F Key Points | KS3 数学：Essential Maths Book 7F 知识点精讲
Welcome to the revision guide for Essential Maths Book 7F, designed to help KS3 students master the foundational topics covered in Year 7. This article breaks down each chapter’s core ideas, from working with whole numbers and fractions to exploring angles and data handling. Use this as your go-to summary for quick recall and exam preparation.
欢迎阅读 Essential Maths Book 7F 复习指南，本文旨在帮助 KS3 阶段学生掌握七年级所涵盖的基础知识。我们将逐章拆解核心概念，从整数运算、分数扩展到角度探索与数据处理，帮助你快速回忆、高效备考。
1. Place Value and Ordering Integers | 位值与整数排序
Understanding place value up to millions allows you to read, write, and compare large numbers. For example, in 5 726 304, the digit 5 represents 5 millions, 7 is in the hundred thousands, and so on. You can use inequality signs such as > (greater than) and < (less than) to order a set of integers.
理解达到百万级的位值有助于读写和比较较大的数。例如在 5 726 304 中，数字 5 表示 5 个百万，7 在十万位上，以此类推。你可以使用 >（大于）和 <（小于）符号对一组整数排序。
- Always start comparing numbers from the leftmost digit. If digits are equal, move to the next column. | 比较数字时始终从最左边的数字开始。如果数字相同，则比较下一位。
- Ascending order means arranging from smallest to largest; descending order means largest to smallest. | 升序排列表示从小到大；降序排列表示从大到小。
Example: 123 456 > 123 455 because 456 > 455
2. Addition and Subtraction of Whole Numbers | 整数加法和减法
When adding or subtracting whole numbers, align the digits by place value – units under units, tens under tens. The column method is the most reliable approach. If a column sum exceeds 9, carry the extra to the next left column. In subtraction, if a digit is smaller than the digit below it, you need to exchange (borrow) from the next place.
整数加减时，将数字按位值对齐——个位对个位，十位对十位。列竖式是最可靠的方法。如果某一列的和超过 9，则向左边一列进位。减法中，如果某一位上的数字小于下方数字，需要从高位借位。
- For mental addition, break numbers into parts: 47 + 38 = 47 + 30 + 8 = 77 + 8 = 85. | 心算加法时可将数字拆分：47 + 38 = 47 + 30 + 8 = 77 + 8 = 85。
- Check subtraction by adding the answer to the subtrahend; you should get the original minuend. | 通过将答案与减数相加来验算减法，应得到原来的被减数。
564 + 278 = 842, 1001 – 456 = 545
3. Multiplication and Division Strategies | 乘法与除法策略
an be tackled by the bus stop method (short division) when dividing by a single digit, or by long division for larger divisors.
多位数乘法可以通过网格法或长乘法竖式将数字分解为更容易计算的部分。例如 34 × 26，可拆成 (30 + 4) × 26 = 30×26 + 4×26。除法中，除以一位数可用短除法（巴士站法），大除数则用长除法。
- imes tables up to 12 × 12 by heart; it makes both multiplication and division much faster. | 熟记 12×12 以内的乘法表，能显著提升乘除计算速度。
- To check a division, multiply the quotient by the divisor and add any remainder. | 验算除法时，将商乘以除数再加上余数。
392 ÷ 7 = 56, because 56 × 7 = 392
4. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a whole number that divides another number exactly. A multiple is the result of multiplying a number by an integer. A prime number has exactly two distinct factors: 1 and itself. The smallest prime is 2. Prime factorisation writes a number as a product of its prime factors, often using factor trees.
因数是可以整除另一个数的整数。倍数是某整数与其他整数相乘的结果。质数是恰好有两个不同因数的数：1 和它本身。最小的质数是 2。质因数分解是使用因数树将一个数写成它的质因数乘积。
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. | 24 的因数：1, 2, 3, 4, 6, 8, 12, 24。
- First five multiples of 7: 7, 14, 21, 28, 35. | 7 的前五个倍数：7, 14, 21, 28, 35。
48 = 2⁴ × 3
5. Fractions: Equivalent, Simplifying and Comparing | 分数：等价、化简和比较
⁄₈. To simplify a fraction, divide numerator and denominator by their highest common factor (HCF). When comparing fractions with different denominators, convert them to a common denominator or compare decimal equivalents.
等价分数表示整体的相同部分，如 ½ = ²⁄₄ = ⁴⁄₈。化简分数时，用分子和分母的最大公因数（HCF）去除。比较不同分母的分数时，先通分为同分母，或比较它们的小数形式。
- Improper fractions have a numerator larger than the denominator; mixed numbers combine a whole number and a proper fraction. | 假分数的分子大于分母；带分数由整数和真分数组成。
- To find a fraction of an amount, divide by the denominator and multiply by the numerator. | 求一个数量的几分之几，先除以分母，再乘以分子。
⅔ of 90 = (90 ÷ 3) × 2 = 60
6. Operations with Fractions | 分数运算
For addition and subtraction of fractions, first make the denominators the same using equivalent fractions. Then add/subtract the numerators and keep the denominator unchanged. For multiplication, multiply numerators together and denominators together. Division by a fraction is equivalent to multiplying by its reciprocal.
分数加减时，首先通过等价转换使分母相同。然后分子相加或相减，分母不变。做分数乘法时，分子相乘、分母相乘。除以一个分数等同于乘以它的倒数。
- Always simplify your final answer if possible. | 最终答案要尽可能化简。
- Convert mixed numbers to improper fractions before multiplying or dividing. | 做乘除运算前先将带分数转化为假分数。
¾ × ⅔ = (3×2)/(4×3) = 6/12 = ½
7. Decimals and Place Value | 小数与位值
Decimals extend the place value system to tenths, hundredths, thousandths, etc. The decimal point separates whole numbers from the fractional part. When ordering decimals, compare digits from left to right, just as with whole numbers. You can add zeros after the last decimal place to help compare without changing value.
小数将位值体系延伸到十分位、百分位、千分位等。小数点将整数部分与小数部分分开。给小数排序时，像整数一样从左往右逐位比较。你可以在最后一位小数后补零以方便比较，而不会改变数值。
- To round a decimal, look at the digit to the right of the required place: if it is 5 or more, round up. | 小数取近似值时，看保留位后一位数字：如果大于或等于 5，则进位。
- Multiplying by 10, 100, 1000 shifts digits left; dividing shifts them right. | 乘以 10、100、1000 将数字向左移动；除法向右移动。
3.45 × 100 = 345
8. Percentages as Fractions and Decimals | 百分数化为分数和小数
ge into a decimal, divide by 100 (move the decimal point two places left). For example, 65% = 65/100 = 13/20 = 0.65.
百分数表示每百分之几，因此百分数化分数只需写成 100 分之几并化简。百分数化小数时除以 100（小数点向左移动两位）。例如 65% = 65/100 = 13/20 = 0.65。
- Common conversions to remember: 50% = ½, 25% = ¼, 75% = ¾, 10% = 0.1, 1% = 0.01. | 需要熟记的常见转换：50% = ½，25% = ¼，75% = ¾，10% = 0.1，1% = 0.01。
- To find a percentage of an amount without a calculator, find 10% first, then scale. | 不用计算器求某数量的百分数时，可先求 10%，再按比例推算。
30% of £80 = 0.30 × 80 = £24
9. Ratio and Proportion | 比与比例
A ratio compares two or more quantities, showing their relative sizes. Ratios can be simplified like fractions by dividing all parts by a common factor. Proportion problems involve finding a missing quantity when the ratio is known. The unitary method—finding the value of one part first—is a powerful strategy.
比用于比较两个或多个数量，显示它们的相对大小。比可以像分数一样通过除以公因数来化简。比例问题涉及在已知比率时求未知量。单位法——先求出一份的量——是强有力的策略。
- If the ratio of boys to girls is 3 : 5, the total number of parts is 3 + 5 = 8. | 若男孩与女孩的比是 3 : 5，总份数为 3 + 5 = 8。
- To share £48 in the ratio 3 : 5, one part = £48 ÷ 8 = £6, so the shares are £18 and £30. | 按 3 : 5 分配 48 英镑，一份为 48 ÷ 8 = 6 英镑，因此分别为 18 英镑和 30 英镑。
10. Introduction to Algebra | 代数入门
and can be collected together; for example, 5x + 2x = 7x, but 3x and 3y are unlike terms.
代数使用字母表示未知数或变量。你可以用运算符号组合数字和字母来构造表达式，如 3a + 2b。同类项具有完全相同的变量部分，可以合并；例如 5x + 2x = 7x，但 3x 和 3y 不是同类项。
- Substitution means replacing a letter with a given number value. | 代入法指用给定的数值替换字母。
- When multiplying, you can omit the multiplication sign: 4 × m = 4m. | 乘法中可省略乘号：4 × m = 4m。
If a = 3 and b = 5, then 2a + b = 6 + 5 = 11
11. Angles, Lines and Symmetry | 角、线与对称
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°. Angles on a straight line sum to 180°, and angles around a point sum to 360°. Vertically opposite angles are equal.
角以度（°）为单位。锐角小于 90°，直角等于 90°，钝角在 90° 到 180° 之间，优角大于 180°。一条直线上的角之和为 180°，围绕一点的所有角之和为 360°。对顶角相等。
- Line symmetry occurs when one half of a shape is the mirror image of the other. | 线对称指图形的一半是另一半的镜像。
- To calculate a missing angle on a straight line, subtract the known angle from 180°. | 求直线上的未知角，从 180° 中减去已知角。
If one angle is 72°, the adjacent angle on a straight line is 180° – 72° = 108°
12. Data Handling and Averages | 数据处理与平均数
Data can be collected, organised into frequency tables, and displayed using bar charts, pictograms, or line graphs. The mode is the most frequent value; the median is the middle value when data are ordered; the mean is calculated by summing all values and dividing by the number of values. The range shows the spread of data: maximum minus minimum.
数据可以被收集、整理为频数表，并用条形图、象形图或折线图展示。众数是出现频率最高的值；中位数是数据排序后位于中间的值；平均数的计算是将所有数值相加再除以数据个数。全距表示数据的波动范围：最大值减最小值。
- When finding the median of an even number of values, take the mean of the two middle numbers. | 当数据个数为偶数时，中位数取中间两个数的平均数。
- Always label axes when drawing graphs and provide a title. | 绘制图表时始终标记坐标轴并提供标题。
For 4, 6, 2, 6, 7: mean = 5, mode = 6, median = 6, range = 5
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