📚 Essential Maths 7S Homework Book: Key Knowledge Points Explained | KS3 数学：Essential Maths 7S 练习册知识点精讲
The Essential Maths 7S Homework Book is a structured practice resource designed for Year 7 students following the Key Stage 3 curriculum. It reinforces core concepts through carefully graded exercises that build fluency in number work, algebra, geometry and data handling. This article provides a focused revision of the key topics covered in the book, explaining the underlying ideas and common methods in clear, bilingual notes to support independent study.
《Essential Maths 7S 练习册》是一套为七年级学生设计的结构化练习资源，紧扣英国 KS3 课程大纲。它通过精心分级编排的习题巩固核心概念，培养学生在算术、代数、几何和数据处理方面的熟练度。本文精选书中核心知识点，以清晰的中英双语要点梳理基本概念和常用方法，帮助自主学习与复习。
1. Number and Place Value | 数与位值
Whole numbers are built from digits 0–9 arranged in place value columns. In Year 7, we work with numbers up to millions and beyond, understanding that each column is ten times larger than the one to its right.
整数由数字 0–9 按位值排列而成。在七年级，我们需要处理数百万甚至更大的数，理解每一个数位比它右边数位大十倍。
The value of the digit 7 in 3 726 154 is 700 000 because it sits in the hundred‑thousands column. Partitioning a number like 5 083 219 helps: 5 000 000 + 80 000 + 3 000 + 200 + 10 + 9.
在 3 726 154 中数字 7 的值是 700 000，因为它位于十万位。像 5 083 219 这样的数可以进行拆分：5 000 000 + 80 000 + 3 000 + 200 + 10 + 9。
For decimals, the place value system continues to the right of the units column. The first decimal place is tenths (1/10), the second is hundredths (1/100) and the third is thousandths (1/1000). So 0.607 means 6 tenths + 0 hundredths + 7 thousandths.
对于小数，位值系统延续到整数个位的右方。第一位小数是十分位 (1/10)，第二位是百分位 (1/100)，第三位是千分位 (1/1000)。因此 0.607 表示 6 个十分之一 + 0 个百分之一 + 7 个千分之一。
2. Negative Numbers | 负数
Negative numbers are numbers less than zero. They are used for temperatures below freezing, debts, or depths below sea level. On a number line, they lie to the left of zero.
负数是小于零的数。它们用于表示冰点以下的温度、负债或海平面以下深度。在数轴上，负数位于零的左侧。
When adding or subtracting with negatives, imagine moving along the number line. For example, 3 − 5 = −2 because you start at 3 and move 5 places to the left. Similarly, −4 + 7 = 3 because starting at −4 and adding 7 means moving right 7 steps.
进行负数加减时，可以想象在数轴上移动。例如，3 − 5 = −2，因为从 3 出发向左移动 5 格。同理，−4 + 7 = 3，因为从 −4 出发加上 7 意味着向右移动 7 步。
Double signs can be simplified: ‘minus a negative’ becomes adding. So 5 − (−3) = 5 + 3 = 8. Multiplying or dividing two numbers with the same sign gives a positive result; with different signs gives a negative result: (−6) × (−4) = 24, whereas (−6) × 4 = −24.
双重符号可以化简：‘减一个负数’变成加法。因此 5 − (−3) = 5 + 3 = 8。同号两数相乘或相除得正；异号得负：(−6) × (−4) = 24，而 (−6) × 4 = −24。
3. Factors, Multiples and Primes | 因数、倍数与质数
A factor of a number divides it exactly without leaving a remainder. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. A multiple is the product of a number and any whole number; the first five multiples of 7 are 7, 14, 21, 28, 35.
一个数的因数能整除该数且没有余数。24 的因数有 1, 2, 3, 4, 6, 8, 12, 24。倍数是一个数与任何整数的乘积；7 的前五个倍数是 7, 14, 21, 28, 35。
Prime numbers have exactly two distinct factors: 1 and themselves. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 1 is not a prime. A composite number has more than two factors.
质数恰好只有两个不同因数：1 和它本身。前十个质数是 2, 3, 5, 7, 11, 13, 17, 19, 23, 29。注意 1 不是质数。合数有多于两个因数。
Every composite number can be written as a product of prime factors using a factor tree. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. The highest common factor (HCF) of two numbers is the largest factor they share; the lowest common multiple (LCM) is the smallest multiple they share.
每个合数都可以用因子树写成质因数的乘积。例如，60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。两个数的最大公因数 (HCF) 是它们共有的最大因数；最小公倍数 (LCM) 是它们共有的最小倍数。
4. Fractions: Simplifying and Equivalent Fractions | 分数：化简与等值分数
A fraction represents a part of a whole. The numerator (top) tells us how many parts we have, the denominator (bottom) tells us how many equal parts the whole is divided into. Equivalent fractions look different but have the same value, e.g. 1/2 = 2/4 = 4/8.
分数表示整体的一部分。分子（上）告诉我们有几个部分，分母（下）告诉我们整体被分成多少等份。等值分数看起来不同但数值相同，如 1/2 = 2/4 = 4/8。
To simplify a fraction, divide the numerator and denominator by their highest common factor. For 18/24, the HCF of 18 and 24 is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. This is the fraction in its simplest form.
化简分数时，用分子分母的最大公因数同时除它们。对于 18/24，18 和 24 的 HCF 是 6，18 ÷ 6 = 3，24 ÷ 6 = 4，得到 3/4。这就是最简分数。
Any improper fraction (numerator larger than denominator) can be converted to a mixed number. 17/5 = 3 remainder 2, so it equals 3 2/5. Conversely, a mixed number can be turned into an improper fraction: 4 1/3 = (4×3 + 1)/3 = 13/3.
任何假分数（分子大于分母）都可以化成带分数。17/5 = 3 余 2，因此等于 3 又 2/5。反过来，带分数可以化成假分数：4 又 1/3 = (4×3 + 1)/3 = 13/3。
5. Adding and Subtracting Fractions | 分数加减法
When the denominators are the same, simply add or subtract the numerators and keep the denominator unchanged: 3/8 + 2/8 = 5/8, and 7/10 − 3/10 = 4/10 which simplifies to 2/5.
当分母相同时，只需将分子相加或相减，分母不变：3/8 + 2/8 = 5/8，7/10 − 3/10 = 4/10 化简得 2/5。
For fractions with different denominators, find a common denominator—the LCM of the denominators. To calculate 2/3 + 1/4, the LCM of 3 and 4 is 12. Convert: 2/3 = 8/12, 1/4 = 3/12. Then add: 8/12 + 3/12 = 11/12.
对于分母不同的分数，要找到公分母——分母的最小公倍数。计算 2/3 + 1/4，3 和 4 的 LCM 是 12。转化：2/3 = 8/12，1/4 = 3/12。然后相加：8/12 + 3/12 = 11/12。
The same process applies to subtraction: 5/6 − 1/4 → LCM of 6 and 4 is 12. 5/6 = 10/12, 1/4 = 3/12, so 10/12 − 3/12 = 7/12. Always finish by simplifying the answer if possible.
减法过程相同：5/6 − 1/4 → 6 和 4 的 LCM 是 12。5/6 = 10/12，1/4 = 3/12，所以 10/12 − 3/12 = 7/12。最后如果可能，一定要化简结果。
6. Decimals and Rounding | 小数与四舍五入
Decimal numbers can be compared digit by digit, starting from the left. To compare 0.75 and 0.8, remember that 0.8 = 0.80, so 0.75 < 0.80. Adding zeros to the right of a decimal does not change its value.
小数可以按数位从左到右逐位比较。比较 0.75 和 0.8 时，记住 0.8 = 0.80，所以 0.75 < 0.80。小数末尾加零不改变其值。
Rounding makes numbers easier to work with. To round to one decimal place, look at the digit in the second decimal place. If it is 5 or more, round up; if it is 4 or less, leave the first decimal unchanged. So 3.67 rounded to 1 decimal place is 3.7, while 3.64 becomes 3.6.
四舍五入让数字更易处理。保留一位小数时，看第二位小数。如果数字是 5 或以上，向前进一；如果是 4 或以下，第一位小数不变。因此 3.67 四舍五入到一位小数为 3.7，而 3.64 变成 3.6。
When multiplying decimals, first ignore the decimal points and multiply as whole numbers. Then count the total number of decimal places in the factors; put that many decimal places in the answer. 0.4 × 0.7: 4 × 7 = 28, and there are 1 + 1 = 2 decimal places, giving 0.28.
小数乘法时，先忽略小数点，当作整数相乘。然后数出两个因数中小数总位数，在乘积中点上同样多位小数。0.4 × 0.7：4 × 7 = 28，共 1+1=2 位小数，得 0.28。
7. Percentages | 百分数
A percentage is a fraction with denominator 100. 35% means 35 out of 100, written as 35/100 or simplified to 7/20. To convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply the fraction by 100%.
百分数是分母为 100 的分数。35% 表示 100 份里的 35 份，写作 35/100 或化简为 7/20。把分数化成百分数，可以找到分母为 100 的等值分数，或将分数乘以 100%。
To find a percentage of a quantity, write the percentage as a fraction or decimal and multiply. 20% of £45 = 20/100 × 45 = 1/5 × 45 = £9. Alternatively, 10% of £45 is £4.50, so 20% is double that: £9.
求一个数量的百分数，先把百分数写成分数或小数再相乘。£45 的 20%：20/100 × 45 = 1/5 × 45 = £9。或者先求 10%：£4.50，那么 20% 就是加倍：£9。
Percentages can also be used to describe increases and decreases. A price increased by 15% means finding 15% of the original and adding it on. The new amount is 115% of the original. To find the original after a percentage change, work backwards using division.
百分数也可用来描述增加和减少。价格上涨 15% 意味着先求原价的 15% 再加上。新价格是原价的 115%。已知变化后的百分数，要倒推原值就用除法。
8. Introduction to Algebra | 代数入门
Algebra uses letters to stand for unknown numbers. An expression like 3a + 2 means ‘3 times a number a, then add 2’. The multiplication sign is usually omitted between a number and a letter: 4 × n is written 4n.
代数用字母代表未知数。像 3a + 2 这样的表达式表示 ‘一个数 a 的 3 倍，再加 2’。数与字母之间的乘号通常省略：4 × n 写作 4n。
Like terms contain exactly the same letters and powers. Only like terms can be combined by adding or subtracting the coefficients. 5x + 2x = 7x, but 3x + 4y cannot be simplified further because the variables are different.
同类项含有完全相同的字母和指数。只有同类项可以通过加减系数来合并。5x + 2x = 7x，但 3x + 4y 不能进一步化简，因为变量不同。
Substitution means replacing letters with numbers. If a = 3 and b = 4, then 2a + 5b = 2×3 + 5×4 = 6 + 20 = 26. Always follow the order of operations: brackets, powers, multiplication/division, addition/subtraction.
代入法就是把字母换成数字。若 a = 3, b = 4，则 2a + 5b = 2×3 + 5×4 = 6 + 20 = 26。务必遵循运算顺序：括号、乘方、乘除、加减。
9. Solving Simple Equations | 解简单方程
An equation states that two expressions are equal. To solve an equation, find the value of the unknown that makes the statement true. We use inverse operations to isolate the unknown on one side of the equation.
方程表示两个表达式相等。解方程就是找到使等式成立的未知数的值。我们使用逆运算将未知数隔离在方程的一边。
For one‑step equations like x + 7 = 15, subtract 7 from both sides: x = 8. For 4y = 32, divide both sides by 4: y = 8. The key ‘golden rule’ is: whatever you do to one side of the equation, you must do to the other.
对于一步方程，如 x + 7 = 15，两边同时减去 7：x = 8。对于 4y = 32，两边同除以 4：y = 8。关键的‘黄金法则’是：对方程一边做什么，另一边的处理必须相同。
Two‑step equations involve two operations. Solve 2p − 5 = 9: first add 5 to both sides to get 2p = 14, then divide by 2 to give p = 7. Always check your solution by substituting it back into the original equation.
两步方程包含两种运算。解 2p − 5 = 9：先两边加 5 得 2p = 14，再除以 2 得 p = 7。始终将解代回原方程验证。
10. Angles | 角度
Angles are measured in degrees (°). An acute angle is less than 90°, a right angle exactly 90°, an obtuse angle between 90° and 180°, and a reflex angle between 180° and 360°. A straight line forms an angle of 180°.
角度以度 (°) 为单位。锐角小于 90°，直角等于 90°，钝角在 90° 到 180° 之间，优角在 180° 到 360° 之间。一条直线构成 180° 角。
Angles around a point sum to 360°. If three angles around a point are 120°, 95° and 85°, the fourth angle is 360 − (120+95+85) = 60°. This rule is helpful for missing angle problems.
绕一点一周的角度和为 360°。如果一点周围三个角分别是 120°、95° 和 85°，第四个角就是 360 − (120+95+85) = 60°。这个法则在求缺失角度时很有用。
Vertically opposite angles are equal when two lines intersect. If one angle is 45°, the angle directly opposite is also 45°. Adjacent angles on a straight line add up to 180°, so the angle next to it is 135°.
两条直线相交时，对顶角相等。如果一个角是 45°，正对着的角也是 45°。直线上的邻角之和为 180°，所以紧邻的角就是 135°。
11. Perimeter and Area | 周长与面积
Perimeter is the total distance around the outside of a shape. For a rectangle of length l and width w, perimeter = 2l + 2w, or 2(l + w). A square with side 5 cm has perimeter 4 × 5 = 20 cm.
周长是形状外边线的总长度。对于长 l、宽 w 的长方形，周长 = 2l + 2w，或 2(l + w)。边长为 5 cm 的正方形周长为 4 × 5 = 20 cm。
Area measures the surface covered. The area of a rectangle is length × width. A rectangle 8 m by 3 m has area 8 × 3 = 24 m². For a triangle, area = ½ × base × height. A triangle with base 6 cm and height 4 cm has area ½ × 6 × 4 = 12 cm².
面积测量覆盖的表面。长方形的面积是长 × 宽。一个 8 m × 3 m 的长方形面积为 8 × 3 = 24 m²。三角形的面积 = ½ × 底 × 高。底 6 cm、高 4 cm 的三角形面积为 ½ × 6 × 4 = 12 cm²。
Compound shapes can be split into rectangles or triangles. Calculate the area of each part and add them together. For perimeter, only count the outer edges, ignoring any internal lines.
组合图形可以分割成矩形或三角形。分别计算各部分面积再相加。求周长时只算外部边界，忽略内部线条。
12. Averages and Range | 平均数与极差
Mean, median, mode and range are used to summarise data sets. The mean is calculated by adding all values and dividing by how many values there are. For the numbers 4, 8, 9, 5, 7, the sum is 33 and there are 5 numbers, so mean = 33 ÷ 5 = 6.6.
平均数、中位数、众数和极差用于概括数据。平均数是将所有数值相加后除以个数。对于数据 4, 8, 9, 5, 7，总和为 33，共 5 个数，所以平均数 = 33 ÷ 5 = 6.6。
The median is the middle value when data is ordered. The set 3, 7, 8, 12, 15 has median 8. If there are an even number of values, the median is the mean of the two middle numbers.
中位数是排序后中间的值。数据集 3, 7, 8, 12, 15 的中位数是 8。如果数据个数为偶数，中位数是中间两个数的平均数。
The mode is the value that appears most often—useful for finding the most common category. Range is the difference between the largest and smallest values, showing how spread out the data is.
众数是出现频率最高的数值——有助于找出最常见类别。极差是最大值与最小值的差，反映数据的离散程度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导