KS3 Maths: Essential Maths 7H Homework Book Key Concepts Explained | KS3 数学：Essential Maths 7H 作业本知识点精讲

📚 KS3 Maths: Essential Maths 7H Homework Book Key Concepts Explained | KS3 数学：Essential Maths 7H 作业本知识点精讲

The Essential Maths 7H Homework Book is designed for students working at the higher end of Key Stage 3. It covers a wide range of fundamental topics, from number skills to algebra and geometry, and provides plenty of practice to build fluency and reasoning. This article breaks down the core concepts from each main area, explaining key ideas and methods in both English and Chinese to support bilingual learners and parents.

《Essential Maths 7H 作业本》专为 KS3 阶段较高水平的学生设计。它涵盖了从数感运算到代数、几何的广泛基础知识，并通过大量练习培养流利度与推理能力。本文逐一梳理各章节的核心概念，用英汉双语解释关键思想和方法，帮助双语学习者与家长加深理解。

1. Place Value and Ordering | 位值与排序

Understanding place value is the foundation of all number work. In KS3, you must be confident with millions, thousandths, and be able to multiply and divide by 10, 100 and 1000 mentally. When you multiply by 10, digits move one place to the left; when dividing, they move one place to the right. This works because our number system is base 10.

理解位值是一切数字运算的基础。KS3 阶段，你需要熟练掌握百万位、千分位，并能心算乘以或除以 10、100、1000。乘以 10 时，所有数字向左移动一位；除以 10 时，向右移动一位。这是因为我们的数制是十进制。

Ordering integers, decimals and fractions often requires converting them to a common form. A number line is a useful tool. Negative numbers can be tricky: remember that −7 is smaller than −2 because it lies further to the left on the number line.

比较整数、小数和分数时，通常需要将它们化成统一的形式。数轴是非常有用的工具。负数可能容易混淆：记住 −7 比 −2 小，因为它在数轴上更靠左。

Phrase	中文
Place value columns	位值列
Tenths, hundredths, thousandths	十分位、百分位、千分位
Ascending / descending order	升序 / 降序

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2. Decimals and Rounding | 小数与四舍五入

Working with decimals involves addition, subtraction, multiplication and division. Always align the decimal points vertically when adding or subtracting. For multiplication, you can ignore the decimal points initially, multiply the numbers, then put the decimal point back so the answer has the same total number of decimal places as the factors combined.

小数的运算包括加、减、乘、除。加减法时，始终要对齐小数点。乘法时，可以先忽略小数点进行整数乘法，然后在结果中从右往左数出因数的小数位数之和，点上小数点。

Rounding to decimal places is common: for example, rounding 3.746 to two decimal places gives 3.75 because the third decimal digit is 6 (five or more, round up). Rounding to significant figures works similarly but the first non‑zero digit is the starting point. For 0.00852, to one significant figure it is 0.009.

四舍五入到指定小数位很常见：例如，将 3.746 保留两位小数得到 3.75，因为第三位是 6（满五进一）。有效数字的四舍五入类似，但第一个非零数字才是起点。如 0.00852 保留一位有效数字为 0.009。

Upper and lower bounds can be introduced here. If a length is given as 6.7 cm to one decimal place, the true length could be anywhere between 6.65 cm and 6.75 cm.

此处可以引入上界和下界的概念。如果一个长度给定为 6.7 cm（精确到一位小数），实际长度可能在 6.65 cm 与 6.75 cm 之间。

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3. Fractions: Adding and Subtracting | 分数的加减法

Adding and subtracting fractions requires a common denominator. To find the common denominator, identify the lowest common multiple (LCM) of the denominators. For example, for 1/4 and 2/5, the LCM of 4 and 5 is 20, so change them to 5/20 and 8/20, then add to get 13/20.

分数的加减法需要公分母。找到公分母就是求分母的最小公倍数（LCM）。例如，1/4 和 2/5，分母 4 和 5 的最小公倍数是 20，所以转换为 5/20 和 8/20，相加得到 13/20。

Mixed numbers should be converted to improper fractions first. Once the sum is found, you can convert back to a mixed number and simplify. Always check if the fraction can be expressed in its simplest form by dividing numerator and denominator by their highest common factor.

带分数应该先化为假分数。计算出结果后，再化回带分数并约简。始终要检查分数能否化成最简形式，即分子分母同除以它们的最大公因数。

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4. Percentages: Increase and Decrease | 百分比的增减问题

A percentage is a fraction out of 100. To find a percentage of an amount, multiply by the percentage and divide by 100. For increase, you can find the percentage amount and add it on, or use a multiplier. For example, increasing £80 by 15%: 100% + 15% = 115%, multiplier 1.15, so £80 × 1.15 = £92.

百分比是以 100 为分母的分数。求一个数的百分之几，乘以百分比再除以 100。求增加量时，可以先算出百分比数值再加，或使用乘数。例如，将 £80 增加 15%：100% + 15% = 115%，乘数为 1.15，因此 £80 × 1.15 = £92。

Decreasing uses the same method: for a 20% discount, the multiplier is 0.80. Reverse percentages come up when you know the final amount after a change and need the original. If a price after a 10% increase is £55, the original was £55 ÷ 1.10 = £50.

减少同理：打八折（20% off）的乘数是 0.80。反向百分比问题则是在已知变化后的量时求原量。比如，一次 10% 提价后的价格为 £55，那么原价是 £55 ÷ 1.10 = £50。

Change	Multiplier
Increase by 5%	1.05
Decrease by 30%	0.70

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5. Algebraic Expressions | 代数表达式

Algebra is about using letters to represent unknowns. An expression like 3a + 2b − a + 5b can be simplified by collecting like terms. Only terms with exactly the same letter combination can be combined. Here, 3a − a = 2a, and 2b + 5b = 7b, giving 2a + 7b.

代数就是用字母表示未知数。像 3a + 2b − a + 5b 这样的式子可以通过合并同类项化简。只有字母部分完全相同的项才能相加减。这里，3a − a = 2a，2b + 5b = 7b，结果是 2a + 7b。

Multiplying terms is straightforward: 4 × 3y = 12y, and p × p = p². When multiplying with different letters, just write them together: 2x × 3y = 6xy. Brackets are expanded using the distributive law: 3(x + 4) = 3x + 12.

项的乘法很简单：4 × 3y = 12y，p × p = p²。不同字母相乘时，直接写在一起：2x × 3y = 6xy。去括号用分配律：3(x + 4) = 3x + 12。

Factorising is the reverse of expanding. For 6x + 9, the highest common factor is 3, so it becomes 3(2x + 3). Always check by expanding back. Simple substitution means replacing letters with given values, for instance if x = 3, then 2x² = 2 × 9 = 18.

因式分解是去括号的逆过程。对于 6x + 9，最大公因数是 3，所以写成 3(2x + 3)。始终要展开验证。简单代入即用给定数值替换字母，例如 x = 3 时，2x² = 2 × 9 = 18。

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6. Equations and Inequalities | 方程与不等式

An equation states that two expressions are equal. To solve, we use inverse operations to isolate the variable. For x + 5 = 12, subtract 5 from both sides to get x = 7. Keep the balance: whatever you do to one side, do to the other.

方程表明两个表达式相等。解方程时，利用逆运算把变量单独留在一边。对于 x + 5 = 12，两边同时减 5 得到 x = 7。保持平衡：对一边做什么运算，对另一边也要做同样的运算。

Two‑step equations might look like 2x − 3 = 7. First add 3: 2x = 10, then divide by 2: x = 5. Equations with brackets should be expanded first, then solved. Letters on both sides require collecting variable terms on one side, constants on the other.

两步方程可能形如 2x − 3 = 7。先加 3：2x = 10，再除以 2：x = 5。含括号的方程先去括号再求解。变量出现在两边的情况下，要把含变量项移到一边，常数项移到另一边。

Inequalities are solved similarly, but with one extra rule: if you multiply or divide by a negative number, you must reverse the inequality sign. For −2x < 6, dividing by −2 gives x > −3. Representation on a number line uses open circles for < or >, closed for ≤ or ≥.

不等式的解法类似，但有一条额外规则：若乘以或除以负数，不等号方向要改变。如 −2x < 6，两边除以 −2 得 x > −3。在数轴上表示时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。

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7. Sequences and nth Term | 数列与第n项

A sequence is a list of numbers following a rule. In KS3, the focus is on linear sequences where the difference between consecutive terms is constant. For 3, 7, 11, 15, …, the term‑to‑term rule is ‘add 4’.

数列是按某种规律排列的一列数。KS3 重点研究线性数列，即相邻两项的差是常数。比如 3, 7, 11, 15, …，递推规律是“每次加 4”。

The position‑to‑term rule, or nth term, lets you find any term directly. For the sequence above, nth term = 4n − 1. You can generate terms: n=1 gives 3, n=2 gives 7, etc. The coefficient of n is the common difference, and the constant is adjusted to make the first term correct.

位量规则，即第 n 项公式，可以直接求出任意一项。上述数列的第 n 项为 4n − 1。你可以生成各项：n=1 得 3，n=2 得 7。n 的系数是公差，常数项的调整使首项正确。

If the sequence is descending, e.g. 10, 7, 4, 1, …, the difference is −3, so nth term = −3n + 13. To check, substitute n=1: −3+13=10. Problems may involve finding whether a given number is a term of a sequence by solving an equation.

如果是递减数列，如 10, 7, 4, 1, …，公差为 −3，所以第 n 项 = −3n + 13。检验：n=1 得 −3+13=10。题目可能会要求判断某个数是否为数列中的项，这需要解方程。

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8. Angles and Lines | 角与线

Basic angle facts are essential: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. When parallel lines are crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles sum to 180°.

基本角度事实非常重要：直线上的角和为 180°，绕一点一周的角和为 360°，对顶角相等。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°）。

In triangles, interior angles sum to 180°. Knowing two angles allows you to find the third. An equilateral triangle has three 60° angles, an isosceles triangle has two equal base angles. The exterior angle of a triangle equals the sum of the two opposite interior angles.

三角形的内角和为 180°。知道两个角就能求出第三个。等边三角形每个角都是 60°，等腰三角形有两个相等的底角。三角形的一个外角等于与它不相邻的两个内角之和。

Bearings are measured clockwise from north and expressed as three‑digit angles. A bearing of 045° means 45° clockwise from north. Always draw a clear diagram, marking north lines and known angles.

方位角从正北方向顺时针测量，用三位数表示。045° 表示从北偏东 45°。解题时务必画出示意图，标出正北线和已知角。

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9. Area and Perimeter | 面积与周长

Perimeter is the total distance around the outside of a shape. For a rectangle, P = 2(l + w). For compound shapes, add all outer side lengths, being careful to work out any missing sides using given dimensions.

周长是图形外边线的总长度。矩形周长 P = 2(l + w)。对于组合图形，把所有的外边长相加，注意利用已知长度算出所有缺失的边长。

Area of a rectangle is A = l × w. The area of a triangle is A = ½ × base × height. The height must be perpendicular to the base. For a parallelogram, A = base × perpendicular height. A trapezium has area A = ½ (a + b)h, where a and b are the parallel sides and h is the perpendicular distance between them.

矩形面积：A = 长 × 宽。三角形面积：A = ½ × 底 × 高，高必须与底垂直。平行四边形面积：A = 底 × 垂直高。梯形面积：A = ½ (a + b)h，其中 a 和 b 是两平行边，h 是它们之间的垂直距离。

Units are crucial: if sides are in cm, area is in cm². For compound shapes, split them into rectangles and triangles, work out each area, and sum them. Converting between units of area involves squaring the length conversion factor: 1 m² = 10 000 cm².

单位很关键：如果边长单位是 cm，面积单位就是 cm²。对于组合图形，可将其拆分成矩形和三角形，分别计算面积再相加。面积单位换算时需要对长度换算因子进行平方：1 m² = 10 000 cm²。

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10. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数与极差

The mean is calculated by summing all data values and dividing by the number of values. It represents the average but is sensitive to outliers. The median is the middle value when data is ordered; it splits the dataset into two halves.

平均数（均值）是将所有数据值求和再除以数据个数。它代表集中趋势，但易受极端值影响。中位数是将数据排序后最中间的值；它将数据集分成两半。

The mode is the most frequent value. Some datasets may have no mode, one mode, or more than one (bimodal). The range is the difference between the largest and smallest values, giving a simple measure of spread.

众数是出现次数最多的值。有些数据集可能没有众数、有一个众数或有多个众数（双众数）。极差是最大值与最小值的差，是描述数据散布情况的简单指标。

When data is given in a frequency table, the mean can be calculated as total (value × frequency) divided by total frequency. The median position is found by (n+1)/2, then locating the interval or value that contains it.

当数据以频数表形式给出时，平均数 = Σ(值 × 频数) ÷ 总频数。中位数的位置由 (n+1)/2 确定，再找出该位置所在的区间或数值。

Data	2,3,3,5,7
Mean	(2+3+3+5+7)/5 = 4
Median	3 (third value)
Mode	3
Range	7−2 = 5

A comparative analysis often requires using two averages or the range to interpret and compare datasets. Always state what the statistic tells you about the data in context.

比较分析常需要利用两种平均数或极差来解释和对比数据集。始终要结合具体情境，说明该统计量反映了数据的什么特征。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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