📚 Essential Maths 7C Homework Book: Key Concepts Explained | Essential Maths 7C 作业本知识点精讲
The Essential Maths 7C Homework Book is designed to reinforce the core mathematical skills required at Key Stage 3, particularly for Year 7 students working at a confident level. This article breaks down the most important topics covered in the book, providing clear explanations, worked examples and useful tips to help you master each concept.
Essential Maths 7C 作业本旨在巩固 KS3 阶段所需的核心数学技能，尤其适合 Year 7 中水平较好的学生。本文将拆解书中涵盖的关键知识点，提供清晰的解释、例题和实用技巧，帮助你掌握每一个概念。
1. Number and Place Value | 数字与位值
Understanding place value is fundamental for all number work. You must be able to read, write and order whole numbers up to at least 10 000 000, and recognise the value of each digit.
理解位值是一切数字运算的基础。你需要能读写并排序至少到一千万的整数，并能识别每一位数字所代表的数值。
In a number like 4 326 579, the digit 4 represents four millions, 3 is three hundred thousands, 2 is twenty thousands, 6 is six thousands, 5 is five hundreds, 7 is seventy, and 9 is nine ones. Using place value columns helps to multiply and divide by 10, 100 or 1000 by shifting digits left or right.
在 4 326 579 这个数中，数字 4 代表四百万，3 是三十万，2 是二万，6 是六千，5 是五百，7 是七十，9 是九个一。借助位值列表，我们可以通过将数字向左或向右移动来乘以或除以 10、100 或 1000。
Decimals work in the same way, with columns to the right of the decimal point representing tenths, hundredths, thousandths, etc. For example, 0.037 has 3 hundredths and 7 thousandths.
小数也是同样的道理，小数点右边的数位依次表示十分位、百分位、千分位等。例如，0.037 中有 3 个百分之一和 7 个千分之一。
2. Negative Numbers | 负数
Negative numbers appear in real-life contexts such as temperature, debts and depths below sea level. You should be confident adding, subtracting, multiplying and dividing with negative values.
负数出现在温度、负债和海平面以下深度等实际情境中。你需要熟练掌握负数的加、减、乘、除运算。
When adding a negative number, think of moving left on a number line. Subtracting a negative number moves right. A useful rule: two like signs become a plus, two unlike signs become a minus. For example, 5 − (−3) = 5 + 3 = 8, and −4 + (−2) = −6.
加上负数时，可以看作是在数轴上向左移动。减去负数则向右移动。一条实用规则：两个同号得正，异号得负。例如，5 − (−3) = 5 + 3 = 8，而 −4 + (−2) = −6。
Multiplication and division follow the sign rule: positive × negative = negative, negative × negative = positive. So (−6) × (−3) = 18, and 20 ÷ (−4) = −5.
乘除法遵循符号法则：正乘负得负，负乘负得正。因此 (−6) × (−3) = 18，而 20 ÷ (−4) = −5。
3. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a whole number that divides exactly into another number. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Common factors are shared by two or more numbers; the highest common factor (HCF) of 24 and 36 is 12.
因数是指能整除另一个整数的数。例如，24 的因数有 1, 2, 3, 4, 6, 8, 12 和 24。公因数是两个或更多数共有的因数；24 和 36 的最大公因数 (HCF) 是 12。
Multiples are the numbers in a multiplication table. The multiples of 7 are 7, 14, 21, 28, 35, … The lowest common multiple (LCM) of 6 and 8 is 24.
倍数就是乘法表中的数。7 的倍数有 7, 14, 21, 28, 35……6 和 8 的最小公倍数 (LCM) 是 24。
Prime numbers have exactly two factors: 1 and themselves. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Every whole number greater than 1 can be written as a product of primes — this is called prime factorisation. For 60, we write 60 = 2² × 3 × 5.
质数恰好只有两个因数：1 和它本身。前十个质数是 2, 3, 5, 7, 11, 13, 17, 19, 23, 29。每一个大于 1 的整数都可以写成质数的乘积，这叫作质因数分解。例如 60 = 2² × 3 × 5。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
Being able to convert fluently between fractions, decimals and percentages is a vital skill. For example, ½ = 0.5 = 50%, ¼ = 0.25 = 25%, and ⅗ = 0.6 = 60%.
能够在分数、小数和百分数之间熟练转换是一项关键技能。例如，½ = 0.5 = 50%，¼ = 0.25 = 25%，而 ⅗ = 0.6 = 60%。
When ordering mixed types, convert them all into the same form. To find a percentage of an amount, change the percentage to a decimal or fraction and multiply. For instance, 15% of £240 = 0.15 × 240 = £36.
在对混合类型进行排序时，先把它们全部转换成同一种形式。要求出一个数的百分比，可以先把百分数化成小数或分数，再做乘法。例如，£240 的 15% = 0.15 × 240 = £36。
Adding and subtracting fractions requires a common denominator. For ⅓ + ¼, use 12 as the common denominator: ⁴/₁₂ + ³/₁₂ = ⁷/₁₂. Multiplying fractions is straightforward: multiply the numerators and the denominators. Dividing fractions uses ‘keep, change, flip’: ⅔ ÷ ¼ = ⅔ × ⁴/₁ = ⁸/₃ = 2 ⅔.
分数的加减需要先通分。对于 ⅓ + ¼，用 12 作为公分母：⁴/₁₂ + ³/₁₂ = ⁷/₁₂。分数乘法很简单：分子相乘，分母相乘。分数除法运用“保留、变号、翻转”口诀：⅔ ÷ ¼ = ⅔ × ⁴/₁ = ⁸/₃ = 2 ⅔。
5. Algebraic Expressions | 代数表达式
Algebra uses letters to represent unknown numbers or quantities. You need to simplify expressions by collecting like terms. For example, 3a + 2b − a + 4b = 2a + 6b.
代数用字母表示未知的数或量。你需要通过合并同类项来化简表达式。例如，3a + 2b − a + 4b = 2a + 6b。
When multiplying terms, write the number first and then the letters in alphabetical order, using indices for repeated letters: p × q × p = p²q. The same rules apply with division, often leaving answers as fractions or decimals.
进行项的乘法时，先写数字，再按字母顺序书写字母，重复的字母用指数表示：p × q × p = p²q。除法也遵循同样的规则，结果通常以分数或小数形式呈现。
Expanding brackets involves multiplying each term inside the bracket by the term outside. For 3(2x + 5), multiply 2x and 5 by 3 to get 6x + 15. This prepares you for solving equations later.
展开括号就是把括号外的项与括号内每一项相乘。例如 3(2x + 5)，将 3 分别乘以 2x 和 5，得到 6x + 15。这为后面解方程做好了准备。
6. Solving Linear Equations | 解线性方程
Solving equations means finding the value of the unknown that makes the equality true. Always aim to isolate the variable using inverse operations, keeping the equation balanced by doing the same to both sides.
解方程就是求出使等式成立的未知数的值。始终要利用逆运算来分离变量，并保持等式平衡——对方程两边做相同的运算。
For a simple one-step equation like x + 8 = 15, subtract 8 from both sides to give x = 7. For two-step equations such as 2y − 5 = 11, add 5 first to get 2y = 16, then divide by 2 to find y = 8.
对于像 x + 8 = 15 这样简单的一步方程，两边同时减去 8，得到 x = 7。对于两步方程，例如 2y − 5 = 11，先加 5 得到 2y = 16，再除以 2，求出 y = 8。
Equations with brackets require expanding first. Solve 4(2n + 3) = 28 → 8n + 12 = 28 → 8n = 16 → n = 2. Always check your answer by substitution.
含有括号的方程需要先展开。解 4(2n + 3) = 28 → 8n + 12 = 28 → 8n = 16 → n = 2。始终要记得代入检验。
7. Angle Properties | 角度性质
Angles are measured in degrees (°) and key facts help you find missing angles without a protractor. Angles on a straight line add up to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.
角以度数 (°) 度量，掌握关键性质可以帮助你不使用量角器就能求出未知角。直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。
In triangles, the three interior angles always total 180°. For a right-angled triangle, the other two angles are acute and sum to 90°. An isosceles triangle has two equal angles opposite the equal sides.
在三角形中，三个内角之和总是 180°。直角三角形中，另外两个角都是锐角，且互余（和为 90°）。等腰三角形两底角相等，对应相等的两边。
Parallel lines create special angle pairs. Alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°. Recognising these patterns saves time in angle problems.
平行线会产生特殊的角对。内错角相等，同位角相等，同旁内角互补（和为 180°）。识别这些关系能节省做角度问题的时间。
8. Perimeter and Area | 周长与面积
Perimeter is the total distance around a 2D shape. Simply add the lengths of all sides. For a rectangle 5 cm by 3 cm, P = 2 × (5 + 3) = 16 cm.
周长是二维图形所有边的总长度，只需将所有边长相加。对于一个 5 cm × 3 cm 的长方形，周长 P = 2 × (5 + 3) = 16 cm。
Area measures the space inside a shape, recorded in square units. The area of a rectangle = length × width. The same rectangle has area A = 5 × 3 = 15 cm². For a triangle, use A = ½ × base × height.
面积衡量形状内部的空间大小，单位为平方单位。长方形的面积 = 长 × 宽。同一个长方形的面积 A = 5 × 3 = 15 cm²。三角形的面积公式是 A = ½ × 底 × 高。
Compound shapes can be split into simpler rectangles and triangles. Calculate the area of each part and add them together. Remember to use the perpendicular height for triangles, not the slant length.
组合图形可以拆分成简单的长方形和三角形。分别计算出各部分的面积，再加起来。记住，三角形的面积必须用垂直高度，而不是斜边长度。
9. Coordinates and Graphs | 坐标与图像
Coordinates are written as (x, y), where x is the horizontal position and y the vertical. The x-axis is horizontal, and the y-axis is vertical. The origin is (0, 0).
坐标记作 (x, y)，x 代表水平位置，y 代表垂直位置。x 轴为水平轴，y 轴为垂直轴，原点为 (0, 0)。
Plotting points accurately on a grid is essential for drawing straight-line graphs. For an equation like y = 2x + 1, choose x-values of −2, −1, 0, 1, 2, calculate the matching y-values, plot the points and join them with a straight line.
在坐标网格上精确描点是绘制直线图像的关键。对于 y = 2x + 1 这样的方程，选取 x 值为 −2, −1, 0, 1, 2，计算出相应的 y 值，描出这些点，再用直线将它们连接起来。
Lines in the form y = mx + c have gradient m and y-intercept c. In y = 2x + 1, the line crosses the y-axis at 1 and rises 2 units for every 1 across. Recognising this helps you sketch graphs quickly.
形式为 y = mx + c 的直线，其斜率为 m，y 轴截距为 c。在 y = 2x + 1 中，直线与 y 轴交于点 1，并且每向右 1 个单位，上升 2 个单位。认识这一点能帮你快速绘制图像草图。
10. Statistics: Averages and Charts | 统计：平均数与图表
The three main averages are mean, median and mode. The mode is the most common value, the median is the middle value when data is ordered, and the mean is found by adding all values and dividing by the number of values.
三种主要的平均数是均数、中位数和众数。众数是最常出现的值，中位数是将数据按顺序排列后位于中间的值，均数则是将所有值相加后除以数值的个数。
For example, in the dataset 3, 4, 4, 7, 10: mode = 4, median = 4, mean = (3+4+4+7+10) ÷ 5 = 28 ÷ 5 = 5.6. Outliers can distort the mean, so the median is often more representative.
例如，在数据集 3, 4, 4, 7, 10 中：众数 = 4，中位数 = 4，均数 = (3+4+4+7+10) ÷ 5 = 28 ÷ 5 = 5.6。离群值可能会扭曲均数，因此中位数往往更具代表性。
Data can be displayed in bar charts, pictograms, line graphs and pie charts. When interpreting, always check the scale, read labels carefully and ask yourself what the chart is showing. A pie chart represents proportions: the whole circle equals 360° or 100%.
数据可以用条形图、象形图、折线图和饼图来展示。解读时，要检查刻度、仔细阅读标签，并思考图表在传达什么信息。饼图用于表示比例：整个圆代表 360° 或 100%。
Drawing accurate charts requires neat labelling, consistent scales and a title. For a bar chart, bars must have equal width and gaps between them. In a pie chart, calculate the angle per category: (category value ÷ total value) × 360°.
绘制准确的图表需要清晰的标注、一致的刻度和标题。条形图中，条宽必须相等，条与条之间留有空隙。饼图则需要计算每一类所对应的角度：(类别值 ÷ 总值) × 360°。
11. Rounding and Estimation | 四舍五入与估算
Rounding makes numbers easier to work with. Numbers can be rounded to the nearest 10, 100, 1000, or to a given number of decimal places. Look at the digit to the right: if it is 5 or more, round up; if less than 5, round down.
四舍五入能让数字更易于使用。数字可以四舍五入到最近的十、百、千，或者指定的小数位数。看紧邻右边的数字：如果是 5 或更大，就“入”；小于 5，则“舍”。
For instance, 3.876 rounded to two decimal places is 3.88. Rounding to one decimal place gives 3.9. Estimation uses rounded numbers to find approximate answers quickly: 5971 ÷ 198 ≈ 6000 ÷ 200 = 30.
例如，3.876 四舍五入到两位小数是 3.88，到一位小数是 3.9。估算正是利用舍入后的数字来快速求得近似结果：5971 ÷ 198 ≈ 6000 ÷ 200 = 30。
Significant figures are another way to round, often used in science. The first non-zero digit is the first significant figure. 0.00452 has three significant figures: 4, 5 and 2. Round 0.00452 to two significant figures → 0.0045.
有效数字是另一种舍入方式，常用于科学领域。第一个非零数字是第一位有效数字。0.00452 有三位有效数字：4, 5, 2。将 0.00452 四舍五入到两位有效数字 → 0.0045。
12. Ratio and Proportion | 比与比例
Ratio compares two or more quantities of the same kind. A ratio of 3:2 means for every 3 parts of the first item, there are 2 parts of the second. Ratios can be simplified like fractions by dividing all parts by the same number.
比用来比较两个或更多同类量。3:2 表示对于第一个物品的 3 份，第二个物品有 2 份。比可以像分数一样，通过将所有部分同时除以同一个数来化简。
Sharing an amount in a given ratio uses the total number of parts. To share £60 in the ratio 3:2, total parts = 3+2=5. One part = £60 ÷ 5 = £12. So the shares are 3×£12 = £36 and 2×£12 = £24.
按给定比例分配总量时，要用到总份数。将 £60 按 3:2 分配，总份数 = 3+2=5。一份 = £60 ÷ 5 = £12。因此两部分分别为 3×£12 = £36 和 2×£12 = £24。
Proportion describes a multiplicative relationship. Direct proportion means as one quantity doubles, the other doubles too. Recognising proportional patterns helps in scaling recipes, converting currencies and reading maps.
比例描述一种倍数关系。正比意味着一个量翻倍，另一个量也跟着翻倍。识别比例模式有助于在调整食谱分量、兑换货币和阅读地图时使用。
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