📚 Essential Maths Book 7C Complete Revision | KS3 数学核心知识点精讲
This revision guide distils the key topics from the Essential Maths Book 7C series, designed for KS3 students aiming to strengthen their mathematical foundations. Covering number operations, algebra, geometry, and data handling, the content is presented in a clear, structured way to support both classroom learning and independent study. Every concept is broken down into manageable steps, with English and Chinese explanations paired for clarity and bilingual reinforcement.
这本复习指南提炼了 Essential Maths Book 7C 系列的核心知识点，专为希望夯实数学基础的 KS3 学生设计。内容涵盖数字运算、代数、几何与数据处理，以清晰、结构化的方式呈现，既辅助课堂学习也适合自主复习。每个概念都被分解为易于掌握的步骤，英文与中文解释成对出现，确保清晰并强化双语理解。
1. Place Value and Rounding | 位值与四舍五入
Place value tells us the value of each digit in a number. In the decimal system, moving left multiplies the place value by 10, while moving right divides it by 10. For example, in 4 702.36, the digit 4 is in the thousands place, 7 is hundreds, 0 is tens, 2 is units, 3 is tenths and 6 is hundredths.
位值告诉我们一个数中每个数字的值。在十进制系统中，向左移动一位数值乘以10，向右移动一位则除以10。例如，在 4 702.36 中，数字4在千位，7在百位，0在十位，2在个位，3在十分位，6在百分位。
Rounding is used to estimate and make numbers easier to work with. To round a decimal to a given place, look at the digit immediately to the right. If that digit is 5 or greater, round up; otherwise, round down. For instance, 3.276 rounded to two decimal places becomes 3.28 because the third decimal digit is 6.
四舍五入用于估算并使数字更易处理。要将一个小数四舍五入到指定位置，查看紧邻右侧的数字。如果该位数字是5或以上，则进位；否则舍去。例如，3.276 四舍五入到两位小数变为 3.28，因为第三位小数是6。
- Significant figures (s.f.) are often used in science; the first non-zero digit is the first significant figure. 0.045 78 to 3 s.f. is 0.045 8.
- 有效数字在科学中常用；第一个非零数字是第一个有效数字。0.045 78 保留三位有效数字为 0.045 8。
2. Negative Numbers | 负数
equivalent to subtracting its positive counterpart: 5 + (−3) = 2. Subtracting a negative number becomes addition: 5 − (−3) = 8.
负数是小于零的数，用负号表示。在数轴上，它们位于零的左侧。加上一个负数等同于减去它的正数形式：5 + (−3) = 2。减去一个负数则变为加法：5 − (−3) = 8。
itive answer, different signs give a negative answer. (−4) × (−2) = 8, and 12 ÷ (−3) = −4.
在乘除法中涉及负数时，记住规则：同号得正，异号得负。(−4) × (−2) = 8，而 12 ÷ (−3) = −4。
| Operation | Sign rule | Example |
| (+) − (−) | Add the positive | 6 − (−2) = 8 |
| (−) × (−) | Positive product | (−3) × (−5) = 15 |
| (+) ÷ (−) | Negative quotient | 20 ÷ (−4) = −5 |
3. Factors, Multiples and Primes | 因数、倍数与质数
A factor divides exactly into a number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. Multiples are the results of multiplying a number by an integer; the first five multiples of 7 are 7, 14, 21, 28, 35.
因数是可以整除某个数的整数，没有余数。例如，12的因数有1、2、3、4、6和12。倍数是一个数乘以整数得到的结果；7的前五个倍数是7、14、21、28、35。
prime factorisation of 60 is 2² × 3 × 5. The highest common factor (HCF) and lowest common multiple (LCM) can be found using prime factorisations or listing methods.
质数只有两个不同的因数：1和它本身。2是唯一的偶质数。合数有超过两个因数。60的质因数分解为 2² × 3 × 5。最高公因数 (HCF) 和最低公倍数 (LCM) 可以通过质因数分解或列表法求得。
- HCF of 24 and 36 is 12; LCM of 24 and 36 is 72.
- 24 和 36 的最高公因数是12；最低公倍数是72。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
compare or add fractions, they must have a common denominator. The sum of 1/3 and 2/5 is found by using a common denominator 15: 5/15 + 6/15 = 11/15.
分数表示整体的部分。等值分数虽然分子分母不同但数值相同，例如 1/2 = 2/4 = 3/6。要比较或相加分数，它们必须有共同的分母。1/3 加 2/5 的和通过公分母15求得：5/15 + 6/15 = 11/15。
r. To convert a decimal to a percentage, multiply by 100. For instance, 3/8 = 0.375 = 37.5%.
分数、小数和百分数之间的转换是一项关键技能。分数化小数，用分子除以分母。小数化百分数，乘以100。例如，3/8 = 0.375 = 37.5%。
Recurring decimals, such as 0.3̅3, can be written as fractions: 0.3̅3 = 1/3. Understanding place value makes these conversions straightforward.
循环小数，如 0.3̅3，可以写成分数：0.3̅3 = 1/3。理解位值使得这些转换变得简单。
5. Algebraic Expressions | 代数表达式
y − 5. Terms are the parts separated by + or − signs. Like terms contain exactly the same variable parts and can be simplified: 4a + 3a = 7a.
代数用字母表示未知数或变量。表达式是变量、数和运算的组合，如 3x + 2y − 5。项是由 + 或 − 号分隔的部分。同类项包含完全相同的变量部分，可以合并：4a + 3a = 7a。
orising is the reverse process, writing an expression as a product: 5x² + 10x = 5x(x + 2).
去括号使用分配律：a(b + c) = ab + ac。例如，3(2x − 4) = 6x − 12。因式分解是逆过程，将表达式写成乘积形式：5x² + 10x = 5x(x + 2)。
iables with given numbers. If x = 3 and y = −2, then 2x² − y = 2(3)² − (−2) = 18 + 2 = 20.
代入求值指用给定的数替换变量。若 x = 3，y = −2，则 2x² − y = 2(3)² − (−2) = 18 + 2 = 20。
6. Solving Linear Equations | 解一元一次方程
r equations have the variable to the power of 1, such as 2x + 3 = 11. Use inverse operations to isolate the variable: subtract 3, then divide by 2, giving x = 4.
方程表明两个表达式相等。解方程意味着找到使等式成立的变量值。一元一次方程中变量的指数为1，如 2x + 3 = 11。使用逆运算分离变量：先减3，再除以2，得 x = 4。
then x = 3. Always check your solution by substituting it back into the original equation.
带括号的方程应先去括号。解 3(2x − 1) = 15：去括号得 6x − 3 = 15，加3得 6x = 18，然后 x = 3。始终将解代回原方程进行检验。
t 2, then multiply by 4: x = 12. Cross-multiplication helps with equations like 2/x = 4/6.
方程中也可以包含分数。解 x/4 + 2 = 5，先减2，再乘以4：x = 12。交叉相乘适用于像 2/x = 4/6 这样的方程。
7. Angles and Polygons | 角与多边形
An angle is formed when two lines meet at a point. Angles are measured in degrees (°) with a protractor. Angles on a straight line add up to 180°, angles around a point total 360°, and vertically opposite angles are equal.
角由两条线交于一点形成。角度用量角器以度(°)为单位测量。直线上的角之和为180°，围绕一点的角之和为360°，对顶角相等。
re equal. In a quadrilateral, interior angles total 360°. Regular polygons have equal sides and angles; the interior angle sum for an n-sided polygon is (n − 2) × 180°.
在三角形中，三个内角之和永远为180°。等边三角形三个角都是60°；等腰三角形的底角相等。在四边形中，内角和为360°。正多边形的边长和角度均相等；n 边形内角和为 (n − 2) × 180°。
Parallel lines cut by a transversal create special angle pairs: corresponding angles are equal, alternate angles are equal, and co‑interior angles sum to 180°. Identifying these helps solve for unknown angles.
平行线被一条截线切割会产生特殊角对：同位角相等，内错角相等，同旁内角之和为180°。识别这些关系有助于求解未知角。
8. Perimeter and Area | 周长与面积
a circle, the perimeter is the circumference: C = 2πr or C = πd, where π ≈ 3.142 or 22/7.
周长是二维图形一周的总距离。对于矩形，周长 P = 2(长 + 宽)。三角形周长是三边之和。对于圆，周长即圆周：C = 2πr 或 C = πd，其中 π ≈ 3.142 或 22/7。
Rectangle area = length × width
矩形面积 = 长 × 宽
Triangle area = ½ × base × height
三角形面积 = ½ × 底 × 高
Parallelogram area = base × perpendicular height
平行四边形面积 = 底 × 垂直高
Trapezium area = ½ × (a + b) × h
梯形面积 = ½ × (上底 + 下底) × 高
Composite shapes can be split into simpler figures. Find the area of each part and add them together. Be careful to use consistent units.
组合图形可以分割成更简单的图形。分别求每一部分的面积再相加。注意使用一致的单位。
9. Ratio and Proportion | 比与比例
A ratio compares quantities of the same kind. The ratio 3:5 means for every 3 parts of one quantity, there are 5 parts of another. Ratios can be simplified by dividing both sides by common factors, just like fractions. 12:18 simplifies to 2:3.
比用于比较同类量。比 3:5 意味着每3份的一种量对应5份的另一种量。比可以通过两边同除公因数来化简，如同分数。12:18 化简得 2:3。
e quantity by the total parts, then multiply by each part of the ratio. For example, share £240 in the ratio 3:5: total parts = 8, one part = £30, so the amounts are £90 and £150.
要按给定比例分配一个量，先求出总份数。将总量除以总份数，再乘以比中的每一项。例如，按 3:5 分配 £240：总份数 = 8，每份 = £30，因此金额分别为 £90 和 £150。
or 4 people requiring 200 g flour can be scaled up to 6 people using the multiplier 6/4 = 1.5, so 300 g flour.
比例描述一个量如何随另一个量变化。正比例意味着一个量翻倍，另一个也翻倍。一份供4人食用的食谱需要200克面粉，可扩展到6人份，乘数 6/4 = 1.5，所以需要300克面粉。
10. Statistics and Probability | 统计与概率
Data can be collected and displayed in bar charts, pie charts, and line graphs. Averages help summarise a dataset: the mean is the sum of values divided by the count; the median is the middle value when ordered; the mode is the most frequent value; the range is the difference between the highest and lowest.
数据可以收集并用柱状图、饼图和折线图展示。平均数有助于概括数据集：平均数（均值）是数值之和除以个数；中位数是排序后中间的值；众数是出现频率最高的值；极差是最大值与最小值之差。
ourable outcomes / total number of outcomes. The probability of rolling a 5 on a fair six-sided die is 1/6.
概率衡量事件发生的可能性，范围从0（不可能）到1（必然）。对于等可能结果，概率 = 有利结果的数量 / 所有可能结果的总数。掷一个公平的六面骰子得到5的概率是 1/6。
Experimental probability is based on trials: the relative frequency is number of successes ÷ number of trials. As more trials are carried out, the experimental probability tends to the theoretical probability.
实验概率基于试验得出：相对频率 = 成功次数 ÷ 试验次数。随着试验次数增加，实验概率趋近于理论概率。
Listing outcomes systematically, such as using sample space diagrams, ensures all combinations are counted. For two dice, there are 36 equally likely outcomes.
系统地列出结果，例如使用样本空间图，可确保所有组合都被计数。两个骰子共有36种等可能结果。
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