📚 Cambridge KS3 Maths Paper 2: Core Topics and Exam Practice | 剑桥KS3数学第二卷：核心主题与考试练习
Cambridge Lower Secondary Checkpoint Mathematics Paper 2 tests a broad range of essential skills from the KS3 curriculum. This article revises the core topics most likely to appear, with clear methods and worked examples to help you build confidence before the exam.
剑桥初中数学第二卷测试KS3课程中一系列核心技能。本文复习最可能出现的核心主题，提供清晰的方法和例题，帮助你在考试前建立信心。
1. Number and Place Value | 数字与位值
In Paper 2 you often need to order positive and negative integers, decimals and fractions on a number line. Place value tells you the value of each digit, so the digit 7 in 0.078 is 7 hundredths, not 7 tenths.
在第二卷中，你经常需要在数轴上排列正负整数、小数和分数。位值告诉你每个数字的值，因此0.078中的数字7是7个百分之一，而不是7个十分之一。
Rounding to a given number of decimal places or significant figures is another common skill. To round 3.456 to 2 decimal places, look at the third decimal digit (6); since it is 5 or more, round 3.45 up to 3.46.
按指定小数位或有效数字四舍五入是另一项常见技能。将3.456四舍五入到两位小数时，看第三位小数（6）；因为大于或等于5，所以把3.45进位为3.46。
- When rounding to decimal places, look one digit to the right of the required place. 按小数位四舍五入时，看所需位置右边的一位数字。
- When rounding to significant figures, start counting from the first non-zero digit. 按有效数字四舍五入时，从第一个非零数字开始计数。
- For negative numbers, remember that -7 is less than -2 because it is further left on the number line. 对于负数，记住 -7 小于 -2，因为它在数轴上更靠左。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
ominator: ¾ = 0.75. To convert a decimal to a percentage, multiply by 100: 0.75 × 100 = 75%.
分数、小数和百分数是表示同一个比例的三种方式。将分数转换为小数，用分子除以分母：¾ = 0.75。将小数转换为百分数，乘以100：0.75 × 100 = 75%。
In Paper 2 you often need to find a percentage of an amount. For example, 15% = 36. You also need to calculate percentage increase or decrease using the formula below.
在第二卷中，你经常需要求一个数量的百分比。例如，240的15%等于0.15 × 240 = 36。你还需要使用下面的公式计算百分比增减。
percentage change = (new value - original value) ÷ original value × 100
- ½ = 0.5 = 50%
- ¼ = 0.25 = 25%
- ¾ = 0.75 = 75%
- ⅓ ≈ 0.333 = 33.3%
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. To simplify a ratio, divide all parts by their highest common factor. For example, 12:18 simplifies to 2:3 by dividing both sides by 6.
比用于比较整体中的各个部分。要化简比，将所有部分除以它们的最大公因数。例如，12:18 两边同时除以6，化简为2:3。
uantity in a given ratio, work out the total number of parts first. For example, to divide £40 in the ratio 3:2, total parts = 5, one part = £40 ÷ 5 = £8, so shares are £24 and £16.
按给定比例分配数量时，先计算总份数。例如，按3:2分配40英镑，总份数为5，一份等于40÷5=8英镑，因此份额分别为24英镑和16英镑。
one part = total amount ÷ sum of ratio parts
- Write the ratio in its simplest form before dividing. 分配前先将比化为最简形式。
- Direct proportion means if one quantity doubles, the other doubles too. 正比例意味着一个量翻倍，另一个量也翻倍。
4. Algebraic Expressions and Substitution | 代数表达式与代入
2a + 4b by collecting like terms: 5a - 2a = 3a and 3b + 4b = 7b, so the answer is 3a + 7b.
在代数中，同类项具有相同的字母和相同的指数。化简 5a + 3b - 2a + 4b，合并同类项：5a - 2a = 3a，3b + 4b = 7b，所以答案是 3a + 7b。
-2)² - 3(-2) + 1 = 8 + 6 + 1 = 15. Use brackets for negative numbers to avoid sign errors.
代入就是用数字替换字母。若 y = 2x² - 3x + 1，求 x = -2 时的 y 值：y = 2(-2)² - 3(-2) + 1 = 8 + 6 + 1 = 15。负数要用括号括起来，以免符号出错。
3(x + 4) = 3x + 12
- Expand brackets by multiplying the term outside by each term inside. 展开括号时，用外面的项乘以括号内的每一项。
- Factorising is the reverse of expanding. 因式分解是展开的逆运算。
5. Linear Equations | 线性方程
. For 2x + 5 = 13, subtract 5 from both sides: 2x = 8, then divide both sides by 2: x = 4.
解线性方程时，按相反顺序撤销运算。对于 2x + 5 = 13，两边同时减5：2x = 8，然后两边同时除以2：x = 4。
For 3x - 2 = x + 6, subtract x from both sides to get 2x - 2 = 6, add 2: 2x = 8, so x = 4.
当未知数出现在等式两边时，先把变量移到同一边。对于 3x - 2 = x + 6，两边同时减去x，得到 2x - 2 = 6；再加2：2x = 8，所以 x = 4。
ax + b = c → x = (c - b) ÷ a
- Always perform the same operation on both sides. 始终在等式两边进行相同的运算。
- Check your answer by substituting it back into the original equation. 将答案代回原方程进行检验。
6. Sequences and Patterns | 数列与规律
A sequence is a list of numbers made by a rule. The term-to-term rule describes how to move from one term to the next. In 5, 8, 11, 14, ... the rule is add 3 each time.
数列是按规则排列的一组数。项到项规则描述如何从一项得到下一项。在 5, 8, 11, 14, ... 中，规则是每次加3。
... the common difference is 3, so the nth term is 3n + 2. Test with n = 1: 3(1) + 2 = 5.
等差数列的第n项是一个线性表达式。对于 5, 8, 11, 14, ...，公差为3，所以第n项是 3n + 2。用 n = 1 检验：3(1) + 2 = 5。
nth term = dn + (a - d)
- d is the common difference and a is the first term. d 是公差，a 是第一项。
- Use the nth term to find the 100th term without listing all terms. 用第n项可以直接求第100项，无需列出所有项。
7. Geometry: Angles and Shapes | 几何：角与图形
Paper 2 frequently asks for missing angles using angle facts. Angles on a straight line total 180°, angles around a point total 360°, and vertically opposite angles are equal.
第二卷经常要求利用角的性质求缺失的角。直线上的角加起来是180°，围绕一个点的角加起来是360°，对顶角相等。
In a triangle, angles total 180°; in a quadrilateral, 360°. For parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles total 180°.
三角形中，内角和为180°；四边形中，内角和为360°。对于平行线
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