📚 Cambridge KS3 Mathematics: Paper 1 Core Skills | 剑桥KS3数学：试卷一核心技能
This bilingual revision guide covers the core skills most commonly tested in Cambridge KS3 Mathematics Paper 1, especially the number, algebra and angle questions that often appear in the early pages of a progression test. Work through each section carefully and practise the examples.
本双语复习指南覆盖剑桥KS3数学试卷一最常考查的核心技能，尤其是阶段性测试前面页面经常出现的数、代数和角的问题。请仔细学习每一节并练习例题。
1. Place Value and Ordering Integers | 数位与整数排序
Place value tells us the value of each digit in a number. In 47 306, the digit 4 stands for 4 ten-thousands, 7 stands for 7 thousands, 3 for 3 hundreds and 6 for 6 ones.
数位告诉我们一个数中每个数字的值。在 47 306 中，数字 4 表示 4 个万，7 表示 7 个千，3 表示 3 个百，6 表示 6 个一。
When ordering integers, write them in a column or use a number line. Negative numbers are smaller than all positive numbers: −5 < −2 < 0 < 3.
对整数排序时，可以竖排书写或使用数轴。负数小于所有正数：−5 < −2 < 0 < 3。
- Compare digits from left to right. | 从左到右逐位比较。
- For negative numbers, the value closer to zero is larger. | 对于负数，越接近零的值越大。
2. Fractions: Simplify and Compare | 分数：化简与比较
A fraction is in simplest form when the numerator and denominator have no common factor other than 1. To simplify 12/18, divide both by 6 to get 2/3.
当分子和分母除 1 外没有其他公因数时，分数就是最简形式。化简 12/18，分子分母同除以 6，得到 2/3。
To compare fractions, rewrite them with a common denominator. For 2/5 and 3/8, use 40 as the common denominator: 16/40 > 15/40, so 2/5 > 3/8.
比较分数时，先将它们改写为同分母。对于 2/5 和 3/8，用 40 作公分母：16/40 > 15/40，所以 2/5 > 3/8。
a/b = (a ÷ k)/(b ÷ k)
3. Operations with Fractions | 分数运算
To add or subtract fractions, first find a common denominator. Keep the denominator and add or subtract the numerators only.
加减分数时，先找到公分母。保持分母不变，只对分子进行加减。
Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
例题：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
To multiply fractions, multiply numerators and denominators separately. To divide, multiply by the reciprocal of the second fraction.
分数相乘时，分子乘分子，分母乘分母。分数相除时，乘以第二个分数的倒数。
a/b × c/d = (a×c)/(b×d)
a/b ÷ c/d = a/b × d/c
4. Decimals and Rounding | 小数与四舍五入
Decimals extend place value to tenths, hundredths and thousandths. In 5.207, the 2 is in the tenths place, 0 is in the hundredths place and 7 is in the thousandths place.
小数将数位扩展到十分位、百分位和千分位。在 5.207 中，2 在十分位，0 在百分位，7 在千分位。
To round 3.276 to 1 decimal place, look at the hundredths digit 7. Since 7 is 5 or more, round up to 3.3.
将 3.276 四舍五入到 1 位小数，看百分位数字 7。因为 7 大于等于 5，所以向上舍入为 3.3。
To round to a given number of significant figures, count from the first non-zero digit and apply the same rounding rule.
将数字四舍五入到指定有效数字位数时，从第一个非零数字开始计数，并应用相同的舍入规则。
5. Percentages of Quantities | 数量的百分数
A percentage is a fraction out of 100. To find 15% of 240, calculate 240 × 15/100 = 36.
百分数就是分母为 100 的分数。求 240 的 15%，计算 240 × 15/100 = 36。
You can also find 1% first: 1% of 240 is 2.4, so 15% is 2.4 × 15 = 36.
也可以先求 1%：240 的 1% 是 2.4，因此 15% 是 2.4 × 15 = 36。
For percentage increase or decrease, use a multiplier. A 15% increase uses 1.15, and a 15% decrease uses 0.85.
计算百分数增减时，使用乘数。增加 15% 用 1.15，减少 15% 用 0.85。
Example: 120 increased by 15% = 120 × 1.15 = 138.
例题：120 增加 15% = 120 × 1.15 = 138。
6. Ratio and Proportion | 比与比例
Ratio compares quantities. The ratio 2:3 means that for every 2 parts of the first quantity, there are 3 parts of the second.
比用来比较数量。比 2:3 表示第一个量每有 2 份，第二个量就有 3 份。
To divide £50 in the ratio 2:3, find the total parts: 2+3=5. One part is £10, so the shares are 2×£10=£20 and 3×£10=£30.
按 2:3 分 50 英镑，先求总份数：2+3=5。一份是 10 英镑，所以份额分别为 2×10=20 英镑和 3×10=30 英镑。
Proportion problems often ask you to scale a recipe or map. Find the unit value first, then multiply by the required number of parts.
比例问题常常要求你按照配方或地图缩放。先求出单位量，再乘以所需的份数。
7. BIDMAS and Negative Numbers | 运算顺序与负数
BIDMAS gives the order: Brackets, Indices, Division and Multiplication, Addition and Subtraction. Work from left to right for division and multiplication.
BIDMAS 给出运算顺序：括号、指数、除法和乘法、加法和减法。除法和乘法按从左到右的顺序计算。
Evaluate 2 + 3 × 4 = 2 + 12 = 14, not 5 × 4 = 20.
计算 2 + 3 × 4 = 2 + 12 = 14，而不是 5 × 4 = 20。
With negative numbers: (−3) + (−4) = −7, (−3) − (−4) = 1, (−3) × (−4) = 12 and (−8) ÷ 2 = −4.
负数运算：(−3) + (−4) = −7，(−3) − (−4) = 1，(−3) × (−4) = 12，(−8) ÷ 2 = −4。
Remember −3² means the negative of 3², so −3² = −9, because the index applies to 3 before the negative sign.
记住 −3² 表示 3² 的相反数，所以 −3² = −9，因为指数先作用于 3，再取负号。
8. Algebraic Expressions | 代数表达式
Like terms have exactly the same variable part. Simplify 3a + 5b + 2a − b = 5a + 4b.
同类项具有完全相同的字母部分。化简 3a + 5b + 2a − b = 5a + 4b。
Expand brackets by multiplying each term inside. 3(x + 2) = 3x + 6.
去括号时将括号内每一项分别相乘。3(x + 2) = 3x + 6。
For double brackets: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² +
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