📚 KS3 Maths: Last-Minute Revision Notes | KS3 数学：考前冲刺笔记
Preparing for your KS3 maths exam can feel overwhelming, but these last-minute revision notes will help you focus on the most important concepts. We’ll break down number skills, algebra, geometry, statistics and probability with clear explanations, worked examples and common pitfalls to avoid. Use this guide to boost your confidence and pick up quick marks.
准备 KS3 数学考试可能会让人感到不知所措，但这些考前冲刺笔记将帮助你聚焦最重要的概念。我们将通过清晰的解释、例题和常见错误提醒，逐一拆解数字运算、代数、几何、统计和概率。利用这份指南增强你的信心，并快速抓住得分点。
1. Number and Place Value | 数与位值
Place value tells us the value of each digit in a number. In 4,372 the 4 represents 4 thousands, 3 is 3 hundreds, 7 is 7 tens and 2 is 2 ones. When comparing whole numbers, start from the leftmost digit. For decimals, line up the decimal points and compare digits from left to right.
位值告诉我们一个数字中每个数码的值。在 4,372 中，4 代表 4 个千，3 是 3 个百，7 是 7 个十，2 是 2 个一。比较整数时，从最左边的数码开始。比较小数时，将小数点对齐，然后从左到右比较数码。
Rounding means reducing the digits in a number while keeping its value close to what it was. To round to the nearest 10, look at the ones digit: 5 or more raises the tens digit by 1. To round to the nearest 100 or 1000, look at the digit immediately to the right of the place you are rounding to. Significant figures (sig figs) are often tested at KS3; the first non-zero digit is the first significant figure.
四舍五入意味着减少数字的位数，同时保持其值接近原数。凑整到最近的 10 时，看个位数字：5 或以上则十位进 1。凑整到最近的 100 或 1000 时，看要凑整的那一位右侧的数字。有效数字在 KS3 中经常考查；第一个非零数字就是第一位有效数字。
Always check whether a question asks for a rounded answer or an exact value. Underline the digit you are rounding to and circle the next digit to help decide whether to round up or down.
务必检查题目要求的是凑整后的答案还是精确值。在你要凑整的那一位下面画线，并在下一位画圈，以帮助判断是向上进一还是舍去。
2. Arithmetic Operations and BIDMAS | 算术运算与运算顺序
The four operations are addition, subtraction, multiplication and division. For mental addition, break numbers into place value parts. For subtraction, use the column method and borrow when the top digit is smaller. Multiplication by 10, 100 or 1000 simply moves digits to the left. Long multiplication and short division are key written methods you must be fluent in.
四种基本运算分别是加、减、乘、除。心算加法时，可将数字按位值拆分。减法可采用竖式计算，当上方数字较小时进行借位。乘以 10、100 或 1000 只需将数字向左移动。长乘法和短除法是你必须熟练掌握的关键笔算方法。
BIDMAS (or BODMAS) gives the order of operations: Brackets first, then Indices (powers), Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). A common mistake is to always do addition before subtraction; they have equal priority and must be tackled from left to right.
BIDMAS（或 BODMAS）给出了运算顺序：先算括号，然后是指数（幂），接着是除法和乘法（从左到右），最后是加法和减法（从左到右）。常见的错误是先做加法再做减法；它们的优先级相同，必须从左到右依次计算。
For example, 3 + 4 × 2 is not 14. Multiplication comes first: 4 × 2 = 8, then 3 + 8 = 11. Adding brackets would change the meaning entirely.
例如，3 + 4 × 2 不等于 14。乘法优先：4 × 2 = 8，然后 3 + 8 = 11。添加括号则会完全改变算式的含义。
3. Fractions, Decimals and Percentages | 分数、小数与百分比
Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same number. Simplify fractions by dividing by the highest common factor. Improper fractions can be converted to mixed numbers and vice versa.
等值分数是通过将分子和分母同时乘以或除以同一个数得到的。用最大公因数约分即可化简分数。假分数可以转化为带分数，反之亦然。
To add or subtract fractions, find a common denominator first. To multiply fractions, multiply the numerators and multiply the denominators. To divide by a fraction, flip the second fraction and multiply (keep-change-flip). Show all working clearly.
分数加减要先找到公分母。分数相乘则分子乘分子，分母乘分母。除以一个分数时，将第二个分数翻转再相乘（保留-改变-翻转）。一定要清晰写出每一步过程。
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅓ | 0.333… (0.3 recurring) | 33⅓% |
| ⅕ | 0.2 | 20% |
Here are the most common conversions between fractions, decimals and percentages. Memorising these will save time in the exam.
这里列出了分数、小数和百分比之间最常见的转换。记住这些可以在考试中节省时间。
To find a percentage of an amount, convert the percentage to a decimal and multiply. For example, 15% of 60 = 0.15 × 60 = 9. To express one quantity as a percentage of another, write it as a fraction and multiply by 100.
求一个数量的百分比时，先将百分比转换成小数再相乘。例如，60 的 15% = 0.15 × 60 = 9。要表示一个量占另一个量的百分比，可先写成分数再乘以 100。
4. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. Simplify ratios by dividing each part by the greatest common factor, just like simplifying fractions. Ratios can be written in the form a:b or a:b:c. Always keep the order of the numbers exactly as given.
比用于比较两个或多个数量。化简比时，将每个部分都除以最大公因数，就像化简分数一样。比可以写成 a:b 或 a:b:c 的形式。务必严格保持题目给出的数字顺序。
Sharing a quantity in a given ratio involves finding the total number of parts, dividing the total quantity by the number of parts, then multiplying by each part of the ratio. For example, share £60 in the ratio 3:2. Total parts = 5, one part = £60 ÷ 5 = £12, so shares are 3×12 = £36 and 2×12 = £24.
按给定比例分配一个数量时，需要先求出总份数，然后用总量除以份数得到一份的量，再乘以比例中的每一份。例如，按 3:2 分配 60 英镑。总份数 = 5，一份 = 60 ÷ 5 = 12 英镑，因此分配额分别为 3×12 = 36 英镑和 2×12 = 24 英镑。
Proportion problems often use scaling. If a recipe for 6 people needs 300 g of flour, for 9 people multiply by 9/6 (or 1.5). Direct proportion means as one quantity doubles, the other doubles. Always state the multiplier clearly.
比例问题常使用标度因子。若一份 6 人份的食谱需要 300 克面粉，那么 9 人份就要乘以 9/6（即 1.5）。正比例意味着一个量翻倍，另一个量也翻倍。清晰地写出乘数非常重要。
5. Algebraic Expressions and Simplifying | 代数表达式与化简
In algebra, letters represent unknown numbers. A term is a combination of numbers and letters multiplied together, like 3x or 5ab. Like terms have exactly the same letter parts and can be collected: 4a + 2a = 6a, but 3x + 2y cannot be simplified further.
在代数中，字母代表未知数。项是数字和字母相乘的组合，如 3x 或 5ab。同类项具有完全相同的字母部分，可以合并：4a + 2a = 6a，但 3x + 2y 无法进一步化简。
To expand brackets, multiply the term outside by every term inside. For example, 3(2x + 5) = 6x + 15. Be careful with negative signs: –2(3x – 4) = –6x + 8. Double brackets like (x+2)(x+3) require the FOIL method: First, Outer, Inner, Last, then simplify by collecting like terms.
展开括号时，用外面的项乘以括号内的每一项。例如，3(2x + 5) = 6x + 15。注意负号：–2(3x – 4) = –6x + 8。像 (x+2)(x+3) 这样的双括号需要使用 FOIL 方法：先乘首项、外项、内项、末项，然后合并同类项化简。
Factorising is the reverse of expanding. Look for the highest common factor of all terms, place it outside the bracket, and write what remains inside. 4x + 8 factorises to 4(x + 2). Always check by expanding mentally.
因式分解是展开的逆过程。找出所有项的最大公因数，将其放在括号外，剩下的部分写在括号内。4x + 8 因式分解为 4(x + 2)。务必通过心算展开来验证。
6. Solving Linear Equations | 解一元一次方程
An equation shows that two expressions are equal. To solve, use inverse operations while keeping the equation balanced. Whatever you do to one side, you must do to the other. Aim to isolate the unknown on one side.
方程表示两个表达式相等。求解时使用逆运算，同时保持等式平衡。对等式一边做什么操作，对另一边也要做同样的操作。目标是将未知数单独留在等式的一边。
For two-step equations like 2x + 3 = 11, first subtract 3 from both sides: 2x = 8, then divide both sides by 2: x = 4. When the unknown appears on both sides, collect the x terms on one side and numbers on the other. For example, 5x – 2 = 3x + 6 → 5x – 3x = 6 + 2 → 2x = 8 → x = 4.
像 2x + 3 = 11 这样的两步方程，首先两边同时减 3：2x = 8，然后两边同时除以 2：x = 4。当未知数出现在两边时，将含 x 的项移到一边，数字移到另一边。例如，5x – 2 = 3x + 6 → 5x – 3x = 6 + 2 → 2x = 8 → x = 4。
Always present your solution clearly, and substitute it back into the original equation to check it works. Look out for equations involving brackets — expand them first before solving.
始终清晰地呈现你的解，并代回原方程检验是否成立。注意带有括号的方程——要先展开括号再求解。
7. Sequences and the nth Term | 数列与第 n 项
A sequence is an ordered list of numbers that follow a rule. In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference. For example, 5, 8, 11, 14,… has a common difference of +3.
数列是按一定规则排列的一列数。在等差数列中，连续两项之间的差是常数，这个差称为公差。例如，5, 8, 11, 14, … 的公差是 +3。
The nth term formula allows you to find any term without listing all previous ones. For an arithmetic sequence, nth term = first term + (n–1) × common difference. Alternatively, write the sequence as a multiple of the difference plus a constant: for 7, 10, 13, 16,… nth term = 3n + 4. To check, substitute n=1, 2, 3 to reproduce the terms.
第 n 项公式让你无需列出所有前面的项就能找到任意一项。对于等差数列，第 n 项 = 首项 + (n–1) × 公差。或者，将数列写成公差的倍数加一个常数：对于 7, 10, 13, 16, …，第 n 项 = 3n + 4。检验时，可代入 n=1, 2, 3 看是否得出原数列的各项。
You may be asked if a certain number is in the sequence. Set the nth term formula equal to that number and solve for n. If n is a positive integer, the number is a term. Always show algebraic working.
你可能会被问到某个数是否在数列中。令第 n 项公式等于该数并解出 n。如果 n 是正整数，则该数是数列中的一项。务必展示代数求解过程。
8. Geometry: Angles and Lines | 几何：角与线
Know your angle types: acute (< 90°), right angle (=90°), obtuse (between 90° and 180°), straight line (180°), and reflex (>180°). Angles on a straight line add up to 180°, and angles around a point total 360°.
牢记角的类型：锐角（小于 90°），直角（等于 90°），钝角（介于 90° 和 180° 之间），平角（180°）和优角（大于 180°）。直线上的角之和为 180°，绕一点一周的角之和为 360°。
Vertically opposite angles are equal. When two parallel lines are cut by a transversal, alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°. Use these facts to find missing angles and always give reasons in brackets.
对顶角相等。当两条平行线被一条横截线所截，内错角相等，同位角相等，同旁内角之和为 180°。利用这些规律求未知角，并始终在括号中注明理由。
The angle sum of a triangle is 180°, and a quadrilateral’s interior angles sum to 360°. In an isosceles triangle, base angles are equal. If you know two angles in a triangle, subtract their sum from 180° to find the third.
三角形内角和为 180°，四边形内角和为 360°。在等腰三角形中，底角相等。如果你知道三角形中的两个角，用 180° 减去它们的和即可求出第三个角。
9. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around a shape — add all the side lengths. Area is the space inside, measured in square units. Volume is the space a 3D object occupies, measured in cubic units. Always write units² for area and units³ for volume.
周长是形状一周的长度——将所有边长相加。面积是内部的区域，以平方单位计量。体积是三维物体所占据的空间，以立方单位计量。写面积时始终标上单位²，写体积时标上单位³。
Area of a rectangle = length × width
长方形面积 = 长 × 宽
Area of a triangle = ½ × base × height
三角形面积 = ½ × 底 × 高
For a parallelogram, use base × perpendicular height. For a trapezium, average the parallel sides and multiply by the height: area = ½(a+b)h. To find areas of compound shapes, split them into rectangles and triangles, find each area separately and add or subtract.
平行四边形的面积用底 × 垂直高。梯形的面积则是取两条平行边的平均值再乘以高：面积 = ½(a+b)h。求组合图形的面积时，将其分割成矩形和三角形，分别计算各部分面积再相加或相减。
Volume of a cuboid = length × width × height or area of cross-section × length. Be prepared to convert between metric units for area and volume: 1 m² = 10,000 cm², 1 m³ = 1,000,000 cm³.
长方体体积 = 长 × 宽 × 高，或者底面积 × 长。要准备好进行面积和体积的公制单位转换：1 m² = 10,000 cm²，1 m³ = 1,000,000 cm³。
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