KS3 Maths: End-of-Term Revision Guide | KS3 数学：期末复习提纲

📚 KS3 Maths: End-of-Term Revision Guide | KS3 数学：期末复习提纲

As the end-of-term exam approaches, a well-organised revision plan can make all the difference. This KS3 maths revision guide covers the key topics you need to master, from number operations to geometry, algebra, statistics and probability. Use it to check your understanding, practise key skills, and build confidence for the test.

随着期末考试的临近，一份条理清晰的复习计划至关重要。本 KS3 数学复习指南涵盖了你需要掌握的核心主题，从数运算到几何、代数、统计和概率。你可以用它来检查理解程度，练习关键技能，并为考试建立信心。

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1. Number: Integers, Fractions, Decimals and Percentages | 数：整数、分数、小数和百分比

Mastering operations with positive and negative integers is the foundation. When adding a negative number, you move left on the number line; subtracting a negative is the same as adding. For multiplication and division, remember that two negatives make a positive: e.g., (-3) × (-4) = 12, and (-15) ÷ (-5) = 3.

掌握正负整数的运算是基础。当加上一个负数时，你是在数轴上向左移动；减去一个负数等同于加上正数。对于乘法和除法，记住两个负数相乘或相除得正：例如 (-3) × (-4) = 12，而 (-15) ÷ (-5) = 3。

Fractions must be simplified, compared and converted. To add or subtract fractions like 2/3 + 1/4, find a common denominator (12): 8/12 + 3/12 = 11/12. When multiplying fractions, multiply numerators and denominators: 2/5 × 3/4 = 6/20 = 3/10. For division, invert the second fraction and multiply: 3/7 ÷ 2/3 = 3/7 × 3/2 = 9/14.

分数需要化简、比较和转换。要加减像 2/3 + 1/4 这样的分数，先求公分母（12）：8/12 + 3/12 = 11/12。分数相乘时，分子乘分子、分母乘分母：2/5 × 3/4 = 6/20 = 3/10。除法时，将除数的分子分母颠倒再相乘：3/7 ÷ 2/3 = 3/7 × 3/2 = 9/14。

Decimals and percentages are closely linked: 0.65 = 65%. To find a percentage of a quantity, multiply by the decimal form (15% of £80 = 0.15 × 80 = £12). For percentage increase or decrease, use the multiplier method – an increase of 20% is equivalent to multiplying by 1.20, while a decrease of 30% means multiplying by 0.70.

小数和百分比紧密相连：0.65 = 65%。求一个数量的百分之几，用小数形式相乘（£80 的 15% = 0.15 × 80 = £12）。对于百分比的增减，使用乘数法——增加 20% 相当于乘以 1.20，而减少 30% 相当于乘以 0.70。

Rounding to a given number of decimal places or significant figures is a key skill. For decimal places, look at the next digit; for significant figures, start counting from the first non-zero digit. Estimation helps you check if an answer is reasonable.

四舍五入到指定的小数位数或有效数字是一项关键技能。对于小数位，看下一位数字；对于有效数字，从第一个非零数字开始计数。估算有助于判断答案是否合理。

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2. Ratio and Proportion | 比与比例

A ratio compares parts of a whole. You can simplify ratios by dividing all parts by their highest common factor. For example, 8:12:20 simplifies to 2:3:5. Three-part ratios work the same way. Always make sure the order matches the wording of the problem.

比用来比较整体的各个部分。你可以通过将所有部分除以它们的最大公因数来化简比。例如，8:12:20 化简为 2:3:5。三部分的比遵循同样规则。务必确保顺序与问题描述一致。

To divide a quantity in a given ratio, first find the total number of parts. If you share £72 between two people in the ratio 3:5, there are 3 + 5 = 8 parts. One part is £72 ÷ 8 = £9, so the shares are 3 × £9 = £27 and 5 × £9 = £45.

按给定比例分割一个数量时，先求出总份数。如果按 3:5 的比例将 £72 分给两人，总份数为 3 + 5 = 8。一份是 £72 ÷ 8 = £9，因此两人分别得到 3 × £9 = £27 和 5 × £9 = £45。

Direct proportion means two quantities increase or decrease at the same rate. Use the unitary method: if 4 tickets cost £26, then one ticket costs £6.50, so 7 tickets cost 7 × £6.50 = £45.50. Set up a clear working to avoid mistakes.

正比例意味着两个量以相同速率增加或减少。使用单位量法：如果 4 张票价格为 £26，那么一张票为 £6.50，因此 7 张票为 7 × £6.50 = £45.50。列出清晰的计算步骤可以避免出错。

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3. Algebra: Expressions and Equations | 代数：表达式与方程

An algebraic expression contains numbers, letters and operation signs but no equals sign. Simplify by collecting like terms: 5a + 2b − 3a + 7b = 2a + 9b. Use the rules of indices correctly: a × a = a² and b⁴ ÷ b² = b².

代数表达式包含数字、字母和运算符号，但没有等号。通过合并同类项进行化简：5a + 2b − 3a + 7b = 2a + 9b。正确使用指数规则：a × a = a²，b⁴ ÷ b² = b²。

Expanding brackets uses the distributive law. For a single bracket, multiply each term inside: 3(2x − 5) = 6x − 15. To expand double brackets such as (x + 4)(x − 2), multiply every term in the first bracket by every term in the second: x² − 2x + 4x − 8 = x² + 2x − 8.

展开括号要用乘法分配律。对于单项乘多项式，将括号外的项与括号内每一项相乘：3(2x − 5) = 6x − 15。展开双括号如 (x + 4)(x − 2)，将第一个括号中的每一项与第二个括号中的每一项相乘：x² − 2x + 4x − 8 = x² + 2x − 8。

Factorising is the reverse of expanding. Always look for the highest common factor: 10x + 15 = 5(2x + 3). For quadratic expressions like x² + 7x + 12, find two numbers that multiply to 12 and add to 7 – here 3 and 4 – so it factorises to (x + 3)(x + 4).

因式分解是展开的逆操作。始终要先寻找最大公因数：10x + 15 = 5(2x + 3)。对于 x² + 7x + 12 这样的二次表达式，找出两个数，其乘积为 12、和为 7——此处是 3 和 4——因此分解为 (x + 3)(x + 4)。

To solve linear equations, keep the balance by doing the same to both sides. For 3x − 7 = 14, add 7 to get 3x = 21, then divide by 3: x = 7. If unknowns appear on both sides, collect them on one side first. Always substitute your answer back to check.

解线性方程时，要在等式两边同时进行相同操作以保持平衡。对于 3x − 7 = 14，两边加 7 得 3x = 21，再除以 3：x = 7。如果未知数出现在等式两边，先把它们移到一边。最后务必代入答案检验。

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4. Linear Graphs | 线性图像

Coordinates are written as (x, y) where x is the horizontal position and y is the vertical. The origin is (0,0). Plotting points accurately is essential before drawing any graph. Use a sharp pencil and label axes clearly.

坐标写作 (x, y)，x 表示水平位置，y 表示垂直位置。原点是 (0,0)。在绘制任何图像前，准确描点是关键。使用削尖的铅笔并清晰标注坐标轴。

To draw the graph of a linear equation like y = 2x + 1, build a table of values. Choose x-values (e.g., -2, -1, 0, 1, 2), calculate the corresponding y-values, plot the points and join them with a straight line. The line extends infinitely in both directions.

要绘制 y = 2x + 1 这样的线性方程图像，先构建数值表。选取 x 值（如 -2、-1、0、1、2），计算出对应的 y 值，描出点并用直线连接。直线向两端无限延伸。

In the form y = mx + c, m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). A positive gradient slopes upwards from left to right; a negative gradient slopes downwards. Parallel lines have the same gradient.

在 y = mx + c 的形式中，m 是斜率（陡度），c 是 y 轴截距（直线与 y 轴的交点）。正斜率从左到右向上倾斜；负斜率向下倾斜。平行线具有相同的斜率。

Real-life graphs such as distance–time or conversion graphs can be interpreted without formulas. A horizontal line indicates a stop, a steeper slope means higher speed. Always read labels to understand what the axes represent.

现实情境中的图像，如距离–时间图或换算图，无需公式即可解读。水平线段表示停止，更陡的坡意味着更高的速度。务必阅读标签以理解坐标轴的含义。

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5. Geometry: Angles and Shapes | 几何：角与图形

Basic angle facts are the building blocks. Angles on a straight line add up to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. Use these to find missing angles without a protractor.

基本角的事实是基石。直线上的角之和为 180°，绕一点一周的角之和为 360°，对顶角相等。利用这些性质可以在不用量角器的情况下求出未知角。

When a transversal crosses parallel lines, identify angle relationships: corresponding angles are equal (F-shape), alternate angles are equal (Z-shape), and interior (co-interior) angles add to 180° (C-shape). Justifying your reasoning with these rules gains marks.

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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