Tag: KS3

📚 Cambridge KS3 Maths Paper 2 Revision: Core Skills | 剑桥KS3数学卷二复习：核心技能

This article reviews the core skills tested in the Cambridge KS3 mathematics Paper 2, focusing on number, algebra, geometry and data handling. Use it alongside your practice paper to spot gaps and improve accuracy.

本文复习剑桥KS3数学卷二考查的核心技能，重点包括数、代数、几何与数据处理。请结合练习卷使用，找出薄弱点并提高正确率。

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1. Place Value and Rounding | 数位与四舍五入

Know place value from millions down to thousandths. When rounding to a given place, look at the next digit; if it is 5 or more, round up.

要掌握百万位到千分位的数位。四舍五入到某一位时，看下一位数字；如果是 5 或更大，就进位。

Example: 12345.678 rounded to two decimal places is 12345.68 because the third decimal digit is 8.

例如：12345.678 保留两位小数后是 12345.68，因为第三位小数是 8。

• Place value chart: millions, hundred thousands, ten thousands, thousands, hundreds, tens, units, tenths, hundredths, thousandths.
• 数位表：百万位、十万位、万位、千位、百位、十位、个位、十分位、百分位、千分位。

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2. Factors, Multiples and Primes | 因数、倍数与质数

A factor divides exactly into a number. A multiple is the product of a number and an integer. Prime numbers have exactly two factors: 1 and themselves.

因数能整除一个数。倍数是一个数与整数相乘的结果。质数只有两个因数：1 和它本身。

Prime factorisation breaks a number into powers of primes. The lowest common multiple (LCM) and highest common factor (HCF) are useful when comparing fractions.

质因数分解把一个数拆成质数的幂。最小公倍数（LCM）和最大公因数（HCF）在比较分数时很有用。

72 = 2³ × 3²

The prime numbers below 20 are 2, 3, 5, 7, 11, 13, 17 and 19.

20 以内的质数有 2、3、5、7、11、13、17 和 19。

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3. Fractions: Four Operations | 分数：四则运算

To add or subtract fractions, write them with a common denominator. To multiply, multiply numerators and multiply denominators. To divide, multiply by the reciprocal.

分数加减要先通分；乘法直接分子乘分子、分母乘分母；除法要乘以倒数。

1/2 + 1/3 = 3/6 + 2/6 = 5/6

Mixed numbers should be changed to improper fractions before multiplying or dividing. Always simplify your final answer where possible.

带分数在乘除前要先化为假分数。最终答案要尽量约简。

• 2/3 × 3/4 = 6/12 = 1/2
• 2/3 × 3/4 = 6/12 = 1/2

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4. Decimals and Percentages | 小数与百分数

Convert a percentage to a decimal by dividing by 100. To find a percentage of an amount, change the percentage to a decimal and multiply.

百分数除以 100 可化为小数。求一个数的百分之几，先把百分数化成小数再相乘。

Fraction	Decimal	Percentage
1/4	0.25	25%
1/2	0.5	50%
3/4	0.75	75%
1/8	0.125	12.5%

Percentage increase or decrease is found by working out the change and dividing by the original amount, then multiplying by 100.

百分数增减题：先算出变化量，除以原量，再乘以 100。

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5. Ratio and Proportion | 比与比例

Share a quantity in a given ratio by dividing by the total number of parts and multiplying by each part. Direct proportion can be modelled with equivalent ratios.

按给定比例分配，先除以总份数，再乘以各部分份数。正比例可以用等值比来建模。

Example: divide £120 in the ratio 3:2. Total parts = 3 + 2 = 5, so one part is £120 ÷ 5 = £24. The shares are £72 and £48.

例如：把 120 英镑按 3:2 分配。总份数 = 3 + 2 = 5，一份为 120 ÷ 5 = 24 英镑。两部分分别是 72 英镑和 48 英镑。

If the ratio of boys to girls is 5:7 and there are 35 girls, the total number of students is 5 + 7 = 12 parts; one part is 35 ÷ 7 = 5, so total = 12 × 5 = 60.

如果男孩与女孩的比是 5:7，女孩有 35 人，总份数为 5 + 7 = 12；一份为 35 ÷ 7 = 5，所以总人数 = 12 × 5 = 60。

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6. Algebraic Expressions | 代数式

Collect like terms by adding or subtracting coefficients. Use the distributive law to expand brackets, then simplify.

合并同类项时只加减系数；用分配律去括号，再化简。

3(x + 4) – 2(x – 1) = 3x + 12 – 2x + 2 = x + 14

Remember to multiply every term inside the bracket by the term outside. Be careful with negative signs.

记住括号外的项要乘括号内每一项。注意负号处理。

• 4a + 3b – 2a + b = 2a + 4b
• 4a + 3b – 2a + b = 2a + 4b

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7. Solving Linear Equations | 解一元一次方程

Use inverse operations in balance steps. Keep the equation balanced by doing the same operation to both sides.

用逆运算逐步解方程；两边同时进行相同运算以保持平衡。

2x + 3 = 11 → 2x = 8 → x = 4

When letters appear on both sides, collect the variable terms on one side first. Then isolate the variable by using inverse operations.

当未知数在等号两边时，先把含未知数的项移到同一边。再用逆运算求出未知数。

Check your answer by substituting it back into the original equation.

把答案代回原方程检验。

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8. Angles and Triangles | 角与三角形

Angles on a straight line add to 180°, angles around a point add to 360°, and the angles in a triangle add to 180°.

平角为 180°，周角为 360°，三角形内角和为 180°。

An equilateral triangle has three 60° angles. An isosceles triangle has two equal sides and two equal base angles. A right-angled triangle has one 90° angle.

等边三角形三个角都是 60°。等腰三角形有两条等边和两个相等的底角。直角三角形有一个 90° 角。

• If a triangle has angles 45° and 60°, the third angle = 180° – 45° – 60° = 75°.
• 如果一个三角形两角为 45° 和 60°，第三个角 = 180° – 45° – 60° = 75°。

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9. Perimeter, Area and Volume | 周长、面积与体积

Perimeter is the distance around a shape. Area of a rectangle = length × width. Volume of a cuboid = length × width × height.

周长是图形一周的长度；长方形面积 = 长 × 宽；长方体体积 = 长 × 宽 × 高。

Area triangle = ½ × base × height

For compound shapes, split them into rectangles or triangles, find each area, then add or subtract as needed.

对组合图形，可拆分成矩形或三角形，分别求面积，再按需要加减。

Volume of a cuboid is found by multiplying the three dimensions. Units must be cubic, such as cm³ or m³.

长方体体积用三条边相乘求得。单位要使用立方单位，例如 cm³ 或 m³。

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10. Coordinates and Graphs | 坐标与图像

Plot points using (x, y). The x-coordinate moves left or right; the y-coordinate moves up or down. Straight-line graphs can be drawn from a table of values.

用 (x, y) 描点；x 坐标控制左右移动，y 坐标控制上下移动。直线图可从数值表画出。

For y = 2x + 1, a table of values might be (0,1), (1,3), (2,5). Plot these points and join them with a straight line.

对于 y = 2x + 1，数值表可以是 (0,1)、(1,3)、(2,5)。描点后用直线连接。

Midpoint of two points is found by averaging the x-coordinates and averaging the y-coordinates.

两点的中点坐标：将两个 x 坐标求平均，再将两个 y 坐标求平均。

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11. Averages and Data Handling | 平均数与数据处理

Mean is the sum divided by the number of values. Median is the middle value when ordered; mode is most frequent; range is highest minus lowest.

平均数 = 总和 ÷ 个数；中位数是排序后中间的数；众数是出现最多的数；极差 = 最大值减最小值。

Example: data 4, 7, 7, 9, 10 has mean 7.4, median 7, mode 7, and range 6.

例如：数据 4、7、7、9、10 的平均数为 7.4，中位数为 7，众数为 7，极差为 6。

When reading bar charts, pictograms or pie charts, check the key or scale carefully before answering frequency questions.

阅读条形图、象形图或饼图时，先仔细查看图例或刻度，再回答频数问题。

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12. Problem-Solving Checklist | 问题解决检查清单

Read carefully, underline key numbers, choose a strategy, show clear working, check units and reasonableness. Use estimation to catch silly mistakes.

仔细读题，圈画关键数字，选择策略，展示清晰步骤，检查单位与合理性。用估算找出低级错误。

• Understand the problem and identify what is asked.
• 理解题意，明确要求什么。
• Plan a method: equation, ratio, table, diagram or trial and improvement.
• 制定方法：方程、比例、表格、图形或试错法。
• Execute the plan step by step and show all working.
• 按步骤执行并展示全部过程。
• Check the answer in the original context.
• 把答案放回原题情境中检查。

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📚 Cambridge KS3 Mathematics: Fractions, Decimals and Percentages | 剑桥 KS3 数学：分数、小数与百分比

Welcome to this Cambridge KS3 revision note. We will build confidence in switching between fractions, decimals and percentages, and in using these forms to solve real exam-style problems. This is one of the most heavily assessed number topics at KS3, so careful practice of the methods below will raise your accuracy and speed.

欢迎阅读本份剑桥 KS3 数学复习笔记。我们将建立分数、小数和百分比之间熟练转换的能力，并学会运用这些形式解决真实的考试型问题。这是 KS3 阶段考查最频繁的“数”主题之一，因此认真练习以下方法将提高你的准确度和速度。

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1. Key Equivalences Between FDP | 分数、小数与百分比的常用等值

The same quantity can be written in three ways: as a fraction, a decimal or a percentage. For example, 1/2 = 0.5 = 50%. Knowing the common equivalences saves time in ordering and comparing questions.

同一个数量可以用三种方式表示：分数、小数或百分数。例如 1/2 = 0.5 = 50%。记住常用等值能帮你在排序和比较题中节省时间。

Fraction 分数	Decimal 小数	Percentage 百分数
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
3/5	0.6	60%
4/5	0.8	80%
1/10	0.1	10%
1/3	0.333…	33⅓%
2/3	0.666…	66⅔%
1/8	0.125	12.5%

You should be able to recall these quickly without a calculator. Rapid recall of these pairs makes later work on percentage increase, probability and pie charts much easier.

你应该能够不借助计算器快速回忆这些等值。快速记忆这些对应关系，会让后面有关百分比增减、概率和饼图的内容简单得多。

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2. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. For a mixed number, keep the whole number part and convert the fractional part only.

要将分数转换为小数，用分子除以分母。对于带分数，保留整数部分，只转换分数部分。

fraction → decimal: numerator ÷ denominator

For example, 7/8 = 7 ÷ 8 = 0.875. You should set out the division carefully if the decimal is recurring, such as 1/3 = 0.333… .

例如，7/8 = 7 ÷ 8 = 0.875。如果小数循环，如 1/3 = 0.333…，你应该认真写出除法过程。

Another method is to change the fraction to an equivalent fraction over 10, 100 or 1000. For example, 3/20 = 15/100 = 0.15. This is especially useful with fractions such as 1/4, 3/5 and 7/25.

另一种方法是把分数改写成分母为 10、100 或 1000 的等值分数。例如 3/20 = 15/100 = 0.15。这种方法对 1/4、3/5 和 7/25 这样的分数尤其有用。

• Divide the numerator by the denominator when exact division is easy. 当可以整除时，用分子除以分母。
• Use equivalent fractions over 10, 100 or 1000 for simple conversions. 对简单转换，使用分母为 10、100 或 1000 的等值分数。

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3. Converting Decimals to Percentages | 小数转百分数

Multiply the decimal by 100, or move the decimal point two places to the right, and add the percent symbol. This method works for all decimals, including values greater than 1.

将小数乘以 100，或将小数点向右移动两位，再加上百分号。这个方法适用于所有小数，包括大于 1 的数。

decimal × 100 = percentage

For example, 0.47 = 47%, 0.03 = 3% and 1.25 = 125%. To see the two-place move clearly, you can first write 0.03 as 03, then insert the percent sign as 3%.

例如 0.47 = 47%，0.03 = 3%，1.25 = 125%。为了看清小数点的两位移动，你可以先把 0.03 写成 03，再加百分号得到 3%。

Be careful with decimals such as 0.7. Since 0.7 is the same as 0.70, it becomes 70%, not 7%. Writing a zero in the hundredths place can help avoid errors.

要小心像 0.7 这样的小数。因为 0.7 等于 0.70，所以它应转换为 70%，而不是 7%。在百分位补写一个零可以避免错误。

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4. Converting Percentages to Fractions | 百分数转分数

Write the percentage as a fraction over 100, then simplify fully. If the percentage contains a decimal part, multiply the numerator and denominator by 10 or 100 to clear the decimal before simplifying.

将百分数写成分母为 100 的分数，然后约分到最简。如果百分数含有小数部分，先给分子和分母同乘 10 或 100 去掉小数点，再约分。

percentage → fraction: percentage ÷ 100

For example, 45% = 45/100 = 9/20. For 37.5%, write 37.5/100, then multiply top and bottom by 10 to get 375/1000, which simplifies to 3/8.

例如 45% = 45/100 = 9/20。对于 37.5%，先写成 37.5/100，然后分子分母同乘 10，得到 375/1000，再约分为 3/8。

You should always give fractions in their simplest form unless a question says otherwise. Simplifying fully is part of the final answer in Cambridge KS3 marking.

除非题目另有说明，你总是要把分数化简到最简形式。完整约分是剑桥 KS3 评卷中最终答案的一部分。

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5. Ordering and Comparing FDP | 分数、小数和百分比的排序与比较

Convert all quantities to the same form, usually decimals, then compare them. After ordering as decimals, rewrite the values in their original forms as required by the question.

将所有数量转换为同一种形式，通常是小数，再进行比较。按小数排好序后，再根据题目要求用原来的形式写出。

For example, order 2/5, 0.45 and 38% from smallest to largest. Convert to decimals: 2/5 = 0.4, 0.45 = 0.45 and 38% = 0.38. The correct order is 0.38, 0.4, 0.45, so the answer is 38%, 2/5, 0.45.

例如，将 2/5、0.45 和 38% 从小到大排列。转换为小数：2/5 = 0.4，0.45 = 0.45，38% = 0.38。正确的顺序是 0.38、0.4、0.45，因此答案是 38%、2/5、0.45。

Do not compare fractions by looking only at numerators or denominators, because fractions with different denominators cannot be compared directly. Always convert first.

不要只通过分子或分母来比较分数，因为分母不同的分数不能直接比较。一定要先转换。

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6. Finding a Fraction of an Amount | 求一个数量的几分之几

To find a fraction of an amount, divide the amount by the denominator, then multiply by the numerator. This works for whole numbers, money, measures and other real-world quantities.

求一个数量的几分之几时，先用分母除这个数量，再乘以分子。这个方法适用于整数、货币、计量单位以及其他现实生活中的量。

a/b of N = N ÷ b × a

For example, 3/5 of 240 = 240 ÷ 5 × 3 = 48 × 3 = 144. For mixed numbers, convert to an improper fraction first, then use the same rule.

例如，240 的 3/5 = 240 ÷ 5 × 3 = 48 × 3 = 144。对于带分数，要先化为假分数，再使用同样的规则。

• Divide by the denominator first. 先除以分母。
• Multiply by the numerator second. 再乘以分子。
• Write the correct units in your answer, such as kg, m or pounds. 答案中要写正确的单位，例如 kg、m 或英镑。

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7. Percentage of an Amount and Percentage Change | 一个数量的百分数与百分比变化

To find a percentage of an amount, convert the percentage to a decimal or fraction, then multiply. For percentage increase or decrease, calculate the change and then add it to or subtract it from the original amount.

求一个数量的百分数时，先将百分数转换为小数或分数，再相乘。求百分比增加或减少时，先算出变化量，再把它加到或从原量中减去。

increase: new = original × (1 + p/100)

decrease: new = original × (1 − p/100)

For example, a 15% decrease on £80 is calculated as £80 × 0.85 = £68. The multiplier 0.85 comes from 1 − 0.15.

例如，£80 减少 15% 的计算方法是 £80 × 0.85 = £68。乘数 0.85 来自 1 − 0.15。

Using multipliers is faster than finding the change first, but both methods are acceptable. Always show your multiplier or change clearly in your working.

使用乘数比先求变化量更快捷，但两种方法都可以接受。解题过程中要清楚地写出乘数或变化量。

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8. Reverse Percentages | 逆推百分数

If a number is given after a percentage change, find the original value by dividing by the multiplier. Do not simply add or subtract the same percentage, because the percentage applies to the original amount, not the final amount.

如果已知百分比变化后的数，要求原数，应除以对应的乘数。不要直接加上或减去同样的百分数，因为百分数是针对原数，而不是最终数。

original = final amount ÷ multiplier

For example, a price after a 20% increase is £96. The multiplier for a 20% increase is 1.2, so the original price = 96 ÷ 1.2 = £80.

例如，某商品价格增加 20% 后为 £96。增加 20% 的乘数是 1.2，因此原价 = 96 ÷ 1.2 = £80。

Check your answer by applying the percentage change forward: £80 × 1.2 = £96, which confirms the reverse calculation.

通过正向百分比变化来检查答案：£80 × 1.2 = £96，这可以验证逆推计算是否正确。

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9. Fractions, Decimals and Percentages in Context | 分数、小数和百分数的实际应用

Many exam questions mix FDP with money, measures

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📚 Cambridge KS3 Mathematics Practice Page p27_2: Core Skills and Worked Examples | 剑桥 KS3 数学 p27_2 练习页：核心技能与例题解析

This revision article supports the Cambridge KS3 Mathematics practice page p27_2. It reviews the core number, algebra, geometry and data skills that most students meet in Lower Secondary, with clear worked examples and paired bilingual explanations.

本文配套剑桥 KS3 数学练习页 p27_2，复习初中阶段最常见的数、代数、几何与数据核心技能，并配有清晰的例题和双语讲解。

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1. Understanding Integers and Negative Numbers | 整数与负数运算

Adding a negative number is the same as subtracting its positive value. For example, 7 + (-4) = 7 – 4 = 3.

加上一个负数等同于减去它的相反数。例如 7 + (-4) = 7 – 4 = 3。

When multiplying or dividing two numbers with the same sign, the result is positive; when the signs are different, the result is negative.

两个同号的数相乘或相除，结果为正；两个异号的数相乘或相除，结果为负。

(-6) × (-3) = 18 and (-8) ÷ 2 = -4

Use a number line if you are unsure: moving left subtracts, moving right adds.

如果不确定，可以使用数轴：向左移动表示减，向右移动表示加。

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2. BIDMAS / Order of Operations | 运算顺序 BIDMAS

BIDMAS tells you the correct order: Brackets, Indices, Division and Multiplication, Addition and Subtraction.

BIDMAS 告诉你正确的运算顺序：括号、指数、除法和乘法、加法和减法。

Always complete division and multiplication from left to right, then addition and subtraction from left to right.

除法和乘法要按照从左到右的顺序完成，加法和减法也要按照从左到右的顺序完成。

20 – 3 × 4 = 20 – 12 = 8

A common mistake is to calculate 20 – 3 first. Because multiplication has a higher priority, you must find 3 × 4 first.

常见的错误是先计算 20 – 3。因为乘法的优先级更高，所以必须先计算 3 × 4。

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3. Fractions: Simplify, Add and Subtract | 分数的化简、加法和减法

To simplify a fraction, divide the numerator and denominator by their highest common factor. For example, 12/16 simplifies to 3/4.

化简分数时，用分子和分母的最大公因数同时去除它们。例如 12/16 化简为 3/4。

To add or subtract fractions, first rewrite them with a common denominator. Then add or subtract the numerators only.

分数加减法需要先将它们化为同分母分数，然后只对分子进行加减。

1/4 + 1/6 = 3/12 + 2/12 = 5/12

When multiplying fractions, multiply the numerators and multiply the denominators. Simplify the result if possible.

分数相乘时，分子乘分子，分母乘分母。如果可能，最后要化简结果。

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4. Decimals and Place Value | 小数与位值

Multiplying by 10 moves every digit one place to the left; dividing by 10 moves every digit one place to the right.

乘以 10 会使每个数字向左移动一位；除以 10 会使每个数字向右移动一位。

When you multiply decimals, first ignore the decimal points and multiply the whole numbers, then put the decimal point back according to the total number of decimal places.

小数相乘时，先忽略小数点并把它们当作整数相乘，然后根据小数位数的总和把小数点放回结果中。

0.6 × 0.2 = 0.12

Check place value carefully: 0.12 has 1 tenth and 2 hundredths, not 12 tenths.

仔细检查位值：0.12 表示 1 个十分位和 2 个百分位，而不是 12 个十分位。

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5. Converting Fractions, Decimals and Percentages | 分数、小数和百分比的互化

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.

把分数化为小数时，用分子除以分母。把小数化为百分数时，乘以 100。

To change a percentage to a fraction, write it over 100 and simplify if possible.

把百分数化为分数时，先写成分母为 100 的分数，然后尽可能化简。

Fraction 分数	Decimal 小数	Percentage 百分数
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
2/5	0.4	40%

Knowing these common conversions by heart saves time in mental calculations and word problems.

熟记这些常见的互化关系可以节省心算和应用题的时间。

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6. Introduction to Algebra: Collecting Like Terms | 代数入门：合并同类项

Like terms contain exactly the same letter or combination of letters. You can add or subtract their coefficients.

同类项含有完全相同的字母或字母组合。你可以对它们的系数进行加减。

3a + 2b – a + 5b = 2a + 7b

Expanding brackets means multiplying each term inside the bracket by the term outside.

去括号是指用括号外的项乘以括号内的每一项。

3(x + 4) = 3x + 12

After expanding, always check whether any like terms can be collected to simplify further.

去括号后，要检查是否可以合并同类项，以便进一步化简。

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7. Solving One-Step and Two-Step Equations | 解一步和两步方程

An equation is like a balance: whatever you do to one side, you must do to the other side.

方程就像一个天平：你对一边做什么，就必须对另一边做同样的操作。

To solve a one-step equation, use the inverse operation. For example, x + 7 = 12 gives x = 12 – 7 = 5.

解一步方程时，使用逆运算。例如 x + 7 = 12，得到 x = 12 – 7 = 5。

For a two-step equation, undo addition or subtraction first, then undo multiplication or division.

解两步方程时，先处理加法或减法，再处理乘法或除法。

2x – 3 = 9 → 2x = 12 → x = 6

Always substitute your answer back into the original equation to check it works.

一定要把答案代回原方程检验它是否正确。

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8. Area and Perimeter of Rectangles | 矩形的面积与周长

Perimeter is the total distance around the outside of a shape. For a rectangle, P = 2(l + w).

周长是图形外边线的总长度。对于矩形，P = 2(l + w)。

Area measures the space inside a shape. For a rectangle, A = l × w.

面积度量图形内部的空间。对于矩形，A = l × w。

For l = 8 cm and w = 5 cm: P = 26 cm, A = 40 cm²

Always write the correct units: cm for perimeter, cm² for area, and m² for larger spaces.

一定要写对单位：周长用 cm，面积用 cm²，较大的空间用 m²。

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9. Angles on a Straight Line and Around a Point | 平角与周角

Angles on a straight line add up to 180°. Angles around a point add up to 360°.

平角上的角度之和为 180°。绕一点一周的角度之和为 360°。

Vertically opposite angles formed by two crossing lines are equal.

两条相交直线形成的对顶角相等。

If one angle is 65° on a straight line, the other is 180° – 65° = 115°

Look for straight lines and full turns in diagrams to set up simple angle equations.

在图中寻找直线和完整的一周，可以帮助建立简单的角度方程。

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10. Mean, Median, Mode and Range | 平均数、中位数、众数和极差

The mean is found by adding all values and dividing by the number of values.

平均数是将所有数值相加后除以数值的个数。

The median is the middle value after the data has been put in order. If there are two middle numbers, find their average.

中位数是将数据排序后位于中间的数。如果有两个中间数，就取它们的平均数。

The mode is the most frequent value, and the range is the largest value minus the smallest value.

众数是出现次数最多的值，极差是最大值减去最小值。

For 4, 7, 8, 10, 10, 12: mean = 8.5, median = 9, mode = 10, range = 8

Ordering the data first makes the median and range much easier to find accurately.

先将数据排序，能更准确地找到中位数和极差。

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11. Probability Scale and Simple Events | 概率刻度与简单事件

Probability is a number from 0 to 1, where 0 means impossible and 1 means certain.

概率是 0 到 1 之间的一个数，0 表示不可能，1 表示必然发生。

For equally likely outcomes, probability equals the number of favourable outcomes divided by the total number of outcomes.

对于等可能的结果，概率等于有利结果的数量除以总结果的数量。

P(red) from 3 red and 5 blue counters = 3/8

A probability closer to 1 means the event is more likely; closer to 0 means it is less likely.

概率越接近 1，事件越可能发生；越接近 0，则越不可能发生。

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12. Exam-Style Tips for p27_2 | p27_2 考试风格提示

Always show your method, even for short questions. Marks are given for correct working, not just the final answer.

即使是简答题也要写出解题过程。评分不仅看最终答案，也看正确的步骤。

Use estimation to check whether your answer is sensible, especially when working with decimals and fractions.

用估算检查答案是否合理，尤其是在处理小数和分数时。

Keep an eye on units, signs and order of operations. Small accuracy errors are often the difference between full marks and lost marks.

注意单位、符号和运算顺序。小的准确性错误常常是满分与失分之间的差别。

If you are stuck on a multi-step problem, break it into smaller parts and write down what you already know.

如果多步问题卡住了，把它拆成几个小部分，并写下你已经知道的条件。

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📚 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。

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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。

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2. Fractions, Decimals and Percentages | 分数、小数和百分数

Fractions, decimals and percentages are three ways of expressing the same proportion. To convert a fraction to a decimal, divide the numerator by the denominator: 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% of 240 is found by 0.15 × 240 = 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%

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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。

When dividing a quantity 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。

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4. Algebraic Expressions and Substitution | 代数表达式与代入

In algebra, like terms have the same letters raised to the same powers. Simplify 5a + 3b – 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。

Substitution means replacing letters with numbers. If y = 2x² – 3x + 1, find y when x = -2: y = 2(-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。

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5. Linear Equations | 线性方程

To solve a linear equation, undo the operations in reverse order. 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。

When the unknown appears on both sides, collect variables on one side first. 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。

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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 nth term of an arithmetic sequence is a linear expression. For 5, 8, 11, 14, … 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。

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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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📚 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。

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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。

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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)

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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

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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。

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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。

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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。

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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。

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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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📚 Solving Linear Equations for Cambridge KS3 Mathematics | 剑桥KS3数学：解线性方程

Linear equations are one of the most important building blocks in Cambridge KS3 mathematics. They appear in almost every algebra topic, from simple number puzzles to real-life word problems. This article, based on the skills covered in p321_1.pdf, explains how to solve one-step, two-step, and more challenging linear equations using the balancing method, and how to check your answer correctly.

线性方程是剑桥KS3数学中最重要的基础内容之一。从简单的数字谜题到现实生活中的应用题，它几乎出现在每一个代数主题中。本文基于 p321_1.pdf 所涵盖的技能，讲解如何使用平衡法解一步方程、两步方程以及更复杂的线性方程，并正确地检验答案。

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1. What is a Linear Equation? | 什么是线性方程？

A linear equation is a mathematical statement that shows two expressions are equal, and the unknown variable has a power of 1. In KS3, the unknown is usually written as x, y, or a letter like a.

线性方程是一种数学语句，它表示两个表达式相等，且未知数的最高次数为 1。在KS3阶段，未知数通常用 x、y 或 a 等字母表示。

• x + 3 = 7 — ‘x’ has power 1, so it is linear.
• 2x − 5 = 9 — the coefficient 2 does not change the power of x.
• x² + 1 = 5 — this is not linear because the power of x is 2.

x + 3 = 7 → x = 7 − 3 → x = 4

x + 3 = 7 → x = 7 − 3 → x = 4

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2. The Balancing Method | 平衡法

Think of an equation as a balance scale: whatever you do to one side, you must do to the other side. This keeps the equation true and allows you to isolate the unknown.

把方程想象成一个天平：你对一边做什么，就必须对另一边做同样的操作。这样方程才能保持成立，并且帮助你分离出未知数。

If we have x + 5 = 12, we can subtract 5 from both sides to keep the balance.

如果我们有 x + 5 = 12，我们可以两边同时减去 5 来保持平衡。

x + 5 − 5 = 12 − 5 → x = 7

x + 5 − 5 = 12 − 5 → x = 7

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3. Solving One-Step Equations | 解一步方程

One-step equations need just one inverse operation to solve. The inverse of addition is subtraction, and the inverse of multiplication is division.

一步方程只需要一步逆运算即可求解。加法的逆运算是减法，乘法的逆运算是除法。

• x + 4 = 10 → subtract 4 → x = 6
• x − 3 = 8 → add 3 → x = 11
• 5x = 20 → divide by 5 → x = 4
• x ÷ 6 = 2 → multiply by 6 → x = 12

The table below summarises how to undo each operation.

下表总结了如何撤销每种运算。

Operation	Inverse Operation
+	−
−	+
×	÷
÷	×

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4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations, such as multiplication and addition. Always undo the addition or subtraction first, then undo the multiplication or division.

两步方程涉及两种运算，例如乘法和加法。一定要先撤销加法或减法，再撤销乘法或除法。

Solve 2x + 3 = 11. First subtract 3 from both sides, then divide both sides by 2.

解方程 2x + 3 = 11。首先两边同时减去 3，然后两边同时除以 2。

2x + 3 = 11 → 2x = 8 → x = 4

2x + 3 = 11 → 2x = 8 → x = 4

This reverse order follows the inverse of the order of operations: undo brackets, then division/multiplication, then addition/subtraction.

这种逆序对应运算顺序的逆过程：先处理括号，再处理除法或乘法，最后处理加法或减法。

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5. Expanding Brackets First | 先展开括号

When an equation contains brackets, expand them first using the distributive law. This turns the equation into a familiar two-step form.

当方程含有括号时，首先使用分配律展开括号。这样方程就变成了熟悉的两步形式。

For 3(x + 2) = 15, expand to get 3x + 6 = 15, then subtract 6 and divide by 3.

对于 3(x + 2) = 15，先展开得到 3x + 6 = 15，然后减去 6，再除以 3。

3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3

3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3

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6. Equations with Unknowns on Both Sides | 两边有未知数

If the unknown appears on both sides, collect the x terms on one side and the number terms on the other. You can do this by adding or subtracting the same x term from both sides.

如果未知数出现在等号两边，就把含 x 的项移到一边，把数字项移到另一边。你可以通过在两边同时加上或减去同一个含 x 的项来实现。

Solve 5x + 2 = 2x + 11. Subtract 2x from both sides, then subtract 2, and finally divide by 3.

解方程 5x + 2 = 2x + 11。两边同时减去 2x，再减去 2，最后除以 3。

5x + 2 = 2x + 11 → 3x + 2 = 11 → 3x = 9 → x = 3

5x + 2 = 2x + 11 → 3x + 2 = 11 → 3x = 9 → x = 3

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7. Equations Involving Fractions | 涉及分数的方程

When an equation contains a fraction, multiply both sides by the denominator first. This removes the fraction and gives a simpler equation.

当方程含有分数时，首先两边同时乘以分母。这样可以去掉分数，得到更简单的方程。

Solve x/4 + 1 = 3. First subtract 1 from both sides, then multiply both sides by 4.

解方程 x/4 + 1 = 3。首先两边同时减去 1，然后两边同时乘以 4。

x/4 + 1 = 3 → x/4 = 2 → x = 8

x/4 + 1 = 3 → x/4 = 2 → x = 8

If the fraction has a numerator such as 2x/5 = 6, multiply by 5 first to get 2x = 30, then divide by 2 to get x = 15.

如果分数的分子是 2x/5 = 6，先乘以 5 得到 2x = 30，再除以 2 得到 x = 15。

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8. Checking Your Answer | 检验答案

After solving an equation, always substitute your answer back into the original equation. If the left side equals the right side, your answer is correct.

解完方程后，一定要把答案代回原方程。如果左边等于右边，说明答案是正确的。

For x = 4 in 2x + 3 = 11, the left side becomes 2(4) + 3 = 8 + 3 = 11, which matches the right side.

对于 2x + 3 = 11 中的 x = 4，左边为 2(4) + 3 = 8 + 3 = 11，与右边相等。

This checking step is especially important in exam questions, where a small arithmetic mistake can cost marks.

在考试题中，检验这一步尤其重要，因为一个小小的计算错误就可能导致失分。

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9. Common Mistakes to Avoid | 常见错误

Many KS3 students lose marks on linear equations because of avoidable errors. Be careful with signs, especially when subtracting a negative number.

许多KS3学生在解线性方程时因为一些可以避免的错误而失分。注意符号，尤其是减去负数的情况。

• Forgetting to do the same operation on both sides.
• Forgetting to divide by the coefficient, so 2x = 8 becomes x = 8 instead of x = 4.
• Sign errors, such as writing x − 5 = 2 so x = 2 − 5 = −3 instead of x = 7.
• Not expanding brackets correctly: 3(x + 2) is not 3x + 2.

Always write each step clearly on a new line. This makes it easier to spot mistakes and to revise your method.

每一步都要清晰地写在新的一行。这样更容易发现错误，也方便检查你的方法。

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10. Exam-Style Practice and Tips | 考试题型与技巧

Exam questions often ask you to solve an equation, then use the value in a word problem. Read the question twice and highlight key information such as ‘total’, ‘difference’, or ‘equals’.

考试题常常要求解出一个方程，然后在应用题中使用这个值。读题两遍，标出关键信息，例如 ‘total’、’difference’ 或 ‘equals’。

Try this Cambridge KS3 style problem: ‘I think of a number. I multiply it by 3, then add 7. The result is 28. Find the number.’

试试这道剑桥KS3风格的题目：“我想了一个数。把它乘以3，再加7。结果是28。求这个数。”

3x + 7 = 28 → 3x = 21 → x = 7

3x + 7 = 28 → 3x = 21 → x = 7

For top marks, show all working, write the answer clearly, and check it in the original equation. These habits will help you in KS3 and beyond.

想要拿高分，就要写出完整的步骤，清晰地写出答案，并在原方程中检验。这些习惯会在KS3及以后的学习中帮助你。

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📚 Cambridge KS3 Maths: Mixed Review – Page 317 Exercise 1 | 剑桥初中数学：第317页练习1 综合复习

Page 317 Exercise 1 in the Cambridge KS3 Mathematics course is a mixed question set that pulls together the main skills from Stage 9. The questions are usually short, but they expect you to show clear working and to switch confidently between number, algebra, geometry, statistics and probability. This article explains the key ideas behind the exercise and gives you a structured way to revise each topic.

剑桥初中数学第317页练习1是一套混合题，集中考查第三学段（Stage 9）的主要数学技能。题目通常不长，但要求写出清晰步骤，并能在数、代数、几何、统计和概率之间灵活切换。本文讲解这套练习背后的核心知识点，并帮助你系统地复习每个主题。

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1. Overview of the Mixed Exercise | 混合练习总览

Mixed exercises such as Page 317 Exercise 1 are designed to test whether you can recognise the correct method without being told which topic is being assessed. You should read each question twice, circle key numbers, and write down the relevant formula or operation before you begin the calculation.

像第317页练习1这样的混合题，目的是考查你能否在没有提示的情况下识别正确方法。每道题应读两遍，圈出关键数字，并在开始计算前写下相关公式或运算步骤。

• Check whether the question asks for length, area, volume, average, probability or an unknown value.
• Write all working in a logical order so that the marker can follow your method.
• Always finish with the correct unit such as cm, cm², cm³, degrees or percent.

• 先判断题目要求的是长度、面积、体积、平均数、概率还是未知数。
• 所有步骤按逻辑顺序书写，让阅卷人能看懂你的方法。
• 最后一定要写上正确单位，如 cm、cm²、cm³、度或百分比。

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2. Number Skills: Fractions, Decimals and Percentages | 数的技能：分数、小数和百分数

Mixed exercises often include conversions between fractions, decimals and percentages. You should memorise the common equivalents such as 0.25 = ¼ = 25% and 0.75 = ¾ = 75%. To compare quantities, change them all into the same form, usually decimals or percentages.

混合练习中常出现分数、小数和百分数的互化。你需要记住常见等值关系，例如 0.25 = ¼ = 25%，0.75 = ¾ = 75%。比较数量时，应把它们化成同一种形式，通常是小数或百分数。

Fraction 分数	Decimal 小数	Percentage 百分数
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%

Percentage change is especially common. Use the following rule:

百分比变化非常常见，使用以下公式：

Percentage change = (new value − old value) ÷ old value × 100%

百分比变化 =（新值 − 旧值）÷ 旧值 × 100%

For example, if a price rises from £40 to £50, the increase is £10, so the percentage increase is 10 ÷ 40 × 100% = 25%.

例如，如果价格从 40 英镑上涨到 50 英镑，增加额为 10 英镑，因此百分比增加率为 10 ÷ 40 × 100% = 25%。

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3. Algebra: Simplifying, Expanding and Solving | 代数：化简、展开与解方程

Algebra questions on a mixed page usually ask you to simplify expressions, expand brackets, or solve linear equations. For example, 3x + 5y − 2x + y simplifies to x + 6y. Expanding 3(x + 4) gives 3x + 12.

混合页上的代数题通常要求化简表达式、展开括号或解一元一次方程。例如，3x + 5y − 2x + y 化简后为 x + 6y。展开 3(x + 4) 得到 3x + 12。

To solve an equation, perform the same inverse operation on both sides. For 2x + 3 = 11, first subtract 3 from both sides to get 2x = 8, then divide by 2 to get x = 4.

解方程时，在等式两边同时进行逆运算。对于 2x + 3 = 11，先在两边减去 3，得到 2x = 8，再除以 2，得到 x = 4。

2x + 3 = 11 → 2x = 8 → x = 4

2x + 3 = 11 → 2x = 8 → x = 4

When questions involve both algebra and area, set up an equation from the words. Define the unknown clearly, build the expression, and solve with full steps.

当题目同时涉及代数和面积时，要根据题意建立方程。清楚地设出未知数，列出表达式，并完整解出结果。

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4. Geometry: Area, Volume and Angles | 几何：面积、体积与角

Geometry problems often require you to choose the correct area or volume formula. The area of a triangle is ½ × base × height, the area of a circle is πr², and the volume of a cuboid is length × width × height.

几何题通常要求选择正确的面积或体积公式。三角形面积为 ½ × 底 × 高，圆的面积为 πr²，长方体体积为长 × 宽 × 高。

Triangle: A = ½ × b × h | Circle: A = πr² | Cuboid: V = l × w × h

三角形：A = ½ × b × h | 圆：A = πr² | 长方体：V = l × w × h

Example: a triangle has base 8 cm and height 5 cm, so its area is ½ × 8 × 5 = 20 cm². Remember that area is measured in square units and volume in cubic units.

示例：一个三角形底为 8 cm，高为 5 cm，面积为 ½ × 8 × 5 = 20 cm²。记住面积用平方单位，体积用立方单位。

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Angles in a triangle add up to 180°.

• 直线上的角之和为 180°。
• 一个点周围的角之和为 360°。
• 三角形内角之和为 180°。

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5. Statistics: Charts and Averages | 统计：图表与平均数

Statistics questions ask you to calculate the mean, median, mode and range, or to read information from a bar chart, pie chart or pictogram. The mean is found by adding all values and dividing by the number of values. The median is the middle value once data is ordered, and the mode is the most frequent value.

统计题要求计算平均数、中位数、众数和极差，或从条形图、饼图、象形图中读取信息。平均数是将所有数值相加后除以数值个数。中位数是数据排序后位于中间的数，众数是出现次数最多的数。

Mean = sum of values ÷ number of values

平均数 = 数值总和 ÷ 数值个数

For the data set 5, 8, 8, 11, 14, the sum is 46, the mean is 46 ÷ 5 = 9.2, the median is the third number 8, the mode is 8, and the range is 14 − 5 = 9.

对于数据集 5、8、8、11、14，总和为 46，平均数为 46 ÷ 5 = 9.2，中位数为第三个数字 8，众数为 8，极差为 14 − 5 = 9。

When reading charts, check the scale carefully. A bar chart may start at 0, but a pictogram may use one symbol to represent several items, so always multiply correctly.

读图时要仔细检查刻度。条形图通常从 0 开始，但象形图中一个符号可能代表多个单位，因此一定要正确乘以倍数。

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6. Probability Basics | 概率基础

Probability measures how likely an event is. It is always a number between 0 and 1, where 0 means impossible and 1 means certain. The basic formula is:

概率衡量一个事件发生的可能性。它始终是 0 到 1 之间的数，0 表示不可能，1 表示必然发生。基本公式为：

Probability = number of favourable outcomes ÷ total number of outcomes

概率 = 有利结果数 ÷ 所有可能结果数

For a fair six-sided dice, the probability of rolling a 6 is 1/6. The probability of rolling an even number is 3/6, which simplifies to 1/2. For mutually exclusive events, add the probabilities: P(A or B) = P(A) + P(B).

对于公平的六面骰子，掷出 6 的概率是 1/6。掷出偶数的概率是 3/6，化简为 1/2。对于互斥事件，概率可以相加：P(A 或 B) = P(A) + P(B)。

If the question asks for the probability of an event not happening, subtract the event probability from 1. For example, P(not 6) = 1 − 1/6 = 5/6.

如果题目问事件未发生的概率，用 1 减去该事件的概率。例如，P(不是 6) = 1 − 1/6 = 5/6。

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7. Common Mistakes to Avoid | 常见错误与避免方法

In mixed exercises, students often lose marks by making small errors that are easy to fix. The most frequent mistakes include forgetting units, mixing area with perimeter, making sign errors when solving equations, using frequency instead of total outcomes in probability, and misreading chart scales.

在混合练习中，学生常因一些容易纠正的小错误而失分。最常见的错误包括忘记单位、混淆面积与周长、解方程时符号出错、在概率题中把频数当作总结果数，以及读错图表刻度。

• After finding a length, write cm or m; after an area, write cm² or m²; after a volume, write cm³ or m³.
• Check that your probability answer is between 0 and 1.
• When using the distributive law, multiply every term inside the bracket, including negative signs.
• Order the data before finding the median.

• 求长度后写上 cm 或 m；求面积后写 cm² 或 m²；求体积后写 cm³ 或 m³。
• 检查概率答案是否在 0 到 1 之间。
• 使用分配律时，括号内每一项都要乘，包括负号。
• 求中位数前先给数据排序。

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8. Worked Example from a Mixed Exercise | 混合练习例题解析

Here is a typical mixed problem: a triangle has base 10 cm and height 4 cm. A rectangle has width 5 cm and unknown length x. The triangle and rectangle have the same area. Find x.

下面是一道典型的混合题：一个三角形底为 10 cm，高为 4 cm。一个矩形宽为 5 cm，长为未知数 x。三角形和矩形面积相等。求 x。

Triangle area = ½ × 10 × 4 = 20 cm²

三角形面积 = ½ × 10 × 4 = 20 cm²

The rectangle area is 5 × x, so the equation is 5x = 20. Dividing both sides by 5 gives x = 4 cm. This example shows how algebra and geometry combine in one question, which is exactly the style used on page 317 mixed exercises.

矩形面积为 5 × x，因此方程为 5x = 20。两边除以 5，得到 x = 4 cm。这个例子展示了代数与几何如何结合在一道题中，这正是第317页混合练习的常见题型。

5x = 20 → x = 4 cm

5x = 20 → x = 4 cm

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9. Exam-Style Practice Tips | 考试风格练习建议

To do well on mixed exercises, practise without looking at topic headings and decide the method yourself. Use the following routine: read the question twice, list the given information, write the formula, substitute the numbers, solve, and state the answer with units.

要在混合练习中取得好成绩，应不看主题标题独立练习，自己判断方法。使用以下流程：题目读两遍，列出已知信息，写出公式，代入数字，解出结果，并带单位写出答案。

• Set a timer for short practice sets to build speed.
• Mark your answers using the mark scheme and note where you lost method marks.
• Redo missed questions after two days to check retention.
• Create a personal error log for each topic.

• 计时完成短篇练习，提升速度。
• 对照评分标准批改答案，记下丢失步骤分的地方。
• 两天后重做错题，检验是否真正掌握。
• 为每个主题建立个人错误记录。

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10. Summary and Revision Checklist | 总结与复习清单

Page 317 Exercise 1 is not about learning new content; it is about retrieving and applying what you already know. Make sure you can convert between fractions, decimals and percentages, simplify and solve expressions, use area and volume formulas, find averages, and compute basic probabilities.

第317页练习1不是学习新内容，而是提取并应用你已经掌握的知识。确保你能熟练转换分数、小数和百分数，化简并解方程，使用面积和体积公式，求平均数，并计算基本概率。

Before attempting the exercise again, use this checklist: units written, working shown, equation balanced, probability between 0 and 1, and answer checked against the question.

再次尝试该练习前，请使用以下清单：单位已写，步骤已展示，方程两边平衡，概率在 0 到 1 之间，答案已与题目核对。

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📚 Master Straight-Line Graphs: y = mx + c for Cambridge KS3 Maths | 掌握直线图像：剑桥KS3数学中的 y = mx + c

A straight-line graph is one of the most important ideas in KS3 mathematics. It links algebra and geometry, showing how an equation like y = mx + c describes a line on a coordinate grid. This article builds your understanding step by step, from plotting points to finding gradients and intercepts.

直线图像是KS3数学中最重要的概念之一。它将代数与几何联系起来，说明像 y = mx + c 这样的方程如何在坐标网格上描述一条直线。本文将循序渐进地帮助你建立从描点到求斜率和截距的理解。

1. What Is a Linear Equation? | 什么是线性方程？

Any equation that can be written in the form y = mx + c is called a linear equation. The word ‘linear’ means that when you draw all the points that satisfy the equation, they form a straight line.

任何可以写成 y = mx + c 形式的方程都称为线性方程。“线性”一词表示当你画出所有满足该方程的点时，它们会形成一条直线。

The highest power of x in a linear equation is 1. This means there are no x² terms, no x³ terms, and no fractions where x is in the denominator.

线性方程中 x 的最高次数是 1。这意味着没有 x² 项、没有 x³ 项，也没有 x 在分母中的分数。

For example, y = 2x + 3, y = −x + 5 and y = ½x − 1 are all linear equations. However, y = x² + 1 and y = 1/x are not linear.

例如，y = 2x + 3、y = −x + 5 和 y = ½x − 1 都是线性方程。然而，y = x² + 1 和 y = 1/x 不是线性的。

y = mx + c

Here, m is the gradient and c is the y-intercept. Understanding these two numbers is the key to drawing and describing straight lines.

这里，m 是斜率，c 是 y 轴截距。理解这两个数字是绘制和描述直线的关键。

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2. Plotting Points and Drawing Lines | 描点与画直线

To draw a straight line, you only need two points, but using a table of values with three or more points makes your graph more reliable.

要画一条直线，你只需要两个点，但使用包含三个或更多点的数值表可以使图像更可靠。

Choose a set of x-values, substitute each one into the equation, and find the matching y-value. Then plot each pair as a coordinate (x, y).

选择一组 x 值，将每个值代入方程，求出对应的 y 值。然后将每一对数值作为坐标 (x, y) 描点。

For y = 2x + 1, we can choose x = −2, −1, 0, 1, 2. The table below shows the calculated y-values.

对于 y = 2x + 1，我们可以选择 x = −2、−1、0、1、2。下表显示了计算出的 y 值。

x	−2	−1	0	1	2
y = 2x + 1	−3	−1	1	3	5

Plot the points (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). Then draw a single straight line through all of them.

描出点 (−2, −3)、(−1, −1)、(0, 1)、(1, 3) 和 (2, 5)。然后画一条穿过所有点的直线。

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3. The Gradient m: Rise over Run | 斜率m：纵增量除以横增量

The gradient of a line measures how steep it is. It tells you how much y changes when x increases by 1.

直线的斜率衡量它的倾斜程度。它告诉你当 x 增加 1 时，y 变化了多少。

m = Δy / Δx = rise / run

Here, Δy is the vertical change and Δx is the horizontal change between two points.

这里，Δy 是两点之间的纵变化，Δx 是横变化。

If m is positive, the line slopes upwards from left to right. If m is negative, the line slopes downwards from left to right.

如果 m 为正，直线从左到右向上倾斜。如果 m 为负，直线从左到右向下倾斜。

For the line y = 3x − 2, the gradient is 3. This means for every 1 unit you move right, the line moves 3 units up.

对于直线 y = 3x − 2，斜率为 3。这意味着每向右移动 1 个单位，直线向上移动 3 个单位。

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4. The y-intercept c: Where the Line Meets the y-axis | y截距c：直线与y轴的交点

The y-intercept is the value of y when x = 0. It is the point where the line crosses the y-axis.

y 轴截距是当 x = 0 时 y 的值。它是直线与 y 轴相交的点。

In the equation y = mx + c, the constant c is always the y-intercept.

在方程 y = mx + c 中，常数 c 始终是 y 轴截距。

For example, in y = 2x + 5, the line crosses the y-axis at (0, 5). In y = −x − 3, the line crosses the y-axis at (0, −3).

例如，在 y = 2x + 5 中，直线在 (0, 5) 处与 y 轴相交。在 y = −x − 3 中，直线在 (0, −3) 处与 y 轴相交。

The y-intercept can be positive, negative, or zero. If c = 0, the line passes through the origin (0, 0).

y 轴截距可以是正数、负数或零。如果 c = 0，直线经过原点 (0, 0)。

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5. Reading y = mx + c | 解读 y = mx + c

Once you know the parts of the equation, you can quickly describe any straight line without drawing a table of values.

一旦你了解了方程的各个部分，就可以快速描述任何直线，而无需绘制数值表。

Look at the number in front of x: that is the gradient m. Look at the constant term on its own: that is the y-intercept c.

看 x 前面的数字：那是斜率 m。看单独的常数项：那是 y 轴截距 c。

Equation	Gradient m	y-intercept c
y = 4x − 7	4	−7
y = −2x + 6	−2	6
y = ¾x	¾	0

Be careful: if the equation is written as y = 5 − 3x, the gradient is −3 and the y-intercept is 5, because it is the same as y = −3x + 5.

注意：如果方程写作 y = 5 − 3x，那么斜率为 −3，y 轴截距为 5，因为它与 y = −3x + 5 相同。

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6. Finding the Gradient from Two Points | 从两点求斜率

You do not always need a full equation to find the gradient. If you know two points on a line, use the formula below.

你并不总是需要完整的方程来求斜率。如果你知道直线上的两个点，可以使用下面的公式。

m = (y₂ − y₁) / (x₂ − x₁)

Here, (x₁, y₁) and (x₂, y₂) are any two different points on the line.

这里，(x₁, y₁) 和 (x₂, y₂) 是直线上任意两个不同的点。

Example: Find the gradient of the line passing through (2, 5) and (4, 11).

示例：求经过 (2, 5) 和 (4, 11) 的直线的斜率。

Substitute the values: m = (11 − 5) / (4 − 2) = 6 / 2 = 3.

代入数值：m = (11 − 5) / (4 − 2) = 6 / 2 = 3。

Always subtract the y-values and x-values in the same order to avoid a sign error.

始终以相同顺序相减 y 值和 x 值，以避免符号错误。

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7. Finding the Equation of a Straight Line | 求直线方程

If you know the gradient and one point on a line, you can find the full equation in the form y = mx + c.

如果你知道直线的斜率和直线上的一个点，就可以求出 y = mx + c 形式的完整方程。

Start by substituting the known gradient and the coordinates of the point into y = mx + c. Then solve for c.

首先将已知斜率和点的坐标代入 y = mx + c。然后解出 c。

Example: A line has gradient 2 and passes through (3, 8). Find its equation.

示例：一条直线的斜率为 2，并且经过点 (3, 8)。求它的方程。

Substitute m = 2, x = 3, y = 8 into y = mx + c: 8 = 2 × 3 + c, so 8 = 6 + c, giving c = 2.

将 m = 2、x = 3、y = 8 代入 y = mx + c：8 = 2 × 3 + c，所以 8 = 6 + c，得到 c = 2。

The equation is therefore y = 2x + 2.

因此方程是 y = 2x + 2。

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8. Parallel Lines and Their Gradients | 平行线及其斜率

Parallel lines never meet. They always have the same gradient but different y-intercepts.

平行线永不相交。它们始终具有相同的斜率，但 y 轴截距不同。

If a line has equation y = 3x + 1, any line parallel to it must also have gradient 3. For example, y = 3x − 4 and y = 3x + 7 are both parallel to it.

如果一条直线的方程是 y = 3x + 1，那么任何与它平行的直线也必须有斜率 3。例如，y = 3x − 4 和 y = 3x + 7 都与它平行。

If two lines have different gradients, they will intersect at exactly one point. If two lines have the same gradient and the same y-intercept, they are the same line.

如果两条直线的斜率不同，它们将恰好相交于一点。如果两条直线的斜率相同且 y 轴截距也相同，它们就是同一条直线。

Checking gradients is a useful way to decide whether lines are parallel in exam questions.

在考试题目中，检查斜率是判断直线是否平行的一种有用方法。

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9. Using Graphs to Solve Problems | 利用图像解决问题

Straight-line graphs can model real situations, such as mobile phone costs, temperature changes, or distance travelled over time.

直线图像可以模拟实际情况，例如手机费用、温度变化或距离随时间的变化。

When two straight lines are drawn on the same set of axes, their point of intersection is the solution to both equations.

当两条直线画在同一坐标系上时，它们的交点就是两个方程的共同解。

Example: The graph of y = x + 2 and y = −x + 6 intersect at (2, 4). This means x = 2 and y = 4 satisfy both equations.

示例：y = x + 2 和 y = −x + 6 的图像相交于点 (2, 4)。这意味着 x = 2 和 y = 4 同时满足两个方程。

You can also read values from a straight-line graph by moving vertically from the x-axis to the line, then horizontally to the y-axis.

你也可以通过从 x 轴垂直移动到直线，再水平移动到 y 轴来读出直线图上的数值。

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10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many marks are lost through small errors. Here are some common mistakes to avoid when working with linear graphs.

许多分数是因为小错误而丢失的。以下是在处理线性图像时需要避免的一些常见错误。

Do not confuse the gradient with the y-intercept. In y = 5x − 3, the gradient is 5 and the y-intercept is −3, not the other way round.

不要将斜率与 y 轴截距混淆。在 y = 5x − 3 中，斜率为 5，y 轴截距为 −3，而不是相反。

When calculating the gradient from two points, make sure you subtract the coordinates in the same order for both y and x.

从两点计算斜率时，请确保 y 和 x 的坐标按相同顺序相减。

Always label your axes and use a ruler to draw a straight line through the plotted points. Check that the line passes through all your calculated coordinates.

始终标注坐标轴，并使用直尺在描出的点上画直线。检查直线是否经过你计算的所有坐标。

Finally, if an equation is not in the form y = mx + c, rearrange it first. For example, 2y = 6x + 4 becomes y = 3x + 2, so the gradient is 3.

最后，如果方程不是 y = mx + c 形式，请先重新整理。例如，2y = 6x + 4 可化为 y = 3x + 2，因此斜率为 3。

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📚 Cambridge KS3 Maths: Straight-Line Graphs and Gradient | 剑桥 KS3 数学：直线图像与斜率

A straight-line graph is one of the most important ideas in Cambridge KS3 mathematics. It connects algebra with coordinate geometry and appears in real-world graphs such as speed-time and distance-time graphs. This article explains how to find the gradient, how to use the equation y = mx + c, and how to draw a straight line from its equation. It is ideal for learners working through Cambridge Lower Secondary Mathematics, including exercise questions like those on page 304, question 1, where you may be asked to read or calculate a gradient from a graph.

直线图像是剑桥 KS3 数学中最重要的概念之一。它将代数与坐标几何联系起来，并出现在现实世界的图像中，如速度-时间图和距离-时间图。本文将解释如何求斜率、如何使用方程 y = mx + c，以及如何根据方程画出直线。本文适合正在学习剑桥初中数学的学习者，包括类似第 304 页第 1 题的练习，题目可能要求你从图像中读取或计算斜率。

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1. What is a straight-line graph? | 什么是直线图像？

A straight-line graph is a set of points that lie on a straight line when plotted on a coordinate grid. Each point has an x-coordinate and a y-coordinate, and all the points satisfy a linear equation such as y = 2x + 1.

直线图像是坐标网格上绘制时位于同一直线上的点集。每个点有一个 x 坐标和一个 y 坐标，并且所有点都满足一个线性方程，例如 y = 2x + 1。

For any straight-line graph, the relationship between x and y can be written in the form y = mx + c, where m and c are constants.

对于任何直线图像，x 和 y 之间的关系都可以写成 y = mx + c 的形式，其中 m 和 c 是常数。

On the coordinate plane, a straight line extends forever in both directions unless we restrict it. We usually draw it by finding at least two points and joining them with a ruler.

在坐标平面上，一条直线会向两端无限延伸，除非我们加以限制。我们通常找到至少两个点，然后用直尺把它们连接起来。

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2. The equation y = mx + c | 方程 y = mx + c

The general equation of a straight line is y = mx + c. In this equation, x and y are variables, while m and c are constants.

直线的一般方程是 y = mx + c。在这个方程中，x 和 y 是变量，而 m 和 c 是常数。

y = mx + c

Here, m represents the gradient, also called the slope of the line. The constant c represents the y-intercept, which is the point where the line crosses the y-axis.

这里，m 表示斜率，也叫做直线的坡度。常数 c 表示 y 轴截距，也就是直线与 y 轴相交的点。

For example, in the equation y = 3x + 2, the gradient is 3 and the y-intercept is 2. In y = −2x + 5, the gradient is −2 and the y-intercept is 5.

例如，在方程 y = 3x + 2 中，斜率是 3，y 轴截距是 2。在 y = −2x + 5 中，斜率是 −2，y 轴截距是 5。

This form is powerful because it allows you to sketch a line directly without needing a table of values every time.

这种形式非常有用，因为它可以让你直接画出直线，而不必每次都列出数值表。

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3. Understanding gradient | 理解斜率

The gradient of a line measures how steep it is. A larger gradient means a steeper line. A smaller gradient means the line is less steep.

斜率衡量一条线的陡峭程度。斜率越大，线越陡；斜率越小，线越平缓。

To calculate the gradient from a graph, choose two points on the line, find how many units the line rises vertically, then divide by how many units it runs horizontally.

要从图像中计算斜率，请选择线上的两点，确定线垂直上升了多少个单位，然后除以水平移动了多少个单位。

gradient = rise ÷ run = change in y ÷ change in x

The rise is the vertical change between the two points, and the run is the horizontal change. Always divide vertical by horizontal, not the other way round.

上升量是两点之间的垂直变化量，水平量是两点之间的水平变化量。一定要用垂直量除以水平量，不要颠倒。

If the line goes up as you move from left to right, the gradient is positive. If it goes down, the gradient is negative.

如果直线从左到右向上倾斜，斜率为正。如果直线从左到右向下倾斜，斜率为负。

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4. Finding the gradient from two points | 从两点求斜率

When you know two points on a line, you can calculate the gradient using the formula below.

当你已知直线上的两个点时，可以使用下面的公式计算斜率。

m = (y₂ − y₁) ÷ (x₂ − x₁)

In this formula, (x₁, y₁) is the first point and (x₂, y₂) is the second point. It does not matter which point you call first, as long as you subtract the coordinates in the same order.

在这个公式中，(x₁, y₁) 是第一个点，(x₂, y₂) 是第二个点。哪个点称为第一个点并不重要，只要两个坐标的相减顺序一致即可。

Worked example: Find the gradient of the line passing through (1, 2) and (4, 8).

示例：求经过点 (1, 2) 和 (4, 8) 的直线的斜率。

Step 1: Label the points as x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 8.

步骤 1：标记点为 x₁ = 1，y₁ = 2，x₂ = 4，y₂ = 8。

Step 2: Substitute into the formula: m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2.

步骤 2：代入公式：m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2。

So the gradient is 2. This means that for every 1 unit the line moves to the right, it moves 2 units up.

因此斜率为 2。这意味着直线每向右移动 1 个单位，就向上移动 2 个单位。

Another example: For the points (0, 5) and (3, 11), the gradient is (11 − 5) ÷ (3 − 0) = 6 ÷ 3 = 2.

另一个示例：对于点 (0, 5) 和 (3, 11)，斜率为 (11 − 5) ÷ (3 − 0) = 6 ÷ 3 = 2。

If the points are (2, 9) and (5, 3), then m = (3 − 9) ÷ (5 − 2) = −6 ÷ 3 = −2, giving a negative gradient.

如果点是 (2, 9) 和 (5, 3)，那么 m = (3 − 9) ÷ (5 − 2) = −6 ÷ 3 = −2，得到负斜率。

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5. Positive and negative gradients | 正斜率与负斜率

A line with a positive gradient rises from left to right. This means that as x increases, y also increases.

斜率为正的直线从左到右上升。这意味着随着 x 增大，y 也增大。

A line with a negative gradient falls from left to right. This means that as x increases, y decreases.

斜率为负的直线从左到右下降。这意味着随着 x 增大，y 减小。

A horizontal line has a gradient of zero because there is no vertical change. Its equation has the form y = c, such as y = 4.

水平线的斜率为零，因为没有垂直变化。它的方程形式为 y = c，例如 y = 4。

A vertical line does not have a gradient that is a real number, because the run is zero. It has

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📚 Cambridge KS3 Maths: Core Skills for p293_1 Practice | 剑桥KS3数学：p293_1 练习核心技能

This bilingual revision guide supports learners working on Cambridge KS3 Mathematics, including focused practice sheets such as p293_1. It consolidates the core skills needed to solve mixed problems confidently: number, algebra, geometry, data handling and probability.

本双语复习指南帮助学习剑桥 KS3 数学的学生，包括 p293_1 一类的专项练习。它巩固混合题所需的数、代数、几何、数据处理和概率等核心技能。

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1. Integers and Order of Operations | 整数与运算顺序

Directed numbers follow simple sign rules. Adding a negative is subtracting; subtracting a negative is adding. For example, 7 + (−2) = 5 and 9 − (−4) = 13.

有向数遵循简单的符号规则。加上负数等于减法；减去负数等于加法。例如 7 + (−2) = 5，9 − (−4) = 13。

7 + (−2) = 5

9 − (−4) = 13

Use BIDMAS/BODMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. For 3 + 4 × 2, multiply first: 3 + 8 = 11.

使用 BIDMAS/BODMAS 规则：括号、指数、除法和乘法、加法和减法。如 3 + 4 × 2，先算乘法：3 + 8 = 11。

3 + 4 × 2 = 11

Always work from left to right for multiplication and division of equal priority. Example: 12 ÷ 3 × 2 = 4 × 2 = 8.

同级的乘法和除法按从左到右的顺序计算。例如 12 ÷ 3 × 2 = 4 × 2 = 8。

12 ÷ 3 × 2 = 8

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

To add or subtract fractions, find a common denominator first. Example: 1/4 + 1/6 = 3/12 + 2/12 = 5/12.

分数加减要先通分。例如 1/4 + 1/6 = 3/12 + 2/12 = 5/12。

1/4 + 1/6 = 3/12 + 2/12 = 5/12

To multiply fractions, multiply numerators and denominators separately: 2/3 × 5/7 = 10/21. To divide by a fraction, multiply by its reciprocal.

分数相乘时，分子乘分子、分母乘分母：2/3 × 5/7 = 10/21。除以一个分数等于乘以它的倒数。

2/3 × 5/7 = 10/21

Convert between fractions, decimals and percentages. For example, 3/8 = 0.375 = 37.5%. To change a fraction to a percentage, divide numerator by denominator and multiply by 100.

分数、小数和百分数可以互相转换。例如 3/8 = 0.375 = 37.5%。将分数化为百分数，用分子除以分母再乘 100。

3/8 = 0.375 = 37.5%

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3. Ratio and Proportion | 比与比例

A ratio compares parts of a whole. If a line is divided in the ratio 3 : 5, the total number of parts is 3 + 5 = 8.

比用于比较整体中的部分。若一条线段按 3 : 5 分配，总份数为 3 + 5 = 8。

3 + 5 = 8

To share £64 in the ratio 3 : 5, find the value of one part: £64 ÷ 8 = £8. The shares are 3 × £8 = £24 and 5 × £8 = £40.

将 £64 按 3 : 5 分配，先求一份：£64 ÷ 8 = £8。两部分的份额为 3 × £8 = £24 和 5 × £8 = £40。

£64 ÷ 8 = £8

Direct proportion means two quantities increase or decrease at the same rate. If 5 pens cost £3, then 1 pen costs £0.60 and 8 pens cost 8 × £0.60 = £4.80.

正比例意味着两个量以相同的速率增加或减少。若 5 支笔 £3，则 1 支 £0.60，8 支为 8 × £0.60 = £4.80。

5 pens → £3, 1 pen → £0.60, 8 pens → £4.80

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4. Algebraic Expressions and Substitution | 代数式与代入求值

Collect like terms to simplify: 3a + 5b − a + 2b = 2a + 7b. Only letters with the same power can be combined.

合并同类项来化简：3a + 5b − a + 2b = 2a + 7b。只有相同字母且相同指数才能合并。

3a + 5b − a + 2b = 2a + 7b

Use the distributive law to expand brackets: 3(x + 4) = 3x + 12 and 2(3y − 5) = 6y − 10.

使用分配律去括号：3(x + 4) = 3x + 12，2(3y − 5) = 6y − 10。

3(x + 4) = 3x +

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