Tag: Year 7

📚 Essential Maths Book 7i Answers: Year 7 Key Topic Worked Solutions | 《Essential Maths 7i》答案解析：Year 7 核心主题例题讲解

This article gives worked examples and concise answers for the core Year 7 topics in Essential Maths Book 7i. Use the explanations to check your method, not just the final answer.

本文针对《Essential Maths 7i》中 Year 7 核心主题，提供典型例题的解答过程和简明答案。请重点核对解题方法，而不只是看最终答案。

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1. Place Value and Number Operations | 位值与数的运算

In Year 7 you must be confident with place value, ordering integers, and the four operations. For example, the number 34 062 is built from 30 000 + 4 000 + 60 + 2. Always line up digits by place value when adding or subtracting decimals.

七年级需要熟练掌握位值、整数排序和四则运算。例如，34 062 由 30 000 + 4 000 + 60 + 2 组成。进行小数加减时，一定要按位值对齐数字。

Worked example: Calculate 2 847 + 356. Add the hundreds: 800 + 300 = 1 100, so carry 1 000. The total is 2 000 + 1 100 + 100 + 3 = 3 203.

例题：计算 2 847 + 356。百位相加：800 + 300 = 1 100，因此进位 1 000。总和为 2 000 + 1 100 + 100 + 3 = 3 203。

Another common Book 7i task is long multiplication: 23 × 46. Work out 20 × 46 = 920 and 3 × 46 = 138, then add to get 1 058.

书中常见的长乘法题目如：23 × 46。先算 20 × 46 = 920，再算 3 × 46 = 138，相加得到 1 058。

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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 result of multiplying by an integer. A prime number has exactly two factors: 1 and itself.

因数能整除某个数；倍数是某个数乘以整数得到的结果。质数只有两个因数：1 和它本身。

Example: Find the HCF and LCM of 24 and 36. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The highest common factor is 12. Multiples of 24 include 24, 48, 72 … and multiples of 36 include 36, 72 … so the LCM is 72.

例题：求 24 和 36 的最大公因数（HCF）与最小公倍数（LCM）。24 的因数有 1、2、3、4、6、8、12、24；36 的因数有 1、2、3、4、6、9、12、18、36。最大公因数是 12。24 的倍数包括 24、48、72……36 的倍数包括 36、72……所以 LCM 是 72。

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3. Fractions: Simplifying and Equivalent Forms | 分数：化简与等值形式

To simplify a fraction, divide the numerator and denominator by their HCF. Equivalent fractions are made by multiplying or dividing both parts by the same non-zero number.

化简分数时，分子和分母同时除以它们的最大公因数。等值分数通过将分子和分母同时乘或除以同一个非零数得到。

Example: Simplify 18/24. The HCF of 18 and 24 is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4. The simplified fraction is 3/4. To write an equivalent fraction, multiply top and bottom: 3/4 = 9/12 = 15/20.

例题：化简 18/24。18 和 24 的最大公因数是 6，所以 18 ÷ 6 = 3，24 ÷ 6 = 4。化简结果为 3/4。要写出等值分数，可以将分子分母同时相乘：3/4 = 9/12 = 15/20。

For addition, use a common denominator: 2/5 + 1/10 becomes 4/10 + 1/10 = 5/10, which simplifies to 1/2.

进行分数加法时要先通分：2/5 + 1/10 变为 4/10 + 1/10 = 5/10，化简为 1/2。

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4. Decimal Operations and Rounding | 小数运算与四舍五入

Decimal calculations follow the same rules as integers, but the decimal point must stay aligned in addition and subtraction. For multiplication, count total decimal places in the question.

小数计算遵循与整数相同的规则，但加减时小数点必须对齐。乘法时，先按整数计算，再根据题目中小数位数总和确定小数点位置。

Example: Work out 3.7 + 2.85. Write 3.70 + 2.85, align tenths and hundredths, and add to get 6.55. For rounding, 3.276 to two decimal places is 3.28 because the third decimal digit is 6, so we round up.

例题：计算 3.7 + 2.85。写成 3.70 + 2.85，将十分位和百分位对齐，相加得到 6.55。保留两位小数时，3.276 约等于 3.28，因为第三位小数是 6，需要进位。

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5. Percentages and Conversions | 百分数与转换

Percentage means ‘out of 100’. To convert a fraction or decimal to a percentage, multiply by 100. To find a percentage of a quantity, change the percentage to a fraction or decimal and multiply.

百分数表示“每一百份中的多少”。将分数或小数转换为百分数时乘以 100。求一个量的百分之几时，先把百分数转为分数或小数，再相乘。

Example: Find 15% of 80. 15% = 15/100 = 0.15, so 0.15 × 80 = 12. Convert 0.45 to a percentage: 0.45 × 100% = 45%. Convert 3/5 to a percentage: 3 ÷ 5 = 0.6, then 0.6 × 100% = 60%.

例题：求 80 的 15%。15% = 15/100 = 0.15，所以 0.15 × 80 = 12。将 0.45 转换为百分数：0.45 × 100% = 45%。将 3/5 转换为百分数：3 ÷ 5 = 0.6，再 0.6 × 100% = 60%。

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

In algebra, letters stand for unknown numbers. Collect like terms by adding or subtracting the coefficients of the same letter. Substitution means replacing a letter with a number and using the correct order of operations.

在代数中，字母表示未知数。合并同类项就是将与同一个字母相乘的系数相加或相减。代入法是指用数字替换字母，并按照正确的运算顺序计算。

Example: Simplify 5a + 3b – 2a + b. Collect like terms: 5a – 2a = 3a and 3b + b = 4b, giving 3a + 4b. If a = 3 and b = -2, evaluate 4a – b. Substitute: 4 × 3 – (-2) = 12 + 2 = 14.

例题：化简 5a + 3b – 2a + b。合并同类项：5a – 2a = 3a，3b + b = 4b，结果为 3a + 4b。若 a = 3、b = -2，求 4a – b 的值。代入：4 × 3 – (-2) = 12 + 2 = 14。

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7. Solving Linear Equations | 解一元一次方程

To solve an equation, use inverse operations to isolate the unknown. Always do the same to both sides of the equation, and check your solution by substituting it back into the original equation.

解方程时使用逆运算将未知数单独留在等号一边。等式两边必须同时进行相同运算，并把解代回原方程检验。

Example: Solve 5x – 3 = 2x + 9. Subtract 2x from both sides: 3x – 3 = 9. Add 3: 3x = 12. Divide by 3: x = 4. Check: 5 × 4 – 3 = 17 and 2 × 4 + 9 = 17, so the answer is correct.

例题：解方程 5x – 3 = 2x + 9。两边同时减去 2x：3x – 3 = 9。两边加 3：3x = 12。两边除以 3：x = 4。检验：5 × 4 – 3 = 17，2 × 4 + 9 = 17，答案正确。

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8. Angles and Properties of Shapes | 角与图形性质

Angles on a straight line add to 180°, angles around a point add to 360°, and interior angles in a triangle add to 180°. Use these facts to find missing angles in diagrams.

直线上的角相加为 180°，一个点周围的角相加为 360°，三角形内角和为 180°。利用这些基本事实可以求出图形中的未知角。

Example: In a triangle, two angles are 68° and 47°. Find the third angle. Sum of known angles: 68° + 47° = 115°. Subtract from 180°: 180° – 115° = 65°. The missing angle is 65°.

例题：三角形中两个角分别为 68° 和 47°，求第三个角。已知角之和：68° + 47° = 115°。用 180° 减去：180° – 115° = 65°。未知角为 65°。

If an angle on a straight line is 132°, the other angle is 180° – 132° = 48°.

如果直线上的一个角为 132°，则另一个角为 180° – 132° = 48°。

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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 is length × width. Volume of a cuboid is length × width × height. Always include the correct units: cm, cm² or cm³.

周长是围绕图形一周的长度。长方形面积 = 长 × 宽。长方体体积 = 长 × 宽 × 高。务必写明正确单位：cm、cm² 或 cm³。

Example: A rectangle has length 12 cm and width 5 cm. Perimeter = 2 × (12 + 5) = 2 × 17 = 34 cm. Area = 12 × 5 = 60 cm². A cuboid with length 4 cm, width 3 cm and height 2 cm has volume = 4 × 3 × 2 = 24 cm³.

例题：一个长方形的长为 12 cm、宽为 5 cm。周长 = 2 × (12 + 5) = 2 × 17 = 34 cm。面积 = 12 × 5 = 60 cm²。一个长方体长 4 cm、宽 3 cm、高 2 cm，体积 = 4 × 3 × 2 = 24 cm³。

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10. Coordinates and Graphs | 坐标与图像

Coordinates are written in the form (x, y), where x is the horizontal position and y is the vertical position. For a linear graph, make a table of

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📚 Essential Maths Book 7C Answers: Key Worked Solutions | 《基础数学 7C》答案解析：核心例题精讲

Essential Maths Book 7C is widely used in Year 7 to stretch students who are confident with Key Stage 3 basics. This article gives worked answers to selected 7C-style question types, focusing on methods rather than just final answers. Use it to check steps, correct mistakes and build independence.

《基础数学 7C》在 Year 7 广泛用于拓展已经掌握 KS3 基础的学生。本文精选 7C 风格题型给出解析答案，重在展示方法而不只是最后结果。你可以用它核对步骤、纠正错误并培养独立解题能力。

1. Order of Operations and Directed Numbers | 运算顺序与正负数

Many 7C exercises test whether you can apply BIDMAS and handle negative numbers correctly. Work inside brackets first, then powers, then division and multiplication from left to right, and finally addition and subtraction. This order is essential when a question combines subtraction with multiplication.

许多 7C 练习考查你能否正确使用 BIDMAS 并处理负数。先算括号，再算乘方，然后从左到右计算除法和乘法，最后计算加法和减法。当一个题目同时包含减法和乘法时，这个顺序至关重要。

Example: Calculate 15 − 3 × (4 + 2) ÷ 9.

例题：计算 15 − 3 × (4 + 2) ÷ 9。

Solution: Brackets first: 4 + 2 = 6. Then multiply and divide left to right: 3 × 6 = 18, then 18 ÷ 9 = 2. Finally subtract: 15 − 2 = 13.

解答：先算括号：4 + 2 = 6。然后从左到右计算乘除：3 × 6 = 18，接着 18 ÷ 9 = 2。最后计算减法：15 − 2 = 13。

For directed numbers, subtracting a negative is the same as adding a positive: 7 − (−3) = 7 + 3 = 10. Multiplying two negatives gives a positive, for example (−4) × (−5) = 20.

对于正负数，减去一个负数等于加上它的相反数：7 − (−3) = 7 + 3 = 10。两个负数相乘得正，例如 (−4) × (−5) = 20。

Common mistake: writing 15 − 3 × 6 as 12 × 6 = 72. The correct order gives 15 − 18 = −3.

常见错误：把 15 − 3 × 6 写成 12 × 6 = 72。正确的顺序应得 15 − 18 = −3。

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

Converting between fractions, decimals and percentages is a core 7C skill. To compare 3/8, 0.36 and 35%, write them all in the same form. This makes ordering and word problems much easier.

在分数、小数和百分数之间进行转换是 7C 的核心技能。要比较 3/8、0.36 和 35%，先把它们写成同一种形式。这样排序和应用题会容易得多。

Example: Write 3/8 as a decimal and a percentage.

例题：把 3/8 写成小数和百分数。

Solution: 3/8 = 3 ÷ 8 = 0.375. As a percentage: 0.375 × 100% = 37.5%.

解答：3/8 = 3 ÷ 8 = 0.375。写成百分数：0.375 × 100% = 37.5%。

Example: Increase 240 g by 15%.

例题：将 240 g 增加 15%。

Solution: 10% of 240 = 24 g, 5% = 12 g, so 15% = 36 g. New amount = 240 + 36 = 276 g.

解答：240 的 10% 是 24 g，5% 是 12 g，所以 15% 是 36 g。新数量 = 240 + 36 = 276 g。

Example: Work out 2/3 + 5/6.

例题：计算 2/3 + 5/6。

Solution: Use a common denominator of 6: 2/3 = 4/6, so 4/6 + 5/6 = 9/6 = 3/2 = 1½.

解答：使用公分母 6：2/3 = 4/6，所以 4/6 + 5/6 = 9/6 = 3/2 = 1½。

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3. Simplifying Algebraic Expressions | 化简代数式

In 7C algebra, you collect like terms and use index notation. Only terms with exactly the same letter part can be combined. Constants are combined separately.

在 7C 代数中，你要合并同类项并使用指数记法。只有字母部分完全相同的项才能合并。常数项单独合并。

Example: Simplify 5a + 3b − 2a + 7b − 4.

例题：化简 5a + 3b − 2a + 7b − 4。

Solution: Collect a terms: 5a − 2a = 3a. Collect b terms: 3b + 7b = 10b. Constant: −4. Answer: 3a + 10b − 4.

解答：合并 a 项：5a − 2a = 3a。合并 b 项：3b + 7b = 10b。常数项：−4。答案：3a + 10b − 4。

Example: Write m × m × m × n × n using index notation.

例题：用指数记法表示 m × m × m × n × n。

Solution: There are three m’s and two n’s, so the answer is m³n².

解答：有三个 m 和两个 n，因此答案为 m³n²。

Example: Expand 3(2x − 5) and simplify 4x + 3(2x − 5).

例题：展开 3(2x − 5) 并化简 4x + 3(2x − 5)。

Solution: 3(2x − 5) = 6x − 15. Then 4x + 6x −

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📚 Essential Maths Book 7 Answers: Year 7 Key Topics Worked Solutions | Year 7 数学 Essential Maths Book 7 答案解析与复习指南

This guide gives worked solutions for the most common Year 7 topics in Essential Maths Book 7. It does not copy the full answer key, but it shows the methods you need to check your own work and build confidence.

本指南提供 Essential Maths Book 7 中 Year 7 最常见主题的解题示例。它不复制完整答案，但展示你检查作业和建立信心所需的方法。

1. Number and Place Value | 数与位值

In Year 7, you must read, write and order whole numbers up to at least 10 000 000. Place value tells you the value of each digit, so in 4 305 721 the digit 4 has value 4 000 000.

在 Year 7，你需要会读、写和比较至少到 10 000 000 的整数。位值表示每个数字的值，因此在 4 305 721 中，数字 4 的值是 4 000 000。

Worked example: Write 7 056 309 in words and find the value of the digit 6.

例题：用文字写出 7 056 309 并找出数字 6 的值。

7 056 309 = seven million fifty-six thousand three hundred and nine; the value of 6 is 6 000.

The answer is easy if you group the digits into millions, thousands and units.

如果把数字按百万、千和个位分组，答案就很容易得出。

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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 result of multiplying by an integer. A prime number has exactly two factors: 1 and itself.

因数能整除一个数。倍数是整数相乘的结果。质数只有两个因数：1 和它本身。

Worked example: Find all the factors of 36 and identify which are prime.

例题：找出 36 的所有因数，并指出哪些是质数。

Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}; prime factors: 2 and 3.

Remember that 1 is not prime because it has only one factor.

请记住 1 不是质数，因为它只有一个因数。

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3. Fractions: Simplifying and Equivalents | 分数：约分与等值分数

To simplify a fraction, divide the numerator and the denominator by their highest common factor. Equivalent fractions show the same value but have different numerators and denominators.

要化简分数，用分子和分母的最大公因数同时除以它们。等值分数表示相同的值，但分子和分母不同。

Worked example: Simplify 18/24 fully.

例题：将 18/24 化为最简分数。

18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4

The highest common factor of 18 and 24 is 6, so dividing both terms by 6 gives 3/4.

18 和 24 的最大公因数是 6，因此分子分母同时除以 6 得到 3/4。

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4. Decimals and Rounding | 小数与四舍五入

When ordering decimals, compare place values from left to right. To round to a given number of decimal places, look at the next digit: if it is 5 or more, round up.

比较小数时，从左到右比较位值。将小数四舍五入到指定小数位时，看下一位数字：如果是 5 或更大，则进位。

Worked example: Round 4.276 to 2 decimal places.

例题：将 4.276 四舍五入到两位小数。

4.276 ≈ 4.28 (to 2 d.p.)

The third decimal place is 6, which is 5 or more, so the second decimal place increases from 7 to 8.

第三位小数是 6，大于或等于 5，因此第二位小数从 7 增加到 8。

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5. Percentages and Conversions | 百分数与转换

Percent means out of 100. To convert a fraction to a percentage, first change it to a decimal, then multiply by 100.

百分数表示每一百份中的多少。要将分数转换为百分数，先把它化为小数，再乘以 100。

Worked example: Write 3/5 as a percentage.

例题：将 3/5 写成百分数。

3/5 = 0.6 = 60%

Because 5 × 20 = 100, you can also multiply the numerator by 20: 3 × 20 = 60, so 3/5 = 60/100 = 60%.

因为 5 × 20 = 100，也可以将分子乘以 20：3 × 20 = 60，所以 3/5 = 60/100 = 60%。

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6. Algebraic Expressions | 代数表达式

Algebra uses letters to stand for unknown numbers. Like terms have the same letter part, so they can be collected together by adding or subtracting the coefficients.

代数用字母表示未知数。同类项具有相同的字母部分，因此可以通过加减系数来合并。

Worked example: Simplify 3a + 5b + 2a – b.

例题：化简 3a + 5b + 2a – b。

3a + 5b + 2a – b = (3a + 2a) + (5b – b) = 5a + 4b

Group the a terms together and the b terms together, then add or subtract their coefficients.

将含 a 的项和含 b 的项分别分组，然后加减它们的系数。

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7. Solving Simple Equations | 解一元一次方程

To solve an equation, find the value of the unknown that makes the statement true. Use inverse operations to undo the steps around the unknown.

要解方程，就要找到使等式成立的未知数的值。使用逆运算来逐步解除未知数周围的运算。

Worked example: Solve x + 5 = 12.

例题：解方程 x + 5 = 12。

x + 5 = 12; x = 12 – 5 = 7

Subtracting 5 from both sides isolates x, so the solution is x = 7.

两边同时减去 5 可以将 x 单独分离出来，因此解为 x = 7。

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

Perimeter is the distance around a shape. Area is the space inside a 2D shape. For a rectangle, perimeter = 2(l + w) and area = l × w.

周长是图形外缘的长度。面积是二维图形内部的空间。对于长方形，周长 = 2(长 + 宽)，面积 = 长 × 宽。

Worked example: A rectangle has length 8 cm and width 5 cm.

例题：一个长方形长 8 厘米，宽 5 厘米。

Perimeter = 2(8 + 5) = 26 cm; Area = 8 × 5 = 40 cm²

Always include the correct units: cm for perimeter, cm² for area, and cm³ for volume.

务必写对单位：周长用 cm，面积用 cm²，体积用 cm³。

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9. Angles and Shapes | 角与图形

Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal.

直线上的角之和为 180°。围绕一点的角之和为 360°。对顶角相等。

Worked example: One angle on a straight line is 65°. Find the missing angle.

例题：直线上一个角为 65°，求另一个角。

Missing angle = 180° – 65° = 115°

Since the two angles lie on a straight line, they must add up to 180°.

因为两个角在同一直线上，它们的和必定为 180°。

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10. Coordinates and Graphs | 坐标与图像

Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. The midpoint of two points is found by averaging the x-values and the y-values.

坐标写为 (x, y)，其中 x 表示水平位置，y 表示垂直位置。两点中点的求法是分别对 x 值和 y 值取平均。

Worked example: Find the midpoint of (2, 5) and (8, 11).

例题：求 (2, 5) 和 (8, 11) 的中点。

Midpoint = ((2 + 8)/2, (5 + 11)/2) = (5, 8)

Add the two x-coordinates and divide by 2, then do the same for the y-coordinates.

将两个 x 坐标相加后除以 2，再对 y 坐标进行同样操作。

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11. Averages and Range | 平均数与极差

The mean is the sum of values divided by how many there are. The median is the middle value when data are ordered. The mode is the most common value. The range is the difference between the largest and smallest values.

平均数是一组数值的总和除以数值的个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。极差是最大值与最小值之差。

Worked example: Find the mean, median, mode and range of 4, 7, 7, 9, 11.

例题：求 4、7、7、9、11 的平均数、中位数、众数和极差。

Mean = (4 + 7 + 7 + 9 + 11) ÷ 5 = 7.6; Median = 7; Mode = 7; Range = 11 – 4 = 7

Order the data first: 4, 7, 7, 9, 11. The middle number is 7, and 7 appears twice.

先将数据排序：4、7、7、9、11。中间的数是 7，且 7 出现了两次。

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12. Checking Answers and Avoiding Errors | 检查答案与避免常见错误

Always check your work by using the inverse operation. For addition, subtract. For subtraction, add. For multiplication, divide.

始终使用逆运算检查作业。加法用减法检查，减法用加法检查，乘法用除法检查。

Worked example: Check whether 42 – 17 = 25 is correct.

例题：检查 42 – 17 = 25 是否正确。

17 + 25 = 42, therefore 42 – 17 = 25 is correct.

Common errors include forgetting to carry in column addition or misreading decimal place value, so write each step clearly.

常见错误包括列竖式加法时忘记进位或看错小数位值，因此每一步都要写清楚。

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📚 Essential Maths 7S Homework Book: Core Skills for Year 7 | Essential Maths 7S 作业册：七年级核心技能

The Year 7 Essential Maths 7S Homework Book covers the key number, algebra, geometry and data topics that form the foundation of lower secondary mathematics. This bilingual revision guide summarises the most important skills you need to practise, with clear examples and English-Chinese explanations.

七年级 Essential Maths 7S 作业册涵盖了数字、代数、几何和数据等核心主题，它们是初中数学的基础。这篇双语复习指南总结了需要重点练习的技能，并配有清晰的示例和中英文讲解。

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1. Place Value and Ordering Integers | 位值与整数排序

In Year 7 you must read, write and order whole numbers up to at least one million. Place value tells us the value of each digit. For example, in 4,306, the digit 4 stands for 4 thousands, 3 stands for 3 hundreds, 0 tens and 6 ones.

七年级需要能读写并排序至少到一百万的整数。位值表示每个数字的值。例如在 4,306 中，数字 4 表示 4 个千，3 表示 3 个百，0 个十，6 个一。

To order integers, compare digits from the left. The symbols < and > mean ‘is less than’ and ‘is greater than’. For example, 5,230 < 5,320 because 2 hundreds < 3 hundreds.

排序整数时从左到右逐位比较。符号 < 和 > 表示 ‘小于’ 和 ‘大于’。例如 5,230 < 5,320，因为 2 个百小于 3 个百。

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2. Decimals and Rounding | 小数与四舍五入

Decimals extend place value into tenths, hundredths and thousandths. To add or subtract decimals, align the decimal points and fill empty places with zeros. For example, 3.4 + 0.25 = 3.65.

小数将位值扩展到十分位、百分位和千分位。加减小数时对齐小数点，空位补零。例如 3.4 + 0.25 = 3.65。

Rounding to one decimal place or two decimal places follows the same rule: look at the next digit. If it is 5 or more, round up; if it is less than 5, round down. So 7.846 rounded to 2 d.p. is 7.85.

四舍五入到一位或两位小数遵循相同规则：看下一位数字。如果大于或等于 5，则进位；如果小于 5，则舍去。所以 7.846 保留两位小数为 7.85。

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3. Fractions, Equivalence and Simplifying | 分数、等价与约分

A fraction represents part of a whole. Equivalent fractions have the same value, such as 1/2, 2/4 and 3/6. To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF).

分数表示整体的一部分。等价分数具有相同的值，例如 1/2、2/4 和 3/6。约分时，用分子和分母的最大公因数（HCF）同时相除。

For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6. Mixed numbers like 2 1/3 combine a whole number and a proper fraction.

例如 18/24 约分后为 3/4，因为 18 和 24 的最大公因数是 6。带分数如 2 1/3 结合了整数和真分数。

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4. Percentage Conversions | 百分比转换

Percentage means ‘out of 100’. To convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply by 100%. For example, 3/5 = 60/100 = 60%.

百分比表示 ‘每一百份’。将分数转换为百分数，可以找到分母为 100 的等价分数，或乘以 100%。例如 3/5 = 60/100 = 60%。

To convert a decimal to a percentage, multiply by 100. So 0.47 = 47%, and 1.25 = 125%. To convert a percentage to a decimal, divide by 100.

将小数转换为百分数时乘以 100。因此 0.47 = 47%，1.25 = 125%。将百分数转换为小数时除以 100。

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5. Negative Numbers and the Number Line | 负数与数轴

Negative numbers are less than zero and are written with a minus sign. On a number line, numbers increase to the right and decrease to the left. For example, -7 is less than -3 because -7 lies further left.

负数小于零，书写时带负号。在数轴上，数字向右增大，向左减小。例如 -7 小于 -3，因为 -7 更靠左。

When adding or subtracting negative numbers, use rules such as a – (-b) = a + b. For example, 5 – (-2) = 5 + 2 = 7, and -3 + 5 = 2.

加减负数时使用规则，例如 a – (-b) = a + b。例如 5 – (-2) = 5 + 2 = 7，而 -3 + 5 = 2。

Multiplying or dividing two negative numbers gives a positive answer; multiplying or dividing a positive and a negative gives a negative answer. For example, -4 × -6 = 24, but -4 × 6 = -24.

两个负数相乘或相除得正数；一个正数与一个负数相乘或相除得负数。例如 -4 × -6 = 24，但 -4 × 6 = -24。

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6. Algebraic Expressions and Simplifying | 代数表达式与化简

In algebra, letters stand for unknown numbers. A term is a number, a letter or a product such as 3a or 5xy. Like terms have the same letter parts, so 2a and 5a can be added to give 7a.

代数中，字母代表未知数。项是数字、字母或乘积，如 3a 或 5xy。同类项具有相同的字母部分，因此 2a 和 5a 可以相加得到 7a。

To simplify an expression, collect like terms. For example, 4x + 3y – 2x + y = 2x + 4y. You can also expand a bracket using the distributive law: 3(x + 5) = 3x + 15.

化简表达式时要合并同类项。例如 4x + 3y – 2x + y = 2x + 4y。还可以使用分配律展开括号：3(x + 5) = 3x + 15。

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7. Solving Simple Equations | 解一元一次方程

An equation shows that two expressions are equal. To solve an equation, find the value of the unknown letter. Use inverse operations to keep the equation balanced.

方程表示两个表达式相等。解方程就是求出未知字母的值。使用逆运算保持等式平衡。

For example, x + 7 = 12, so x = 12 – 7 = 5. For 3x = 18, divide both sides by 3 to get x = 6. Always check by substituting the answer back into the original equation.

例如 x + 7 = 12，所以 x = 12 – 7 = 5。对于 3x = 18，两边同时除以 3 得 x = 6。始终通过将答案代回原方程进行检验。

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8. Angles and Triangles | 角度与三角形

Angles are measured in degrees (°). Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are equal when two lines cross.

角度以度（°）为单位。直线上相邻角之和为 180°，围绕一个点的角度之和为 360°。两条直线相交时对顶角相等。

In a triangle, the interior angles always add up to 180°. An equilateral triangle has three 60° angles, while an isosceles triangle has two equal angles.

三角形内角和始终为 180°。等边三角形有 3 个 60° 角，等腰三角形有两个相等的角。

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📚 Essential Maths Book 7S Answers: Year 7 Core Skills Review Guide | 七年级 Essential Maths Book 7S 答案解析与核心技能复习指南

This revision guide reviews typical answers from Essential Maths Book 7S and shows how to check each result using clear, repeatable methods. Use it after you have attempted the questions yourself, not before, so that mistakes become learning points. The focus is on method, common errors, and quick self-checking rather than simply copying results.

本复习指南复习 Essential Maths Book 7S 中的典型答案，并展示如何用清晰、可重复的方法核对每个结果。请先自己尝试题目，再使用本指南，这样错误才能成为学习要点。本指南侧重解题方法、常见错误和快速自查，而非简单抄写结果。

1. How to Use the Answer Section | 如何使用答案解析

Always do your working in pencil and write the final answer in pen. Mark your own work using a different colour. If an answer is wrong, classify the error as reading (R), method (M), or calculation (C). This classification helps target revision. Keep a short error log with the question number, your answer, and the correct answer.

始终用铅笔写过程，用钢笔写最终答案。用不同颜色批改自己的作业。如果答案错误，将错误分为读题错误 R、方法错误 M 或计算错误 C。这种分类有助于有针对性复习。用简短的错题本记录题号、你的答案和正确答案。

Before you mark a question, ask yourself three things: did I copy the numbers correctly, did I use the correct order of operations, and is my answer sensible for the size of the numbers? This quick habit prevents small mistakes from becoming lost marks.

批改题目之前，先问自己三件事：数字抄对了吗，运算顺序正确吗，答案的大小是否合理。这个快速习惯可以防止小错误导致丢分。

Self-check step	Question to ask
1	Did I copy the numbers and symbols correctly?
2	Did I follow the correct order of operations?
3	Is my answer a sensible size for the question?

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2. Whole Numbers and Place Value | 整数与位值

Place value is the foundation for all written methods. When marking, compare the digits in

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📚 Essential Maths Book 7S Compact Core Skills for Year 7 | Year 7 数学核心技能精讲

The Essential Maths Book 7S offers a structured Year 7 course covering number, geometry, algebra and measures. This bilingual revision guide summarises the key skills you need to master, with clear examples for quick checking.

《Essential Maths Book 7S》提供系统的七年级数学课程，涵盖数、几何、代数和度量。本双语复习指南总结你需要掌握的核心技能，并配有清晰的示例供快速检查。

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1. Place Value and the Decimal System | 数位与十进制

In Year 7, you must read, write and order whole numbers up to at least one million. Each digit has a place value: units, tens, hundreds, thousands, ten-thousands, hundred-thousands and millions.

在七年级，你需要读写并排序至少到一百万以内的整数。每个数字都有一个数位：个位、十位、百位、千位、万位、十万位和百万位。

For example, in 3 405 278, the digit 4 means four hundred thousand, and the digit 7 means seventy. Decimal places extend this idea to tenths, hundredths and thousandths after the decimal point.

例如，在 3 405 278 中，数字 4 表示四十万，数字 7 表示七十。小数点后的数位则把这一概念扩展到十分位、百分位和千分位。

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2. Ordering and Rounding Integers | 整数排序与四舍五入

To order integers, compare digits from the left. For example, 56 293 is greater than 56 239 because 9 tens is greater than 3 tens.

比较整数时，从最左边的数字开始。例如，56 293 大于 56 239，因为 9 个十大于 3 个十。

When rounding to the nearest 10, 100 or 1000, look at the digit immediately to the right of the place you are rounding to. If that digit is 5 or more, round up; if it is 4 or less, round down.

四舍五入到最近的 10、100 或 1000 时，要看保留位右边的一位数字。如果该位是 5 或更大，就向上进位；如果是 4 或更小，就舍去。

Example: 4 672 rounded to the nearest 100 is 4 700, because the tens digit is 7.

示例：4 672 四舍五入到最近的 100 是 4 700，因为十位数字是 7。

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3. Addition and Subtraction | 加法与减法

For column addition and subtraction, align digits by place value. Carry or exchange correctly when a column total exceeds 9 or when subtracting a larger digit from a smaller one.

列竖式加减法时，要将相同数位对齐。当某一列的和超过 9，或需要从较小数字中减去较大数字时，要正确进位或借位。

Always use estimation to check answers. For example, 6 498 + 2 305 is about 6 500 + 2 300 = 8 800, so the exact answer 8 803 is reasonable.

始终使用估算来检查答案。例如，6 498 + 2 305 大约是 6 500 + 2 300 = 8 800，因此精确答案 8 803 是合理的。

6 498 + 2 305 = 8 803

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4. Multiplication and Division Methods | 乘法与除法

Strong times-table recall is essential for Year 7. Use short multiplication for a two- or three-digit number by one digit, and long multiplication for two multi-digit numbers.

熟练的乘法表记忆是七年级的基础。用短乘法计算两位数或三位数乘一位数，用长乘法计算两个多位数相乘。

For division, use short division for a single-digit divisor and long division for a two-digit divisor. Check a division by multiplying the quotient by the divisor and adding any remainder.

除法中，除数是一位数时用短除法，除数是两位数时用长除法。用商乘除数再加上余数来检验除法。

Example: 342 × 6 = 2 052 and 2 052 ÷ 6 = 342.

示例：342 × 6 = 2 052，并且 2 052 ÷ 6 = 342。

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5. Factors, Multiples and Prime Numbers | 因数、倍数与质数

A factor divides exactly into a number. A multiple is the product of a number and any integer. A prime number has exactly two factors: 1 and itself.

因数能整除一个数。倍数是某个数与任意整数的乘积。质数只有两个因数：1 和它本身。

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Note that 1 is not prime.

前十个质数是 2、3、5、7、11、13、17、19、23 和 29。注意 1 不是质数。

To find prime factors, keep dividing by the smallest prime until you reach 1. For example, 12 = 2 × 2 × 3.

求质因数时，不断除以最小的质数，直到得到 1。例如，12 = 2 × 2 × 3。

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6. Understanding Fractions | 分数的理解

A fraction is written as numerator over denominator. The denominator tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.

分数写作分子在分母之上。分母表示整体被平均分成多少份；分子表示取了多少份。

Equivalent fractions have the same value but different numerators and denominators. To simplify a fraction, divide the numerator and denominator by their highest common factor.

等值分数数值相同，但分子和分母不同。化简分数时，用分子和分母的最大公因数同时除以它们。

Example: 12/18 = 2/3 because both 12 and 18 are divided by 6.

示例：12/18 = 2/3，因为 12 和 18 都除以 6。

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7. Decimals and Conversion | 小数与转换

Decimals are another way to write fractions with denominators 10, 100, 1000 and so on. The first decimal place is tenths, the second is hundredths, and the third is thousandths.

小数是表示分母为 10、100、1000 等的分数的另一种方式。小数点后第一位是十分位，第二位是百分位，第三位是千分位。

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 = 0.75.

将分数转换为小数，用分子除以分母。例如，3/4 = 0.75。

When rounding decimals, look at the digit after the required place. For example, 2.386 rounded to two decimal places is 2.39.

小数四舍五入时，看保留位后一位。例如，2.386 保留两位小数是 2.39。

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8. Percentages | 百分数

A percentage means ‘out of 100’. To convert a percentage to a fraction, write it over 100 and simplify. To convert to a decimal, divide by 100.

百分数表示“每 100 份中的多少份”。将百分数转换为分数，把它写在 100 之上并化简；转换为小数时除以 100。

Common conversions to remember: 50% = 1/2 = 0.5, 25% = 1/4 = 0.25,

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📚 Year 7 Maths Essential Maths 7H Homework Answers | Year 7 数学 Essential Maths 7H 作业答案

Welcome to this Year 7 Maths homework support article. It covers the key skills in Essential Maths 7H, with worked examples and answers for the most common homework questions. Each section gives a typical question, the correct answer, and the step-by-step method you need to show in your exercise book.

欢迎阅读本篇 Year 7 数学作业辅导文章。本文涵盖 Essential Maths 7H 的核心技能，针对最常见的作业题提供典型例题、正确答案和分步解题方法，帮助你规范书写过程。

1. Place Value and Ordering Integers | 位值与整数排序

In Year 7, you must be able to read, write and order whole numbers and decimals. Homework questions often ask you to arrange a set of numbers in ascending or descending order, or to write a number in words.

在 Year 7，你需要能够读写整数与小数并比较大小。作业题常要求将一组数字按升序或降序排列，或将数字写成文字形式。

Model question: Write these numbers in order from smallest to largest: 905, 509, 950, 590, 59.

示例题： 将下列各数从小到大排列：905、509、950、590、59。

Answer: 59, 509, 590, 905, 950. Compare the hundreds digits first. The only two-digit number is 59, so it is the smallest. Among the hundreds, 5 is less than 9, so 509 and 590 come before 905 and 950. Then compare the tens digits to order 509 before 590 and 905 before 950.

答案： 59、509、590、905、950。先比较百位数字。唯一的两位数是 59，因此它最小。在三位数中，百位 5 小于 9，所以 509 和 590 排在 905 和 950 之前。再比较十位，509 在 590 前，905 在 950 前。

Another typical question: Write 6 084 in words.

另一道典型题： 用英文写出 6 084。

Answer: Six thousand and eighty-four. In British English, use ‘and’ before the tens or units when the hundreds part is present, but do not use ‘and’ between thousands and hundreds.

答案： Six thousand and eighty-four。英式英语中，当百位后有十位或个位时使用 ‘and’，但千位与百位之间不用 ‘

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📚 Essential Maths Book 7F: Core Topics for Year 7 | 《Essential Maths Book 7F》Year 7 核心主题精讲

Essential Maths Book 7F is a widely used foundation-level textbook for Year 7 students. It builds confidence in whole numbers, decimals, fractions, simple algebra, geometry and statistics. This article summarises the key topics and shows how to master each skill step by step.

《Essential Maths Book 7F》是一本广泛使用的 Year 7 基础数学教材。它帮助学生建立整数、小数、分数、简单代数、几何与统计等方面的信心。本文梳理核心主题，并逐步讲解如何掌握每项技能。

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1. Place Value and Ordering Numbers | 数位与数字排序

Place value is the foundation of all number work. Each digit has a value depending on its position. In the number 5,306 the 5 is in the thousands place, the 3 is in the hundreds place, the 0 is in the tens place and the 6 is in the units place. For decimals, the first place after the decimal point is tenths, then hundredths, then thousandths.

数位是所有数字运算的基础。每个数字的值取决于它的位置。在 5,306 中，5 在千位，3 在百位，0 在十位，6 在个位。对于小数，小数点后第一位是十分位，然后是百分位、千分位。

To order numbers, compare the digits from left to right. For example, 0.72 is greater than 0.7 because 2 hundredths is greater than 0 hundredths. Using a place-value grid can help you line up the digits neatly.

排序数字时，要从左到右逐位比较。例如 0.72 大于 0.7，因为 2 个百分之一大于 0 个百分之一。使用数位表可以帮助你把数字整齐排列。

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2. Addition and Subtraction of Whole Numbers | 整数加减法

Addition and subtraction are essential arithmetic skills. You should be able to add and subtract whole numbers using mental methods and written column methods. When using column addition, line up the units, tens and hundreds carefully. Carry or exchange when a column total is 10 or more.

加法和减法是基本的算术技能。你需要能够使用心算和竖式方法进行整数的加减。使用竖式加法时，要仔细对齐个位、十位和百位。当某一列的和达到或超过 10 时，就要进位或交换。

For subtraction, start from the right and borrow from the next column if the top digit is too small. Always check your answer by adding: if a − b = c, then c + b should equal a.

减法要从右向左计算，如果上方的数字太小，就向相邻列借位。最后要用加法验算：如果 a − b = c，那么 c + b 应该等于 a。

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3. Multiplication and Division Strategies | 乘除法策略

Multiplication is repeated addition. You should know times tables up to 12 × 12 because they make mental arithmetic much faster. The grid method and long multiplication help with larger numbers, such as 23 × 14.

乘法是重复相加。你应该熟记 12 × 12 以内的乘法表，因为它们能让心算快得多。网格法和长乘法可以帮助计算较大的数字，例如 23 × 14。

Division is sharing into equal groups. Short division is used for dividing by a single digit, while long division is used for larger divisors. The answer can include a remainder, a decimal or a fraction.

除法是分成相等的组。短除法用于除以一位数，长除法用于除数较大的情况。答案可以包含余数、小数或分数。

dividend ÷ divisor = quotient

被除数 ÷ 除数 = 商

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

A factor is a number that divides exactly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is the result of multiplying a number by an integer, so the multiples of 5 are 5, 10, 15, 20 and so on.

因数是可以正好整除另一个数的数。例如 12 的因数有 1、2、3、4、6 和 12。倍数是一个数乘以整数得到的结果，所以 5 的倍数是 5、10、15、20 等。

A prime number has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19. Remember that 1 is not prime because it has only one factor.

质数恰好有两个不同的因数：1 和它本身。前几个质数是 2、3、5、7、11、13、17 和 19。请记住 1 不是质数，因为它只有一个因数。

The highest common factor (HCF) is the largest factor shared by two numbers, while the lowest common multiple (LCM) is the smallest multiple they share. These ideas help when simplifying fractions and solving problems.

最大公因数（HCF）是两个数共有的最大因数，最小公倍数（LCM）是它们共有的最小倍数。这些概念有助于化简分数和解决问题。

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5.

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📚 Essential Maths Book 7C: Year 7 Mastery Guide | 核心数学 7C 七年级精通指南

Essential Maths Book 7C covers the core skills that Year 7 students need to secure before moving into Key Stage 3 extension work. This guide rehearses the main topics, from number operations to early algebra and geometry, with clear examples and paired explanations.

《核心数学 7C》涵盖了七年级学生在进入第三学段拓展学习之前必须掌握的核心技能。本指南复练主要专题，从数的运算到初步代数和几何，配有清晰的例子和中英对照讲解。

1. Place Value and Decimals | 位值与小数

Place value is the value given to a digit based on its position in a number. In a decimal number such as 347.528, the digit 3 means 300, the digit 4 means 40, and the digit 7 means 7 units. The digits after the point are tenths, hundredths and thousandths.

位值是根据数字在数中的位置赋予它的值。在像 347.528 这样的小数中，数字 3 表示 300，数字 4 表示 40，数字 7 表示 7 个一。小数点后的数字分别表示十分位、百分位和千分位。

Multiplying by 10 moves every digit one place to the left, and dividing by 10 moves every digit one place to the right. This means 0.6 × 10 = 6 and 45 ÷ 10 = 4.5.

乘以 10 时每一位数字向左移动一位，除以 10 时每一位数字向右移动一位。因此 0.6 × 10 = 6，45 ÷ 10 = 4.5。

To compare decimals, look at the place value columns from left to right. For example, 0.56 is greater than 0.529 because the hundredths digit 6 is greater than 2.

比较小数时，从左到右依次看位值列。例如 0.56 大于 0.529，因为百分位上的 6 大于 2。

347.528 = 300 + 40 + 7 + 0.5 + 0.02 + 0.008

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2. Fractions, Decimals and Equivalence | 分数、小数与等值

A fraction such as ¾ can also be written as the decimal 0.75. The top number is the numerator and the bottom number is the denominator. The denominator tells how many equal parts the whole is divided into.

分数如 ¾ 也可以写成小数 0.75。上面的数是分子，下面的数是分母。分母表示整体被分成多少等份。

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the place value digits as the numerator and simplify if possible. For example, 0.6 = 6/10 = 3/5.

要将分数转换为小数，用分子除以分母。要将小数转换为分数，把位值数字写成分子，并尽可能化简。例如 0.6 = 6/10 = 3/5。

Equivalent fractions have the same value. You can multiply or divide the numerator and denominator by the same non-zero number to find equivalent forms. For example, 2/3 = 4/6 = 8/12.

等值分数具有相同的值。你可以将分子和分母同时乘以或除以同一个非零数来得到等值形式。例如 2/3 = 4/6 = 8/12。

¾ = 0.75 = 75%

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3. Percentages of Amounts | 求一个数的百分比

A percentage is a number out of 100. The symbol % means ‘out of 100’. To find a percentage of an amount, first find 1% by dividing by 100, then multiply by the percentage needed.

百分数表示每一百份中的多少份。符号 % 表示“每一百份中”。要求一个数的百分比，先除以 100 得到 1%，再乘以所需的百分数。

For example, to find 15% of 60, find 1% of 60 (0.6) and multiply by 15 to get 9. Alternatively, write the percentage as a decimal: 15% = 0.15, so 0.15 × 60 = 9.

例如，要求 60 的 15%，先求 60 的 1%（即 0.6），再乘以 15 得到 9。也可以把百分数写成小数：15% = 0.15，因此 0.15 × 60 = 9。

To increase or decrease an amount by a percentage, calculate the percentage part and then add or subtract it. For example, increasing 80 by 10% gives 80 + 8 = 88.

要按百分数增加或减少一个量，先算出百分数对应的部分，然后加上或减去它。例如，把 80 增加 10% 得到 80 + 8 = 88。

15% of 60 = 0.15 × 60 = 9

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4. Algebraic Notation and Substitution | 代数记号与代入

In algebra, letters stand for unknown or changing numbers. We write 3 × a as 3a, and a × a as a². The number in front of a letter is called the coefficient.

在代数中，字母代表未知或变化的数。我们把 3 × a 写成 3a，把 a × a 写成 a²。字母前面的数称为系数。

Substitution means replacing letters with numbers. If a = 4 and b = 2, then 3a + 2b = 3 × 4 + 2 × 2 = 16. Always follow the order of operations: work out multiplication before addition.

代入是指用数字替换字母。如果 a = 4，b = 2，那么 3a + 2b = 3 × 4 + 2 × 2 = 16。始终遵循运算顺序：先算乘除后算加减。

Like terms contain exactly the same letters and powers. You can simplify expressions by collecting like terms. For example, 5x + 3y − 2x = 3x + 3y.

同类项含有完全相同的字母和幂。你可以通过合并同类项来化简式子。例如 5x + 3y − 2x = 3x + 3y。

3a + 2b = 3 × 4 + 2 × 2 = 16

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5. Solving Equations | 解方程

An equation is a statement that two expressions are equal. To solve an equation, keep it balanced by doing the same operation to both sides. The solution is the value of the unknown that makes the equation true.

方程是表示两个式子相等的语句。解方程时，要对两边做相同的运算，保持方程平衡。方程的解是使方程成立的未知数的值。

For x + 3 = 10, subtract 3 from both sides: x = 7. For 4x = 20, divide both sides by 4: x = 5. Always check your answer by substituting it back into the original equation.

对于 x + 3 =

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📚 Essential Maths 7S Homework Answers: Year 7 Practice Guide | Essential Maths 7S 作业答案解析：七年级练习指南

This guide gives selected answers and step-by-step explanations for question types found in Essential Maths 7S. It is designed to help Year 7 students check their reasoning, correct common errors, and build confidence before moving to the next topic.

本指南提供 Essential Maths 7S 中常见题型的精选答案和逐步解析，帮助七年级学生检查解题思路、纠正常见错误，并在进入下一个主题前建立信心。

Each section follows the same pattern as the homework book: short worked examples, answers in bold, and a clear method you can reuse. Always revise the method behind an answer, not just the final number.

每节都遵循与作业本相同的模式：简短的示例、加粗答案和可反复使用的清晰方法。始终要复习答案背后的方法，而不只是最后的数字。

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1. How to Use Essential Maths 7S Homework Answers | 如何使用 Essential Maths 7S 作业答案

The answers below are selected examples that mirror the style of Essential Maths 7S. Always attempt each question on your own first, then compare your working with the answer. Check whether your mistake is in the calculation, the method, or the reading of the question.

下面的作业答案是从 Essential Maths 7S 风格中挑选出来的示例。先自己尝试每一道题，再对照答案检查。看看错误是出在计算、方法，还是题意理解。

Use estimation to test whether an answer is sensible. For example, when adding 48 and 29, expect about 80, so an answer of 77 is reasonable. Copying answers without working will not build the fluency needed for tests.

用估算来检查答案是否合理。例如计算 48 + 29 时，结果应在 80 左右，所以答案是 77 就合理。只抄答案而不写过程，无法培养考试所需的熟练度。

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2. Number Facts and Place Value | 数字知识与位值

Place value tells us the value of each digit. In 37,406, the digit 7 is in the thousands column, so its value is 7000.

位值告诉我们每个数字的值。在 37,406 中，数字 7 位于千位，所以它的值是 7000。

Write 6 thousands, 2 hundreds, 4 tens and 9 ones as one number: 6249.

把 6 个千、2 个百、4 个十和 9 个一写成一个数：6249。

Digit	Place value	Value
5	ten thousands	50000
2	thousands	2000
7	hundreds	700
3	tens	30
8	ones	8

Order these numbers from smallest to largest: 3104, 3014, 3140, 3041. The correct order is 3014, 3041, 3104, 3140.

将这些数字从小到大排列：3104、3014、3140、3041。正确顺序是 3014、3041、3104、3140。

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3. Negative Numbers | 负数

Negative numbers appear below zero on the number line. To add a positive number to a negative number, move right: -7 + 12 = 5.

负数在数轴上位于零以下。给负数加上一个正数，向右移动：-7 + 12 = 5。

To subtract a positive number from a negative number, move left: -3 – 8 = -11.

从负数中减去一个正数，向左移动：-3 – 8 = -11。

When multiplying or dividing two negatives, the answer is positive: (-4) × (-6) = 24 and (-20) ÷ (-5) = 4.

两个负数相乘或相除时，结果为正：(-4) × (-6) = 24，(-20) ÷ (-5) = 4。

When the signs are different, the answer is negative: (-6) × 3 = -18.

当符号不同时，结果为负：(-6) × 3 = -18。

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4. Fractions Basics | 分数基础

To simplify a fraction, divide the numerator and denominator by their highest common factor. 18/24 simplifies to 3/4 because both 18 and 24 divide by 6.

化简分数时，分子和分母同时除以它们的最大公因数。18/24 化简为 3/4，因为 18 和 24 都能被 6 整除。

Equivalent fractions multiply or divide both parts by the same number: 3/7 = 12/28 because 3 × 4 = 12 and 7 × 4 = 28.

等价分数是把分子和分母同时乘以或除以同一个数：3/7 = 12/28，因为 3 × 4 = 12，7 × 4 = 28。

To add fractions, use a common denominator: 2/5 + 1/10 = 4/10 + 1/10 = 5/10 = 1/2.

分数加法要使用公分母：2/5 + 1/10 = 4/10 + 1/10 = 5/10 = 1/2。

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5. Decimals and Rounding | 小数与四舍五入

Round 3.476 to 2 decimal places. The second decimal place is 7, and the following digit is 6, so we round up: 3.48.

将 3.476 保留两位小数。第二位小数是

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📚 Essential Maths 7H Answers: How to Use Them for Year 7 Success | Essential Maths 7H 答案：如何利用答案提升 Year 7 数学成绩

This article explains how to use the Essential Maths 7H answer booklet effectively. It does not replace the official answers but shows Year 7 students how to check work, understand methods, and avoid common errors.

本文介绍如何有效使用 Essential Maths 7H 答案册。本文不替代官方答案，而是帮助 Year 7 学生学会检查作业、理解方法并避免常见错误。

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1. What Is Essential Maths 7H? | 什么是 Essential Maths 7H？

Essential Maths 7H is a widely used Year 7 textbook for higher-attaining students. It covers number, algebra, geometry, ratio, and statistics. The answer booklet gives final answers, but understanding the steps is more important than seeing the result.

Essential Maths 7H 是一本广泛使用的 Year 7 高年级学生教材，内容涵盖数、代数、几何、比和统计。答案册提供最终答案，但理解步骤比看到结果更重要。

Many schools use the 7H book to stretch pupils who coped well with primary mathematics. The question style gradually becomes more challenging, so answer checking needs a clear routine.

许多学校使用 7H 教材来拓展小学数学基础较好的学生。题目风格逐渐变难，因此核对答案需要清晰的流程。

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2. Why You Should Not Just Copy Answers | 为什么不要直接抄答案

Copying answers can feel quick, but it hides gaps in understanding. Use answers only after you have attempted a question. Mark your work in a different colour and write a short note when you get something wrong.

直接抄答案看似快速，但会掩盖知识漏洞。只有在你尝试过题目后才能使用答案。用不同颜色的笔批改，并在出错时写一条简短注释。

If you are stuck, write down what you tried before looking at the answer. This helps you compare your method with the correct one and learn from the difference.

如果卡住了，先写下你尝试过的方法，然后再看答案。这有助于你将你的方法与正确方法进行比较，并从差异中学习。

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3. How to Mark Your Work Efficiently | 如何高效批改作业

Mark one exercise at a time. Write the correct answer next to wrong ones. Circle mistakes you do not understand. Then redo the question without looking at the answer.

一次批改一个练习。把正确答案写在错误答案旁边。圈出你不理解的错误，然后不看答案重做题目。

Use a simple marking code: T for ‘thinking error’, C for ‘calculation error’, and U for ‘unit missing’. This helps you see patterns over time.

使用简单的批改符号：T 表示 ‘思考错误’，C 表示 ‘计算错误’，U 表示 ‘缺少单位’。这有助于你发现长期存在的模式。

Try to mark your work after a short break rather than immediately after finishing. A fresh mind spots errors more easily.

尽量在完成后休息片刻再批改，而不是刚写完就批改。头脑清醒时更容易发现错误。

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4. Number and Place Value: Answer Check | 数与位值：答案核对技巧

For place-value questions, check zeros and decimal points carefully. Multiplying by 10 moves digits one place to the left. Example: 3.06 × 10 = 30.6, not 30.06.

在位值题中，仔细检查零和小数点。乘以 10 会让数字向左移动一位。例如：3.06 × 10 = 30.6，而不是 30.06。

3.06 × 10 = 30.6

Always use estimation to see if an answer is reasonable. For 48 × 19, estimate 50 × 20 = 1000, so the exact answer should be close to 1000.

始终用估算判断答案是否合理。对于 48 × 19，估算为 50 × 20 = 1000，因此精确答案应接近 1000。

With negative numbers, use a number line mentally. Example: -5 + 12 = 7. If the sign looks wrong, recheck the direction of movement.

涉及负数时，在脑海中想象数轴。例如：-5 + 12 = 7。如果符号看起来不对，重新检查移动的方向。

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5. Fractions, Decimals and Percentages: Common Answer Formats | 分数、小数和百分比：常见答案格式

Always simplify fractions unless the question says otherwise. Show equivalent working clearly. Example: 0.75 = 75/100 = 3/4. When comparing fractions, convert them to the same denominator or to decimals.

除非题目另有说明，否则分数始终要化简。清晰展示等值过程。例如：0.75 = 75/100 = 3/4。比较分数时，将它们化为同分母或化为小数。

0.75 = 75/100 = 3/4

For percentage increase, find the increase first, then divide by the original amount and multiply by 100. Example: increase from 20 to 25 is 5 ÷ 20 × 100 = 25%.

计算百分比增加时，先求增加量，再除以原始量并乘以 100。例如：从 20 增加到 25，增加率为 5 ÷ 20 × 100 = 25%。

When adding or subtracting fractions, make sure the denominators are the same before you combine the numerators. Example: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.

加减分数时，先确保分母相同，再合并分子。例如：1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2。

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6. Algebra Basics: Substitution and Simplifying | 基础代数：代入与化简

When substituting, put negative numbers in brackets. This prevents sign errors. Example: if a = 3 and b = -2, then 2a – b = 2(3) – (-2) = 6 + 2 = 8.

代入时，将负数放入括号内，这样可以避免符号错误。例如：若 a = 3 且 b = -2，则 2a – b = 2(3) – (-2) = 6 + 2 = 8。

2a − b = 2(3) − (−2) = 6 + 2 = 8

Write one step per line when simplifying expressions. This makes it easier to compare your working with the answer booklet and spot mistakes.

化简表达式时每行只写一步。这样更容易将你的过程与答案册对照，并发现错误。

When collecting like terms, only add or subtract terms with exactly the same letters and powers. Example: 3x + 2y + 5x = 8x + 2y.

合并同类项时，只加减字母和次数完全相同的项。例如：3x + 2y + 5x = 8x + 2y。

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7. Geometry and Measures: Units and Formula Layout | 几何与度量：单位与公式布局

Always include units in final answers. For area, write cm² or m². For perimeter, write cm or m. Show the formula first, then substitute, then calculate.

最终答案必须包含单位。面积要写 cm² 或 m²，周长要写 cm 或 m。先写出公式，再代入，最后计算。

Area = length × width = 5 cm × 3 cm = 15 cm²

For triangle area, remember the half. Example: base 8 m, height 5 m gives Area = ½ × 8 × 5 = 20 m².

三角形面积不要忘记乘二分之一。例如：底 8 m，高 5 m，面积 = ½ × 8 × 5 = 20 m²。

When converting units, check the direction. 1 m = 100 cm, so 2.5 m = 250 cm. A common error is writing 25 cm instead of 250 cm.

换算单位时要注意方向。1 m = 100 cm，所以 2.5 m = 250 cm。常见错误是把 250 cm 写成 25 cm。

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8. Ratio and Proportion: Showing Steps | 比和比例：展示步骤

Simplify ratios like fractions by dividing all parts by the highest common factor. Example: 12 : 18 = 2 : 3 because both sides divide by 6.

化简比的方法与分数类似，所有部分都除以最大公因数。例如：12 : 18 = 2 : 3，因为两边都除以 6。

12 : 18 = (12 ÷ 6) : (18 ÷ 6) = 2 : 3

For sharing in a ratio, find the total number of parts first, then the value of one part. Example: share £40 in the ratio 3 : 5 gives total parts 8, one part £5.

按比例分配时，先求总份数，再求一份的值。例如：按 3 : 5 分配 £40，总份数为 8，一份为 £5。

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📚 Essential Maths 7 Higher Answers: Using Answer Keys to Master Year 7 Mathematics | 《Essential Maths 7 Higher》答案解析：掌握 Year 7 数学的答案使用指南

Answer keys are not just for marking a question right or wrong. When you use the Essential Maths 7 Higher answers carefully, you can see exactly where your method breaks down, learn the standard notation expected at Key Stage 3, and build the self-checking habits that make Year 8 and GCSE work much easier.

答案并不只是用来判断一道题对错。当你仔细使用 Essential Maths 7 Higher 的答案时，你可以准确看到自己的解题方法在哪里出了问题，学会 Key Stage 3 要求的标准书写方式，并养成自我检查的习惯，这会让你在 Year 8 和 GCSE 阶段的学习轻松很多。

1. What Is Essential Maths 7 Higher? | 什么是 Essential Maths 7 Higher？

Essential Maths 7 Higher is a widely used Key Stage 3 textbook written to challenge pupils who are working above the expected standard in Year 7. It covers number, algebra, geometry, ratio, statistics and probability, often with multi-step questions that require a clear written method.

Essential Maths 7 Higher 是一本广泛使用的 Key Stage 3 教材，专为 Year 7 中高于预期水平的学生编写。它涵盖数字、代数、几何、比、统计与概率，题目常常涉及多个步骤，需要清晰的书面解题过程。

The accompanying answer book gives final answers and, in many cases, brief worked solutions. However, it is designed to support learning, not to replace thinking. Simply copying answers does not improve your ability to solve unfamiliar problems.

配套答案册给出最终答案，很多情况下还给出简要的解题步骤。但它的作用是辅助学习，而不是代替思考。简单地抄答案并不能提高你解决陌生问题的能力。

The Higher book moves faster than the Core book. It introduces formal algebra, directed numbers, angle properties, and proportional reasoning earlier. Therefore, the answer key is especially useful for catching small notation errors that can become big problems later.

Higher 教材比 Core 教材进度更快。它更早引入正式的代数、有向数、角的性质和比例推理。因此，答案册对发现小的书写错误特别有用，这些错误以后可能会变成大问题。

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2. Why Answers Matter: From Checking to Deep Learning | 答案为何重要：从核对到深度学习

When you mark your own work, you engage in a loop: attempt, compare, diagnose, and correct. This loop strengthens memory far more than passive reading or copying.

当你自己批改作业时，你经历了一个循环：尝试、比较、诊断、订正。这个循环比被动阅读或抄写更能强化记忆。

An answer key also helps you recognise what examiners call “method marks”. In Year 7, marks are often awarded for showing a correct substitution, a correct first step, or a correct conversion even if the final answer is wrong.

答案还能帮助你识别考官所说的 “方法分”。在 Year 7，即使最终答案错了，只要展示了正确的代入、正确的第一步或正确的单位换算，也常常能得到分数。

Always mark in a different colour pen, and never change a wrong answer without writing the correct method beside it. If you are still unsure why an answer is correct, ask your teacher before moving on.

批改时始终使用不同颜色的笔；不要只改答案，还要在旁边写出正确的解题方法。如果你仍然不明白为什么那个答案是对的，先问老师，再继续往下做。

Teachers often expect students to mark homework before the next lesson. This saves time, but it only works if you understand why an answer is right or wrong. Highlight any corrected question so that the teacher can see where you struggled.

老师通常希望学生在下一节课之前批改家庭作业。这可以节省时间，但只有当你明白答案为什么对或错时才有效。把订正过的题目标出来，老师就能看到你在哪里遇到了困难。

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3. Getting Ready: How to Self-Mark Effectively | 做好准备：如何有效自批

Before opening the answer book, complete every question in full. Beware of the illusion of fluency: recognising a solution when you see it is very different from producing it under time pressure.

在打开答案册之前，要完整地做完每一道题。小心 “熟练度错觉”：看到答案时能认出来，和在时间压力下自己写出答案是完全不同的。

Mark in stages: first check the final answer, then compare the method. If the final answer is wrong, re-read the question to find where you misread a word or ignored a unit. Then re-attempt the question without looking at the answer.

分阶段批改：先核对最终答案，再比较方法。如果最终答案错误，重新读题，找出你误解了哪个词或忽略了哪个单位。然后不看答案，重新做一遍。

A useful rule is “five-minute retry”: if you made a mistake, wait at least five minutes, then solve the question again on a blank page. This tests whether you have truly understood the correction or merely copied it.

一个实用的方法是 “五分钟重做法”：如果你做错了，至少等五分钟，然后在空白页上重新解这道题。这能检验你是真正理解了订正内容，还是只是抄了一遍。

Use a ruler to cover the answer column if the answer book has answers on the same page.

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📚 Year 7 Essential Maths 7 Core Answers: Worked Solutions for Key Topics | Year 7 数学 Essential Maths 7 Core 核心答案解析

This revision guide gives clear, model answers for the core question types in Year 7 Essential Maths 7 Core. Use it to check your method, not just your final answer, because marks are awarded for working.

本复习指南为 Year 7 Essential Maths 7 Core 的核心题型提供清晰、规范的答案示范。请用它来检查解题方法，而不只是核对最终答案，因为评分时会看重解题过程。

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1. Place Value and Rounding | 位值与四舍五入

Write the value of the digit 7 in 37 452. The digit 7 is in the thousands column, so its value is 7 000.

写出 37 452 中数字 7 的值。数字 7 位于千位，因此它的值是 7 000。

In 5 308, the value of the digit 5 is 5 000 because it is in the thousands column. In 4 623, the digit 6 has a value of 600 because it is in the hundreds column.

在 5 308 中，数字 5 的值是 5 000，因为它位于千位。在 4 623 中，数字 6 的值是 600，因为它位于百位。

Round 2 648 to the nearest 10, 100 and 1 000. To the nearest 10 the answer is 2 650, to the nearest 100 it is 2 600, and to the nearest 1 000 it is 3 000.

将 2 648 四舍五入到最接近的 10、100 和 1 000。到最近的 10 是 2 650，到最近的 100 是 2 600，到最近的 1 000 是 3 000。

Order 3 142, 2 986 and 3 099 from smallest to largest. The correct order is 2 986, 3 099, 3 142.

将 3 142、2 986 和 3 099 从小到大排列。正确顺序是 2 986、3 099、3 142。

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2. Addition and Subtraction of Integers | 整数加减法

Calculate 5 426 + 3 879. Add the ones: 6 + 9 = 15, carry the 1. Continue with tens, hundreds and thousands columns to get 9 305.

计算 5 426 + 3 879。个位相加：6 + 9 = 15，进 1。继续计算十位、百位和千位，得到 9 305。

Calculate 7 003 – 2 648. You need to borrow from the thousands column because the hundreds and tens columns have zeros. The answer is 4 355.

计算 7 003 – 2 648。由于百位和十位是 0，需要从千位借位。答案是 4 355。

A library has 1 250 fiction books and 875 non-fiction books. The total is 1 250 + 875 = 2 125 books. The difference between the two types is 1 250 – 875 = 375 books.

一个图书馆有 1 250 本

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📚 Year 7 Essential Maths Core | Year 7 核心数学精讲

This revision guide covers the essential topics in the Year 7 Core Mathematics course. It is designed to help you build strong foundations in number, algebra, geometry and data handling. Work through each section carefully and practise the examples before attempting exam-style questions.

本复习指南涵盖 Year 7 核心数学课程的重要主题。它旨在帮助你在数、代数、几何和数据处理方面打下坚实基础。请认真学习每一节，并在尝试考试型题目之前练习示例。

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1. Place Value and Ordering | 位值与排序

Place value tells us the value of each digit in a number. In Year 7, you must be confident reading and writing whole numbers up to 10 000 000 and decimals to at least three decimal places.

位值告诉我们一个数中每个数字的值。在 Year 7，你必须能够熟练读写最大到 10 000 000 的整数以及至少三位小数。

To order numbers, compare the digits from left to right. For example, 0.704 is greater than 0.69 because 7 tenths is more than 6 tenths.

要比较大小，就从左到右比较数字。例如，0.704 大于 0.69，因为 7 个十分之一比 6 个十分之一多。

Rounding is also a key skill. To round 47 562 to the nearest thousand, look at the hundreds digit: 5 means we round up, so the answer is 48 000.

四舍五入也是一项关键技能。将 47 562 四舍五入到最近的千位时，要看百位数字：5 意味着向前进一，所以答案是 48 000。

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2. Addition and Subtraction | 加法和减法

Column addition and subtraction are the standard written methods in Year 7. Always line up the digits by place value and carry or borrow carefully.

竖式加法和减法是 Year 7 的标准书面方法。始终按位值对齐数字，并仔细进位或借位。

Estimation helps you check your work. If you add 489 and 312, a sensible estimate is 500 + 300 = 800, so the exact answer 801 is reasonable.

估算帮助你检查答案。如果你计算 489 加 312，合理的估算是 500 + 300 = 800，所以精确答案 801 是合理的。

For subtraction, remember that 8000 − 3 = 7997. When zeros are involved, regrouping must be done step by step across place values.

对于减法，记住 8000 − 3 = 7997。当涉及零时，必须逐步跨位值进行借位。

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3. Multiplication and Division | 乘法和除法

Multiplication tables up to 12 × 12 are essential. Long multiplication and short division allow you to calculate larger products and quotients without a calculator.

12 × 12 以内的乘法表是必备基础。长乘法和短除法使你能够不用计算器计算较大的乘积和商。

Factors are numbers that divide exactly into another number. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

因数是能整除另一个数的数。24 的因数有 1、2、3、4、6、8、12 和 24。

A prime number has exactly two factors: 1 and itself. The first five prime numbers are 2, 3, 5, 7 and 11.

质数恰好有两个因数：1 和它本身。前五个质数是 2、3、5、7 和 11。

When multiplying by 10, 100 or 1000, digits move one, two or three places to the left. For example, 3.6 × 100 = 360.

当乘以 10、100 或 1000 时，数字向左移动一位、两位或三位。例如，3.6 × 100 = 360。

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4. Negative Numbers | 负数

Negative numbers are numbers less than zero. They appear on the left side of a number line, and they are used in temperature, height above sea level and money.

负数是小于零的数。它们出现在数轴的左侧，广泛用于温度、海拔高度和金钱债务。

Adding a negative number is the same as subtracting its positive value. For example, 5 + (−3) = 5 − 3 = 2.

加上一个负数等于减去它的正值。例如，5 + (−3) = 5 − 3 = 2。

Subtracting a negative number is the same as adding. So (−4) − (−6) = −4 + 6 = 2.

减去一个负数等于加上它的正值。所以 (−4) − (−6) = −4 + 6 = 2。

When multiplying or dividing, two same signs give a positive answer, while different signs give a negative answer: (−3) × (−4) = 12, but (−3) × 4 = −12.

乘法或除法中，同号得正，异号得负： (−3) × (−4) = 12，但 (−3) × 4 = −12。

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

Fractions, decimals and percentages are three ways of representing parts of a whole. You must be able to convert between them fluently.

分数、小数和百分数是表示整体的一部分的三种方式。你必须能够在它们之间熟练转换。

To simplify a fraction, divide the numerator and the denominator by their highest common factor. For example, 12/18 simplifies to 2/3.

要化简分数，就将分子和分母同时除以它们的最大公因数。例如，12/18 化简为 2/3。

Equivalent fractions have the same value. 1/2, 2/4 and 5/10 are all equivalent.

等价分数具有相同的值。1/2、2/4 和 5/10 都是等价分数。

To convert a fraction to a decimal, divide the top by the bottom. 3/8 = 0.375. To convert a decimal to a percentage, multiply by 100: 0.375 × 100 = 37.5%.

要将分数转换为小数，用分子除以分母。3/8 = 0.375。要将小数转换为百分数，乘以 100：0.375 × 100 = 37.5%。

Fraction 分数	Decimal 小数	Percentage 百分数
1/4	0.25	25%
3/5	0.6	60%
7/10	0.7	70%

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6. Ratio and Proportion | 比和比例

Ratio compares the sizes of two or more quantities. It can be written with a colon, such as 3:2, and must be simplified by dividing both sides by common factors.

比用于比较两个或多个量的大小。它可以用冒号书写，例如 3:2，并且必须通过同时除以公因数来化简。

To share £80 in the ratio 3:5, first add the parts: 3 + 5 = 8. Then one part is £80 ÷ 8 = £10. The shares are 3 × £10 = £30 and 5 × £10 = £50.

要将 80 英镑按 3:5 的比例分配，先求总份数：3 + 5 = 8。然后一份是 80 ÷ 8 = 10 英镑。分配结果是 3 × 10 = 30 英镑和 5 × 10 = 50 英镑。

Proportion problems often involve scaling recipes or maps. If 3 apples cost 90p, then 5 apples cost 5 × 30p = 150p, because one apple costs 30p.

比例问题常涉及按比例调整配方或地图距离。如果 3 个苹果售价 90 便士，那么 5 个苹果售价 5 × 30 = 150 便士，因为一个苹果售价 30 便士。

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7. Introduction to Algebra | 代数入门

Algebra uses letters to represent unknown numbers or changing values. In Year 7, you learn to write expressions, substitute values and simplify by collecting like terms.

代数使用字母表示未知数或变化的量。在 Year 7，你将学习书写代数式、代入数值以及通过合并同类项进行化简。

An expression like 3a + 2b − a + 5b simplifies to 2a + 7b. Only like terms can be combined.

像 3a + 2b − a + 5b 这样的代数式可以化简为 2a + 7b。只有同类项才能合并。

Substitution means replacing letters with numbers. If a = 4 and b = −2, then 2a + 3b = 2 × 4 + 3 × (−2) = 8 − 6 = 2.

代入就是用数字替换字母。如果 a = 4 且 b = −2，那么 2a + 3b = 2 × 4 + 3 × (−2) = 8 − 6 = 2。

Writing expressions follows rules: ‘5 more than n’ is written n + 5, and ‘3 lots of x’ is written 3x.

书写代数式遵循规则：’比 n 大 5′ 写作 n + 5，’x 的 3 倍’ 写作 3x。

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8. Solving Equations | 解方程

An equation shows that two expressions are equal. To solve it, find the value of the unknown letter that makes the equation true.

方程表示两个代数式相等。解方程就是找出使方程成立的未知字母的值。

Use inverse operations to isolate the letter. For x + 7 = 15, subtract 7 from both sides: x = 15 − 7, so x = 8.

使用逆运算来分离字母。对于 x + 7 = 15，两边同时减去 7：x = 15 − 7，所以 x = 8。

For 3x = 21, divide both sides by 3: x = 21 ÷ 3, so x = 7. Always substitute your answer back to check it works.

对于 3x = 21，两边同时除以 3：x = 21 ÷ 3，所以 x = 7。一定要把答案代回原方程进行检验。

Two-step equations require undoing addition before multiplication. For 2x + 3 = 11, first subtract 3: 2x = 8, then divide by 2: x = 4.

两步方程需要先消去加法，再处理乘法。对于 2x + 3 = 11，先减去 3：2x = 8，然后除以 2：x = 4。

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

Angles are measured in degrees. You need to identify acute, right, obtuse, straight and reflex angles from diagrams and descriptions.

角以度为单位测量。你需要从图形和描述中识别锐角、直角、钝角、平角和优角。

Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are equal.

直线上的角之和为 180°，一个点周围的角之和为 360°。对顶角相等。

The angles inside a triangle always total 180°. If two angles are 65° and 45°, the third angle is 180° − 65° − 45° = 70°.

三角形内角和总是 180°。如果两个角分别是 65° 和 45°，第三个角就是 180° − 65° − 45° = 70°。

Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites. Their interior angles always add up to 360°.

四边形包括正方形、长方形、平行四边形、菱形、梯形和风筝形。它们的内角和总是 360°。

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10. Perimeter, Area and Coordinates | 周长、面积与坐标

Perimeter is the distance around the outside of a shape. For a rectangle, add the lengths of all four sides, or use 2 × (length + width).

周长是图形外边缘的总长度。对于长方形，将四条边的长度相加，或使用公式 2 × (长 + 宽)。

Area is the amount of space inside a shape. The area of a rectangle is length × width. A rectangle 7 cm by 4 cm has area 28 cm² and perimeter 22 cm.

面积是图形内部的空间大小。长方形的面积等于长 × 宽。一个长 7 厘米、宽 4 厘米的长方形，面积是 28 平方厘米，周长是 22 厘米。

Compound shapes can be split into rectangles. Find each rectangle’s area and add them together to get the total area.

复合图形可以分割成若干个长方形。分别求出每个长方形的面积，再把它们相加得到总面积。

Coordinates are written as (x, y). The x-coordinate gives the horizontal position, and the y-coordinate gives the vertical position. Translating a point 3 units right and 2 units up changes (4, 5) to (7, 7).

坐标写作 (x, y)。x 坐标表示水平位置，y 坐标表示垂直位置。将一个点向右平移 3 个单位、向上平移 2 个单位，(4, 5) 就变成 (7, 7)。

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

There are three main averages. The mode is the most common value, the median is the middle value when data is ordered, and the mean is the sum of all values divided by how many there are.

有三种主要的平均数。众数是最常出现的值，中位数是将数据排序后位于中间的值，平均数（均值）是所有数值之和除以数值的个数。

The range is the difference between the largest and smallest values. For the data set 3, 7, 8, 8, 12, the mode is 8, the median is 8, the mean is (3 + 7 + 8 + 8 + 12) ÷ 5 = 7.6, and the range is 12 − 3 = 9.

极差是最大值与最小值之差。对于数据集 3、7、8、8、12，众数是 8，中位数是 8，平均数是 (3 + 7 + 8 + 8 + 12) ÷ 5 = 7.6，极差是 12 − 3 = 9。

Data can be displayed in bar charts, pictograms, line graphs and pie charts. Always read the scale carefully and check that totals make sense.

数据可以用条形图、象形图、折线图和饼图来展示。一定要仔细读取刻度，并检查总数是否合理。

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12. Probability | 概率

Probability measures how likely an event is to happen. It always lies between 0 and 1, where 0 means impossible and 1 means certain.

概率用来衡量一个事件发生的可能性大小。它总是在 0 和 1 之间，0 表示不可能，1 表示必然发生。

For equally likely outcomes, probability = number of favourable outcomes ÷ total number of possible outcomes. The probability of rolling a 4 on a fair six-sided dice is 1/6.

对于等可能的结果，概率 = 有利结果数 ÷ 所有可能结果数。在一个均匀六面骰子上掷出 4 的概率是 1/6。

If a bag contains 3 red and 5 blue counters, the probability of picking a red counter is 3/8, and the probability of picking a blue counter is 5/8.

如果一个袋子里有 3 个红色和 5 个蓝色筹码，那么抽到红色筹码的概率是 3/8，抽到蓝色筹码的概率是 5/8。

To find an expected number of outcomes, multiply the probability by the number of trials. In 200 rolls of a dice, you would expect about 200 × 1/6 ≈ 33 rolls of a 6.

要求期望结果数，就用概率乘以试验次数。掷骰子 200 次，你大约会期望掷出 200 × 1/6 ≈ 33 次 6。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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