Tag: Year 8

📚 Essential Maths 8C Homework Answers | 八年级数学 Essential Maths 8C 作业答案

Welcome to this comprehensive guide on Essential Maths 8C homework answers. This article is designed for Year 8 students who are using the Essential Maths 8C textbook and need clear, step-by-step explanations for their homework problems. We will cover the main topics, provide worked examples, and highlight common mistakes so that you can master each concept with confidence.

欢迎阅读这份 Essential Maths 8C 作业答案的全面指南。本文面向使用 Essential Maths 8C 教材的八年级学生，为你的作业问题提供清晰、分步的讲解。我们将涵盖主要知识点，给出典型例题，并指出常见错误，帮助你自信地掌握每个概念。

1. Understanding Essential Maths 8C | 了解 Essential Maths 8C

Essential Maths 8C is a core textbook used in many Year 8 mathematics courses. The “8C” typically indicates the level or tier (often the third book in a series, targeting students aiming for higher attainment). The book covers topics such as algebraic manipulation, linear equations, proportional reasoning, geometry of shapes, and handling data. Homework exercises at the end of each chapter are designed to reinforce classroom learning and build problem-solving skills.

Essential Maths 8C 是许多八年级数学课程的核心教材。其中的 “8C” 通常表示级别（通常是系列中的第三册，面向目标是更高学业水平的学生）。本书涵盖代数运算、线性方程、比例推理、几何图形以及数据处理等主题。每章末的课后练习旨在巩固课堂所学，并培养解决问题的技能。

2. Key Topics Covered in Essential Maths 8C | Essential Maths 8C 涵盖的关键主题

The textbook is organised into distinct units. Typical units for Year 8 include: numbers and place value, fractions and percentages, introduction to algebra, solving equations, linear graphs, perimeter and area, angles and polygons, and statistical charts. In the homework, you will often be asked to combine these skills in multi-step problems.

本书按照不同单元组织。八年级的典型单元包括：数与位值、分数与百分比、代数入门、解方程、线性图象、周长与面积、角度与多边形，以及统计图表。在作业中，你经常需要综合运用这些技能解决多步问题。

• Algebra: simplifying expressions, expanding brackets, solving equations.

  代数：化简表达式、展开括号、解方程。

• Number: ratio, proportion, fractions, percentages, and standard form.

  数：比例、比率、分数、百分比和标准形式。

• Geometry: angle rules, triangle properties, area and volume.

  几何：角度法则、三角形性质、面积和体积。

• Statistics: interpreting bar charts, pie charts, and calculating averages.

  统计：解读条形图、饼图，并计算平均数。

3. Solving Linear Equations | 解线性方程

One of the most important skills in Year 8 is solving linear equations. A linear equation has the form ax + b = c, where x is the unknown. To solve it, you must isolate x by performing inverse operations on both sides. Always remember: whatever you do to one side, do the same to the other.

八年级最重要的技能之一就是解线性方程。线性方程形如 ax + b = c，其中 x 是未知数。解方程时，你必须通过在等式两边进行逆运算来隔离 x。永远记住：对一边做了什么，也要对另一边做同样的操作。

Worked example 1: Solve 3x + 5 = 20.

例题 1：解方程 3x + 5 = 20。

• Subtract 5 from both sides: 3x = 15.

  两边同时减 5：3x = 15。

• Divide both sides by 3: x = 5.

  两边同时除以 3：x = 5。

Answer: x = 5.

答案：x = 5。

Check your solution by substituting x = 5 back into the original equation: 3(5) + 5 = 20. This confirms the answer is correct.

将 x = 5 代回原方程进行检验：3(5) + 5 = 20。这确认答案是正确的。

For equations with unknowns on both sides, collect like terms first. For example, 5x – 3 = 2x + 9. Subtract 2x from both sides to get 3x – 3 = 9, then add 3 to get 3x = 12, so x = 4.

对于未知数在两边的方程，先合并同类项。例如 5x – 3 = 2x + 9。两边同时减 2x 得到 3x – 3 = 9，再加 3 得到 3x = 12，因此 x = 4。

4. Working with Fractions and Percentages | 分数与百分比运算

Fractions and percentages appear frequently in Essential Maths 8C homework. You need to switch between fractions, decimals, and percentages fluently, and apply operations such as addition, subtraction, multiplication, and division. Always simplify fractions to their lowest terms.

分数和百分比经常出现在 Essential Maths 8C 的作业中。你需要熟练地在分数、小数和百分比之间转换，并应用加、减、乘、除等运算。务必把分数化简为最简形式。

When adding or subtracting fractions, find a common denominator first. For instance, ½ + ⅓ = 3⁄6 + 2⁄6 = 5⁄6.

进行分数加减运算时，先找到公分母。例如，½ + ⅓ = 3⁄6 + 2⁄6 = 5⁄6。

To find a percentage of a quantity, convert the percentage to a decimal and multiply. For example, 15% of 80 = 0.15 × 80 = 12.

要求一个数量的百分比，先把百分比转换为小数再相乘。例如，80 的 15% = 0.15 × 80 = 12。

Word problems involving percentages often ask for percentage increase or decrease. The formula is: percentage change = (change ÷ original) × 100%.

涉及百分比的文字题经常要求计算百分比的增加或减少。公式是：百分比变化 = (变化量 ÷ 原量) × 100%。

Percentage change = (Change ÷ Original) × 100%

百分比变化 = (变化量 ÷ 原量) × 100%

5. Geometry and Angles | 几何与角度

Geometry is a major component of Year 8 mathematics. You need to know that angles on a straight line sum to 180°, angles around a point sum to 360°, and angles in a triangle sum to 180°. These rules are essential for solving many homework questions.

几何是八年级数学的重要组成部分。你需要知道平角之和为 180°，周角之和为 360°，三角形内角之和为 180°。这些规则对于解决许多作业问题至关重要。

For example, if a triangle has angles 40° and 75°, the third angle is 180° – 40° – 75° = 65°.

例如，如果一个三角形的两个角分别是 40° 和 75°，则第三个角为 180° – 40° – 75° = 65°。

Another common topic is area and perimeter of 2D shapes. The area of a rectangle is length × width. For a triangle, area = ½ × base × height. For a circle, area = πr², where r is the radius and π ≈ 3.14 or 22/7.

另一个常见主题是二维图形的面积和周长。矩形的面积是长 × 宽。三角形的面积 = ½ × 底 × 高。圆的面积 = πr²，其中 r 是半径，π ≈ 3.14 或 22/7。

Area of triangle = ½ × base × height | 三角形面积 = ½ × 底 × 高

Circumference of circle = 2πr | 圆的周长 = 2πr

6. Interpreting Data and Graphs | 数据与图表解读

Data handling and statistics are covered in many Year 8 lessons. You must be able to read information from tables, construct and interpret bar charts, line graphs, and pie charts, and calculate the mean, median, mode, and range of a data set.

数据处理和统计在八年级的许多课程中都有涉及。你必须能够从表格中读取信息，绘制并解读条形图、折线图和饼图，并计算一组数据的平均数、中位数、众数和极差。

The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.

平均数是所有数值之和除以数值的个数。中位数是数据按大小排列后的中间值。众数是出现次数最多的值。极差是最大值与最小值之差。

For example, given the data set {3, 7, 7, 8, 10}:

例如，给定数据集 {3, 7, 7, 8, 10}：

• Mean: (3 + 7 + 7 + 8 + 10) ÷ 5 = 35 ÷ 5 = 7.

  平均数：(3 + 7 + 7 + 8 + 10) ÷ 5 = 35 ÷ 5 = 7。

• Median: 7 (the middle value).

  中位数：7（中间的值）。

• Mode: 7 (appears twice).

  众数：7（出现两次）。

• Range: 10 – 3 = 7.

  极差：10 – 3 = 7。

7. Common Pitfalls and How to Avoid Them | 常见错误与避免方法

Many students make avoidable mistakes in their homework. Here are some common pitfalls and advice to help you stay on track:

很多学生在作业中会犯一些可以避免的错误。以下是一些常见错误和帮助你避免它们的建议：

• Sign errors in algebra: Always write down the operation you are performing. When you move a term to the other side, change its sign.

  代数中的符号错误：写下你正在进行的运算。把一个项移到另一边时，要改变它的符号。

• Forgetting to multiply by the denominator when working with fractions: Never skip steps.

  处理分数时忘记乘以分母：不要跳过步骤。

• Using the wrong formula for area or volume: Keep a formula sheet and refer to it until you memorise the formulas.

  使用错误的面积或体积公式：保留一张公式表，并查阅它直到你记住公式。

• Reading the graph incorrectly: Check the axis scales carefully before reading values.

  阅读图形错误：读取数值前仔细检查坐标轴刻度。

• Not showing your working: Even if you get the correct answer, a clear method earns full marks and helps you spot mistakes.

  不写出解题过程：即使答案正确，清晰的过程能帮你得满分，也有助于发现错误。

8. Practice Problems and Answers | 练习题与答案

To help you apply the skills you have learned, here are some typical homework problems with full solutions. Attempt each problem before reading the answer.

为了帮助你运用所学技能，这里有一些典型的作业题及完整解答。在查看答案之前，请先尝试解答每一个问题。

Let’s check the first problem in detail. Subtract 7 from both sides: 2x = 8. Then divide by 2: x = 4. Substituting back: 2(4) + 7 = 15. Correct.

我们详细检查第一个问题。两边减 7：2x = 8。然后除以 2：x = 4。代回原式：2(4) + 7 = 15。正确。

9. Tips for Self-Checking | 自我检查技巧

Before submitting your homework, you should always check your answers. One effective method is to substitute your answers back into the original equations. For geometry, verify that the sum of angles is correct. For statistics, recompute the mean using a different method, such as adding all values again.

在提交作业之前，你应该始终检查答案。一种有效的方法是将答案代回原方程。对于几何题，验证角度之和是否正确。对于统计题，可以使用不同方法重新计算平均数，例如重新加一遍所有数值。

Another tip is to work with a partner. Explain your solution to a friend or family member. If you can explain it clearly, you probably understand it. If not, review the steps.

另一个技巧是与同伴一起学习。向朋友或家人解释你的解答。如果你能清晰地解释，说明你可能已经理解了。如果不能，请复习步骤。

10. Conclusion | 结论

This guide has covered the core skills needed for Essential Maths 8C homework: solving equations, working with fractions and percentages, applying geometry rules, and interpreting data. Remember that mathematics is learned by doing. The more problems you solve, the more confident you will become. Always show your working, check your answers, and ask your teacher when you are stuck.

本指南涵盖了 Essential Maths 8C 作业所需的核心技能：解方程、分数与百分比运算、应用几何法则以及解读数据。请记住，数学是通过实践来学习的。你解决的题目越多，就会越自信。始终写出解题过程，检查答案，遇到困难时请教老师。

We hope these answers and explanations support your learning. Good luck with your homework and your Year 8 mathematics studies!

我们希望这些答案和讲解能够支持你的学习。祝你作业顺利，八年级数学学习成功！

Published by TutorHao | Mathematics Revision Series | aleveler.com

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📚 Year 8 Essential Maths Book 8C Answers: Smart Study Guide | Year 8 Essential Maths Book 8C 答案：聪明学习指南

Essential Maths Book 8C is a widely used resource for Year 8 students who are building stronger problem-solving skills. This guide explains how to use the answer key responsibly, highlights the main topic areas in Book 8C, and offers revision strategies to help you turn a simple mark into real understanding. It is not a replacement for your own working, but a tool for checking and learning from mistakes.

Essential Maths Book 8C 是许多 Year 8 学生用来提高解题能力的常用教材。本指南将说明如何正确使用答案、梳理 Book 8C 的主要知识板块，并提供复习策略，帮助你从简单的对答案变成真正的理解。答案不是用来替代你自己的计算过程，而是用来检查错误并从中学习的工具。

1. Understanding the Role of Answers in Year 8 Maths | 理解答案在 Year 8 数学中的作用

An answer key should be used after you have attempted a question fully. When you compare your working with the given answer, look for differences in method, not just the final number. In Year 8, marks are often awarded for clear steps, so the answer alone does not tell you whether your method would earn full marks in a test.

答案应该在你完整尝试一道题之后再使用。当你把自己的解题过程与答案对比时，要注意方法上的差异，而不仅仅是最后的数字。在 Year 8 阶段，很多分数是给清晰步骤的，所以只看答案并不能判断你的方法在考试中是否能拿满分。

• Attempt every question before checking the answer.
• Write down your working, not just the final result.
• Use the answer to diagnose where your method went wrong.

• 每道题都要先自己做，再看答案。
• 写出解题步骤，不要只写最后结果。
• 用答案来诊断你的方法错在哪里。

2. How to Use Compressed Answer Files Safely and Effectively | 如何安全高效使用压缩答案文件

Compressed answer files are often shared as .zip or .rar folders to save space. Always download from your school’s official platform or a trusted publisher website. Inside, you may find PDF or image files organised by chapter. Keep the file on one device only and avoid re-uploading it to public sites, as this may break copyright rules.

压缩答案文件通常以 .zip 或 .rar 格式共享，以节省空间。请始终从学校官方平台或可信的出版社网站下载。文件内可能是按章节整理的 PDF 或图片。请只在一个设备上保存该文件，不要重新上传到公开网站，因为这可能违反版权规定。

• Open the file only after you have finished a whole exercise.
• Use a printed or split-screen copy to compare your steps.
• Do not share the file publicly; use it for personal study only.

• 完成整个练习后再打开文件。
• 用打印版或分屏视图对比你的步骤。
• 不要公开分享文件，仅供个人学习使用。

3. Number Skills: Fractions, Decimals and Percentages | 数字技能：分数、小数与百分数

Book 8C extends earlier work on fractions, decimals and percentages. You need to convert fluently between the three forms, for example 3/4 = 0.75 = 75%. Questions often ask you to find a percentage of an amount, increase or decrease a quantity, or compare fractions by finding a common denominator.

Book 8C 在之前分数、小数和百分数的基础上进一步拓展。你需要熟练地在三种形式之间转换，例如 3/4 = 0.75 = 75%。题目经常要求你求一个数的百分之几、增加或减少一个量，或者通过找公分母来比较分数。

Fraction to percentage: multiply by 100. Example: 3/8 × 100 = 37.5%

分数转百分数：乘以 100。例如：3/8 × 100 = 37.5%

• Always simplify fractions unless the question says otherwise.
• Use a calculator for multi-step percentage problems, but show your method.
• Check that percentages in a real-life context make sense, e.g. a sale price should be lower than the original.

• 除非题目另有要求，分数都要化简。
• 多步百分数问题可以使用计算器，但要写出步骤。
• 检查实际情况中的百分数是否合理，例如打折后的价格应低于原价。

4. Algebra: Simplifying Expressions and Solving Equations | 代数：化简表达式与解方程

Algebra in Year 8 focuses on collecting like terms, expanding single brackets, and solving linear equations. A typical task might be to simplify 4x + 3y – 2x + y, which becomes 2x + 4y. When solving an equation such as 5x – 3 = 2x + 9, move the variable terms to one side and the numbers to the other.

Year 8 的代数重点是合并同类项、展开单项括号和解一元一次方程。一个典型任务是化简 4x + 3y – 2x + y，得到 2x + 4y。解方程如 5x – 3 = 2x + 9 时，把含变量的项移到一边，数字移到另一边。

Solve: 5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4

解：5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4

• Do the same operation to both sides of an equation.
• Substitute your solution back into the original equation to check.
• Keep negative signs with the correct term when collecting like terms.

• 方程两边要同时进行相同的运算。
• 把解代回原方程进行检查。
• 合并同类项时注意负号要跟着正确的项。

5. Ratio, Proportion and Rates of Change | 比、比例与变化率

Ratio questions in Book 8C include sharing an amount in a given ratio, simplifying ratios, and using proportion to scale recipes or maps. For example, sharing £120 in the ratio 3:5 means dividing by 8 total parts, so one part is £15, giving £45 and £75. Rates of change often appear as speed, price per item, or unit cost.

Book 8C 中的比和比例题包括按给定比例分配数量、化简比，以及用比例来调整食谱或地图比例尺。例如，按 3:5 分配 £120，需要除以总共 8 份，所以一份是 £15，得到 £45 和 £75。变化率常以速度、单价或单位成本的形式出现。

Speed = distance ÷ time. Example: 150 km in 2 hours → 75 km/h

速度 = 路程 ÷ 时间。例如：2 小时行 150 km → 75 km/h

• Write ratios in their simplest form using whole numbers.
• Find the value of one part before sharing an amount.
• Label units clearly in rate problems, e.g. £/kg or m/s.

• 用整数写出最简形式的比。
• 分配数量前先求出一份的值。
• 在变化率问题中清楚标注单位，如 £/kg 或 m/s。

6. Geometry: Angles, Area and Volume | 几何：角度、面积与体积

Geometry in Book 8C covers angle rules on parallel lines, properties of triangles and quadrilaterals, and area and volume formulas. You may need to find missing angles using the fact that angles on a straight line add to 180° and angles around a point add to 360°. Area of a trapezium and volume of a prism are common exam topics.

Book 8C 的几何部分涵盖平行线上的角度规则、三角形和四边形的性质，以及面积和体积公式。你可能需要利用平角为 180°、周角为 360° 来求缺失的角。梯形面积和棱柱体积是常见的考试题目。

Area of trapezium = 1/2 × (a + b) × h

梯形面积 = 1/2 × (a + b) × h

• Give a reason for every angle fact you use, such as ‘alternate angles are equal’.
• Convert all measurements to the same unit before finding area or volume.
• Sketch a diagram if the question does not provide one.

• 每使用一个角度规则都要说明理由，例如“内错角相等”。
• 求面积或体积前，把所有测量值换算成相同单位。
• 如果题目没有配图，自己画一个草图。

7. Statistics: Averages and Charts | 统计：平均数与图表

Statistical work in Year 8 includes calculating the mean, median, mode and range, and drawing or interpreting bar charts, pie charts and line graphs. The mean is found by adding all values and dividing by the number of values. The median is the middle value when data are ordered.

Year 8 的统计内容包括计算平均数、中位数、众数和极差，以及绘制或解读条形图、饼图和折线图。平均数是将所有数值相加后除以数值的个数。中位数是将数据排序后位于中间的值。

Mean = sum of values ÷ number of values

平均数 = 所有数值之和 ÷ 数值个数

• Order the data before finding the median.
• Check that your pie chart sectors add up to 360°.
• Use the range to describe how spread out the data are.

• 求中位数前要先排序。
• 检查饼图的扇形角度总和是否为 360°。
• 用极差描述数据的分散程度。

8. Probability: Simple Events and Expected Outcomes | 概率：简单事件与期望结果

Probability in Book 8C involves writing probabilities as fractions, decimals or percentages, and using the probability scale from 0 to 1. A probability of 0 means impossible, and 1 means certain. You also need to calculate expected outcomes by multiplying the probability by the number of trials.

Book 8C 中的概率包括用分数、小数或百分数表示概率，并使用从 0 到 1 的概率标尺。概率为 0 表示不可能，1 表示必然发生。你还需要通过概率乘以试验次数来计算期望结果。

Expected frequency = probability × number of trials

期望频数 = 概率 × 试验次数

• Probabilities of all possible outcomes must add to 1.
• Simplify probability fractions where possible.
• An expected frequency is not a guarantee; it is an average over many trials.

• 所有可能结果的概率之和必须为 1。
• 概率分数要尽可能化简。
• 期望频数不是必然结果，而是多次试验中的平均值。

9. Common Mistakes to Avoid When Marking Your Work | 订正时的常见错误

Many students look at an answer, see that the final number matches, and move on. This misses valuable learning. Common errors include copying the answer without understanding, ignoring units, misreading the question, and making sign errors in algebra. When you mark your work, write a short note next to any mistake explaining what went wrong.

许多学生看到答案的最后数字与自己一致，就直接跳过。这样做会错失宝贵的学习机会。常见错误包括抄答案而不理解、忽略单位、误读题目，以及代数中的符号错误。订正时，在旁边写一句简短的说明，解释错在哪里。

10. Building a Revision Plan with Book 8C | 用 Book 8C 制定复习计划

A good revision plan uses Book 8C in short, focused sessions. Aim for three 30-minute blocks per week. In each block, complete one exercise, mark it with the answer key, and then redo any questions you got wrong. Keep a simple log of topics you find difficult so you can return to them before a test.

一个好的复习计划是短时间、集中地用 Book 8C 练习。目标是每周三次、每次 30 分钟。每次完成一个练习，用答案订正，然后重做做错的题目。简单记录你觉得困难的知识点，以便考试前再次复习。

• Week 1: Number and ratio topics.
• Week 2: Algebra and equations.
• Week 3: Geometry, statistics and probability.
• Final review: mixed questions from all chapters.

• 第 1 周：数字与比的知识。
• 第 2 周：代数与方程。
• 第 3 周：几何、统计和概率。
• 最终复习：所有章节的混合题。

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📚 Mastering Year 8 Maths: Essential Maths Book 8C Core Review | 掌握八年级数学：Essential Maths Book 8C 核心复习

Welcome to the TutorHao Year 8 Maths revision guide based on Essential Maths Book 8C. This article condenses the most important topics from the book into a clear, bilingual summary. You will revisit number skills, algebra, geometry, and statistics with worked examples and exam-style reminders.

欢迎阅读 TutorHao 根据《Essential Maths Book 8C》编写的八年级数学复习指南。本文将该书最重要的主题浓缩为清晰的中英双语总结。您将重新回顾数字技能、代数、几何和统计，并配有例题和考试提示。

1. Number Skills and Place Value | 数字技能与位值

In Book 8C, number work starts with place value, powers of 10 and ordering negative numbers. You must be able to read and write large numbers, identify the value of each digit, and use index notation such as 10² = 100.

在 Book 8C 中，数字部分从位值、10 的幂和负数排序开始。你必须能够读写大数、识别每个数字的值，并使用指数符号，例如 10² = 100。

10² = 100, 10³ = 1000, 10⁰ = 1

For negative numbers, remember that the further left a number is on the number line, the smaller it is. So −7 < −2 because −7 lies to the left of −2.

对于负数，请记住在数轴上越靠左的数字越小。因此 −7 < −2，因为 −7 位于 −2 的左侧。

When rounding to a given number of decimal places, look at the next digit. If it is 5 or more, round up. For example, 3.146 rounded to 2 decimal places is 3.15.

当四舍五入到指定小数位时，查看下一位数字。如果是 5 或更大，则向上进位。例如，3.146 四舍五入到两位小数是 3.15。

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

You need to convert fluently between fractions, decimals and percentages. Key facts include ½ = 0.5 = 50%, ¼ = 0.25 = 25% and ¾ = 0.75 = 75%.

你需要熟练地在分数、小数和百分比之间进行转换。关键事实包括 ½ = 0.5 = 50%、¼ = 0.25 = 25% 和 ¾ = 0.75 = 75%。

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

要加减分数，首先要找到公分母。例如，2/3 + 1/6 = 4/6 + 1/6 = 5/6。

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

To multiply fractions, multiply the numerators and multiply the denominators. To divide by a fraction, multiply by its reciprocal.

要乘以分数，将分子相乘并将分母相乘。要除以一个分数，乘以其倒数。

3. Ratio and Proportion | 比与比例

A ratio compares quantities. Always simplify a ratio by dividing all parts by their highest common factor. For example, 12:18 simplifies to 2:3.

比用于比较数量。始终通过将所有部分除以它们的最大公因数来简化比。例如，12:18 简化为 2:3。

To divide a quantity in a given ratio, first add the parts. For example, divide £60 in the ratio 2:3. The total parts are 2 + 3 = 5, so one part is £60 ÷ 5 = £12. The shares are 2 × £12 = £24 and 3 × £12 = £36.

要按给定比例分配数量，首先将各部分相加。例如，按 2:3 分配 60 英镑。总份数为 2 + 3 = 5，因此一份为 60 ÷ 5 = 12 英镑。份额分别为 2 × 12 = 24 英镑和 3 × 12 = 36 英镑。

2:3 → 2 + 3 = 5 parts → £60 ÷ 5 = £12 per part

Direct proportion means that as one quantity increases, the other increases at the same rate. Use the unitary method: if 5 pens cost £2, then one pen costs £2 ÷ 5 = £0.40, so 20 pens cost 20 × £0.40 = £8.

正比例意味着当一个量增加时，另一个量以相同的速率增加。使用单位法：如果 5 支笔花费 2 英镑，那么一支笔花费 2 ÷ 5 = 0.40 英镑，因此 20 支笔花费 20 × 0.40 = 8 英镑。

4. Algebraic Expressions and Formulae | 代数表达式与公式

In algebra, you collect like terms. Terms with the same letter combinations can be added or subtracted. For example, 3a + 2b − a + 5b = 2a + 7b.

在代数中，你要合并同类项。具有相同字母组合的项可以相加或相减。例如，3a + 2b − a + 5b = 2a + 7b。

3a + 2b − a + 5b = 2a + 7b

To expand brackets, multiply the term outside by every term inside. For example, 3(x + 2) = 3x + 6. To factorise, do the reverse: 4x + 8 = 4(x + 2).

要展开括号，将外面的项乘以里面的每一项。例如，3(x + 2) = 3x + 6。要因式分解，则反向操作：4x + 8 = 4(x + 2)。

Substitution means replacing letters with numbers. If a = 3 and b = 5, then 2a + b = 2 × 3 + 5 = 11.

代入意味着用数字替换字母。如果 a = 3 且 b = 5，则 2a + b = 2 × 3 + 5 = 11。

5. Solving Linear Equations | 解一元一次方程

To solve a linear equation, use the balance method: do the same operation to both sides. For example, solve 4x − 3 = 9. Add 3 to both sides to get 4x = 12, then divide both sides by 4 to get x = 3.

要解一元一次方程，使用平衡法：对方程两边进行相同的运算。例如，解 4x − 3 = 9。两边加 3 得到 4x = 12，然后两边除以 4 得到 x = 3。

4x − 3 = 9 → 4x = 12 → x = 3

If the equation contains brackets, expand them first. For example, 2(x − 5) = 6 becomes 2x − 10 = 6, so 2x = 16 and x = 8.

如果方程包含括号，请先展开它们。例如，2(x − 5) = 6 变为 2x − 10 = 6，因此 2x = 16，x = 8。

Always check your answer by substituting it back into the original equation.

始终通过将答案代回原方程来检验。

6. Sequences, Functions and Graphs | 数列、函数与图像

A sequence is a list of numbers following a rule. The nth term of a linear sequence is often written as an + b, where a is the common difference.

数列是按照规则排列的一组数字。线性数列的第 n 项通常写作 an + b，其中 a 是公差。

For example, the sequence 4, 7, 10, 13, … has a common difference of 3, so the nth term is 3n + 1. Check: when n = 1, 3 × 1 + 1 = 4.

例如，数列 4、7、10、13… 的公差为 3，因此第 n 项为 3n + 1。检验：当 n = 1 时，3 × 1 + 1 = 4。

Sequence: 4, 7, 10, 13 → nth term = 3n + 1

Coordinates are written as (x, y). To plot a linear graph such as y = 2x + 1, choose x-values, calculate y-values, and plot the points. The points lie on a straight line.

坐标写作 (x, y)。要绘制线性图像，例如 y = 2x + 1，选择 x 值，计算 y 值，然后描点。这些点位于一条直线上。

7. Angles and Polygons | 角与多边形

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°，围绕一个点的角之和为 360°。对顶角相等。

The interior angles of a triangle add up to 180°, and the interior angles of a quadrilateral add up to 360°.

三角形的内角之和为 180°，四边形的内角之和为 360°。

For any polygon with n sides, the sum of interior angles is (n − 2) × 180°. For example, a hexagon has 6 sides, so the sum is (6 − 2) × 180° = 720°.

对于任何有 n 条边的多边形，内角之和为 (n − 2) × 180°。例如，六边形有 6 条边，因此内角之和为 (6 − 2) × 180° = 720°。

Interior angle sum = (n − 2) × 180°

The exterior angles of any polygon add up to 360°. In a regular polygon, each exterior angle equals 360° ÷ n.

任何多边形的外角之和为 360°。在正多边形中，每个外角等于 360° ÷ n。

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

Perimeter is the distance around a shape. For a rectangle, perimeter P = 2(l + w), where l is length and w is width.

周长是围绕一个形状的距离。对于矩形，周长 P = 2(l + w)，其中 l 是长，w 是宽。

Area formulas you must know: rectangle A = l × w, triangle A = ½ × base × height, parallelogram A = base × height, trapezium A = ½ × (a + b) × h, where a and b are the parallel sides.

你必须掌握的面积公式：矩形 A = 长 × 宽，三角形 A = ½ × 底 × 高，平行四边形 A = 底 × 高，梯形 A = ½ × (a + b) × h，其中 a 和 b 是平行边。

A(triangle) = ½ × b × h, A(trapezium) = ½ × (a + b) × h

Volume measures space inside a 3D shape. For a cuboid, volume V = length × width × height. Compound shapes can be split into rectangles or known shapes.

体积测量三维形状内部的空间。对于长方体，体积 V = 长 × 宽 × 高。组合图形可以分割成矩形或已知形状。

9. Transformations and Symmetry | 变换与对称

Transformations include reflection, rotation, translation and enlargement. Always describe a transformation fully: for rotation give the centre, angle and direction; for enlargement give the centre and scale factor.

变换包括反射、旋转、平移和放大。始终完整描述变换：旋转需给出中心、角度和方向；放大需给出中心和比例因子。

A reflection flips a shape over a mirror line. A rotation turns a shape about a fixed point. A translation slides a shape by a vector, and an enlargement changes its size.

反射使图形沿镜像线翻转。旋转使图形绕固定点转动。平移使图形按向量滑动，放大则改变其大小。

A shape has line symmetry if it can be folded so both halves match. Rotational symmetry is the number of times a shape fits onto itself in one full turn.

如果一个图形可以折叠使两半重合，则它具有线对称。旋转对称是图形在完整旋转一周中与自身重合的次数。

10. Statistics and Probability | 统计与概率

For a data set, the mean is the sum of values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value, and the range is the largest value minus the smallest.

对于一组数据，平均数等于数值之和除以数值个数。中位数是数据排序后的中间值。众数是出现最频繁的值，极差是最大值减最小值。

Mean = sum of values ÷ number of values

Probability measures how likely an event is. It always lies between 0 (impossible) and 1 (certain). Probability = number of favourable outcomes ÷ total number of outcomes.

概率衡量事件发生的可能性。它始终介于 0（不可能）和 1（必然）之间。概率 = 有利结果数 ÷ 总结果数。

For example, if a bag has 3 red and 5 blue marbles, the probability of picking a red marble is 3/8.

例如，如果一个袋子里有 3 个红色和 5 个蓝色弹珠，则取出红色弹珠的概率是 3/8。

P(red) = 3/8 = 0.375 = 37.5%

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📚 Year 8 Essential Maths Book 8: Core Topics | Year 8 数学核心考点精讲

This article is a focused revision guide for Year 8 students using Essential Maths Book 8. It summarises the most important topics in Number, Algebra, Geometry, Statistics and Probability, with worked examples and key formulae.

本文是使用《Essential Maths Book 8》的 Year 8 学生专用复习指南，总结数字、代数、几何、统计与概率中最核心的知识点，并提供例题与关键公式。

1. Number Sense and Place Value | 数感与位值

Year 8 students need to be confident with large numbers and decimals. Place value tells us the value of each digit based on its position. For example, in 3 456.78, the digit 3 means 3 thousands, 4 means 4 hundreds, 5 means 5 tens, 6 means 6 ones, 7 means 7 tenths and 8 means 8 hundredths.

Year 8 学生需要熟练处理大数与小数。位值表示每个数字根据其位置所代表的值。例如在 3 456.78 中，数字 3 表示 3 个千，4 表示 4 个百，5 表示 5 个十，6 表示 6 个一，7 表示 7 个十分之一，8 表示 8 个百分之一。

Rounding and estimation are key skills. To round 27 486 to the nearest thousand, look at the hundreds digit (4): since it is less than 5, round down to 27 000.

四舍五入与估算也是关键技能。将 27 486 四舍五入到最接近的千位时，看百位数字（4）：因为小于 5，因此向下舍入到 27 000。

Standard form is introduced for very large or small numbers. For example, 56 000 = 5.6 × 10⁴. This makes comparing numbers easier.

对非常大或非常小的数引入科学计数法。例如 56 000 = 5.6 × 10⁴。这样可以更容易比较数字的大小。

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

Converting between fractions, decimals and percentages is essential. A fraction like 3/4 means 3 ÷ 4 = 0.75, which is 75%. To convert a percentage to a fraction, write it over 100 and simplify: 40% = 40/100 = 2/5.

分数、小数与百分比之间的转换是基本技能。分数 3/4 表示 3 ÷ 4 = 0.75，即 75%。把百分比转换为分数时，先写成百分之几并化简：40% = 40/100 = 2/5。

When adding fractions, find a common denominator. For 2/3 + 1/4, the common denominator is 12: 2/3 = 8/12 and 1/4 = 3/12, so 8/12 + 3/12 = 11/12.

分数相加时，先找到公分母。例如 2/3 + 1/4，公分母为 12：2/3 = 8/12，1/4 = 3/12，因此 8/12 + 3/12 = 11/12。

To multiply fractions, multiply numerators and denominators separately: 2/5 × 3/7 = 6/35. To divide by a fraction, multiply by its reciprocal: 2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15.

分数相乘时，分子与分母分别相乘：2/5 × 3/7 = 6/35。除以一个分数等于乘以它的倒数：2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15。

3. Negative Numbers and Order of Operations | 负数与运算顺序

Negative numbers appear in temperature, bank balances and coordinates. Adding a negative is the same as subtracting: 5 + (−3) = 5 − 3 = 2. Subtracting a negative is the same as adding: 5 − (−3) = 5 + 3 = 8.

负数出现在温度、银行余额和坐标中。加上一个负数等于减去它的绝对值：5 + (−3) = 5 − 3 = 2。减去一个负数等于加上它的绝对值：5 − (−3) = 5 + 3 = 8。

When multiplying or dividing two negative numbers, the result is positive: (−4) × (−6) = 24 and (−12) ÷ (−3) = 4. When signs are different, the result is negative: (−4) × 6 = −24.

两个负数相乘或相除，结果为正：(−4) × (−6) = 24，(−12) ÷ (−3) = 4。当符号不同时，结果为负：(−4) × 6 = −24。

For mixed calculations, follow BIDMAS/BODMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction. In 2 +

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📚 Essential Maths Book 8i Compressed Core Revision | 《基础数学 8i》压缩核心复习

Welcome to this compressed revision guide for Essential Maths Book 8i. It brings together the key skills that Year 8 students need to master, from number operations and fractions to algebra, geometry, statistics and probability. Each section gives clear rules, worked examples and common error alerts so you can revise efficiently.

欢迎使用《基础数学 8i》压缩复习指南。本指南汇集了 Year 8 学生必须掌握的核心技能，包括数的运算、分数、代数、几何、统计与概率。每个专题都给出清晰规则、例题和常见错误提醒，帮助你高效复习。

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

The order of operations is remembered as BIDMAS or BODMAS: Brackets, Indices (or Orders), Division and Multiplication from left to right, then Addition and Subtraction from left to right. For example, 3 + 4 × 2 = 3 + 8 = 11, not 14.

运算顺序可用 BIDMAS 或 BODMAS 记忆：先算括号，再算指数（或乘方），然后从左到右计算乘除，最后从左到右计算加减。例如 3 + 4 × 2 = 3 + 8 = 11，而不是 14。

With negative numbers, adding a negative is the same as subtracting: 5 + (−3) = 2. Multiplying or dividing two negatives gives a positive, while a negative and a positive gives a negative. Example: (−2) × (−6) = 12 but (−2) × 6 = −12.

涉及负数时，加上一个负数等于减去它的相反数：5 + (−3) = 2。两个负数相乘或相除得正，一负一正相乘或相除得负。例如 (−2) × (−6) = 12，而 (−2) × 6 = −12。

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

To add or subtract fractions, rewrite them with a common denominator. For example, 3/4 + 1/6 = 9/12 + 2/12 = 11/12. Never add denominators: 1/2 + 1/3 is not 2/5.

分数加减时，必须先化为同分母。例如 3/4 + 1/6 = 9/12 + 2/12 = 11/12。不能直接把分子和分母分别相加：1/2 + 1/3 不等于 2/5。

To multiply fractions, multiply the numerators and multiply the denominators. To divide, multiply by the reciprocal. For example, 2/3 × 5/7 = 10/21, and 2/3 ÷ 5/7 = 2/3 × 7/5 = 14/15.

分数相乘时，分子乘分子，分母乘分母；分数相除时，乘以除数的倒数。例如 2/3 × 5/7 = 10/21，2/3 ÷ 5/7 = 2/3 × 7/5 = 14/15。

A percentage is a fraction out of 100. To convert a fraction to a percentage, multiply by 100. For instance, 3/5 = 3 ÷ 5 × 100 = 60%.

百分数是分母为 100 的分数。将分数化为百分数，可先除以分母再乘以 100。例如 3/5 = 3 ÷ 5 × 100 = 60%。

3. Ratio and Proportion | 比与比例

A ratio compares quantities in the same unit. Simplify a ratio by dividing all parts by their highest common factor. For example, 12:18 simplifies to 2:3 by dividing by 6.

比用于比较同类量。化简比时，把各项同时除以最大公因数。例如 12:18 同时除以 6，化简为 2:3。

To share an amount in a given ratio, first find the total number of parts. For 2:3, there are 5 parts. If £60 is shared in ratio 2:3, one part is £60 ÷ 5 = £12, so the shares are £24 and £36.

按比例分配时，先求总份数。例如 2:3 共有 5 份。把 60 英镑按 2:3 分配，一份是 60 ÷ 5 = 12 英镑，因此两份为 24 英镑，三份为 36 英镑。

A proportion is an equation stating that two ratios are equal. Use cross multiplication to solve missing values, such as 3/4 = x/20 gives x = 15.

比例是一个等式，表示两个比相等。解未知项时可用交叉相乘。例如 3/4 = x/20，得 x = 15。

4. Algebraic Expressions and Equations | 代数表达式与方程

Like terms have exactly the same letter part. Combine them by adding or subtracting the coefficients. Example: 3x + 2y − x + 4y = 2x + 6y. Expanding brackets means multiplying each term inside: 3(x + 2) = 3x + 6.

同类项是指字母部分完全相同的项，合并时只加减系数。例如 3x + 2y − x + 4y = 2x + 6y。去括号就是把括号外的数乘给括号内每一项：3(x + 2) = 3x + 6。

An equation has an equals sign. Keep it balanced by doing the same operation to both sides. Solve 2x + 3 = 11 by subtracting 3, then dividing by 2: x = 4.

方程含有等号，解方程时要保持两边平衡，即两边同时进行相同运算。解 2x + 3 = 11，先两边减 3，再两边除以 2，得 x = 4。

Check your solution by substituting the value back into the original equation: 2(4) + 3 = 11, so the answer is correct.

得到解后，应把数值代回原方程检验：2(4) + 3 = 11，等式成立，说明答案正确。

5. Linear Graphs and Sequences | 线性图像与数列

A coordinate pair (x, y) is plotted from the origin. To draw a linear graph such as y = 2x + 1, choose x values, calculate y, then plot points and join them with a straight line.

坐标 (x, y) 从原点出发读取。画 y = 2x + 1 这样的线性图像时，先选若干个 x 值，算出对应的 y 值，再描点并用直线连接。

The y-intercept is the value of y when x = 0, and the gradient is the coefficient of x. In y = 2x + 1, the gradient is 2 and the y-intercept is 1.

y 轴截距是 x = 0 时的 y 值，

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📚 Essential Maths Book 8S Answers: Year 8 Core Solutions and Revision | Year 8 数学 Essential Maths Book 8S 答案解析与复习指南

Essential Maths Book 8S is a widely used Year 8 workbook that builds fluency in number, algebra, geometry, and data handling. This article summarises the main answer types, shows worked examples, and explains how to use the compressed answer set for effective revision without copying blindly.

《Essential Maths Book 8S》是 Year 8 常用的数学练习册，帮助学生在数、代数、几何和数据处理方面打牢基础。本文总结主要答案类型，给出解题范例，并说明如何有效利用精简版答案进行复习，而不是盲目抄写。

1. Understanding the Book 8S Structure | 了解 Book 8S 结构

Book 8S is organised into short exercise sets covering Number, Algebra, Shape and Space, and Handling Data. Each exercise usually begins with a worked example, then practice questions, and a summary box at the end. The answer file lists final answers in the same order, with diagrams marked clearly.

本书按短练习编排，涵盖数、代数、图形与空间、数据处理。每个练习开头通常有例题，接着是练习题，末尾有小结框。答案文件一般按相同顺序列出最终答案，图形标注清楚。

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

Year 8 integer questions often mix negative numbers with brackets and powers. The key rule is BIDMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. For example, 3 + 4 × (−2) = 3 + (−8) = −5, not −14.

Year 8 整数题常把负数与括号、乘方混合。关键规则是 BIDMAS：括号、指数、乘除、加减。例如 3 + 4 × (−2) = 3 + (−8) = −5，而不是 −14。

3 + 4 × (−2) = 3 − 8 = −5

When squaring negative numbers, (−3)² = 9 but −3² = −9 because the negative sign is not inside the bracket.

乘方时要注意，(−3)² = 9，但 −3² = −9，因为负号不在括号内。

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

Book 8S expects students to convert freely between 1/4, 0.25 and 25%, and to add or subtract fractions with different denominators. To add 2/3 and 1/4, find the LCM of 3 and 4, which is 12: 8/12 + 3/12 = 11/12.

本书要求学生能在 1/4、0.25 和 25% 之间自由转换，并能对异分母分数进行加减。计算 2/3 + 1/4 时，先求 3 和 4 的最小公倍数 12：8/12 + 3/12 = 11/12。

2/3 + 1/4 = 8/12 + 3/12 = 11/12

Percentage increase and decrease questions often appear in the answer set. A 15% increase on £60 is found by multiplying 60 × 1.15 = 69.

百分数增减题在答案集中很常见。£60 增加 15%，用 60 × 1.15 = 69 计算。

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

Simplifying expressions such as 5a + 3b − 2a + 7b gives 3a + 10b. The answer file shows only the final simplified form, so you must check that like terms are collected correctly.

化简如 5a + 3b − 2a + 7b 得到 3a + 10b。答案文件只显示最终化简结果，因此你要检查同类项是否合并正确。

5a + 3b − 2a + 7b = 3a + 10b

Substitution questions require replacing letters with given values. If a = 3 and b = −2, then 4a + 5b = 4×3 + 5×(−2) = 12 − 10 = 2.

代入题要求把字母替换为给定数值。如果 a = 3，b = −2，那么 4a + 5b = 4×3 + 5×(−2) = 12 − 10 = 2。

4a + 5b = 12 − 10 = 2

5. Linear Equations | 一元一次方程

Solving equations like 3x + 4 = 19 follows inverse operations: subtract 4 from both sides, then divide by 3. The Book 8S answer set gives x = 5, but you need to show both steps to gain marks in class.

解如 3x + 4 = 19 的方程需要逆运算：两边先减 4，再除以 3。Book 8S 答案集给出 x = 5，但在课堂上要展示两个步骤才能得分。

3x + 4 = 19 → 3x = 15 → x = 5

Equations with variables on both sides, such as 5x − 2 = 2x + 7, require collecting x-terms first: 3x = 9, so x = 3.

两边都含未知数的方程如 5x − 2 = 2x + 7，要先合并 x 项：3x = 9，所以 x = 3。

6. Ratio and Proportion | 比与比例

Ratio questions often ask you to share an amount in a given ratio. To divide £72 in the ratio 2:3:4, the total number of parts is 9, so each part is £8. The shares are £16, £24, and £32.

比例题常要求按给定比例分配数量。把 £72 按 2:3:4 分配，总份数是 9，每份 £8，因此份额为 £16、£24 和 £32。

72 ÷ (2+3+4) = 8; 2×8 = 16, 3×8 = 24, 4×8 = 32

Direct proportion can be solved by finding the unit rate. If 5 pens cost £3.50, one pen costs £0.70, so 8 pens cost £5.60.

正比例可通过先求单位量解决。如果 5 支笔 £3.50，一支 £0.70，那么 8 支 £5.60。

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

Book 8S geometry answers cover angles on a straight line, around a point, and in triangles. If two angles in a triangle are 65° and 47°, the third angle is 180° − 65° − 47° = 68°.

Book 8S 几何答案涉及平角、周角和三角形内角。如果三角形两个角分别是 65° 和 47°，第三个角是 180° − 65° − 47° = 68°。

180° − 65° − 47° = 68°

Alternate and corresponding angles are tested with parallel lines. When a transversal crosses parallel lines, corresponding angles are equal, and alternate interior angles are equal.

平行线中的同位角和内错角是考点。当一条截线穿过平行线时，同位角相等，内错角也相等。

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

Area formulas for rectangles, triangles, and parallelograms appear frequently. The area of a triangle with base 8 cm and height 5 cm is (8 × 5) ÷ 2 = 20 cm².

矩形、三角形和平行四边形的面积公式常考。底为 8 cm、高为 5 cm 的三角形面积是 (8 × 5) ÷ 2 = 20 cm²。

A = ½ × 8 × 5 = 20 cm²

Volume of a cuboid is length × width × height. A box 4 cm by 3 cm by 5 cm has volume 60 cm³, and surface area 2(4×3 + 4×5 + 3×5) = 94 cm².

长方体体积 = 长 × 宽 × 高。一个 4 cm × 3 cm × 5 cm 的盒子体积为 60 cm³，表面积为 2(4×3 + 4×5 + 3×5) = 94 cm²。

9. Coordinates and Graphs | 坐标与图像

Plotting coordinates in all four quadrants is a core Year 8 skill. A point (−3, 4) lies in the second quadrant because x is negative and y is positive. The answer set labels axes and scale clearly.

在四个象限中绘制坐标是 Year 8 的核心技能。点 (−3, 4) 位于第二象限，因为 x 为负、y 为正。答案集清楚标注坐标轴和刻度。

Point (−3, 4): quadrant II

Straight-line graphs from equations like y = 2x + 1 are found by substituting x-values. For x = 0, 1, 2, the points are (0,1), (1,3), (2,5), which lie on a straight line.

像 y = 2x + 1 这样的直线图可通过代入 x 值求得。当 x = 0、1、2 时，点为 (0,1)、(1,3)、(2,5)，它们在同一直线上。

10. Statistics and Probability | 统计与概率

The mean, median, mode, and range are calculated from small data sets. For 3, 7, 7, 9, 12: mean = (3+7+7+9+12) ÷ 5 = 7.6, median = 7, mode = 7, range = 9.

平均数、中位数、众数和极差从小数据集中

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📚 Essential Maths Book 8i Answers | 《核心数学8i》答案解析与复习指南

This article provides a structured set of worked answers and revision notes for the main topics in Essential Maths Book 8i. It is designed to help Year 8 students check their working, understand common mistakes, and build confidence for class tests and end-of-year assessments.

本文为《Essential Maths Book 8i》的主要章节提供系统的解题答案与复习要点，帮助八年级学生检查计算过程、理解常见错误，并为课堂测验和年终考试建立信心。

1. Number and BIDMAS | 数与运算顺序

Always follow the order of operations: brackets, powers (or indices), multiplication and division from left to right, then addition and subtraction from left to right. The acronym BIDMAS is a useful checklist.

务必遵循运算顺序：先括号，再幂（指数），然后从左到右进行乘除，最后从左到右进行加减。助记词 BIDMAS 是一个有用的检查表。

Worked example: 5 + 2² × 3. The power comes first: 2² = 4. Next multiply: 4 × 3 = 12. Finally add: 5 + 12 = 17.

示例：5 + 2² × 3。先算幂：2² = 4。再算乘法：4 × 3 = 12。最后相加：5 + 12 = 17。

Common mistake: working from left to right as 5 + 2 = 7, then 7² = 49, then 49 × 3 = 147 is wrong, because addition must wait until multiplication is complete.

常见错误：从左到右先算 5 + 2 = 7，再平方得 49，再乘以 3 得 147 是错误的，因为加法必须等乘法完成后再进行。

2. Negative Numbers and Powers | 负数与幂

For addition and subtraction, think of a number line. Adding a negative means moving left, so 4 + (−7) = −3. Subtracting a negative means moving right, so 5 − (−2) = 7.

加减负数时，可以想象一条数轴。加上负数相当于向左移动，所以 4 + (−7) = −3。减去负数相当于向右移动，所以 5 − (−2) = 7。

For multiplication and division: same signs give a positive answer, different signs give a negative answer. Example: (−6) × 3 = −18; (−12) ÷ (−4) = 3.

乘法和除法规则：同号得正，异号得负。例如 (−6) × 3 = −18；(−12) ÷ (−4) = 3。

With powers, only the base directly attached to the index is raised. So (−2)² = (−2) × (−2) = 4, but −2² = −(2²) = −4. This distinction is very common in Book 8i exercises.

幂运算中，只有紧靠指数的底数才被乘方。因此 (−2)² = (−2) × (−2) = 4，而 −2² = −(2²) = −4。这个区别在 Book 8i 练习中非常常见。

3. Fractions and Mixed Numbers | 分数与带分数

To add or subtract fractions, first find a common denominator. For 3/4 + 1/6, the lowest common denominator is 12, so 3/4 = 9/12, 1/6 = 2/12, and the sum is 11/12.

分数的加减法要先找到公分母。例如 3/4 + 1/6，最小公分母是 12，因此 3/4 = 9/12，1/6 = 2/12，和为 11/12。

To multiply fractions, multiply the numerators and multiply the denominators. For 2/5 × 3/7, the answer is (2×3)/(5×7) = 6/35. To divide, multiply by the reciprocal: 2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15.

分数相乘时，分子乘分子，分母乘分母。例如 2/5 × 3/7 = (2×3)/(5×7) = 6/35。分数相除时，乘以倒数：2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15。

Always simplify your final answer. Example: 6/8 = 3/4; 15/25 = 3/5.

最终答案一定要化简。例如 6/8 = 3/4；15/25 = 3/5。

4. Percentages and Conversions | 百分数与换算

To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100. For 3/8, 3 ÷ 8 = 0.375, so the percentage is 37.5%.

将分数转换为百分数，用分子除以分母再乘以 100。例如 3/8，3 ÷ 8 = 0.375，所以百分数为 37.5%。

To find a percentage of a quantity, convert the percentage to a decimal or fraction. Example: 15% of 80 = 0.15 × 80 = 12. To increase 80 by 15%, find 12 and add: 80 + 12 = 92.

求一个数量的百分数，先将百分数转换为小数或分数。例如 80 的 15% = 0.15 × 80 = 12。若要在 80 上增加 15%，先求出 12，再加：80 + 12 = 92。

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

Simplify expressions by collecting like terms. Example: 3a + 5b − 2a + 4b = (3a − 2a) + (5b + 4b) = a + 9b.

通过合并同类项来化简代数式。例如 3a + 5b − 2a + 4b = (3a − 2a) + (5b + 4b) = a + 9b。

When expanding a single bracket, multiply the term outside by each term inside. Example: 4(2x − 3) = 8x − 12.

展开单项括号时，将括号外的项分别乘以括号内的每一项。例如 4(2x − 3) = 8x −

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📚 Essential Maths Book 8 Support (Feb 21): Building Confidence in Year 8 Maths | 八年级数学基础支持（2021年2月版）：建立数学信心

This revision guide walks through the core skills in the Essential Maths Book 8 Support edition (February 2021). It is written for Year 8 learners who need clear steps, simple examples and regular practice to strengthen their mathematical understanding.

本复习指南带你梳理《Essential Maths Book 8 Support》（2021年2月版）中的核心技能。它专为需要清晰步骤、简单示例和定期练习来夯实数学基础的八年级学生编写。

1. Understanding Place Value and Integers | 理解位值与整数

Whole numbers are made from digits placed in columns: units, tens, hundreds, thousands and beyond. In Year 8 support work, you must be able to read, write, order and compare large numbers fluently, such as 340 715 or 1 208 046.

整数由位于不同数位的数字组成：个位、十位、百位、千位等。在八年级基础练习中，你需要能够熟练地读、写、排列和比较大数，例如 340 715 或 1 208 046。

For example, in 340 715 the digit 4 has a value of 40 000 because it is in the ten-thousands column. The digit 7 has a value of 700, because it is in the hundreds column. Writing the number in a place-value table helps prevent mistakes with zeros.

例如，在 340 715 中，数字 4 的值是 40 000，因为它位于万位。数字 7 的值是 700，因为它位于百位。把数字写进数位表有助于避免零带来的错误。

Integers include positive and negative whole numbers. We often use a number line to compare them. Remember that on a horizontal number line, numbers increase to the right, so -3 < 2 and 5 > -1.

整数包括正整数和负整数。我们常用数轴来比较它们。记住，在水平数轴上，数字向右逐渐增大，所以 -3 < 2，而 5 > -1。

2. Order of Operations and Directed Numbers | 运算顺序与有向数

The order of operations helps us calculate correctly when a problem has more than one operation. In the UK, many students remember this as BODMAS or BIDMAS: Brackets, Orders (or Indices), Division and Multiplication, Addition and Subtraction.

当一个算式包含多个运算时，运算顺序能帮助我们正确地计算。在英国，许多学生用 BODMAS 或 BIDMAS 来记忆：括号、幂（或指数）、除法和乘法、加法和减法。

Division and multiplication have equal priority, so we work from left to right. The same is true for addition and subtraction. This rule prevents different people getting different answers to the same question.

除法和乘法具有相同的优先级，因此我们按从左到右的顺序计算。加法和减法也是如此。这条规则可以避免不同的人对同一个算式得出不同的答案。

For example, to evaluate 3 + 4 × 5, we multiply first: 4 × 5 = 20, then add 3 to get 23. If we incorrectly added first, we would get 3 + 4 = 7 and then 7 × 5 = 35, which is wrong.

例如，计算 3 + 4 × 5 时，我们先算乘法：4 × 5 = 20，再加 3 得到 23。如果错误地先算加法，就会得到 3 + 4 = 7，再算 7 × 5 = 35，这是错误的。

Directed numbers include positive and negative values. Adding a negative number is the same as subtracting its positive value: 6 + (-2) = 6 – 2 = 4. Subtracting a negative number is the same as adding: 5 – (-3) = 5 + 3 = 8.

有向数包括正数和负数。加上一个负数等于减去它的正数：6 + (-2) = 6 – 2 = 4。减去一个负数等于加上它的正数：5 – (-3) = 5 + 3 = 8。

3. Fractions: Simplifying and Calculating | 分数：化简与运算

Equivalent fractions represent the same part of a whole even though they look different. To simplify a fraction, divide the numerator and denominator by their highest common factor. For example, 12/18 simplifies to 2/3 because both 12 and 18 can be divided by 6.

等值分数表示相同的部分，但它们看起来不同。化简分数时，把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，因为 12 和 18 都可以除以 6。

To add or subtract fractions, they must have the same denominator. If the denominators are different, find an equivalent fraction for each one first. For example, 1/4 + 2/4 = 3/4, but 1/3 + 1/4 needs a common denominator of 12: 4/12 + 3/12 = 7/12.

分数相加或相减时，它们必须有相同的分母。如果分母不同，先为每个分数找出等值分数。例如，1/4 + 2/4 = 3/4，但 1/3 + 1/4 需要公分母 12：4/12 + 3/12 = 7/12。

To multiply fractions, multiply the numerators together and multiply the denominators together. To divide by a fraction, multiply by its reciprocal. For example, 2/3 × 3/5 = 6/15 = 2/5, and 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2.

分数相乘时，分子乘分子，分母乘分母。除以一个分数时，乘以它的倒数。例如，2/3 × 3/5 = 6/15 = 2/5，而 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2。

Fraction multiplication: a/b × c/d = (a × c) / (b × d)

分数乘法：a/b × c/d = (a × c) / (b × d)

4. Decimals and Percentages | 小数与百分数

Decimal numbers extend place value into tenths, hundredths and thousandths. It is useful to know the value of each decimal digit. For example, in 3.476, the digit 4 is in the tenths column, the 7 is in the hundredths column and the 6 is in the thousandths column.

小数把位值延伸到十分位、百分位和千分位。了解每个小数位的值很有用。例如，在 3.476 中，数字 4 在十分位，数字 7 在百分位，数字 6 在千分位。

Rounding decimals is a key skill. To round 3.476 to two decimal places, look at the third decimal digit. Since it is 6, we round up, so 3.476 rounds to 3.48.

小数取整是一项关键技能。把 3.476 保留两位小数时，看第三位小数。由于它是 6，我们要进位，所以 3.476 约等于 3.48。

Fractions, decimals and percentages are three ways of expressing the same amount. You should know common conversions by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/10 = 0.1 = 10% and 1/5 = 0.2 = 20%.

分数、小数和百分数是表示相同数量的三种方式。你应该熟记常见换算：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/10 = 0.1 = 10%，1/5 = 0.2 = 20%。

To find a percentage of an amount, convert the percentage to a fraction or decimal and multiply. For example, 20% of 60 = 0.2 × 60 = 12. To increase an amount by a percentage, work out the increase and then add it to the original amount.

求一个数的百分之几时，把百分数转换为分数或小数再相乘。例如，60 的 20% = 0.2 × 60 = 12。要把一个数增加某个百分比，先算出增加量，再加到原来的数上。

5. Ratio and Proportion | 比与比例

A ratio compares two or more quantities. Ratios can be simplified by dividing each part by the same number. For example, 8:12 simplifies to 2:3 because both parts can be divided by 4.

比用于比较两个或多个数量。比可以通过把每一部分都除以同一个数来化简。例如，8:12 可以化简为 2:3，因为两部分都可以除以 4。

When sharing an amount in a ratio, first find the total number of parts. Then divide the amount by the total number of parts to find the value of one part. Finally multiply by the number of parts for each share.

按比例分配一个数量时，先求总份数。然后用总量除以总份数，得到一份的值。最后乘以每部分对应的份数。

For example, to share £60 in the ratio 2:3, the total number of parts is 2 + 3 = 5. One part is £60 ÷ 5 = £12. The two shares are 2 × £12 = £24 and 3 × £12 = £36.

例如，按 2:3 的比例分配 60 英镑，总份数是 2 + 3 = 5。每份是 £60 ÷ 5 = £12。两部分分别是 2 × £12 = £24 和 3 × £12 = £36。

Proportion is used when two quantities change in the same ratio. If 3 pens cost £1.50, then 6 pens cost double, which is £3.00. Finding the unit rate first is often the easiest strategy.

当两个量以相同比例变化时，就会用到比例。如果 3 支笔售价 £1.50，那么 6 支笔的价格是双倍，即 £3.00。先求出单位价格通常是最简单的策略。

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

Algebra uses letters to represent unknown numbers or changing quantities. An expression such as 3a + 2b combines numbers and letters using operations. You cannot solve an expression, but you can simplify or evaluate it.

代数用字母表示未知数或变化的量。像 3a + 2b 这样的表达式用运算把数字和字母组合在一起。表达式无法求解，但可以化简或求值。

Like terms have exactly the same letter part. Only like terms can be collected together. For example, 3a + 2a = 5a, but 3a + 2b cannot be simplified because the letters are different.

同类项具有完全相同的字母部分。只有同类项可以合并。例如，3a + 2a = 5a，但 3a + 2b 不能化简，因为字母不同。

Substitution means replacing a letter with a number and then calculating the result. If x = 4, then the value of 3x + 2 is 3 × 4 + 2 = 12 + 2 = 14.

代入是指用数字替换字母，然后计算结果。如果 x = 4，那么 3x + 2 的值就是 3 × 4 + 2 = 12 + 2 = 14。

When working with expressions, use brackets carefully. If y = 3, then 2(y + 5) = 2 × (3 + 5) = 2 × 8 = 16. Always show each step clearly in your working.

处理表达式时要小心使用括号。如果 y = 3，那么 2(y + 5) = 2 × (3 + 5) = 2 × 8 = 16。解题时每一步都要写清楚。

7. Solving Linear Equations | 解线性方程

An equation shows that two expressions are equal. To solve an equation, we find the value of the unknown letter that makes the equation true. Keep the equation balanced by doing the same operation to both sides.

方程表示两个表达式相等。解方程就是求出使方程成立的未知字母的值。通过对方程两边进行相同的运算来保持等式平衡。

Use inverse operations to isolate the letter. If the equation is x + 5 = 12, subtract 5 from both sides: x + 5 – 5 = 12 – 5, so x = 7. If the equation is 3x = 18, divide both sides by 3 to get x = 6.

使用逆运算把字母单独留在一边。如果方程是 x + 5 = 12，两边都减去 5：x + 5 – 5 = 12 – 5，所以 x = 7。如果方程是 3x = 18，两边都除以 3，得到 x = 6。

For equations with the letter on both sides, collect the letter terms on one side and the number terms on the other. For example, 2x + 1 = x + 6. Subtract x from both sides to get x + 1 = 6, then subtract 1 from both sides to get x = 5.

对于字母在等号两边的方程，把含字母的项移到一边，数字项移到另一边。例如，2x + 1 = x + 6。两边都减去 x，得到 x + 1 = 6，再两边都减去 1，得到 x = 5。

Balanced equation: if