📚 Year 8 Essential Maths Book 8H Answers: Compressed Revision Guide | Year 8 数学 Essential Maths Book 8H 答案：压缩复习指南
This compressed revision guide helps Year 8 students use the Essential Maths Book 8H answers in a smarter way. Instead of copying final answers, you will learn how to check working, spot common errors, and revise the key skills that appear most often in higher-tier Year 8 mathematics.
这份压缩复习指南帮助 Year 8 学生更聪明地使用 Essential Maths Book 8H 的答案。与其直接抄写最终答案，你将学会如何检查解题过程、发现常见错误，并复习 Year 8 高阶数学中最常出现的关键技能。
1. What '8H' Means and How to Use Answers | “8H”代表什么以及如何高效使用答案
The 'H' in Book 8H stands for Higher. This means the book is written for students who are working above the expected Year 8 standard. The questions often involve multi-step reasoning, algebra with fractions, and harder geometry problems.
Book 8H 中的 “H” 代表 Higher，即高阶。这意味着该书是为高于 Year 8 预期水平的学生编写的。题目通常涉及多步推理、含分数的代数以及更难的几何问题。
You should always attempt a question fully before looking at the answer. When you check, compare your method as well as your final result. If only the final answer is wrong, re-trace the last two lines of working first.
你应该在查看答案之前先完整尝试一道题。检查时，既要比较最终结果，也要比较解题方法。如果只有最终答案错误，请先重新检查最后两行解题过程。
Use the answers as a self-marking tool, not as a shortcut. A good routine is: attempt, mark, correct in green pen, then redo one similar question without help.
把答案当作自我批改工具，而不是走捷径。一个良好的流程是：先尝试、再批改、用绿笔订正，然后不看答案重新做一道类似题目。
2. Number Skills: Fractions, Decimals and Percentages | 数字技能：分数、小数和百分比
Year 8 higher number work often asks you to convert fluently between fractions, decimals and percentages without a calculator. Mastery of these conversions saves time in ratio, probability and algebra questions.
Year 8 高阶数字部分经常要求你在不使用计算器的情况下，熟练地在分数、小数和百分比之间进行转换。掌握这些转换可以节省在比、概率和代数题目中的时间。
Example: Convert 3/8 to a decimal and then to a percentage.
例题：将 3/8 转换为小数，再转换为百分比。
3 ÷ 8 = 0.375 → 0.375 × 100 = 37.5%
Another common skill is finding a percentage increase or decrease. For example, a coat costing £64 is reduced by 15%. Find the new price.
另一个常见技能是求百分比增加或减少。例如，一件外套原价 64 英镑，降价 15%。求新价格。
15% of £64 = 0.15 × 64 = £9.60 → New price = £64 − £9.60 = £54.40
Always write the percentage as a decimal multiplier when using a calculator. A 15% decrease uses the multiplier 0.85, so £64 × 0.85 = £54.40 directly.
使用计算器时，始终把百分比写成小数乘数。降价 15% 使用乘数 0.85，因此直接计算 £64 × 0.85 = £54.40。
3. Algebra: Simplifying and Solving Linear Equations | 代数：化简与解一元一次方程
Higher Year 8 algebra includes collecting like terms, expanding brackets, and solving equations with unknowns on both sides. You must keep each line balanced by doing the same operation to both sides.
Year 8 高阶代数包括合并同类项、展开括号以及求解未知数在等式两边的方程。你必须对等式两边进行相同的运算，保持每行平衡。
Example: Solve 5x + 3 = 2x + 18.
例题：解方程 5x + 3 = 2x + 18。
5x + 3 = 2x + 18
5x − 2x = 18 − 3
3x = 15 → x = 5
ets, remember to multiply every term inside. For example, 3(2x − 4) = 6x − 12, not 6x − 4.
展开括号时，记得乘以括号内的每一项。例如，3(2x − 4) = 6x − 12，而不是 6x − 4。
5, the left side is 5(5) + 3 = 28 and the right side is 2(5) + 18 = 28, so it is correct.
始终将解代回原方程进行检验。当 x = 5 时，左边为 5(5) + 3 = 28，右边为 2(5) + 18 = 28，因此答案正确。
4. Ratio, Proportion and Rates | 比、比例和比率
Ratio questions in Book 8H often combine two or more ratios, or link ratio to fractions and totals. Always find the value of one part first when you know the total or the difference.
Book 8H 中的比的问题通常结合两个或多个比，或将比与分数和总量联系起来。当你知道总量或差值时，总是先求出一份的值。
Example: The ratio of boys to girls in a school is 3 : 5. There are 96 girls. Find the total number of pupils.
例题：一所学校男生与女生的比是 3 : 5。女生有 96 人。求学生总人数。
5 parts = 96 → 1 part = 96 ÷ 5 = 19.2
Total parts = 3 + 5 = 8 → Total pupils = 8 × 19.2 = 153.6
19.2 is possible but indicates the numbers may come from a model rather than a real count.
答案 153.6 对于学校人数来说并不现实，这表明实际比的问题通常使用整数份数。在这种情况下，检查原始数字：如果 5 份 = 96，那么 96 ÷ 5 = 19.2 是可能的，但这表明这些数字可能来自模型而非真实计数。
stance = speed × time. Example: A car travels at 52 km/h for 2.5 hours. Find the distance.
对于速率问题，使用公式 路程 = 速度 × 时间。例题：一辆汽车以 52 公里/小时行驶 2.5 小时。求路程。
Distance = 52 × 2.5 = 130 km
5. Geometry: Angles and Polygons | 几何：角与多边形
Year 8 higher geometry includes angle rules on parallel lines, interior and exterior angles of polygons, and bearings. You need to give reasons for each angle statement in a multi-step question.
Year 8 高阶几何包括平行线上的角度规则、多边形的内角和外角以及方位角。在多步题目中，你需要为每个角度结论给出理由。
Example: Find the sum of the interior angles of a decagon.
例题：求十边形的内角和。
Sum of interior angles = (n − 2) × 180°
For n = 10: (10 − 2) × 180° = 8 × 180° = 1440°
lar, each interior angle is 1440° ÷ 10 = 144°, and each exterior angle is 360° ÷ 10 = 36°.
如果该多边形是正多边形，则每个内角为 1440° ÷ 10 = 144°，每个外角为 360° ÷ 10 = 36°。
Always show the angle rule you use, such as 'angles on a straight line sum to 180°' or 'vertically opposite angles are equal'. This is where many marks are earned in a mark scheme.
始终写出你使用的角度规则，例如“平角之和为 180°”或“对顶角相等”。这是许多评分标准中的得分点。
6. Area, Perimeter and Volume | 面积、周长和体积
Higher Year 8 problems often mix area and perimeter, or ask you to find the volume of prisms and composite shapes. Write the formula first, substitute, then calculate.
Year 8 高阶问题通常将面积和周长混合，或要求你求棱柱和组合图形的体积。先写出公式，代入数值，再计算。
Example: Find the area of a trapezium with parallel sides 6 cm and 10 cm, and height 4 cm.
例题：求梯形面积，上底 6 cm，下底 10 cm，高 4 cm。
Area = ½ × (a + b) × h
Area = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 cm²
For volume of a prism, multiply the cross-sectional area by the length. Example: A triangular prism has cross-sectional area 12 cm² and length 9 cm. Find its volume.
对于棱柱体积，用横截面积乘以长度。例题：一个三棱柱的横截面积为 12 cm²，长度为 9 cm。求其体积。
Volume = 12 × 9 = 108 cm³
Remember to use square units for area and cubic units for volume. Mixing these up is a common Book 8H error.
记住面积使用平方单位，体积使用立方单位。混淆这两者是 Book 8H 中的常见错误。
7. Linear Graphs and Coordinates | 线性图像与坐标
+ c. You need to be confident working with negative coordinates and fractional gradients.
Year 8 高阶图像部分包括绘制一次函数图像、求斜率和理解 y = mx + c。你需要能熟练处理负坐标和分数斜率。
Example: Complete the table for y = 2x − 1, then plot the points for x = −2, −1, 0, 1, 2.
例题：完成 y = 2x − 1 的表格，然后绘制 x = −2、−1、0、1、2 对应的点。
| x | −2 | −1 | 0 | 1 | 2 |
| y = 2x − 1 | −5 | −3 | −1 | 1 | 3 |
Plot these points on a coordinate grid and join them with a straight line. The line crosses the y-axis at −1, so the y-intercept is −1. The gradient is 2 because y increases by 2 for every 1 increase in x.
在坐标网格上绘制这些点，并用直线连接。该直线与 y 轴交于 −1，因此 y 截距为 −1。斜率为 2，因为 x 每增加 1，y 增加 2。
To find the gradient between two points, use the formula:
要计算两点之间的斜率，使用公式：
Gradient = change in y ÷ change in x
For example, between (1, 3) and (4, 9), the gradient is (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2.
例如，在点 (1, 3) 和 (4, 9) 之间，斜率为 (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2。
8. Probability and Venn Diagrams | 概率与韦恩图
Probability questions in 8H often ask for theoretical probability, expected frequency, or combined events. Always simplify fractions and give your answer as a fraction, decimal or percentage depending on the question.
8H 中的概率题通常要求理论概率、期望频数或组合事件。始终化简分数，并根据题目要求以分数、小数或百分比形式给出答案。
Example: A bag contains 4 red, 3 blue and 5 green counters. One counter is picked at random. Find the probability it is not blue.
例题：一个袋子里有 4 个红色、3 个蓝色和 5 个绿色筹码。随机取出一个。求它不是蓝色的概率。
Total counters = 4 + 3 + 5 = 12
Not blue = 4 + 5 = 9 → P(not blue) = 9/12 = 3/4
For Venn diagrams, start by placing the intersection value first, then fill in the remaining sections. The total outside the circles must also be considered if it is given.
对于韦恩图，首先填入交集值，然后填写其余部分。如果给出，还必须考虑圆圈外部的总数。
Example: 30 students were asked if they like football or basketball. 18 like football, 12 like basketball, and 5 like both. Draw a Venn diagram and find the number who like neither.
例题：30 名学生被问及是否喜欢足球或篮球。18 人喜欢足球，12 人喜欢篮球，5 人两者都喜欢。画出韦恩图并求两者都不喜欢的人数。
Football only = 18 − 5 = 13
Basketball only = 12 − 5 = 7
Neither = 30 − (13 + 5 + 7) = 5
9. Statistics: Averages and Range | 统计：平均数与极差
Higher Year 8 statistics includes calculating the mean from a frequency table, finding the median from a list, and comparing data sets using the mean and range.
Year 8 高阶统计包括从频数表计算平均数、从列表求中位数，以及使用平均数和极差比较数据集。
Example: Find the mean, median, mode and range of 4, 7, 9, 7, 3, 10.
例题：求 4、7、9、7、3、10 的平均数、中位数、众数和极差。
Mean = (4 + 7 + 9 + 7 + 3 + 10) ÷ 6 = 40 ÷ 6 = 6.67 (to 2 d.p.)
Ordered: 3, 4, 7, 7, 9, 10 → Median = (7 + 7) ÷ 2 = 7
Mode = 7
Range = 10 − 3 = 7
When comparing two data sets, write one sentence about the average and one about the spread. For example, 'Set A has a higher mean, so it is generally larger. Set B has a smaller range, so it is more consistent.'
比较两个数据集时，写一句关于平均水平的句子，再写一句关于离散程度的句子。例如，“集合 A 的平均数更高，因此总体较大。集合 B 的极差更小，因此更稳定。”
10. Common Mistakes and How to Mark Your Own Work | 常见错误与如何自我批改
When using Book 8H answers, do not just tick the final answer. Look for these frequent mistakes: sign errors in algebra, forgetting units, rounding too early, and misreading the question.
使用 Book 8H 答案时，不要只对最终答案打勾。请留意这些常见错误：代数中的符号错误、忘记单位、过早四舍五入以及误读题目。
egative number or moving terms across the equals sign. Remember that 4 − (−3) = 4 + 3 = 7.
符号错误通常发生在减去负数或将项移过等号时。记住 4 − (−3) = 4 + 3 = 7。
For units, write cm, m, cm², cm³, £, or % after every answer in geometry, speed and money questions. A missing unit can lose a mark in a test.
对于单位，在几何、速度和金钱题中，每个答案后都要写 cm、m、cm²、cm³、£ 或 %。缺少单位可能在考试中丢分。
When marking, use a coloured pen and write a short note beside each error, such as 'sign error' or 'forgot to square'. This helps you remember what to check next time.
批改时，使用彩色笔并在每个错误旁写简短的注释，例如“符号错误”或“忘记平方”。这有助于你下次记住需要检查什么。
11. Exam-Style Practice: One Compressed Mini Test | 考试型练习：一份压缩迷你测试
Try this short set of questions that mirrors the style of Essential Maths Book 8H. Attempt all questions first, then check the answers given below.
尝试以下这组与 Essential Maths Book 8H 风格相似的简短题目。先尝试所有题目，然后对照下方答案进行检查。
- 1. Convert 0.625 to a fraction in its simplest form.
- 2. Solve 4(x − 3) = 2x + 8.
- 3. A map uses the scale 1 : 25000. Two towns are 8 cm apart on the map. Find the real distance in km.
- 4. Find the exterior angle of a regular 15-sided polygon.
- 5. A spinner has 2 red, 3 blue and 5 green sections. Find the probability it lands on blue or green.
- 1. 将 0.625 化为最简分数。
- 2. 解方程 4(x − 3) = 2x + 8。
- 3. 地图比例尺为 1 : 25000。地图上两镇相距 8 cm。求实际距离（公里）。
- 4. 求正十五边形的一个外角。
- 5. 一个转盘有 2 个红色、3 个蓝色和 5 个绿色区域。求它落在蓝色或绿色区域的概率。
Answers:
答案：
1. 0.625 = 625/1000 = 5/8
2. 4x − 12 = 2x + 8 → 2x = 20 → x = 10
3. 8 cm × 25000 = 200000 cm = 2000 m = 2 km
4. Exterior angle = 360° ÷ 15 = 24°
5. P(blue or green) = (3 + 5) ÷ 10 = 8/10 = 4/5
12. Next Steps for Year 8 Higher Learners | Year 8 高阶学习者下一步
Once you can confidently mark your own work using the compressed answer guide, move on to timed practice. Use past paper questions or extension tasks that require you to explain your reasoning in full sentences.
一旦你能自信地使用压缩答案指南自我批改，就可以进行限时练习。使用历年真题或需要你用完整句子解释推理过程的拓展任务。
Focus on your weakest area from this guide. If algebra equations are difficult, practise 10 similar equations every day for one week. If geometry angle reasons are weak, write out the reasons from memory before starting a question.
重点关注本指南中你最薄弱的领域。如果代数方程有困难，每天练习 10 道类似方程，持续一周。如果几何角度理由薄弱，在开始做题前先凭记忆写出理由。
Ask your teacher for the next extension material, such as Essential Maths Book 9H or IGCSE Foundation-level questions. Transition slowly and do not rush ahead without mastering 8H first.
向老师索取下一阶段的拓展材料，例如 Essential Maths Book 9H 或 IGCSE 基础级题目。请循序渐进，不要在没有掌握 8H 之前急于超前。
Finally, keep an error log. Every time you make a mistake, write the topic, the mistake, and the correct method in one sentence. Review this log before any test.
最后，坚持记录错题本。每当你犯错时，用一句话写下题目类型、错误内容和正确方法。在任何测试前复习这个错题本。
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