📚 Essential Maths Book 8 Support Answers | 八年级数学核心练习答案解析
This revision article supports Year 8 learners using Essential Maths Book 8 by breaking down core topics and showing clear, step-by-step answers to typical support-level questions. Each section includes a worked example, a focused explanation, and common mistakes to avoid.
本文为使用 Essential Maths Book 8 的八年级学生提供复习支持，将核心主题分解并展示典型基础题的清晰分步答案。每节包括示例题、重点讲解和常见错误提醒。
1. Number and Place Value | 数字与位值
Year 8 place value work extends whole numbers to large integers and introduces rounding, estimation, and standard form ideas in simpler contexts. Understanding the value of each digit is essential before moving to decimals and powers.
八年级的位值学习将整数扩展到大数，并引入四舍五入、估算以及简单情境下的标准形式概念。在进入小数和幂之前，理解每个数字的位值是关键。
Worked example: Round 4,762 to the nearest hundred. Look at the tens digit, which is 6, so the hundreds digit rounds up from 7 to 8.
示例：将 4,762 四舍五入到最接近的百位。看十位数字是 6，因此百位数字从 7 进位到 8。
4,762 ≈ 4,800
Another common question asks for the value of a digit, such as the 5 in 35,208. Since the 5 is in the thousands column, its value is 5,000.
另一个常见问题是求某个数字的值，例如 35,208 中的 5。因为 5 在千位，所以它的值是 5,000。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Year 8 students must convert fluently between fractions, decimals, and percentages. They also add, subtract, multiply, and divide fractions with different denominators.
八年级学生必须熟练地在分数、小数和百分数之间进行转换。他们还需要对不同分母的分数进行加、减、乘、除运算。
Worked example: Convert 3/5 to a decimal and a percentage. Divide 3 by 5 to get 0.6, then multiply by 100 to get 60%.
示例：将 3/5 转换为小数和百分数。用 3 除以 5 得到 0.6，再乘以 100 得到 60%。
3/5 = 0.6 = 60%
When adding 2/3 + 1/4, find a common denominator of 12. The sum is 8/12 + 3/12 = 11/12.
计算 2/3 + 1/4 时，找到公分母 12。和为 8/12 + 3/12 = 11/12。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
| 1/4 | 0.25 | 25% |
| 2/5 | 0.4 | 40% |
| 7/10 | 0.7 | 70% |
3. Ratio and Proportion | 比与比例
Ratio questions often involve sharing a quantity in a given ratio or simplifying ratios. Support-level answers must show each division step clearly.
比例题通常涉及按给定比例分配数量或化简比。基础题答案必须清晰地展示每一步除法。
Worked example: Share £60 in the ratio 3:2. First add the parts: 3 + 2 = 5. Then divide £60 by 5 to get £12 per part. Multiply: 3 × £12 = £36 and 2 × £12 = £24.
示例：按 3:2 的比例分配 60 英镑。首先将比例项相加：3 + 2 = 5。然后用 60 英镑除以 5，得到每份 12 英镑。再相乘：3 × 12 = 36 英镑，2 × 12 = 24 英镑。
£60 in 3:2 = £36 : £24
To simplify a ratio such as 18:24, find the highest common factor. Both 18 and 24 divide by 6, so the simplest form is 3:4.
化简 18:24 这样的比时，找到最大公因数。18 和 24 都能被 6 整除，因此最简形式为 3:4。
4. Algebraic Expressions and Substitution | 代数表达式与代入
Year 8 algebra focuses on collecting like terms, expanding single brackets, and substituting values into expressions. Answers must keep the equals sign balanced and show substitution clearly.
八年级代数的重点是合并同类项、展开单项括号以及将数值代入表达式。答案必须保持等号平衡，并清晰展示代入过程。
Worked example: Simplify 5a + 3b – 2a + 7b. Collect the a terms: 5a – 2a = 3a. Collect the b terms: 3b + 7b = 10b. So the answer is 3a + 10b.
示例：化简 5a + 3b – 2a + 7b。合并 a 项：5a – 2a = 3a。合并 b 项：3b + 7b = 10b。因此答案是 3a + 10b。
5a + 3b – 2a + 7b = 3a + 10b
Substitution example: If x = 4 and y = 3, evaluate 2x + 5y. Replace x with 4 and y with 3: 2 × 4 + 5 × 3 = 8 + 15 = 23.
代入示例：若 x = 4，y = 3，求 2x + 5y 的值。将 x 替换为 4，y 替换为 3：2 × 4 + 5 × 3 = 8 + 15 = 23。
5. Solving Linear Equations | 解一元一次方程
Solving equations at Year 8 includes one-step, two-step, and equations with brackets. Support answers should show inverse operations in order and a final check.
八年级解方程包括一步方程、两步方程和含括号的方程。基础题答案应展示逆运算的顺序以及最后的检验。
Worked example: Solve 3x + 5 = 20. Subtract 5 from both sides: 3x = 15. Then divide both sides by 3: x = 5.
示例：解方程 3x + 5 = 20。两边同时减去 5：3x = 15。然后两边同时除以 3：x = 5。
3x + 5 = 20 → x = 5
For equations with brackets such as 2(x – 3) = 10, expand first: 2x – 6 = 10. Add 6 to both sides: 2x = 16. Divide by 2: x = 8.
对于含括号的方程如 2(x – 3) = 10，首先展开：2x – 6 = 10。两边加 6：2x = 16。再除以 2：x = 8。
6. Angles and Shapes | 角与图形
Angle rules such as angles on a straight line, angles around a point, and vertically opposite angles are essential in Year 8. Students also classify triangles and quadrilaterals by their properties.
角规则，如直线上的角、围绕一点的角以及对顶角，在八年级非常重要。学生还要根据性质对三角形和四边形进行分类。
Worked example: If two angles on a straight line are 65° and y, find y. Angles on a straight line add to 180°, so y = 180° – 65° = 115°.
示例：如果直线上的两个角分别是 65° 和 y，求 y。直线上的角和为 180°，因此 y = 180° – 65° = 115°。
65° + y = 180° → y = 115°
In a triangle, the three interior angles always add to 180°. If two angles are 50° and 70°, the third angle is 180° – 50° – 70° = 60°.
在三角形中，三个内角之和总是 180°。如果两个角分别是 50° 和 70°，则第三个角为 180° – 50° – 70° = 60°。
7. Area and Perimeter | 面积与周长
Year 8 students calculate the area and perimeter of rectangles, triangles, parallelograms, and compound shapes. Correct units are important: perimeter uses linear units, and area uses square units.
八年级学生计算矩形、三角形、平行四边形和组合图形的面积与周长。正确的单位很重要：周长使用长度单位，面积使用平方单位。
Worked example: A rectangle has length 8 cm and width 5 cm. Perimeter = 2 × (8 + 5) = 26 cm. Area = 8 × 5 = 40 cm².
示例：一个矩形的长为 8 cm，宽为 5 cm。周长 = 2 × (8 + 5) = 26 cm。面积 = 8 × 5 = 40 cm²。
P = 2(l + w) = 26 cm, A = l × w = 40 cm²
The area of a triangle is half the base times the height. For a triangle with base 10 cm and height 6 cm, area = ½ × 10 × 6 = 30 cm².
三角形的面积等于底乘以高的一半。对于底为 10 cm、高为 6 cm 的三角形，面积 = ½ × 10 × 6 = 30 cm²。
8. Volume and Surface Area | 体积与表面积
Volume and surface area of cubes and cuboids are common support-level topics. Volume is measured in cubic units, while surface area is the total area of all faces in square units.
立方体和长方体的体积与表面积是常见的基础题主题。体积使用立方单位，而表面积是所有面的总面积，使用平方单位。
Worked example: A cuboid is 4 cm long, 3 cm wide, and 2 cm high. Volume = 4 × 3 × 2 = 24 cm³. Surface area = 2(4×3 + 4×2 + 3×2) = 2(12 + 8 + 6) = 52 cm².
示例：一个长方体长 4 cm、宽 3 cm、高 2 cm。体积 = 4 × 3 × 2 = 24 cm³。表面积 = 2(4×3 + 4×2 + 3×2) = 2(12 + 8 + 6) = 52 cm²。
V = l × w × h = 24 cm³, SA = 2(lw + lh + wh) = 52 cm²
Remember that a cube has equal side lengths. If one edge is 5 cm, volume = 5³ = 125 cm³ and surface area = 6 × 5² = 150 cm².
请记住，立方体的所有棱长相等。如果一条棱长为 5 cm，体积 = 5³ = 125 cm³，表面积 = 6 × 5² = 150 cm²。
9. Statistics and Averages | 统计与平均数
Year 8 statistics covers mean, median, mode, and range, as well as reading bar charts and pie charts. Support answers must show the ordered list before finding the median.
八年级统计内容包括平均数、中位数、众数和极差，以及读取条形图和饼图。基础题答案在求中位数前必须展示排序后的列表。
Worked example: Find the mean, median, mode, and range of 4, 7, 7, 9, 13. Mean = (4+7+7+9+13) ÷ 5 = 40 ÷ 5 = 8. Median = 7. Mode = 7. Range = 13 – 4 = 9.
示例：求 4、7、7、9、13 的平均数、中位数、众数和极差。平均数 = (4+7+7+9+13) ÷ 5 = 40 ÷ 5 = 8。中位数 = 7。众数 = 7。极差 = 13 – 4 = 9。
Mean = 8, Median = 7, Mode = 7, Range = 9
When a data set has an even number of values, the median is the mean of the two middle numbers. For 2, 4, 6, 8, the median is (4 + 6) ÷ 2 = 5.
当数据集有偶数个值时，中位数是中间两个数的平均数。对于 2、4、6、8，中位数为 (4 + 6) ÷ 2 = 5。
10. Probability | 概率
Probability in Year 8 is written as a fraction, decimal, or percentage, and always lies between 0 and 1. Students learn to calculate the probability of a single event from equally likely outcomes.
八年级的概率用分数、小数或百分数表示，并且始终在 0 到 1 之间。学生学习根据等可能结果计算单个事件的概率。
Worked example: A bag contains 3 red, 2 blue, and 5 green counters. The total number of counters is 10. The probability of picking a red counter is 3/10 = 0.3 = 30%.
示例：一个袋子里有 3 个红色、2 个蓝色和 5 个绿色计数器。计数器总数为 10。抽到红色计数器的概率是 3/10 = 0.3 = 30%。
P(red) = 3/10 = 0.3 = 30%
The probability of an event not happening is 1 minus the probability that it does happen. If P(rain) = 0.4, then P(no rain) = 1 – 0.4 = 0.6.
事件不发生的概率等于 1 减去它发生的概率。如果 P(下雨) = 0.4，那么 P(不下雨) = 1 – 0.4 = 0.6。
11. Negative Numbers and Order of Operations | 负数与运算顺序
Year 8 students must handle negative numbers in addition, subtraction, multiplication, and division. They also use BIDMAS or BODMAS to evaluate multi-step expressions correctly.
八年级学生必须能够在加、减、乘、除中处理负数。他们还需要使用 BIDMAS 或 BODMAS 正确计算多步表达式。
Worked example: Evaluate -6 + 4 × 3. According to the order of operations, multiply first: 4 × 3 = 12. Then add: -6 + 12 = 6.
示例：计算 -6 + 4 × 3。根据运算顺序，先算乘法：4 × 3 = 12。然后相加：-6 + 12 = 6。
-6 + 4 × 3 = -6 + 12 = 6
When multiplying or dividing two negative numbers, the result is positive. For example, -8 × -3 = 24 and -12 ÷ -4 = 3.
当两个负数相乘或相除时，结果为正数。例如，-8 × -3 = 24，-12 ÷ -4 = 3。
12. Coordinates and Graphs | 坐标与图像
Coordinates are written as ordered pairs (x, y). Year 8 students plot points in all four quadrants and draw straight-line graphs from simple linear equations.
坐标写作有序数对 (x, y)。八年级学生在四个象限中描点，并根据简单的一次方程绘制直线图像。
Worked example: Plot the points A(3, 2), B(-2, 4), and C(0, -5). A is in the first quadrant, B is in the second quadrant, and C lies on the y-axis below the origin.
示例：描出点 A(3, 2)、B(-2, 4) 和 C(0, -5)。A 在第一象限，B 在第二象限，C 位于原点下方的 y 轴上。
A(3, 2), B(-2, 4), C(0, -5)
For the graph of y = 2x – 1, choose x-values such as -1, 0, 1, and 2. Substituting gives the points (-1, -3), (0, -1), (1, 1), and (2, 3), which lie on a straight line.
对于 y = 2x – 1 的图像，选择 x 值如 -1、0、1 和 2。代入后得到点 (-1, -3)、(0, -1)、(1, 1) 和 (2, 3)，它们都在同一条直线上。
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