📚 KS3 Maths: Essential Maths 7H Homework Answers Explained | KS3 数学：Essential Maths 7H 作业题型解析
This article breaks down typical homework question types found in the Essential Maths 7H course for Year 7 higher level students. You will learn how to approach each topic, avoid common mistakes, and check your answers effectively. The explanations cover the core mathematical skills needed to build confidence and achieve full marks in your homework tasks.
本文拆解了 Essential Maths 7H 课程中常见的作业题型，面向七年级较高水平的学生。你将学会如何处理每个主题、避开常见错误并有效检查答案。这些解析涵盖所需的核心数学技能，帮助你建立自信并在作业中取得满分。
1. Place Value and Decimal Operations | 位值与小数运算
Questions on place value often ask you to multiply or divide decimals by powers of 10. For instance, 4.56 × 100 shifts the decimal point two places to the right, giving 456. Similarly, 78.3 ÷ 1000 moves it three places left, resulting in 0.0783.
位值相关的题目经常要求对小数进行乘以或除以10的幂的操作。例如，4.56 × 100 将小数点向右移动两位，得到 456。同样，78.3 ÷ 1000 小数点左移三位，结果为 0.0783。
In homework answers you might see a table with missing digits under headings Thousands, Hundreds, Tens, Units, Tenths, Hundredths. Make sure you align the digits correctly. For the number 23.07, the digit 2 occupies the Tens column, 3 is Units, 0 is Tenths and 7 is Hundredths.
在作业答案中你可能会看到一个表格，表头为千位、百位、十位、个位、十分位、百分位，需要填入缺失数字。确保数字对齐正确。对于 23.07，数字 2 在十位列，3 在个位列，0 在十分位列，7 在百分位列。
When comparing decimals like 0.6 and 0.57, always add trailing zeros so both have the same number of decimal places: 0.60 > 0.57. This prevents the common error of thinking 0.57 is larger because 57 is greater than 6.
比较像 0.6 和 0.57 这样的小数时，务必在后补零使其小数位数相同：0.60 > 0.57。这能避免因 57 比 6 大而错误地认为 0.57 更大的常见错误。
2. Working with Negative Numbers | 负数的运算
Adding a negative number is the same as subtracting its absolute value: 5 + (−3) = 5 − 3 = 2. Subtracting a negative is equivalent to adding: 4 − (−2) = 4 + 2 = 6. Many homework exercises use temperature or bank balance contexts to practise these rules.
加上一个负数等同于减去其绝对值：5 + (−3) = 5 − 3 = 2。减去一个负数等同于加上正数：4 − (−2) = 4 + 2 = 6。很多作业练习以温度或银行余额为背景来巩固这些规则。
When multiplying or dividing two numbers with the same sign, the answer is positive: (−7) × (−4) = 28. If the signs are different, the result is negative: (−6) × 3 = −18, and 15 ÷ (−5) = −3.
两个同号数相乘或相除，答案为正：(−7) × (−4) = 28。如果异号，结果为负：(−6) × 3 = −18，15 ÷ (−5) = −3。
In homework answers, pay close attention to brackets. The expression −5² is interpreted as −(5²) = −25, whereas (−5)² = 25. This is a classic pitfall in Essential Maths 7H.
作业答案中要特别注意括号。表达式 −5² 被解读为 −(5²) = −25，而 (−5)² = 25。这是 Essential Maths 7H 中的一个经典陷阱。
3. Simplifying Algebraic Expressions | 代数式的化简
Collect like terms by adding or subtracting coefficients. For example, 5a + 3b − 2a + 7b simplifies to 3a + 10b. Remember that a on its own means 1a, and −a means −1a.
通过加减系数来合并同类项。例如，5a + 3b − 2a + 7b 化简为 3a + 10b。记住单独的 a 表示 1a，−a 表示 −1a。
When multiplying terms, write the number first and then the letters alphabetically: 4 × y × z = 4yz. For powers, x × x is written as x², and p × p × p = p³. Recognise that y × y² = y³.
项相乘时，把数字写在前面，字母按字母表顺序书写：4 × y × z = 4yz。幂的表示：x × x 写作 x²，p × p × p = p³。要能识别 y × y² = y³。
Typical homework tasks might give you a formula like P = 4s + 2 and ask for its value when s = 3. Simply substitute: P = 4×3 + 2 = 14. Ensure you show the substitution step clearly in your answers.
典型作业题可能给出如 P = 4s + 2 的公式并要求当 s = 3 时求值。只需代入：P = 4×3 + 2 = 14。务必在答案中清晰写出代入步骤。
4. Solving Two-Step Equations | 解两步方程
To solve 3x + 5 = 20, first undo the addition by subtracting 5 from both sides, giving 3x = 15. Then divide both sides by 3, so x = 5. Always perform inverse operations in reverse order (PEMDAS backwards).
解方程 3x + 5 = 20，首先通过两边减5抵消加法，得到 3x = 15。然后两边除以3，得出 x = 5。始终按逆序执行逆运算（PEMDAS 的逆向）。
When the equation has a fraction, such as x/4 = 12, multiply both sides by 4: x = 48. For equations with brackets like 2(x + 3) = 14, either expand or divide both sides by 2 first: x + 3 = 7, so x = 4.
当方程含有分数，如 x/4 = 12，两边乘以4：x = 48。对于含括号的方程如 2(x + 3) = 14，可以先展开或两边先除以2：x + 3 = 7，解得 x = 4。
Homework answers often require checking by substituting back into the original equation. If 2(4 + 3) = 2×7 = 14, the solution is correct. Always include this verification in your final answer.
作业答案常要求代回原方程检验。如果 2(4 + 3) = 2×7 = 14，则解正确。务必在最终答案中包含此验证。
5. Angles on a Straight Line and at a Point | 直线上的角与周角
Essential Maths 7H emphasises angle rules. The sum of angles on a straight line is 180°. So if one angle is 72°, the adjacent angle must be 180° − 72° = 108°.
Essential Maths 7H 强调角度规则。直线上各角之和为 180°。因此，若一角为 72°，相邻角必然是 180° − 72° = 108°。
Angles around a point add up to 360°. Vertically opposite angles are equal. When two lines cross, if one angle is 45°, the opposite angle is also 45°, and the other two are each 135°.
环绕一点的各角之和为 360°。对顶角相等。当两直线相交，若一角为 45°，对顶角也为 45°，另外两角各为 135°。
Questions might present a diagram with algebraic expressions for angles, e.g. an angle labelled 3x and the adjacent one 2x on a straight line. You would set up 3x + 2x = 180, giving 5x = 180, x = 36°. Then the angles are 108° and 72°.
题目可能给出含代数表达式的角度图，例如直线上一个角标为 3x，相邻角标为 2x。列出 3x + 2x = 180，得 5x = 180，x = 36°。然后两角分别为 108° 和 72°。
6. Converting Fractions, Decimals and Percentages | 分数、小数和百分比的转换
To convert a fraction to a decimal, divide the numerator by the denominator. 3/8 becomes 0.375. To turn a decimal into a percentage, multiply by 100: 0.375 × 100 = 37.5%. Reverse the process by dividing by 100.
分数转小数，用分子除以分母。3/8 变成 0.375。小数转百分比，乘以100：0.375 × 100 = 37.5%。逆向操作则除以100。
Common equivalences you must memorise: ½ = 0.5 = 50%, ⅓ ≈ 0.333… = 33.3%, ¼ = 0.25 = 25%, ⅕ = 0.2 = 20%, and ¾ = 0.75 = 75%. Use these benchmarks to compare mixed fractions quickly.
必须熟记的常见等价关系：½ = 0.5 = 50%，⅓ ≈ 0.333… = 33.3%，¼ = 0.25 = 25%，⅕ = 0.2 = 20%，¾ = 0.75 = 75%。利用这些基准值快速比较带分数。
Homework answers sometimes ask for which is larger: 0.45 or ⅖? Convert ⅖ to 0.4, so 0.45 is larger. Always convert all values to the same form before comparing.
作业答案有时会问哪个更大：0.45 还是 ⅖？将 ⅖ 转换为 0.4，所以 0.45 更大。始终将所有值转换为同一种形式后再比较。
7. Ratio and Proportion Word Problems | 比例和比率应用题
When sharing £360 in the ratio 2:3:4, first find the total number of parts: 2+3+4 = 9. One part is £360 ÷ 9 = £40. Then the shares are 2×40 = £80, 3×40 = £120, and 4×40 = £160.
按比例 2:3:4 分享 £360 时，先求总份数：2+3+4 = 9。一份为 £360 ÷ 9 = £40。然后分配额为 2×40 = £80，3×40 = £120，4×40 = £160。
For direct proportion recipes, e.g. if 4 muffins need 120 g flour, then 10 muffins need (10/4) × 120 = 300 g. The unitary method (find for 1) is also reliable: 1 muffin needs 30 g, so 10 need 300 g.
对于正比例食谱问题，例如 4 个松饼需 120 克面粉，那么 10 个需要 (10/4) × 120 = 300 克。单位法（先求单个）也同样可靠：1 个需 30 克，所以 10 个需 300 克。
Check your homework ratios by reducing them to simplest form. 24:36 simplifies to 2:3 by dividing by 12. Answers should always be given in their lowest terms with integer parts.
通过化为最简形式来检查作业中的比例。24:36 除以 12 化简为 2:3。答案始终应以最简整数比形式给出。
8. Calculating Area of Triangles and Parallelograms | 三角形和平行四边形的面积计算
The area of a triangle is ½ × base × height. For a triangle with base 8 cm and perpendicular height 5 cm, area = ½ × 8 × 5 = 20 cm². Ensure you use the perpendicular height, not the slant length.
三角形面积公式为 ½ × 底 × 高。若三角形底为 8 cm，垂直高为 5 cm，面积 = ½ × 8 × 5 = 20 cm²。务必使用垂直高，而非斜边长。
Area of a parallelogram = base × perpendicular height. A parallelogram with base 9 m and perpendicular height 4 m has area = 36 m². Do not confuse with the formula for a rectangle; the height must be perpendicular.
平行四边形面积 = 底 × 垂直高。底为 9 m、垂直高为 4 m 的平行四边形面积为 36 m²。切勿与长方形公式混淆；高必须是垂直距离。
Compound shapes require splitting into rectangles and triangles, finding individual areas, then adding. Watch out for missing lengths that need to be deduced from given dimensions before calculating.
复合图形需拆分为长方形和三角形，分别求面积再相加。注意计算前要先从已知尺寸推导出缺失的边长。
9. Mean, Median and Mode from Frequency Tables | 频数表中的平均数、中位数和众数
From a frequency table, the mode is the value with the highest frequency. The median position is (n+1)/2 where n is total frequency. For grouped data, find the interval containing the median by cumulative frequency.
由频数表，众数是频数最高的值。中位数的位置为 (n+1)/2，n 是总频数。对于分组数据，利用累计频数确定包含中位数的组距。
To calculate the mean from a frequency table, use Mean = (sum of (value × frequency)) / (total frequency). For example, if 3 appears 2 times, 5 appears 3 times, total sum = (3×2)+(5×3) = 6+15 = 21, total frequency = 5, mean = 4.2.
要用频数表计算平均数，使用 平均数 = (各值×频数之和) / (总频数)。例如，3 出现 2 次，5 出现 3 次，总和 = (3×2)+(5×3) = 6+15 = 21，总频数 = 5，平均数为 4.2。
Always show an extra column for ‘value × frequency’ in your working. This makes it easier to check and reduces arithmetic mistakes. Essential Maths 7H marking often requires the working steps.
在解题过程中始终额外列出一栏“值×频数”。这样便于检查并减少算术错误。Essential Maths 7H 的评分常要求写出解题步骤。
10. Coordinates and Plotting Linear Graphs | 坐标与线性图绘制
Coordinates are written as (x, y). The x-coordinate tells you how far right (positive) or left (negative) to go; the y-coordinate tells you how far up or down. (3, -4) means 3 right, 4 down.
坐标写作 (x, y)。x 坐标表示向右（正）或向左（负）移动多远；y 坐标表示向上或向下移动多远。(3, -4) 表示右移3，下移4。
To plot the graph of y = 2x + 1, make a table of x values and work out y. For x = 0, y = 1; for x = 1, y = 3; for x = 2, y = 5. Plot these points and draw a straight line through them.
绘制 y = 2x + 1 的图像，先列出 x 值表并计算 y。当 x = 0，y = 1；x = 1，y = 3；x = 2，y = 5。描出这些点并穿过它们画直线。
Graphs of the form y = c are horizontal lines, and x = c are vertical lines. Recognise that points like (4, 2) and (4, 5) lie on the vertical line x = 4. This is frequently tested in 7H homework.
形如 y = c 的图像是水平线，x = c 是垂直线。要能识别像 (4, 2) 和 (4, 5) 这样位于垂直线 x = 4 上的点。这在 7H 作业中经常考查。
11. Basic Probability | 基础概率
Probability is calculated as (number of favourable outcomes) / (total number of possible outcomes). The probability of rolling a 3 on a fair six-sided die is 1/6. Probabilities are expressed as fractions, decimals or percentages between 0 and 1.
概率计算为 (有利结果数) / (所有可能结果数)。掷一个公平六面骰子得到 3 的概率是 1/6。概率用 0 到 1 之间的分数、小数或百分数表示。
The probability scale: an event certain to happen has probability 1; an impossible event has probability 0. ‘Even chance’ corresponds to 0.5, ½ or 50%. Use language like ‘likely’, ‘unlikely’ along with numbers.
概率标尺：必然发生的事件概率为 1；不可能事件概率为 0。“等可能机会”对应 0.5、½ 或 50%。使用“可能”、“不太可能”等语言并配合数字。
For combined events without bias, list outcomes systematically using a sample space or two-way table. The sum of all mutually exclusive probabilities is always 1. When adding fractions, ensure denominators match.
对于无偏好的组合事件，使用样本空间或双向表系统地列出结果。所有互斥事件的概率之和始终为 1。分数相加时要确保分母相同。
12. Key Tips for Checking Homework Answers | 检查作业答案的要点
Always re-read the question to see if the answer is sensible. For example, an angle found as 120° on a straight line means the other should be 60°; if your sum doesn’t give 180°, find the error.
始终重新读题，看答案是否合理。例如，直线上找出一个角为 120°，那么另一角应为 60°；若总和不是 180°，就要找出错误。
Use estimation to verify calculations: 48.9 × 11 is about 50 × 10 = 500. The exact answer is 537.9, close to the estimate. If you had a wildly different value, redo the working.
用估算验证计算：48.9 × 11 大约是 50 × 10 = 500。精确答案为 537.9，接近估算值。如果相差悬殊，就要重算。
Finally, when writing final answers, include units (cm, m², £) and simplify fractions. If the question is a word problem, phrase your answer as a sentence. In Essential Maths 7H, clear presentation counts for method marks.
最后，书写最终答案时要包含单位（cm、m²、£）并化简分数。如果是应用题，用完整的句子给出答案。在 Essential Maths 7H 中，清晰的呈现可以获得方法分。
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