📚 KS3 Maths: Essential Maths 7C Homework Answers Explained | KS3数学：Essential Maths 7C 作业答案知识点精讲
This guide breaks down the key mathematical topics covered in Essential Maths 7C, providing clear explanations and worked examples to help students check their homework answers and build a deeper understanding. From operations with negative numbers to solving equations and calculating perimeter, each section targets the core skills needed at KS3 level.
本指南拆解了《Essential Maths 7C》中涵盖的核心数学主题，提供清晰的解释与示例解答，帮助学生检查作业答案并建立更深层次的理解。从负数运算到解方程和周长计算，每个部分都针对KS3阶段所需的关键技能。
1. Place Value and Number Types | 位值与数的类型
In Essential Maths 7C, you often write numbers in words or figures and identify digit values. For example, in 8.407, the digit 4 is in the tenths column, so it represents 4/10 or 0.4.
在Essential Maths 7C中，你经常需要将数字写成文字或数字形式，并识别数字的位值。例如，在8.407中，数字4位于十分位，因此表示4/10或0.4。
Understanding the difference between integers, decimals and fractions is tested. The number 3.5 is not an integer; it is a decimal. The number 7 is an integer and also a rational number.
理解整数、小数和分数之间的区别是考试内容。3.5不是整数，而是小数。数字7是整数，也是有理数。
When rounding numbers, look at the next digit to the right. To round 26.78 to one decimal place, check the hundredths digit 8, so it rounds up to 26.8.
进行四舍五入时，看右侧的下一位数字。将26.78四舍五入到一位小数，检查百分位上的8，因此向上舍入为26.8。
2. Order of Operations (BIDMAS) | 运算顺序（BIDMAS 法则）
BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction. It reminds us that calculations must be done in this order. Always solve brackets first.
BIDMAS代表括号、指数、除法、乘法、加法、减法。它提醒我们必须按照这个顺序进行计算。始终先算括号。
For example, to evaluate 3 + 6 × (5 − 2)², we first calculate the bracket (5 − 2 = 3), then the index 3² = 9, then multiplication 6 × 9 = 54, and finally addition 3 + 54 = 57.
例如，计算 3 + 6 × (5 − 2)²，我们先算括号 (5 − 2 = 3)，然后指数 3² = 9，再乘法 6 × 9 = 54，最后加法 3 + 54 = 57。
Division and multiplication have the same priority and are worked from left to right. So 18 ÷ 3 × 2 = 6 × 2 = 12, not 18 ÷ 6.
除法和乘法具有相同的优先级，按从左到右的顺序计算。因此 18 ÷ 3 × 2 = 6 × 2 = 12，而不是 18 ÷ 6。
10 − 2 × 3 + 4 = 10 − 6 + 4 = 8
3. Fractions, Decimals and Percentages | 分数、小数与百分比
To add fractions with different denominators, find the least common multiple (LCM). For 1/3 + 1/4, the LCM is 12, so we rewrite: 4/12 + 3/12 = 7/12.
要对不同分母的分数进行加法，需要找到最小公倍数（LCM）。对于 1/3 + 1/4，LCM为12，所以改写为：4/12 + 3/12 = 7/12。
Converting a fraction to a percentage: Multiply by 100. For example, 3/5 as a percentage is 3 ÷ 5 × 100 = 60%.
将分数转换为百分比：乘以100。例如，3/5 作为百分比是 3 ÷ 5 × 100 = 60%。
In 7C, you also compare fractions by writing them with a common denominator. To compare 5/8 and 2/3, rewrite as 15/24 and 16/24, so 2/3 is larger.
在7C中，你还需要通过通分来比较分数。要比较 5/8 和 2/3，改写为 15/24 和 16/24，所以 2/3 更大。
Multiplying fractions: multiply top by top and bottom by bottom. 2/5 × 3/4 = 6/20 = 3/10.
分数乘法：分子乘分子，分母乘分母。2/5 × 3/4 = 6/20 = 3/10。
4. Working with Negative Numbers | 负数运算
Adding a negative number is the same as subtracting its positive. For instance, 7 + (−3) = 7 − 3 = 4.
加上一个负数等同于减去它的正数。例如，7 + (−3) = 7 − 3 = 4。
Subtracting a negative number turns into addition. 5 − (−2) = 5 + 2 = 7. A common mistake is to treat it as 5 − 2.
减去一个负数变为加法。5 − (−2) = 5 + 2 = 7。常见的错误是把它当成 5 − 2 来算。
When multiplying or dividing two negatives, the result is positive. (−4) × (−6) = 24 and (−12) ÷ (−3) = 4. If only one number is negative, the answer is negative.
当两个负数相乘或相除时，结果为正数。(−4) × (−6) = 24 且 (−12) ÷ (−3) = 4。如果只有一个负号，结果为负。
(−3) × 5 = −15
5. Algebraic Expressions and Simplifying | 代数表达式与化简
Like terms have the same letter parts. You can simplify 3a + 5a to 8a. Terms with different letters, like 3a + 4b, cannot be combined.
同类项具有相同的字母部分。可以将 3a + 5a 化简为 8a。字母不同的项，如 3a + 4b，不能合并。
When expanding brackets, multiply each term inside. 4(2x + 3) = 4 × 2x + 4 × 3 = 8x + 12. Be careful with negatives: −2(3y − 5) = −6y + 10.
去括号时，要将括号外的因数乘以括号内的每一项。4(2x + 3) = 4 × 2x + 4 × 3 = 8x + 12。注意负号：−2(3y − 5) = −6y + 10。
Substitution means replacing letters with numbers. If x = 3, then 5x² becomes 5 × (3)² = 5 × 9 = 45.
代入法是指用数字替换字母。如果 x = 3，那么 5x² 变成 5 × (3)² = 5 × 9 = 45。
6. Solving Linear Equations | 解一元一次方程
To solve an equation like x + 7 = 15, subtract 7 from both sides: x = 8. Always keep the equation balanced by doing the same to both sides.
解类似 x + 7 = 15 的方程时，两边同时减去7：x = 8。始终在方程两边作相同的运算以保持平衡。
For two-step equations such as 3x − 4 = 11, first add 4 to get 3x = 15, then divide by 3 to find x = 5.
对于两步方程 3x − 4 = 11，首先加4得到 3x = 15，然后除以3求出 x = 5。
When there are x terms on both sides, eliminate the smaller term. Solve 5x + 2 = 2x + 14: subtract 2x → 3x + 2 = 14, then subtract 2 → 3x = 12, so x = 4.
当 x 项出现在两边时，消去较小的那一项。解 5x + 2 = 2x + 14：减去 2x → 3x + 2 = 14，再减去2 → 3x = 12，因此 x = 4。
7. Angles and Properties of Shapes | 角与图形的性质
Angles on a straight line add up to 180°. If one angle is 65°, the missing angle is 180° − 65° = 115°.
直线上的角之和为180°。若一个角是65°，则另一个角为 180° − 65° = 115°。
In a triangle, the three interior angles sum to 180°. A right‑angled triangle has one 90° angle, so the other two acute angles add up to 90°.
三角形三个内角之和为180°。直角三角形有一个90°角，因此另外两个锐角之和为90°。
Vertically opposite angles are equal. When two lines cross, the angles opposite each other are the same size.
对顶角相等。当两条直线相交时，互为对顶角的两个角大小相同。
In 7C, you might calculate missing angles using these rules without a protractor, setting up simple equations, e.g. in a triangle, if two angles are 50° and 60°, the third is 180° − 110° = 70°.
在7C中，你可能需要运用这些规则计算缺失的角而无需量角器，建立简单方程，例如在三角形中，两个角为50°和60°，第三个角为 180° − 110° = 70°。
8. Perimeter, Area and Volume | 周长、面积与体积
The perimeter is the total distance around a shape. For a rectangle of length l and width w, perimeter = 2l + 2w. A rectangle 5 cm by 3 cm has a perimeter of 2×5 + 2×3 = 16 cm.
周长是围绕图形的总距离。对于长为 l、宽为 w 的矩形，周长 = 2l + 2w。一个5厘米长、3厘米宽的矩形周长为 2×5 + 2×3 = 16 cm。
Area of a rectangle = length × width. The same 5 cm × 3 cm rectangle has area = 15 cm². Remember to use square units.
矩形面积 = 长 × 宽。同样的 5 cm × 3 cm 矩形面积为 15 cm²。注意使用平方单位。
The area of a triangle is (base × height) ÷ 2. A triangle with base 6 cm and height 4 cm has area (6 × 4) ÷ 2 = 12 cm².
三角形面积 = (底 × 高) ÷ 2。底为6 cm、高为4 cm的三角形面积为 (6 × 4) ÷ 2 = 12 cm²。
Volume of a cuboid = length × width × height. A 4 cm × 3 cm × 2 cm cuboid has volume 24 cm³.
长方体体积 = 长 × 宽 × 高。一个 4 cm × 3 cm × 2 cm 的长方体体积为 24 cm³。
9. Statistics and Data Interpretation | 统计与数据解读
Mean, median, mode and range are the key averages and spread measures. The mean is the sum of values divided by the number of values. For the set 3, 7, 8, 2, 10, the mean is (3+7+8+2+10)÷5 = 6.
平均数、中位数、众数和极差是主要的平均值与离散度量。平均数是所有数值之和除以数值的个数。对于数据集 3, 7, 8, 2, 10，平均数为 (3+7+8+2+10)÷5 = 6。
The median is the middle number when sorted. In 2, 3, 7, 8, 10, the median is 7. If there is an even count, median is the average of the two middle numbers.
中位数是将数据排序后位于中间的数。在2, 3, 7, 8, 10中，中位数为7。如果数据个数为偶数，中位数是中间两个数的平均数。
Bar charts and pictograms are used to represent data. Always read the key for pictograms carefully. A pictogram where one circle represents 4 books means 3 circles show 12 books.
条形图和象形图用于表示数据。务必仔细阅读象形图的图例。若一个圆圈代表4本书，则3个圆圈表示12本书。
10. Ratio and Proportion | 比和比例
Ratios compare quantities. The ratio of red to blue marbles 4:2 can be simplified to 2:1 by dividing both sides by the common factor 2.
比用来比较数量。红球与蓝球的比 4:2 可以通过两边除以公因数2简化为 2:1。
To share an amount in a given ratio, add the total parts. Share £30 in the ratio 2:3: total parts = 5, so one part = £30÷5 = £6, then the shares are 2×£6 = £12 and 3×£6 = £18.
按给定比例分配金额时，先求总份数。按 2:3 分配 £30：总份数 = 5，一份 = £30÷5 = £6，因此分配额为 2×£6 = £12 和 3×£6 = £18。
Proportion problems often involve recipes. If 500 g of flour serves 4 people, to serve 6 people multiply by the scale factor 6/4 = 1.5, so flour needed = 500×1.5 = 750 g.
比例问题常涉及食谱。若500克面粉供4人食用，供6人食用需乘以比例因子 6/4 = 1.5，因此所需面粉为 500×1.5 = 750 g。
11. Number Sequences and Patterns | 数列与模式
A sequence follows a rule. The sequence 5, 9, 13, 17, … increases by 4 each time, so the term‑to‑term rule is “add 4”. The next term is 21.
数列遵循一定的规律。数列5, 9, 13, 17, … 每次增加4，因此递推规则是“加4”。下一项是21。
The nth term of a linear sequence is often written as an expression. For the pattern 2, 5, 8, 11, …, the difference is 3, so the nth term is 3n − 1. Check: when n=1, 3×1−1=2.
线性数列的第n项通常写作一个表达式。对于模式2, 5, 8, 11, …，差为3，因此第n项为 3n − 1。检验：n=1时，3×1−1=2。
Square numbers (1, 4, 9, 16, …) and triangular numbers are also introduced. The next square after 25 is 36, as 6² = 36.
书中还介绍了平方数（1, 4, 9, 16, …）和三角形数。25之后的下一个平方数是36，因为 6² = 36。
12. Symmetry and Transformations | 对称与变换
A shape has line symmetry if it can be folded along a line so that both halves match exactly. A square has four lines of symmetry.
如果一个图形可以沿一条线对折且两部分完全重合，则它具有线对称性。正方形有四条对称轴。
Reflection is a transformation flipping a shape over a mirror line. The image and the original are the same distance from the mirror line but on opposite sides.
反射是一种将图形沿对称轴翻转的变换。镜像与原图形到对称轴的距离相等但位于相反侧。
Rotation turns a shape around a centre point. A half turn is a rotation of 180°. In 7C, you track coordinates after transformations: reflecting point (2,3) in the x‑axis gives (2,−3).
旋转是围绕一个中心点转动图形。半圈旋转即180°。在7C中，你需要追踪变换后的坐标：点 (2,3) 关于x轴反射后得到 (2,−3)。
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