📚 7C Homework: 04 June 2008 | 7C 数学作业：2008年6月4日
This article reviews the key mathematical skills covered in the 7C homework set for 4 June 2008. It is designed to help Year 7 students consolidate their understanding of arithmetic, algebra, geometry, averages and probability before moving into the final weeks of the summer term.
本文回顾2008年6月4日布置的7C数学作业所涉及的核心数学技能。旨在帮助七年级学生在夏季学期最后几周之前，巩固对算术、代数、几何、平均数和概率的理解。
1. Integer Arithmetic and Order of Operations | 整数运算与运算顺序
Many questions in the 7C homework focus on adding, subtracting, multiplying and dividing positive and negative integers. The key rule is that two signs together follow a simple pattern: same signs give a positive, and different signs give a negative.
7C作业中的许多题目关注正负整数的加、减、乘、除运算。关键规则是两个符号连在一起时遵循一个简单模式：同号得正，异号得负。
When a calculation mixes several operations, you must apply the correct order of operations. In the UK this is often remembered as BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Multiplication and division have equal priority and are worked from left to right, as are addition and subtraction.
当一个计算包含多种运算时，必须使用正确的运算顺序。在英国这通常记为BIDMAS：括号、指数、除法、乘法、加法、减法。乘法和除法优先级相同，从左到右计算；加法和减法也是一样。
- Example 1: (−6) + (−4) = −10 because the two negative signs keep the answer negative.
- 例1：(−6) + (−4) = −10，因为两个负号保持结果为负。
- Example 2: (−5) × (−3) = 15 because a negative times a negative gives a positive.
- 例2：(−5) × (−3) = 15，因为负乘负得正。
- Example 3: 20 − 4 × 3 + 2² = 20 − 12 + 4 = 12.
- 例3：20 − 4 × 3 + 2² = 20 − 12 + 4 = 12。
BIDMAS: Brackets → Indices → Division → Multiplication → Addition → Subtraction
BIDMAS：括号 → 指数 → 除法 → 乘法 → 加法 → 减法
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Equivalent fractions are fractions that represent the same part of a whole. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.
等价分数表示同一个整体中相同的部分。你可以将分子和分母同时乘以或除以同一个非零数来得到等价分数。
To compare fractions, rewrite them with a common denominator. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. A percentage to a fraction is written with denominator 100 and then simplified.
要比较分数，可将它们改写成同分母。要将分数转换为小数，用分子除以分母。要将小数转换为百分数，乘以100。百分数转换为分数时以100为分母，然后化简。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
/2. For multiplication, multiply the numerators and denominators directly. For division, multiply by the reciprocal.
分数的加减需要同分母。例如，1/3 + 1/6 变成 2/6 + 1/6 = 3/6 = 1/2。乘法可直接将分子相乘、分母相乘。除法则乘以倒数。
3. Algebraic Expressions and Like Terms | 代数表达式与同类项
An algebraic expression uses letters to stand for unknown or variable numbers. In the expression 3x + 2y − 5, the terms are 3x, 2y and −5. The number in front of a letter is called the coefficient.
代数表达式用字母表示未知数或变量。在表达式 3x + 2y − 5 中，项为 3x、2y 和 −5。字母前面的数字叫作系数。
Like terms contain exactly the same letter part. You can collect like terms by adding or subtracting their coefficients. Unlike terms cannot be combined. For example, 4a + 3b − 2a + 5b simplifies to 2a + 8b.
同类项含有完全相同的字母部分。可以通过加减其系数来合并同类项。非同类项不能合并。例如，4a + 3b − 2a + 5b 化简为 2a + 8b。
3y = 6xy. When dividing, simplify by cancelling common factors, for example 8x² ÷ 2x = 4x.
代数项相乘时，将数字和字母分别相乘。表达式 2x × 3y = 6xy。相除时，通过约去公因数来化简，例如 8x² ÷ 2x = 4x。
4a + 3b − 2a + 5b = (4 − 2)a + (3 + 5)b = 2a + 8b
4a + 3b − 2a + 5b = (4 − 2)a + (3 + 5)b = 2a + 8b
4. Solving Linear Equations | 解一元一次方程
A linear equation is an equation where the unknown appears only to the first power. The goal is to isolate the unknown on one side of the equation by performing the same operation on both sides.
一元一次方程是指未知数只出现一次幂的方程。目标是通过在等式两边执行相同的运算，将未知数单独移到等式的一边。
to get x = 5. Always check your answer by substituting it back into the original equation.
例如，解方程 3x + 5 = 20，首先两边同时减去5，得到 3x = 15。然后两边同时除以3，得到 x = 5。一定要将答案代回原方程进行检验。
For example, 5x − 4 = 2x + 8 becomes 3x = 12 after subtracting 2x and adding 4, so x = 4.
如果未知数出现在等式两边，则将含变量项移到一边、数字项移到另一边。例如，5x − 4 = 2x + 8 经减去 2x 并加上 4 后得到 3x = 12，所以 x = 4。
3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5
3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5
5. Coordinates and Straight-Line Graphs | 坐标与直线图
Coordinates are written as ordered pairs (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The origin is the point (0, 0). Positive x values move right, negative x move left, positive y move up and negative y move down.
坐标写成有序数对 (x, y)，其中 x 是距原点的水平距离，y 是垂直距离。原点是点 (0, 0)。正的 x 值向右，负的 x 向左，正的 y 向上，负的 y 向下。
ues for a rule such as y = 2x + 1, plot the points, and join them to form a straight line.
绘制直线图时，准确描点至关重要。七年级常见任务是完成如 y = 2x + 1 的函数的表格、描点，并将它们连接成一条直线。
e rule. For example, when x = 3 in y = 2x + 1, y = 2 × 3 + 1 = 7, giving the point (3, 7).
要求 y 的值，将 x 值代入函数式中。例如，在 y = 2x + 1 中，当 x = 3 时，y = 2 × 3 + 1 = 7，得到点 (3, 7)。
6. Angles on Lines and Around Points | 直线与点周围的角
Angles are measured in degrees. A right angle is 90°, a straight angle is 180°, and a full turn is 360°. These facts are used to find missing angles on a straight line, around a point, and in right angles.
角以度为单位。直角为90°，平角为180°，周角为360°。这些事实用于求直线上的角、点周围的角以及直角中的缺失角。
ht line is 65°, the other must be 180° − 65° = 115°. Angles around a point add up to 360°.
直线上的角之和为180°。因此，如果直线上一个角是65°，另一个角必定是 180° − 65° = 115°。点周围的角之和为360°。
Vertically opposite angles are formed when two straight lines cross. They are always equal. For example, if one angle at an intersection is 70°, the angle directly opposite it is also 70°.
对顶角由两条直线相交形成。它们总是相等。例如，如果交点处一个角是70°，则正对的角也是70°。
Straight line: a + b = 180° | Point: a + b + c + d = 360°
直线：a + b = 180° | 点周围：a + b + c + d = 360°
7. Perimeter, Area and Units | 周长、面积与单位
h). For a regular polygon, perimeter is the number of sides multiplied by the side length.
周长是图形外部一周的距离。对于矩形，周长 = 2 × (长 + 宽)。对于正多边形，周长是边数乘以边长。
triangle, area = ½ × base × height. The units of area are squared, such as cm², m² or mm².
面积测量图形内部的空间。对于矩形，面积 = 长 × 宽。对于三角形，面积 = ½ × 底 × 高。面积的单位是平方单位，如 cm²、m² 或 mm²。
= 100 cm, 1 km = 1000 m. For area, you square the conversion factor, so 1 m² = 10,000 cm².
公制单位之间的换算是常见技能。长度换算包括：1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m。对于面积，换算因子需要平方，因此 1 m² = 10,000 cm²。
Rectangle area = l × w | Triangle area = ½ × b × h
矩形面积 = 长 × 宽 | 三角形面积 = ½ × 底 × 高
8. Averages and Range | 平均数与极差
The mean is found by adding all the values together and dividing by the number of values. The median is the middle value when the data are arranged in order. The mode is the value that appears most often. The range is the difference between the largest and smallest values.
平均数是将所有数值相加后除以数值的个数。中位数是将数据按顺序排列后位于中间的值。众数是出现次数最多的值。极差是最大值与最小值之差。
use it is the third value. The mode is 7 because it occurs twice. The range is 13 − 4 = 9.
对于数据集 4、7、7、9、13，平均数为 (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8。中位数是7，因为它是第三个值。众数是7，因为它出现了两次。极差是 13 − 4 = 9。
s the mean of those two values. For the data 2, 5, 8, 11, the median is (5 + 8) ÷ 2 = 6.5.
当有两个中间值时，中位数是这两个值的平均数。对于数据 2、5、8、11，中位数为 (5 + 8) ÷ 2 = 6.5。
Mean = sum of values ÷ number of values
平均数 = 数值总和 ÷ 数值个数
9. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It is expressed as a fraction, decimal or percentage between 0 and 1. A probability of 0 means impossible, and a probability of 1 means certain.
概率衡量事件发生的可能性大小。它用0到1之间的分数、小数或百分数表示。概率为0表示不可能，概率为1表示必然发生。
example, the probability of rolling an even number on a fair six-sided dice is 3/6 = 1/2.
事件发生的概率等于有利结果的数量除以所有可能结果的总数。例如，掷一个公平的六面骰子得到偶数的概率是 3/6 = 1/2。
to 1. If the probability of rain is 0.3, then the probability of no rain is 1 − 0.3 = 0.7.
一个实验中所有可能结果的概率之和为1。如果下雨的概率是0.3，那么不下雨的概率就是 1 − 0.3 = 0.7。
P(event) = favourable outcomes ÷ total outcomes
P(事件) = 有利结果数 ÷ 总结果数
10. Checking Your Homework | 作业检查清单
Before submitting your homework, check that every answer is written in the correct place and that all working out is shown clearly. Teachers often award method marks even if the final answer is slightly wrong, so never leave a question blank if you can show a valid method.
在提交作业之前，检查每个答案是否写在正确的位置，以及是否有清晰的解题过程。即使最终答案稍微有误，老师通常也会给步骤分，所以只要能写出有效过程，就不要留空题。
Read each question twice: first to understand what is being asked, and second to check that your answer matches the question. For word problems, underline key numbers and keywords such as total, difference, per, average and probability.
每道题读两遍：第一遍理解题目问什么，第二遍检查答案是否切题。对于应用题，在关键数字和关键词下划线，如总共、差、每、平均数和概率。
to 50 × 50 = 2500, so an answer near 2496 looks reasonable, but 496 would show a mistake.
用估算检验答案是否合理。例如，48 × 52 接近 50 × 50 = 2500，所以接近2496的答案看起来合理，而496则说明有错误。
Finally, go back to any questions you found difficult and review the relevant examples in your class notes or this article. Consistent practice is the most reliable way to build confidence in mathematics.
最后，回到你觉得困难的题目，回顾课堂笔记或本文中的相关示例。持续练习是建立数学信心最可靠的方法。
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