📚 Year 7 Essential Maths Core | Year 7 核心数学精讲
This revision guide covers the essential topics in the Year 7 Core Mathematics course. It is designed to help you build strong foundations in number, algebra, geometry and data handling. Work through each section carefully and practise the examples before attempting exam-style questions.
本复习指南涵盖 Year 7 核心数学课程的重要主题。它旨在帮助你在数、代数、几何和数据处理方面打下坚实基础。请认真学习每一节，并在尝试考试型题目之前练习示例。
1. Place Value and Ordering | 位值与排序
Place value tells us the value of each digit in a number. In Year 7, you must be confident reading and writing whole numbers up to 10 000 000 and decimals to at least three decimal places.
位值告诉我们一个数中每个数字的值。在 Year 7，你必须能够熟练读写最大到 10 000 000 的整数以及至少三位小数。
To order numbers, compare the digits from left to right. For example, 0.704 is greater than 0.69 because 7 tenths is more than 6 tenths.
要比较大小，就从左到右比较数字。例如，0.704 大于 0.69，因为 7 个十分之一比 6 个十分之一多。
Rounding is also a key skill. To round 47 562 to the nearest thousand, look at the hundreds digit: 5 means we round up, so the answer is 48 000.
四舍五入也是一项关键技能。将 47 562 四舍五入到最近的千位时，要看百位数字：5 意味着向前进一，所以答案是 48 000。
2. Addition and Subtraction | 加法和减法
Column addition and subtraction are the standard written methods in Year 7. Always line up the digits by place value and carry or borrow carefully.
竖式加法和减法是 Year 7 的标准书面方法。始终按位值对齐数字，并仔细进位或借位。
89 and 312, a sensible estimate is 500 + 300 = 800, so the exact answer 801 is reasonable.
估算帮助你检查答案。如果你计算 489 加 312，合理的估算是 500 + 300 = 800，所以精确答案 801 是合理的。
= 7997. When zeros are involved, regrouping must be done step by step across place values.
对于减法，记住 8000 − 3 = 7997。当涉及零时，必须逐步跨位值进行借位。
3. Multiplication and Division | 乘法和除法
e essential. Long multiplication and short division allow you to calculate larger products and quotients without a calculator.
12 × 12 以内的乘法表是必备基础。长乘法和短除法使你能够不用计算器计算较大的乘积和商。
Factors are numbers that divide exactly into another number. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
因数是能整除另一个数的数。24 的因数有 1、2、3、4、6、8、12 和 24。
A prime number has exactly two factors: 1 and itself. The first five prime numbers are 2, 3, 5, 7 and 11.
质数恰好有两个因数：1 和它本身。前五个质数是 2、3、5、7 和 11。
0 or 1000, digits move one, two or three places to the left. For example, 3.6 × 100 = 360.
当乘以 10、100 或 1000 时，数字向左移动一位、两位或三位。例如，3.6 × 100 = 360。
4. Negative Numbers | 负数
Negative numbers are numbers less than zero. They appear on the left side of a number line, and they are used in temperature, height above sea level and money.
负数是小于零的数。它们出现在数轴的左侧，广泛用于温度、海拔高度和金钱债务。
e number is the same as subtracting its positive value. For example, 5 + (−3) = 5 − 3 = 2.
加上一个负数等于减去它的正值。例如，5 + (−3) = 5 − 3 = 2。
Subtracting a negative number is the same as adding. So (−4) − (−6) = −4 + 6 = 2.
减去一个负数等于加上它的正值。所以 (−4) − (−6) = −4 + 6 = 2。
nswer, while different signs give a negative answer: (−3) × (−4) = 12, but (−3) × 4 = −12.
乘法或除法中，同号得正，异号得负： (−3) × (−4) = 12，但 (−3) × 4 = −12。
5. Fractions, Decimals and Percentages | 分数、小数和百分数
Fractions, decimals and percentages are three ways of representing parts of a whole. You must be able to convert between them fluently.
分数、小数和百分数是表示整体的一部分的三种方式。你必须能够在它们之间熟练转换。
To simplify a fraction, divide the numerator and the denominator by their highest common factor. For example, 12/18 simplifies to 2/3.
要化简分数，就将分子和分母同时除以它们的最大公因数。例如，12/18 化简为 2/3。
Equivalent fractions have the same value. 1/2, 2/4 and 5/10 are all equivalent.
等价分数具有相同的值。1/2、2/4 和 5/10 都是等价分数。
om. 3/8 = 0.375. To convert a decimal to a percentage, multiply by 100: 0.375 × 100 = 37.5%.
要将分数转换为小数，用分子除以分母。3/8 = 0.375。要将小数转换为百分数，乘以 100：0.375 × 100 = 37.5%。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
| 7/10 | 0.7 | 70% |
6. Ratio and Proportion | 比和比例
Ratio compares the sizes of two or more quantities. It can be written with a colon, such as 3:2, and must be simplified by dividing both sides by common factors.
比用于比较两个或多个量的大小。它可以用冒号书写，例如 3:2，并且必须通过同时除以公因数来化简。
To share £80 in the ratio 3:5, first add the parts: 3 + 5 = 8. Then one part is £80 ÷ 8 = £10. The shares are 3 × £10 = £30 and 5 × £10 = £50.
要将 80 英镑按 3:5 的比例分配，先求总份数：3 + 5 = 8。然后一份是 80 ÷ 8 = 10 英镑。分配结果是 3 × 10 = 30 英镑和 5 × 10 = 50 英镑。
aps. If 3 apples cost 90p, then 5 apples cost 5 × 30p = 150p, because one apple costs 30p.
比例问题常涉及按比例调整配方或地图距离。如果 3 个苹果售价 90 便士，那么 5 个苹果售价 5 × 30 = 150 便士，因为一个苹果售价 30 便士。
7. Introduction to Algebra | 代数入门
Algebra uses letters to represent unknown numbers or changing values. In Year 7, you learn to write expressions, substitute values and simplify by collecting like terms.
代数使用字母表示未知数或变化的量。在 Year 7，你将学习书写代数式、代入数值以及通过合并同类项进行化简。
An expression like 3a + 2b − a + 5b simplifies to 2a + 7b. Only like terms can be combined.
像 3a + 2b − a + 5b 这样的代数式可以化简为 2a + 7b。只有同类项才能合并。
ng letters with numbers. If a = 4 and b = −2, then 2a + 3b = 2 × 4 + 3 × (−2) = 8 − 6 = 2.
代入就是用数字替换字母。如果 a = 4 且 b = −2，那么 2a + 3b = 2 × 4 + 3 × (−2) = 8 − 6 = 2。
Writing expressions follows rules: '5 more than n' is written n + 5, and '3 lots of x' is written 3x.
书写代数式遵循规则：'比 n 大 5' 写作 n + 5，'x 的 3 倍' 写作 3x。
8. Solving Equations | 解方程
An equation shows that two expressions are equal. To solve it, find the value of the unknown letter that makes the equation true.
方程表示两个代数式相等。解方程就是找出使方程成立的未知字母的值。
s to isolate the letter. For x + 7 = 15, subtract 7 from both sides: x = 15 − 7, so x = 8.
使用逆运算来分离字母。对于 x + 7 = 15，两边同时减去 7：x = 15 − 7，所以 x = 8。
th sides by 3: x = 21 ÷ 3, so x = 7. Always substitute your answer back to check it works.
对于 3x = 21，两边同时除以 3：x = 21 ÷ 3，所以 x = 7。一定要把答案代回原方程进行检验。
before multiplication. For 2x + 3 = 11, first subtract 3: 2x = 8, then divide by 2: x = 4.
两步方程需要先消去加法，再处理乘法。对于 2x + 3 = 11，先减去 3：2x = 8，然后除以 2：x = 4。
9. Geometry: Angles and Shapes | 几何：角与图形
Angles are measured in degrees. You need to identify acute, right, obtuse, straight and reflex angles from diagrams and descriptions.
角以度为单位测量。你需要从图形和描述中识别锐角、直角、钝角、平角和优角。
Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°，一个点周围的角之和为 360°。对顶角相等。
ways total 180°. If two angles are 65° and 45°, the third angle is 180° − 65° − 45° = 70°.
三角形内角和总是 180°。如果两个角分别是 65° 和 45°，第三个角就是 180° − 65° − 45° = 70°。
Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites. Their interior angles always add up to 360°.
四边形包括正方形、长方形、平行四边形、菱形、梯形和风筝形。它们的内角和总是 360°。
10. Perimeter, Area and Coordinates | 周长、面积与坐标
a shape. For a rectangle, add the lengths of all four sides, or use 2 × (length + width).
周长是图形外边缘的总长度。对于长方形，将四条边的长度相加，或使用公式 2 × (长 + 宽)。
rectangle is length × width. A rectangle 7 cm by 4 cm has area 28 cm² and perimeter 22 cm.
面积是图形内部的空间大小。长方形的面积等于长 × 宽。一个长 7 厘米、宽 4 厘米的长方形，面积是 28 平方厘米，周长是 22 厘米。
Compound shapes can be split into rectangles. Find each rectangle's area and add them together to get the total area.
复合图形可以分割成若干个长方形。分别求出每个长方形的面积，再把它们相加得到总面积。
Coordinates are written as (x, y). The x-coordinate gives the horizontal position, and the y-coordinate gives the vertical position. Translating a point 3 units right and 2 units up changes (4, 5) to (7, 7).
坐标写作 (x, y)。x 坐标表示水平位置，y 坐标表示垂直位置。将一个点向右平移 3 个单位、向上平移 2 个单位，(4, 5) 就变成 (7, 7)。
11. Averages and Data Handling | 平均数与数据处理
There are three main averages. The mode is the most common value, the median is the middle value when data is ordered, and the mean is the sum of all values divided by how many there are.
有三种主要的平均数。众数是最常出现的值，中位数是将数据排序后位于中间的值，平均数（均值）是所有数值之和除以数值的个数。
the median is 8, the mean is (3 + 7 + 8 + 8 + 12) ÷ 5 = 7.6, and the range is 12 − 3 = 9.
极差是最大值与最小值之差。对于数据集 3、7、8、8、12，众数是 8，中位数是 8，平均数是 (3 + 7 + 8 + 8 + 12) ÷ 5 = 7.6，极差是 12 − 3 = 9。
Data can be displayed in bar charts, pictograms, line graphs and pie charts. Always read the scale carefully and check that totals make sense.
数据可以用条形图、象形图、折线图和饼图来展示。一定要仔细读取刻度，并检查总数是否合理。
12. Probability | 概率
Probability measures how likely an event is to happen. It always lies between 0 and 1, where 0 means impossible and 1 means certain.
概率用来衡量一个事件发生的可能性大小。它总是在 0 和 1 之间，0 表示不可能，1 表示必然发生。
s ÷ total number of possible outcomes. The probability of rolling a 4 on a fair six-sided dice is 1/6.
对于等可能的结果，概率 = 有利结果数 ÷ 所有可能结果数。在一个均匀六面骰子上掷出 4 的概率是 1/6。
If a bag contains 3 red and 5 blue counters, the probability of picking a red counter is 3/8, and the probability of picking a blue counter is 5/8.
如果一个袋子里有 3 个红色和 5 个蓝色筹码，那么抽到红色筹码的概率是 3/8，抽到蓝色筹码的概率是 5/8。
ber of trials. In 200 rolls of a dice, you would expect about 200 × 1/6 ≈ 33 rolls of a 6.
要求期望结果数，就用概率乘以试验次数。掷骰子 200 次，你大约会期望掷出 200 × 1/6 ≈ 33 次 6。
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