📚 KS3 Maths: Essential Maths 7 Core Key Points Explained | KS3 数学：Essential Maths 7 Core 知识点精讲
Essential Maths 7 Core builds a strong foundation for Key Stage 3 learners, covering number, algebra, geometry, and data handling. This article walks you through the most important topics, with clear English explanations paired with Chinese translations to support bilingual learning. Use this as your go-to revision guide to solidify understanding and boost confidence.
《Essential Maths 7 Core》为 KS3 阶段的学生打下扎实的数学基础，涵盖数、代数、几何和数据处理。本文带你梳理核心知识点，采用清晰的英文讲解配以中文翻译，助力双语学习。把它作为你的复习宝典，巩固理解、建立信心。
1. Place Value and Ordering Numbers | 位值与数字排序
Place value tells us the value of each digit in a number based on its position. In the number 3 527 416, the digit 3 represents 3 million, 5 means 5 hundred thousands, and so on. Understanding place value is essential for reading, writing, and comparing large numbers.
位值告诉我们每个数字在数中的价值取决于它的位置。例如 3 527 416 这个数，数字 3 代表 3 个百万，5 代表 5 个十万，以此类推。理解位值是读写和比较较大数字的基础。
argest using the symbols < (less than), > (greater than) and = (equal). For example, –4 < –1 < 0 < 3. When working with negative numbers, remember that –5 is smaller than –2 because it lies further left on the number line.
我们可以使用符号 <（小于）、>（大于）和 =（等于）对整数进行排序。例如 –4 < –1 < 0 < 3。处理负数时，记住 –5 比 –2 小，因为它在数轴上更靠左。
The digits in a decimal also have place value. The first digit after the decimal point is tenths, then hundredths, then thousandths. So 0.37 has 3 tenths and 7 hundredths.
小数中的数字同样有位值。小数点后第一位是十分位，第二位是百分位，第三位是千分位。因此 0.37 表示 3 个十分之一和 7 个百分之一。
2. Addition and Subtraction Methods | 加法和减法方法
ten; tens: 5 + 7 + 1 = 13, write 3, carry 1 hundred; hundreds: 4 + 2 + 1 = 7, result 734.
列竖式加法是可靠的笔算方法。按位值对齐数字，从最右边的个位列开始。每列相加，满十向前一列进一。例如 456 + 278：个位 6 + 8 = 14，写 4 进 1 个十；十位 5 + 7 + 1 = 13，写 3 进 1 个百；百位 4 + 2 + 1 = 7，结果是 734。
Similarly, column subtraction often requires borrowing. For 602 – 247, we cannot take 7 from 2 in the ones column, so we borrow from the tens column. The process continues until each column is subtracted correctly.
同理，列竖式减法则常需要借位。例如 602 – 247，个位的 2 不能减去 7，所以要向十位借 1。借位过程一直进行，直到每列正确相减。
lations. For instance, to find 198 + 47, think 200 + 47 = 247, then subtract 2 to get 245.
心算策略如数对分解、凑整补偿可以加快计算。比如计算 198 + 47，可以先算 200 + 47 = 247，再减 2 得 245。
3. Multiplication and Division Strategies | 乘法和除法策略
tal. Multiples of a number are found by multiplying it by integers. The lowest common multiple (LCM) is the smallest number that is a multiple of two or more numbers. Factors are numbers that divide exactly into another number, and the highest common factor (HCF) is the largest factor shared by two or more numbers.
熟记 12 × 12 以内的乘法表至关重要。一个数的倍数由这个数乘以整数得到。最小公倍数 (LCM) 是两个或多个数共有的倍数中最小的一个。因数是能整除某个数的数，最大公因数 (HCF) 是两个或多个数共有的最大因数。
ying as needed: 7 × 4 = 28 (carry 2), 2 × 4 + 2 = 10 (carry 1), 1 × 4 + 1 = 5, giving 508.
进行短乘法时，将数字按位值排列。计算 127 × 4 时，每位分别乘以 4，必要时进位：7 × 4 = 28（进 2），2 × 4 + 2 = 10（进 1），1 × 4 + 1 = 5，得到 508。
26, then 26 ÷ 5 = 5 remainder 1, carry to units to make 15, 15 ÷ 5 = 3. The answer is 153.
短除法设定类似。计算 765 ÷ 5，从左到右看 5 能容纳在每个数位中几次。7 ÷ 5 = 1 余 2，将 2 带到十位成 26，26 ÷ 5 = 5 余 1，带到个位成 15，15 ÷ 5 = 3。答案是 153。
4. Understanding Fractions | 分数理解
fy a fraction, divide both numerator and denominator by their HCF. 8/12 simplifies to 2/3.
分数表示整体的一部分或一个除法。在 3/4 中，3 是分子，4 是分母。等值分数具有相同的值，例如 1/2 = 2/4 = 4/8。化简分数时，分子分母同时除以它们的最大公因数。8/12 化简得 2/3。
. 1¾. You can convert between them: 7/4 = 1¾, because 4 goes into 7 once with 3 leftovers.
假分数的分子大于分母，如 7/4。带分数包含整数和分数，如 1¾。两者可互相转换：7/4 = 1¾，因为 4 除 7 得 1 余 3。
To compare fractions, it is helpful to rewrite them with a common denominator. For 2/3 and 3/5, the LCM of 3 and 5 is 15. Converting gives 10/15 and 9/15, so 2/3 > 3/5.
比较分数时，化成同分母会更容易。比如 2/3 和 3/5，3 和 5 的最小公倍数是 15。转化后得 10/15 和 9/15，因此 2/3 > 3/5。
5. Decimals and Their Operations | 小数及其运算
e them with the decimal points lined up, adding zeros as placeholders: 3.25 + 1.70 = 4.95.
小数将位值延伸到十分位、百分位和千分位。小数的运算规则与整数相同，但一定要对齐小数点。如添加 3.25 和 1.7 时，对齐小数点，用 0 占位：3.25 + 1.70 = 4.95。
100 = 56; 0.56 × 1000 = 560. Dividing by powers of 10 moves the decimal point to the left.
将小数乘以 10、100 或 1000，小数点向右移动相应位数。0.56 × 10 = 5.6；0.56 × 100 = 56；0.56 × 1000 = 560。除以 10 的幂，小数点向左移动。
2 × 3 = 6 → one decimal place in each factor gives two decimal places in the answer: 0.06.
两个小数相乘时，可以先忽略小数点，按整数相乘，再给结果加上小数点，使小数位数等于两个因数小数位数之和。例如 0.2 × 0.3 → 2 × 3 = 6 → 每个因数各有一位小数，所以答案应有两位小数：0.06。
6. Introduction to Percentages | 百分比入门
A percentage is a fraction with a denominator of 100. The symbol % means 'per cent' or 'out of 100'. So 25% . To change 3/5 to a percentage, find an equivalent fraction out of 100: 3/5 = 60/100 = 60%.
百分比是分母为 100 的分数。符号 % 表示“每一百”或“百分之”。因此 25% = 25/100 = 1/4。分数、小数和百分比的互相转换是核心技能。将 3/5 化为百分数，可先转换为同分母为 100 的分数：3/5 = 60/100 = 60%。
To convert a decimal to a percentage, multiply by 100. 0.7 = 0.7 × 100% = 70%. Conversely, 8% as a decimal is 8 ÷ 100 = 0.08.
将小数化为百分数，乘以 100 即可。0.7 = 0.7 × 100% = 70%。反过来，8% 写成小数是 8 ÷ 100 = 0.08。
Finding a percentage of a quantity often involves using a simple fraction or a multiplier. To find 20% of 45, think 10% is 4.5, so 20% is 9. Or multiply 45 by 0.2.
求一个数量的百分之几，常用简单分数或乘数。比如 45 的 20%：10% 是 4.5，所以 20% 是 9。或者直接用 45 × 0.2 计算。
7. Basic Algebraic Expressions | 基本代数表达式
number 3 is called the coefficient of a. We can simplify expressions by collecting like terms. For example, 4y + 5 + 2y – 3 = 6y + 2.
代数用字母表示未知数或变量。像 3a + 2 这样的表达式含有一个字母项 (3a) 和一个常数 (2)。数字 3 是 a 的系数。我们可以通过合并同类项来化简表达式。例如 4y + 5 + 2y – 3 = 6y + 2。
ication sign. So x × 5 is written as 5x. Division is expressed as a fraction: a ÷ 3 = a/3.
乘法书写时数字在前、字母在后，省略乘号。所以 x × 5 写成 5x。除法用分数表示：a ÷ 3 = a/3。
e expression 2x + 3 becomes 2(4) + 3 = 11. Always follow the order of operations (BIDMAS).
代入法是将变量替换成给定数字。若 x = 4，那么表达式 2x + 3 变成 2(4) + 3 = 11。始终遵循运算顺序（BIDMAS：括号、指数、乘除、加减）。
8. Solving Simple Equations | 解简单方程
An equation states that two expressions are equal. To solve an equation, we find the unknown value that makes the statement true. The key is to keep the equation balanced by performing the same operation on both sides.
方程表明两个表达式相等。解方程就是找到使等式成立的未知数的值。关键是通过在等号两边执行相同操作来保持平衡。
For x + 5 = 12, subtract 5 from both sides: x = 7. For x – For 2x = 16, divide both sides by 2: x = 8. For x/4 = 5, multiply both sides by 4: x = 20.
对于 x + 5 = 12，两边减 5：x = 7。对于 x – 3 = 9，两边加 3：x = 12。对于 2x = 16，两边除以 2：x = 8。对于 x/4 = 5，两边乘以 4：x = 20。
the solution is correct. Building these habits early is crucial for more complex algebra.
务必把答案代回原方程检验。若 2 × 8 = 16，解正确。尽早养成这些习惯对后续更复杂的代数学习至关重要。
9. Angles and Shapes | 角度与图形
Angles are measured in degrees (°) using the symbol °. Key angle types include acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), and reflex (greater than 180° but less than 360°).
角度用度 (°) 衡量，符号为 °。关键角型包括锐角（小于 90°）、直角（恰好 90°）、钝角（介于 90° 与 180° 之间）和优角（大于 180° 但小于 360°）。
Angles on a straight line add up to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In the diagram of intersecting lines, if one angle is 70°, the opposite angle is also 70°, and the adjacent angles are 110° each.
直线上的角之和为 180°，绕一点一周的角之和为 360°，对顶角相等。在相交直线图形中，若有一个角是 70°，其对顶角也是 70°，相邻的两个角各为 110°。
In a triangle, the interior angles always add up to 180°. If two angles are 50° and 60°, the third is 70°. An equilateral triangle has three 60° angles.
三角形内角和恒为 180°。若已知两角为 50° 和 60°，则第三个角是 70°。等边三角形的每个角都是 60°。
10. Perimeter and Area | 周长和面积
h). For a compound shape, add all the outer side lengths, being careful to identify missing lengths from given measurements.
周长是图形外周边界的总长度。长方形的周长 = 2 ×（长 + 宽）。对于组合图形，将所有外边长加起来，注意从已知尺寸中找出缺失的边长度。
length by width: A = l × w. If a rectangle is 5 cm long and 3 cm wide, its area is 15 cm².
面积衡量二维图形内部的空间，通常以平方单位表示，如 cm² 或 m²。长方形的面积等于长乘以宽：A = l × w。若长方形的长为 5 cm，宽为 3 cm，其面积为 15 cm²。
½ × base × height. A triangle with base 6 cm and height 4 cm has area ½ × 6 × 4 = 12 cm².
还可以通过数格子的方法估算不规则图形的面积。三角形的面积 = ½ × 底 × 高。底为 6 cm、高为 4 cm 的三角形，面积是 ½ × 6 × 4 = 12 cm²。
s. Converting between units is often required, such as 1 m = 100 cm, so 1 m² = 10 000 cm².
务必核对单位：周长是长度，面积是平方单位。经常需要进行单位换算，例如 1 m = 100 cm，所以 1 m² = 10 000 cm²。
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