📚 Year 7 OCR Further Mathematics: Core Concepts Review | Year 7 OCR 进阶数学：核心知识点梳理
Welcome to this comprehensive review of the core topics in Year 7 Further Mathematics following the OCR approach. This article will guide you through the key concepts you need to master, from number skills to geometry and data handling. Each section is designed to reinforce your understanding with clear explanations and examples.
欢迎阅读这篇针对Year 7 OCR进阶数学核心知识点的全面梳理。本文将带你回顾需要掌握的各个关键概念，从数字运算到几何与数据处理。每个部分都配有清晰的解释和示例，以巩固你的理解。
1. Number and Place Value | 数字与数位
Understanding place value allows you to read, write, and compare large numbers confidently. In Year 7, you extend your knowledge to millions, billions, and decimals.
理解数位能够让你自信地读、写和比较较大的数字。在七年级，你需要将知识扩展到百万、十亿以及小数。
The value of a digit depends on its position. For example, in 3,456,789, the digit '3' represents 3 million, '4' represents 400 thousand, and so on. You should be able to order and round numbers to a given place, using rules such as: if the next digit is 5 or more, round up; otherwise round down.
一个数字的值取决于它所在的位置。例如，在 3,456,789 中，数字“3”代表 3 百万，“4”代表 40 万，以此类推。你应当能够将数字按大小排序，并按照规则（下一位数字大于或等于5则进位，否则舍去）四舍五入到指定位数。
You will also work with decimal place value up to thousandths. Ordering decimals is a common exam skill: align the decimal points and compare digits from left to right.
你还将处理精确到千分位的小数位值。比较小数的大小是一项常见考试技能：对齐小数点，同时从左到右逐位比较。
2. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a number that divides exactly into another number. Multiples are the results of multiplying a number by an integer. Prime numbers have exactly two distinct factors: 1 and itself.
因数是指能够整除另一个数的数。倍数是指一个数与整数相乘得到的结果。质数正好有两个不同的因数：1 和它本身。
g factor trees, is also a key skill. For instance, 60 = 2² × 3 × 5. This helps with simplifying fractions and finding LCMs.
你需要能够找出两个或多个数的最大公因数（HCF）和最小公倍数（LCM）。使用因子树进行质因数分解也是一项关键技能。例如，60 = 2² × 3 × 5。这有助于化简分数和求最小公倍数。
Identify prime numbers up to 100 using the sieve of Eratosthenes. Recognise that 1 is not prime. Composite numbers have more than two factors.
使用埃拉托色尼筛法找出100以内的质数。注意1不是质数。合数有两个以上的因数。
3. Fractions, Decimals and Percentages | 分数、小数与百分比
ctions. Converting between fractions, decimals, and percentages is essential. For example, ⅗ = 0.6 = 60%.
你将学习等值分数、带分数和假分数。在分数、小数和百分比之间进行转换是必不可少的。例如，⅗ = 0.6 = 60%。
Operations with fractions include addition, subtraction, multiplication, and division. Remember to find a common denominator when adding or subtracting. When multiplying, multiply numerators together and denominators together. For division, invert the second fraction and multiply.
分数的运算包括加法、减法、乘法和除法。进行加减运算时，记得先找到公分母。乘法时，分子相乘，分母相乘。除法时，将第二个分数颠倒再相乘。
a percentage of another, and calculate percentage increase and decrease. Example: increase £40 by 15% gives 40 × 1.15 = £46.
百分比用于比较比例。你需要能够求出一个数量的百分之几，用一个数表示另一个数的百分比，并计算百分比增减。例如：将 40 英镑增加 15% 得到 40 × 1.15 = 46 英镑。
4. Negative Numbers | 负数
Negative numbers are numbers less than zero. You will use them in contexts such as temperature, bank balances, and elevations. Compare and order negative numbers with positive numbers on a number line.
负数是小于零的数。你将在温度、银行余额和海拔等情境中使用它们。要能在数轴上比较负数与正数的大小并排序。
e the rule: same signs add and keep the sign, different signs subtract and use the sign of the larger absolute value for addition of two negatives. For multiplication and division, same signs give a positive result, different signs give a negative result.
负数的计算：加上一个负数相当于减去该数的绝对值（例如 5 + (−3) = 2）。减去一个负数相当于加上该数的绝对值（5 − (−3) = 8）。使用以下法则：同号相加，符号不变；异号相减，用绝对值较大的符号。对于乘除法，同号得正，异号得负。
in temperature or changes in elevation. Practice writing expressions like −7 − (−3) = −4.
现实生活中的问题常涉及温度升降或海拔变化。练习书写像 −7 − (−3) = −4 这样的表达式。
5. Algebraic Expressions and Equations | 代数表达式与方程
Algebra involves using letters to represent unknown numbers. You will learn to simplify expressions by collecting like terms. For instance, 3a + 5 − a + 2 simplifies to 2a + 7.
代数是使用字母表示未知数的数学分支。你将学习通过合并同类项来化简表达式。例如，3a + 5 − a + 2 化简为 2a + 7。
16, subtract 4 from both sides to get 3x = 12, then divide both sides by 3 to find x = 4.
你需要能够将数值代入公式和表达式。解一步和两步方程是一项核心技能。例如，解方程 3x + 4 = 16，两边减去 4 得到 3x = 12，然后两边除以 3，得 x = 4。
em on a number line. Multiply out brackets using the distributive law: 3(y + 4) = 3y + 12.
会使用括号并理解运算顺序（BIDMAS/BODMAS）。你还可能遇到如 x > 2 这样的简单不等式，并能在数轴上表示出来。使用乘法分配律去括号：3(y + 4) = 3y + 12。
6. Sequences and Patterns | 序列与规律
A sequence is an ordered set of numbers following a rule. You will generate terms of a sequence given the nth term or a term-to-term rule. For example, the sequence with nth term 2n + 1 gives 3, 5, 7, 9 …
序列是按照一定规则排列的一组有序数字。你需要根据第 n 项或项到项的规则生成序列的项。例如，第 n 项为 2n + 1 的序列给出 3, 5, 7, 9 …
Recognising arithmetic sequences (with a common difference) and geometric sequences (with a common ratio) is expected. You can also explore number patterns in triangle numbers, square numbers, and Fibonacci-like sequences.
你需要认识等差数列（有公差）和等比数列（有公比）。还可以探究三角形数、平方数以及类似斐波那契数列的数字模式。
(a − d), where d is the difference and a is the first term. For the sequence 4, 7, 10, 13, …, d = 3, a = 4, so nth term = 3n + 1.
找出线性数列的第 n 项是一项关键技能：如果一次差为常数，第 n 项可表示为 dn + (a − d)，其中 d 为公差，a 为首项。对于序列 4, 7, 10, 13, …, d = 3, a = 4，因此第 n 项为 3n + 1。
7. Geometry: Angles and Shapes | 几何：角与图形
You should be able to identify different types of angles: acute (< 90°), right angle (90°), obtuse (between 90° and 180°), and reflex (> 180°). Measure and draw angles with a protractor.
你应该能识别不同类型的角：锐角（小于 90°），直角（90°），钝角（介于 90° 和 180° 之间）和优角（大于 180°）。会使用量角器测量和绘制角度。
Use angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In parallel lines, alternate angles and corresponding angles are equal, and co-interior angles sum to 180°.
运用角度事实：直线上的邻角之和为 180°，围绕一点的角度之和为 360°，对顶角相等。在平行线中，内错角相等，同位角相等，同旁内角之和为 180°。
Classify triangles and quadrilaterals by their properties. Understand the angle sum of a triangle (180°) and a quadrilateral (360°). Use symmetry and basic congruence. The exterior angle of a triangle equals the sum of the two opposite interior angles.
根据性质对三角形和四边形进行分类。理解三角形内角和为 180°，四边形内角和为 360°。会运用对称和基本的全等概念。三角形的外角等于其两个不相邻内角的和。
8. Perimeter, Area and Volume | 周长、面积与体积
w). For composite shapes, add all the outer lengths. Be careful with missing side lengths.
周长是图形边界的总长度。对于矩形，P = 2(l + w)。对于复合图形，将所有外部边长相加。注意找出缺失的边长。
erpendicular height, and area of a trapezium is ½(a + b)h. Remember to use the same units.
面积测量图形内部的空间。公式：矩形面积 A = l × w，三角形面积 A = ½ × b × h。平行四边形的面积为底 × 垂直高，梯形面积为 ½(a + b)h。记得使用统一的单位。
a as the total area of all faces. Convert between mm², cm², m² and mm³, cm³, m³ correctly.
由立方体构成的三维立体体积：数单位立方体的数量。对于长方体，V = l × w × h。你还可能探索表面积，即所有面的总面积。要正确换算平方毫米、平方厘米、平方米和立方毫米、立方厘米、立方米。
9. Coordinates and Graphs | 坐标与图形
Coordinates are written as (x, y) and plotted on a Cartesian grid. You should be confident working in all four quadrants, handling positive and negative coordinates.
坐标写成 (x, y) 形式，并在笛卡尔网格上描点。你需要能够熟练地在所有四个象限中处理正负坐标。
Drawing and interpreting straight-line graphs of simple linear functions, such as y = 2x +
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