📚 Year 7 AQA Further Maths: Core Concepts Breakdown | 核心知识点梳理
Welcome to your essential revision guide for Year 7 AQA Further Maths. This article breaks down all the core topics you'll encounter, from number operations to algebra and geometry, with clear explanations and practical examples. Mastering these concepts will build a strong foundation for higher-level mathematics.
欢迎阅读 Year 7 AQA 进阶数学的核心复习指南。本文梳理了你会接触到的所有核心主题，涵盖数的运算、代数、几何等，并配以清晰的解释和实用的例子。掌握这些概念将为更高层次的数学打下坚实的基础。
1. Number Properties and Operations | 数的性质与运算
Prime numbers are whole numbers greater than 1 that have exactly two factors: 1 and the number itself. The first few primes are 2, 3, 5, 7, 11, 13.
质数是大于 1 且只有两个因数（1 和它本身）的整数。最初的几个质数是 2, 3, 5, 7, 11, 13。
A factor is a number that divides another number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.
因数是能够整除另一个数且不留余数的数。例如，12 的因数有 1, 2, 3, 4, 6 和 12。
A multiple is the product of a number and any integer. The first five multiples of 6 are 6, 12, 18, 24, 30.
倍数是某数与任一整数的乘积。6 的前五个倍数是 6, 12, 18, 24, 30。
The highest common factor (HCF) of two numbers is the largest factor they share. The lowest common multiple (LCM) is the smallest multiple they share.
两个数的最大公因数（HCF）是它们共有的最大因数。最小公倍数（LCM）是它们共有的最小倍数。
racting a negative is equivalent to addition. For example, 5 + (-3) = 2, and 4 - (-2) = 6.
负数将数轴延伸到零的左侧。加上一个负数等同于减法，减去一个负数等同于加法。例如，5 + (-3) = 2，4 - (-2) = 6。
The order of operations, remembered by BIDMAS or BODMAS, instructs us to calculate in this order: Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
运算顺序（记住 BIDMAS 或 BODMAS）要求按以下顺序计算：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
Squares and cubes are powers: 5² = 25 and 4³ = 64. The square root of 25 is √25 = 5, and the cube root of 64 is ³√64 = 4.
平方与立方是乘方运算：5² = 25，4³ = 64。25 的平方根是 √25 = 5，64 的立方根是 ³√64 = 4。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
express parts of a whole. Equivalent fractions have the same value, e.g. 1/2 = 2/4 = 3/6.
分数表示整体的一部分。等值分数具有相同的值，例如 1/2 = 2/4 = 3/6。
(numerator larger than denominator) can be written as mixed numbers, such as 11/4 = 2 3/4.
假分数（分子大于分母）可以写成带分数，如 11/4 = 2 3/4。
tor and then add or subtract the numerators. For instance, 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
加减分数时，先找到公分母，再将分子相加或相减。例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12。
Decimals are based on tenths, hundredths, thousandths. Multiplying by powers of 10 moves the decimal point to the right; dividing moves it to the left.
小数以十分位、百分位、千分位为基础。乘以 10 的幂会将小数点右移；除以 10 的幂会将小数点左移。
Percentages mean 'out of 100' o convert a fraction to a percentage, multiply by 100. For example, 3/5 = (3/5) × 100 = 60%.
百分数表示“每一百份”。将分数化为百分数需要乘以 100。例如，3/5 = (3/5) × 100 = 60%。
Essential conversions are summarised below:
下面总结了必备的转换关系：
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333... | 33.3% |
| 1/5 | 0.2 | 20% |
Working fluently with these three forms is essential for solving ratio and probability problems later on.
熟练地在这三种形式之间切换对后续解决比和概率问题至关重要。
3. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra uses letters to represent unknown numbers or variables. An expression is a combination of numbers, variables and operation signs, such as 3x + 2.
代数用字母表示未知数或变量。表达式是数字、变量和运算符号的组合，例如 3x + 2。
To simplify an expression, collect like terms (terms with the same letter and power). For example, 5a + 3b - 2a + 4b simplifies to 3a + 7b.
化简表达式时，要合并同类项（具有相同字母和指数的项）。例如，5a + 3b - 2a + 4b 化简为 3a + 7b。
4x + 2y - x + 5y = 3x + 7y
the bracket by the term outside. For instance, 3(x + 4) = 3x + 12 and 2(3x - 5) = 6x - 10.
展开括号是指用括号外的项乘以括号内的每一项。例如，3(x + 4) = 3x + 12，2(3x - 5) = 6x - 10。
Factorising is the reverse: writing an expression as a product of a common factor and a bracket. For example, 6x + 9 can be factorised as 3(2x + 3).
因式分解是展开的反过程：将表达式写成公因式与括号的乘积。例如，6x + 9 可因式分解为 3(2x + 3)。
cing a letter with a number. If a = 2 and b = 5, then 3a + 2b = 3(2) + 2(5) = 6 + 10 = 16.
代入法是用数字替换字母。如果 a = 2，b = 5，那么 3a + 2b = 3(2) + 2(5) = 6 + 10 = 16。
d writing simple formula is another key skill, such as the area of a rectangle: A = l × w.
使用和书写简单公式也是一项关键技能，比如长方形的面积公式：A = l × w。
4. Solving Linear Equations | 解线性方程
A linear equation is an equation where the unknown appears only to the power of 1. The goal is to find the value of the variable that makes the equation true.
线性方程是未知数的指数仅为 1 的方程。目标是找到使方程成立的变量的值。
s and keep the equation balanced. For x + 7 = 15, subtract 7 from both sides to get x = 8.
求解时需要运用逆运算并保持方程两边平衡。对于 x + 7 = 15，两边同时减去 7 得到 x = 8。
f the equation has a multiplier, divide both sides. For 4x = 20, divide by 4 to get x = 5.
如果方程含有乘数，就将两边同时除以该数。对于 4x = 20，两边除以 4 得到 x = 5。
ract 2x from both sides to obtain 3x + 3 = 12, then subtract 3, divide by 3 to find x = 3.
当未知数出现在方程两边时，先消去系数较小的项。对于 5x + 3 = 2x + 12，两边减去 2x 得到 3x + 3 = 12，再减去 3，除以 3 得到 x = 3。
Always check your solution by substituting it back into the original equation.
始终将解代回原方程进行检验。
simplify. For 2(3x - 1) = 10, expand to 6x - 2 = 10, add 2 then divide by 6 to get x = 2.
有些方程含有括号。先展开，再化简。对于 2(3x - 1) = 10，展开得 6x - 2 = 10，加 2 然后除以 6 得到 x = 2。
Solve: 4(x + 5) = 3x + 23 → 4x + 20 = 3x + 23 → x = 3
5. Sequences and Patterns | 数列与规律
A sequence is an ordered list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant.
数列是按照某种规则排列的一串数。在等差数列中，相邻两项的差是一个常数。
For the sequence 3, 7, 11, 15, ..., the term-to-term rule is 'add 4' ion-to-term rule (nth term) is 4n - 1, because when n=1, 4(1)-1=3; n=2 gives 7, and so on.
对于数列 3, 7, 11, 15, ...，项与项之间的规则是“加 4”。位置到项的规则（第 n 项）是 4n - 1，因为当 n=1 时，4(1)-1=3；n=2 时结果为 7，依此类推。
nth term of 5, 8, 11, 14, ... is 3n + 2
To find the nth term of an arithmetic sequence, find the common difference (this becomes the coefficient of n) and then adjust by adding or subtracting to match the first term.
要找等差数列的第 n 项，首先找到公差（这就成为 n 的系数），然后通过加减一个数使首项吻合。
Sequences can also be generated from a rule given in words, such as 'start at 2 and multiply by 3 each time': 2, 6, 18, 54... This is a geometric sequence.
数列也可以由文字描述的规则生成，例如“从
Published by TutorHao | Year 7 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图