📚 Core Knowledge Summary for Year 9 WJEC Mathematics | Year 9 WJEC 数学：核心知识点梳理
This article provides a structured overview of the core topics in the Year 9 WJEC Mathematics curriculum, helping students build a strong foundation for GCSE studies. Key areas include number operations, algebra, geometry, statistics and probability, with an emphasis on problem-solving and reasoning skills.
本文系统梳理了 Year 9 WJEC 数学课程的核心知识点，帮助学生为 GCSE 学习打下坚实基础。主要涵盖数的运算、代数、几何、统计与概率等领域，强调问题解决和推理能力的培养。
1. Working with Numbers | 数的运算
to right), Addition and Subtraction (left to right). For example, 3 + 4 × 2 = 11, not 14.
运算顺序遵循 BIDMAS 规则：括号、指数、乘除（从左到右）、加减（从左到右）。例如 3 + 4 × 2 = 11，而不是 14。
With negative numbers: adding a negative is the same as subtracting its positive; subtracting a negative is adding. Multiplying or dividing two negatives gives a positive result, while a positive and a negative give a negative.
负数运算规则：加一个负数等于减去它的正数；减去一个负数等于加上它的正数。两个负数相乘或相除结果为正，一正一负相乘或相除结果为负。
s: 1 and itself. Prime factor decomposition uses factor trees to break a number into a product of primes, e.g., 60 = 2² × 3 × 5.
因子是能整除某数的数。倍数是乘法表的结果。质数恰好有两个不同的因子：1 和它本身。质因数分解利用因子树将一个数分解为质因数的乘积，如 60 = 2² × 3 × 5。
Highest Common Factor (HCF) is the largest factor shared by two or more numbers. Lowest Common Multiple (LCM) is the smallest multiple they share. These are useful for simplifying fractions and solving problems.
最大公因数 (HCF) 是两个或多个数共有的最大因子。最小公倍数 (LCM) 是它们共有的最小倍数。这些概念有助于化简分数和解决问题。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent fractions are obtained by multiplying or dividing the numerator and denominator by the same non-zero number. To simplify a fraction, divide both by their HCF.
等值分数可通过分子分母同乘或同除一个非零数得到。化简分数时，将分子分母同时除以它们的最大公因数。
To add or subtract fractions, write them with a common denominator first. For multiplication, multiply the numerators and multiply the denominators. For division, multiply by the reciprocal of the divisor.
分数的加减法需要先通分，化成同分母再计算。乘法直接分子乘分子、分母乘分母。除法则是乘以除数的倒数。
Converting between fractions, decimals and percentages: 1/2 = 0.5 = 50%; 1/4 = 0.25 = 25% tage of a quantity, multiply the quantity by the percentage written as a decimal (e.g., 20% of 80 = 0.2 × 80 = 16).
分数、小数和百分数的转换：1/2 = 0.5 = 50%；1/4 = 0.25 = 25%。分数化小数用分子除以分母。求一个数的百分数，用它乘以百分数对应的小数（如 80 的 20% = 0.2 × 80 = 16）。
Increasing or decreasing by a percentage: to increase 200 by 15%, multiply 200 by 1.15. For a decrease of 15%, multiply by 0.85.
百分比增减：将 200 增加 15%，用 200 × 1.15；减少 15%，则用 200 × 0.85。
3. Powers and Roots | 幂与方根
2³ = 2 × 2 × 2 = 8. The square root √ undoes squaring, and the cube root ³√ undoes cubing.
幂表示底数自乘的次数：5² = 5 × 5 = 25，2³ = 2 × 2 × 2 = 8。平方根 √ 是平方的逆运算，立方根 ³√ 是立方的逆运算。
multiplied and divided using index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.
对于任何非零数 a，a⁰ = 1。利用指数律可进行幂的乘除：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。
Simplify 3⁴ × 3² = 3⁶, and 5⁷ ÷ 5³ = 5⁴. Always apply the law that matches the operation.
示例：3⁴ × 3² = 3⁶，5⁷ ÷ 5³ = 5⁴。要始终运用与运算相对应的指数律。
Estimating roots: √50 lies between √49=7 and √64=8, so roughly 7.1. Recognising perfect squares and cubes helps with estimation.
估算方根：√50 介于 √49=7 和 √64=8 之间，因此大约是 7.1。熟悉完全平方数和完全立方数有助于估算。
4. Ratio and Proportion | 比率与比例
Ratios compare quantities. To simplify a ratio, divide all parts by their HCF. For example, 8:12 simplifies to 2:3. Ratios can be expressed in the form 1:n by dividing both parts by the first number, e.g., 2:5 becomes 1:2.5.
比率用于比较数量。化简比率时，将所有部分除以它们的最大公因数。例如 8:12 化简为 2:3。可将比率表示为 1:n 的形式，即两部分同除以第一个数，如 2:5 化为 1:2.5。
shares. Divide the total quantity by this number, then multiply by each part. E.g., share £60 in the ratio 2:3 gives 2 parts = £24 and 3 parts = £36.
按给定比例分配数量时，先将比的部分相加得到总份数，再用总数量除以总份数，然后分别乘以各部分。例如按 2:3 分配 £60，2 份得 £24，3 份得 £36。
Proportion problems involve scaling quantities up or down. Use the unitary method: find the value of one unit first, then multiply. Scale diagrams and maps use ratios such as 1:50 000 to represent real distances.
比例问题涉及数量的按比例放大或缩小。常用单位法：先求出一个单位的值，再乘以所需份数。比例尺图及地图使用如 1:50000 的比率表示实际距离。
5. Algebra Basics | 代数基础
Algebra uses letters to stand for unknown numbers. An expression like 3a + 2b - a collects to 2a + 2b by combining like terms (terms with the same variable and power).
代数是使用字母表示未知数的数学分支。表达式如 3a + 2b - a 可通过合并同类项（变量和指数均相同的项）化简为 2a + 2b。
12. If there is a negative outside, remember to multiply carefully: -2(x - 5) = -2x + 10.
展开括号是将括号内每一项与括号外的项相乘：4(x + 3) = 4x + 12。括号外如果是负数，注意符号变化：-2(x - 5) = -2x + 10。
ite it outside a bracket. For example, 6x + 9 = 3(2x + 3). Always check by expanding back.
因式分解是展开的逆过程：找出公因子并将其写在括号外。例如 6x + 9 = 3(2x + 3)。务必通过展开来验证。
ven numbers, then calculating. If a = 3 and b = 4, then 2a + 3b = 2×3 + 3×4 = 6 + 12 = 18.
代入法是将字母替换为给定的数值，再进行计算。若 a = 3，b = 4，则 2a + 3b = 2×3 + 3×4 = 6 + 12 = 18。
6. Equations and Inequalities | 方程与不等式
2x + 5 = 13, subtract 5 then divide by 2, giving x = 4. Always check by substituting back.
方程表示两个表达式相等。解方程时使用逆运算：对于 2x + 5 = 13，先减 5 再除以 2，得 x = 4。务必代入原方程检验。
th sides, bring all variable terms to one side and constants to the other: 5x - 3 = 2x + 9 ⇒ 5x - 2x = 9 + 3 ⇒ 3x = 12 ⇒ x = 4.
当未知数出现在等号两边时，将所有含未知数的项移到一边，常数项移到另一边：5x - 3 = 2x + 9 ⇒ 5x - 2x = 9 + 3 ⇒ 3x = 12 ⇒ x = 4。
. Solve like equations, except reverse the inequality sign when multiplying or dividing by a negative number.
不等式使用符号 <（小于）、>（大于）、≤（小于等于）、≥（大于等于）。解法与方程类似，但乘以或除以负数时，需要改变不等号方向。
r < or >, and a closed circle for ≤ or ≥. For integer solutions, list the possible values.
在数轴上表示解集：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。整数解可直接列出所有可能的值。
7. Graphs and Sequences | 图像与数列
A sequence is a list of numbers following a rule. The term-to-term rule describes how to get from one term to the next. The position-to-term rule (nth term) allows you to find any term directly.
数列是按一定规则排列的一列数。项对项规则描述如何从一项得到下一项。位置对项规则（第 n 项）可直接求出任意项。
For a linear sequence with constant difference a, the nth term is an + b, where b is found by checking the 0th term or adjusting. E.g., 5, 8, 11, 14… has nth term 3n + 2.
对于公差为 a 的线性数列，第 n 项公式为 an + b，其中 b 可通过反向推算第 0 项或调整求得。例如 5, 8, 11, 14… 的第 n 项为 3n + 2。
s gradient m and y-intercept c. The gradient measures steepness; lines with the same gradient are parallel.
线性图像为直线。通过填写 x 与 y 的数值表来绘制。方程 y = mx + c 中，m 是斜率，c 是 y 轴截距。斜率表示倾斜程度，斜率相等的直线互相平行。
The midpoint of a line segment between (x₁,y₁) and (x₂,y₂) is found by averaging the coordinates: ((x₁+x₂)/2, (y₁+y₂)/2). Real-life graphs show relationships like distance–time or conversion graphs.
线段 (x₁,y₁) 到 (x₂,y₂) 的中点坐标是将端点坐标取平均值：((x₁+x₂)/2, (y₁+y₂)/2)。实际情境图包括距离–时间图和转换图。
8. Geometry and Measures | 几何与测量
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. Recognise alternate, corresponding and allied (co-interior
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