📚 Year 7 WJEC Further Mathematics: Core Knowledge Points | WJEC 7年级进阶数学：核心知识点梳理
This article provides a comprehensive overview of the core topics in the Year 7 WJEC Further Mathematics course. It is designed to help students consolidate key concepts and prepare for assessments. We break down each area with clear explanations and examples, ensuring you can tackle more challenging problems with confidence.
本文全面梳理了 WJEC 7 年级进阶数学课程的核心知识点，旨在帮助学生巩固关键概念并为评估做好准备。我们将每个领域拆解为清晰解释与实例，确保你能自信应对更具挑战性的问题。
1. Number Theory and Operations | 数论与运算
Place value extends to millions and decimals, enabling you to read, write and order very large or very small numbers accurately.
位值概念延伸到百万位和小数，使你能够准确读、写和排列非常大或非常小的数。
Factors, multiples and prime numbers form the backbone of number theory; a prime number has exactly two distinct factors, while composite numbers have more.
因数、倍数和质数是数论的基础；质数恰好有两个不同的因数，而合数则有多个。
The order of operations (BIDMAS/BODMAS) ensures consistent evaluation: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
运算顺序（括号、指数、乘除、加减）保证了一致性：先算括号，再指数，然后乘除（从左到右），最后加减（从左到右）。
operations with negative numbers follow strict rules, e.g., −3 × −4 = 12 and −5 + 2 = −3.
负数的整数运算遵循严格规则，例如 −3 × −4 = 12，−5 + 2 = −3。
Rounding to a specified number of decimal places or significant figures is crucial for estimating and checking answers.
按指定的小数位数或有效数字进行舍入，对于估算和检验答案至关重要。
2. Algebraic Expressions and Simplification | 代数表达式与化简
A term is a single number, variable or product of numbers and variables; like terms have the same variable parts and can be combined.
项是单个数字、变量或数字与变量的乘积；同类项具有相同的变量部分，可以合并。
Simplifying expressions such as 4a + 3b − 2a + b gives 2a + 4b by collecting like terms.
通过合并同类项化简表达式，如 4a + 3b − 2a + b 得到 2a + 4b。
The distributive law allows us to expand brackets:
分配律允许我们展开括号：
a(b + c) = ab + ac
a(b + c) = ab + ac
g reverses this process by taking out the highest common factor, e.g., 6x + 9 = 3(2x + 3).
因式分解则反过来提取最大公因数，例如 6x + 9 = 3(2x + 3)。
Substituting numbers into algebraic expressions and formulas helps evaluate real‑life situations, such as using a formula for the perimeter of a rectangle.
将数字代入代数表达式和公式有助于解决实际问题，比如使用长方形周长公式。
3. Linear Equations | 线性方程
A linear equation in one variable can be solved by performing the same operation on both sides to isolate the unknown.
一元线性方程可以通过对两边进行相同运算来求解，从而求出未知数。
For example, solving 2x + 5 = 13: subtract 5 (2x = 8), then divide by 2 (x = 4).
例如，解方程 2x + 5 = 13：两边减 5 得 2x = 8，再除以 2 得 x = 4。
Equations may involve brackets or fractions; always clear them first using expanding or multiplying both sides by the denominator.
方程可能包含括号或分数；通常先用展开或两边同乘分母来消去它们。
Forming equations from word problems is a key skill: represent the unknown with a variable, write the relationship and solve.
根据文字题列方程是一项关键技能：用变量表示未知数，写出关系式并求解。
Checking your solution by substituting back into the original equation is essential to avoid errors.
将解代入原方程进行检验，这是避免错误的关键步骤。
4. Fractions, Decimals and Percentages | 分数、小数与百分比
Equivalent fractions simplify calculations; multiplying or dividing the numerator and denominator by the same non‑zero number retains the value.
等值分数可简化计算；分子分母同乘或同除一个非零数，分数值不变。
Adding and subtracting fractions require a common denominator, while multiplying fractions is straightforward: multiply numerators and denominators.
分数加减需要公分母，而分数乘法直接分子相乘、分母相乘。
Converting between fractions, decimals and percentages is a fundamental skill: ⅗ = 0.6 = 60%.
分数、小数和百分比之间的转换是基本技能：⅗ = 0.6 = 60%。
a percentage of a quantity, multiply the quantity by the percentage as a decimal; e.g., 15% of 200 = 200 × 0.15 = 30.
求一个量的百分比，用这个量乘以百分比化成的小数；例如 200 的 15% = 200 × 0.15 = 30。
Percentage increase and decrease problems model real‑life situations like discounts and interest.
百分比的增减问题模拟了折扣、利息等现实情境。
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities; it can be simplified by dividing all parts by a common factor.
比用来比较两个或多个量；可通过用公因数去除所有部分来化简。
Sharing a quantity in a given ratio means dividing it into parts according to the ratio; for example, split £100 in the ratio 3:2 gives £60 and £40.
按给定比例分配一个量，意味着将其分成与比相对应的份数；例如，按 3:2 分 100 英镑得到 60 和 40 英镑。
e or decrease at the same rate; the graph of y = kx is a straight line through the origin.
正比例表示两个量以相同速率增减；y = kx 的图像是一条过原点的直线。
Understanding scale in maps and diagrams requires ratio reasoning: a scale of 1:100 means 1 cm on paper represents 100 cm in reality.
理解地图和图表中的比例尺需要比例的推理：比例尺 1:100 意味着纸上 1 cm 代表实际 100 cm。
6. Geometry: Angles, Triangles and Quadrilaterals | 几何：角、三角形与四边形
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.
直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。
In triangles, interior angles add up to 180°; isosceles triangles have two equal base angles, and equilateral triangles have all angles 60°.
三角形内角和为 180°；等腰三角形有两个相等的底角，等边三角形的每个角都是 60°。
Quadrilaterals have angle sum 360°; special types include parallelograms, rectangles, squares and trapeziums, each with their own properties.
四边形内角和为 360°；特殊的四边形包括平行四边形、矩形、正方形和梯形，各自具有不同的性质。
Parallel lines cut by a transversal produce equal corresponding angles and alternate angles, which help solve unknown angle problems.
平行线与截线产生相等的同位角及内错角，可用来求解未知角。
Basic constructions using a compass and protractor include angle bisectors and perpendicular bisectors, building geometric reasoning.
使用圆规和量角器进行基本作图，包括角平分线和垂直平分线，培养几何推理能力。
7. Perimeter, Area and Volume | 周长、面积与体积
The perimeter of a shape is the total distance around its edges, measured in units such as cm or m.
周长是一个图形边缘的总长，以厘米或米为单位。
Area formulae for rectangles and triangles: rectangle A = l × w, triangle A = ½ × b × h.
长方形和三角形的面积公式：长方形 A = 长 × 宽，三角形 A = ½ × 底 × 高。
For compound shapes, split them into simpler rectangles and triangles, calculate each area and sum them.
对于组合图形，将其拆分为简单的长方形和三角形，计算各自面积再相加。
Volume measures the space inside a 3D shape; for a cuboid, V = length × width × height.
体积测量三维图形内部空间；对于长方体，体积 = 长 × 宽 × 高。
rea adds the areas of all faces of a 3D solid; for a cube with side s, surface area = 6s².
表面积是把立体图形所有面的面积相加；边长为 s 的立方体，表面积 = 6s²。
8. Statistics: Collecting and Representing Data | 统计：数据收集与表示
Data can be collected through surveys, experiments or observations; designing clear data collection tables helps organise raw information.
数据可通过调查、实验或观察收集；设计清晰的数据收集表有助于整理原始信息。
Bar charts, pictograms and line graphs display discrete and continuous data effectively; each needs a title, labelled axes and consistent scales.
条形图、象形图和折线图能有效展示离散与连续数据；每张图都必须有标题、坐标轴标签和一致的刻度。
Frequency tables and grouped frequency tables summarise data; the modal class is the group with the highest frequency.
频数表和分组频数表可汇总数据；众数所在组是频数最高的组。
Mean, median, mode and range are measures of average and spread; the mean is calculated by sum of values ÷ number of values.
平均数、中位数、众数和极差是集中趋势和离散程度的度量；平均数 = 数值总和 ÷ 数值个数。
represent proportions: each sector angle = (category frequency ÷ total frequency) × 360°.
饼图用来表示比例：每部分的圆心角 =（某类频数 ÷ 总频数）× 360°。
9. Probability Basics | 概率基础
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain).
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。
kely outcomes, probability = (number of favourable outcomes) ÷ (total number of outcomes).
对于等可能的结果，概率 =（有利结果的数量）÷（可能结果的总数）。
The probability of an event not happening is 1 minus the probability that it does happen.
事件不发生的概率等于 1 减去它发生的概率。
Sample space diagrams list all possible outcomes and help find probabilities of combined events like rolling two dice.
样本空间图列出所有可能结果，有助于求两个事件组合的概率，如掷两个骰子。
Experimental probability is based on actual trials, while theoretical probability is what we expect; the two converge with more trials.
实验概率基于实际试验，理论概率则是期望值；试验次数越多，两者越接近。
10. Sequences, Functions and Graphs | 数列、函数与图像
A sequence is a list of numbers following a rule; the term‑to‑term rule describes how to move from one term to the next.
数列是按规则排列的一列数；逐项规则描述如何从前一项得到后一项。
For linear sequences, the nth term takes the form an + b, where a is the common difference and b is the zero‑th term.
对于线性数列，第 n 项的形式为 an + b，其中 a 是公差，b 是第零项。
Coordinates are written (x, y); plotting points and joining them reveals the graph of a relationship, often a straight line for linear functions.
坐标写作 (x, y)；描点并连线可显示函数关系的图像，线性函数图像通常是直线。
A function machine takes an input, applies a rule and produces an output; this helps understand substitution and inverse operations.
函数机器接受输入，按规则运算后得到输出；这有助于理解代入与逆运算。
Distance–time graphs show how distance changes over time; the slope (steepness) indicates speed.
距离–时间图展示距离随时间的变化；斜率（陡度）表示速度。
11. Further Algebra: Inequalities and Simple Formulae | 进阶代数：不等式与简单公式
than or equal to), > (greater than), ≥ (greater than or equal to) to compare expressions.
不等式使用符号 <（小于）、≤（小于或等于）、>（大于）、≥（大于或等于）来比较表达式。
Solving inequalities works like equations, but multiplying or dividing by a negative number reverses the inequality sign.
解不等式的方法与解方程类似，但当两边同乘或同除一个负数时，需要反转不等号。
e: open circles for < or >, closed circles for ≤ or ≥, and shading the range of solutions.
在数轴上表示不等式：用空心圆表示 < 或 >，实心圆表示 ≤ 或 ≥，然后涂出解的区间。
he subject is an essential algebraic skill; for example, make t the subject of v = u + at.
变换简单公式的主项是一项重要的代数技能；例如，将 v = u + at 改写为用 t 表示。
roblems with unknowns on both sides, such as 5x − 3 = 2x + 9, develops systematic equation‑solving strategies.
求解未知数在两边的问题，如 5x − 3 = 2x + 9，能发展系统的解方程策略。
12. Problem Solving and Reasoning | 问题解决与推理
Mathematical reasoning involves spotting patterns, making conjectures and testing them with examples and counterexamples.
数学推理包括发现模式、提出猜想，并用实例和反例进行检验。
Breaking a multi‑step problem into smaller parts, drawing diagrams and using trial and improvement are powerful strategies.
将多步问题分解为小部分、画图、使用试错改进法，都是强有力的解题策略。
Word problems often combine several topics: for example a ratio question may also involve converting percentages or using fractions.
文字题常常结合多个知识点：例如一道比例题可能还需转换百分数或使用分数。
Communicating your method clearly using correct mathematical language earns marks in assessments and solidifies understanding.
用正确的数学语言清晰地阐述解题方法，在评估中能获得分数，并巩固理解。
Regular practice of mixed topic questions builds confidence and identifies areas that need more revision, especially under timed conditions.
经常练习混合主题的题目可以建立自信，并找出需要更多复习的薄弱点，尤其是在限时条件下。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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