📚 Year 7 WJEC Further Mathematics: Core Knowledge Review | Year 7 WJEC 进阶数学：核心知识点梳理
In Year 7, students begin to deepen their mathematical understanding, moving from simple arithmetic to more advanced ideas such as algebraic manipulation, geometric reasoning, ratio, proportion and data handling. This article summarises the core knowledge areas for WJEC Further Mathematics at this level, so , helping you can consolidate essential skills, spot the links between topics, and prepare confidently for assessments.
在七年级，学生开始加深对数学的理解，从简单的算术转向更具挑战性的概念，如代数运算、几何推理、比与比例以及数据处理。本文梳理了 WJEC 进阶数学在这一阶段的核心知识点，帮助你巩固关键技能、看清各主题之间的联系，并为评估做好准备。
1. Number Skills and BIDMAS | 数字运算与运算顺序
l rule: Brackets, Indices, Division and Multiplication (worked from left to right), then Addition and Subtraction (worked from left to right). Without it, an expression such as 3 + 5 × 2 can easily be misinterpreted.
所有进阶数学的学习都建立在扎实的数字运算之上，其中最关键的是运算顺序。BIDMAS（也写作 BODMAS）是一条重要规则：括号、指数、除法与乘法（从左到右）、加法与减法（从左到右）。没有它，像 3 + 5 × 2 这样的表达式就容易被误解。
3 + 5 × 2 = 3 + 10 = 13
(3 + 5) × 2 = 8 × 2 = 16
Index notation is part of the same rule set. Students should be able to evaluate powers, apply the index laws, and recognise that a number to the power zero equals one.
指数记法属于同一套规则。学生应能计算乘方、运用指数运算法则，并认识到任何非零数的零次幂都等于 1。
- 2³ × 2⁴ = 2⁷ = 128
- 2⁵ ÷ 2² = 2³ = 8
- (3²)³ = 3⁶ = 729
- 5⁰ = 1
Year 7 pupils also need to be confident with negative numbers. Adding a negative is the same as subtraction; subtracting a negative is the same as addition. For multiplication and division, two negatives make a positive.
七年级学生还需要熟练处理负数。加上一个负数相当于减法；减去一个负数相当于加法。在乘除运算中，两个负数相乘或相除得到正数。
- (−3) + (−5) = −8
- 6 − (−2) = 6 + 2 = 8
- (−4) × (−3) = 12
- (−12) ÷ 4 = −3
- (−2)³ = −8, but (−2)² = 4
2. Fractions, Decimals and Percentages | 分数、小数与百分数
ctions, decimals and percentages is a key skill at this level. For example, 1/2 = 0.5 = 50% quivalent fractions and compare them by converting to a common denominator or to decimals.
分数、小数与百分数之间的相互转换是这一阶段的关键技能。例如，1/2 = 0.5 = 50%。学生必须能够化简分数、寻找等值分数，并通过通分或化为小数来比较大小。
1/2 = 0.5 = 50% 3/4 = 0.75 = 75% 1/5 = 0.2 = 20% 1/8 = 0.125 = 12.5%
Some fractions produce terminating decimals, while others produce recurring decimals. Recognising the difference helps with rounding and with deciding on an exact or approximate answer.
有些分数化成有限小数，有些则化成循环小数。识别这一差别有助于取近似值，也有助于判断何时需要精确答案、何时可以估算。
- 1/4 = 0.25 (terminating / 有限小数)
- 1/3 = 0.333... = 0.3̇ (recurring / 循环小数)
- 2/7 = 0.285714285714... (recurring / 循环小数)
To compare two fractions, either rewrite them with a common denominator, or convert both to decimals. For instance, 3/5 and 5/8 become 24/40 and 25/40, so 5/8 is slightly larger.
比较两个分数时，可以通分，也可以都化为小数。例如 3/5 和 5/8 化为 24/40 和 25/40，因此 5/8 略大一些。
3. Calculating with Fractions and Percentages | 分数的四则运算与百分比计算
Adding and subtracting fractions requires a common denominator. Multiplying fractions is simpler: multiply the numerators together and the denominators together. Dividing by a fraction is the same as multiplying by its reciprocal.
分数加减法需要先通分。分数乘法更简单：分子乘分子，分母乘分母。除以一个分数，相当于乘以它的倒数。
2/3 + 1/4 = 8/12 + 3/12 = 11/12
3/4 × 2/5 = 6/20 = 3/10
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
le, 2 1/2 becomes 5/2, and 1 1/3 becomes 4/3, so 2 1/2 ÷ 1 1/3 = 5/2 × 3/4 = 15/8 = 1 7/8.
带分数在乘除之前应先化为假分数。例如 2 1/2 化为 5/2，1 1/3 化为 4/3，因此 2 1/2 ÷ 1 1/3 = 5/2 × 3/4 = 15/8 = 1 7/8。
Percentages are used to describe parts of a whole. Finding a percentage of an amount, and working out the original amount after a percentage change, are both examinable skills.
百分比用来描述整体中的一部分。求某个量的百分之几，以及已知变化后的结果反推原量，都是可考查的技能。
- 15% of 80 = 0.15 × 80 = 12
- Increase 40 by 10%: 40 × 1.1 = 44
- Decrease 60 by 15%: 60 × 0.85 = 51
- After a 20% rise a price is £96, so the original price was 96 ÷ 1.2 = £80
4. Ratio, Proportion and Percentage Change | 比、比例与百分比变化
A ratio compares quantities of the same kind. Ratios are simplified in the same way as fractions, by dividing every part by a common factor.
比用于比较同类量之间的关系。比的化简方式与分数相同，即把每一部分同时除以公因数。
- 12 : 18 = 2 : 3 (divide both parts by 6)
- 45 : 30 : 15 = 3 : 2 : 1 (divide all parts by 15)
Sharing in a given ratio uses the idea of total parts. If £60 is shared in the ratio 2 : 3, there are 5 parts, each worth £12, so the shares are £24 and £36.
按比分配要用到"总份数"的思想。若把 60 英镑按 2 : 3 分配，共 5 份，每份 12 英镑，因此两部分分别为 24 英镑和 36 英镑。
2 + 3 = 5 parts → 60 ÷ 5 = 12 → 2 × 12 = 24 and 3 × 12 = 36
Direct proportion means that as one quantity multiplies, the other multiplies by the same factor. Recipes, scale drawings and unit conversions all rely on this idea.
正比例意味着一个量按某个倍数变化时，另一个量也按相同倍数变化。配方问题、比例尺作图和单位换算都依赖这一思想。
- If 4 pens cost £6, then 10 pens cost 6 ÷ 4 × 10 = £15
- A scale of 1 : 50 means 1 cm on the drawing represents 50 cm in real life
- 1 km = 1000 m, 1 m = 100 cm, 1 hour = 60 minutes
5. Algebraic Notation, Simplifying and Substitution | 代数记号、化简与代入求值
Algebra uses letters to represent unknown or variable numbers. Students should be able to identify terms, coefficients, constants and like terms, and combine like terms correctly.
代数用字母表示未知数或变量。学生应能识别项、系数、常数和同类项，并正确地合并同类项。
- 3a + 5a = 8a
- 7b − 2b = 5b
- 4x + 3y − x + 2y = 3x + 5y
- 6p² + 2p² = 8p² (but 6p² + 2p cannot be simplified)
hile the second means 4x × 4x = 16x². Reading notation precisely prevents many lost marks.
常见错误是把 4x² 与 (4x)² 混淆。前者表示 4 × x × x，后者表示 4x × 4x = 16x²。准确读懂记号可以避免许多失分。
Substitution means replacing letters with given values and evaluating carefully, respecting BIDMAS and signs throughout.
代入求值就是用给定的数值替换字母，并遵守运算顺序与符号规则，仔细计算。
If x = 3 and y = −2: 5x − 2y = 15 − (−4) = 19
Index laws also apply to algebraic terms, and they follow exactly the same rules as numerical powers.
指数运算法则同样适用于代数式，其规则与数字幂完全一致。
- a³ × a⁴ = a⁷
- a⁶ ÷ a² = a⁴
- (a²)³ = a⁶
- a⁰ = 1
6. Expanding Brackets and Factorising | 展开括号与因式分解
Expanding brackets means multiplying out. Every term inside the bracket must be multiplied by the term outside, and signs must be handled with care.
展开括号就是做乘法。括号内每一项都必须与括号外的项相乘，同时要小心处理符号。
- 3(x + 4) = 3x + 12
- −2(3x − 5) = −6x + 10
- x(x + 7) = x² + 7x
Double brackets are expanded using the distributive law, often remembered as FOIL: First, Outer, Inner, Last. The result should then be simplified by collecting like terms.
双括号的展开运用分配律，常记作 FOIL：首项、外项、内项、末项相乘，然后合并同类项化简。
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
(x + 4)² = x² + 8x + 16 (x + 5)(x − 5) = x² − 25
Factorising is the reverse process: writing an expression as a product of factors. Always look for a common factor first, and check your answer by expanding it back.
因式分解是相反的过程：把一个表达式写成若干因式的乘积。首先要寻找公因式，最后通过展开检查答案是否正确。
- 6x + 9 = 3(2x + 3)
- x² + 7x = x(x + 7)
- 4a² − 8a = 4a(a − 2)
7. Linear Equations and Inequalities | 一元一次方程与不等式
An equation states that two expressions are equal. To solve it, keep the equation balanced by performing the same operation on both sides until the unknown stands alone.
方程表示两个表达式相等。解方程时要保持等式两边平衡：对两边做同样的运算，直到未知数单独留在一边。
3x + 5 = 20 → 3x = 15 → x = 5
2(x − 3) = 10 → x − 3 = 5 → x = 8
x/4 + 2 = 5 → x/4 = 3 → x = 12
Always substitute your answer back into the original equation to check it. This habit catches sign errors and arithmetic slips before they cost marks.
一定要把答案代回原方程检验。这个习惯能在因符号错误或计算失误丢分之前把它们找出来。
Inequalities are solved in almost the same way, but there is one crucial extra rule: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
不等式的解法几乎相同，但有一条关键附加规则：当两边同时乘以或除以负数时，不等号方向必须反转。
- 2x + 1 < 9 → 2x < 8 → x < 4
- −2x > 8 → x < −4
- On a number line, an open circle shows < or >, and a filled circle shows ≤ or ≥
8. Sequences and the nth Term | 数列与第 n 项
A sequence is an ordered list of numbers following a rule. In an arithmetic sequence the difference between consecutive terms is constant, and this difference is called the common difference.
数列是按某种规则排列的一列数。在等差数列中，相邻两项之差恒定，这个差称为公差。
3, 7, 11, 15, 19, ... common difference = 4 nth term = 4n − 1
ork out what must be added or subtracted so that the rule gives the first term when n = 1.
求等差数列的第 n 项时，先找出公差 d，再确定需要加减多少，使 n = 1 时该规则恰好给出首项。
- Check: 4 × 1 − 1 = 3 ✓ and 4 × 4 − 1 = 15 ✓
- Sequence 20, 17, 14, 11, ... has d = −3 and nth term = 23 − 3n
Some special sequences appear repeatedly in questions and are worth memorising, including square numbers, triangular numbers and the Fibonacci sequence.
有些特殊数列在题目中反复出现，值得记忆，包括平方数、三角形数和斐波那契数列。
- Square numbers: 1, 4, 9, 16, 25, ... with nth term n²
- Triangular numbers: 1, 3, 6, 10, 15, ... with nth term n(n + 1) ÷ 2
- Fibonacci: 1, 1, 2, 3, 5, 8, 13, ... where each term is the sum of the previous two
9. Coordinates, Graphs and Straight Lines | 坐标、图像与直线
Coordinates are written as ordered pairs (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The axes divide the plane into four quadrants, and signs change from quadrant to quadrant.
坐标写成有序数对 (x, y)，其中 x 是到原点的水平距离，y 是到原点的竖直距离。坐标轴把平面分成四个象限，各象限内坐标的正负号不同。
- Quadrant I: (+, +) Quadrant II: (−, +)
- Quadrant III: (−, −) Quadrant IV: (+, −)
- Midpoint of (2, 3) and (8, 11) is ((2 + 8) ÷ 2, (3 + 11) ÷ 2) = (5, 7)
gradient (steepness) and c is the y-intercept, the point where the line crosses the y-axis.
直线的图像可写成 y = mx + c 的形式，其中 m 是斜率（倾斜程度），c 是 y 轴截距，即直线与 y 轴交点的纵坐标。
y = 2x + 1 passes through (0, 1), (1, 3), (2, 5), (3, 7)
ave the form x = k and their gradient is undefined, because there is no horizontal change.
水平线形如 y = k，斜率为 0。竖直线形如 x = k，其斜率不存在，因为水平方向没有变化量。
- y = 3 is horizontal and crosses the y-axis at 3
- x = −2 is vertical and passes through every point with x-coordinate −2
10. Angles, Perimeter and Area | 角、周长与面积
Angle facts follow directly from the geometry of lines and shapes. Angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal.
角的性质直接来自直线与图形的几何关系。直线上的邻补角之和为 180°，围绕一点的角之和为 360°，对顶角相等。
- Angles in a triangle add to 180°
- Angles in a quadrilateral add to 360°
- With parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°
Perimeter is the total distance around a shape, while area measures the space inside it. Compound shapes should be split into rectangles, triangles or circles, and the parts added together.
周长是图形外边界的总长度，面积则度量图形内部的空间大小。复合图形应拆分为长方形、三角形或圆，再把各部分相加。
- Rectangle: perimeter = 2(l + w), area = l × w
- Triangle: area = ½ × base × height
- Circle: circumference = 2πr, area = πr²
- Cuboid: volume = l × w × h
Units matter. Length is measured in mm, cm, m or km; area in mm², cm² or m²; and volume in cm³ or m³. Always check that units are consistent before calculating.
单位很重要。长度用毫米、厘米、米或千米；面积用平方毫米、平方厘米或平方米；体积用立方厘米或立方米。计算前务必确认单位统一。
11. Statistics and Probability | 统计与概率
Statistics involves collecting, representing and interpreting data. The three averages - mean, median and mode - describe a data set in different ways, while the range measures its spread.
统计涉及数据的收集、表示与解释。平均数、中位数和众数这三种"平均值"从不同角度描述数据，而极差衡量数据的离散程度。
: 4, 7, 7, 9, 13 → mean = 40 ÷ 5 = 8, median = 7, mode = 7, range = 13 − 4 = 9
The mean is affected by extreme values, whereas the median is more resistant to them. Choosing the most suitable average is often worth a mark in itself.
平均数会受到极端值影响，中位数则更稳健。选择最合适的平均值本身往往就值得拿一分。
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Probabilities can be written as fractions, decimals or percentages.
概率衡量事件发生的可能性，取值范围从 0（不可能）到 1（必然）。概率可写成分数、小数或百分数。
P(event) = number of favourable outcomes ÷ total number of outcomes
P(not A) = 1 − P(A)
- Rolling a fair die, P(even) = 3/6 = 1/2
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