📚 Year 7 WJEC Maths: Quick Reference Formula & Theorem Guide | WJEC 七年级数学：公式定理速查手册
This quick reference handbook brings together the most important formulas, theorems, and key facts you need for Year 7 WJEC Mathematics. Each topic is presented with a clear explanation in English followed by its Chinese counterpart, helping you to understand and memorise the essential mathematical rules. Keep this guide handy for homework, revision and classroom quizzes.
这本速查手册汇集了 WJEC 七年级数学中最重要的公式、定理与关键知识点。每个主题先以英文清晰解释，再配以对应中文，帮助你理解并记忆核心数学规则。做作业、复习和课堂测验时，请将它放在手边随时查阅。
1. Number and Place Value | 数与位值
The value of a digit depends on its place in the number. Starting from the right, places are units, tens, hundreds, thousands and so on. For example, in the number 357, 3 is in the hundreds place so its value is 300.
一个数字的值取决于它在数字中的位置。从右向左依次是个位、十位、百位、千位等等。例如，在 357 中，3 在百位上，因此它的值是 300。
We use place value to compare and order numbers. To round a number to the nearest 10, look at the units digit: if it is 5 or more, round up; otherwise round down.
我们使用位值来比较和排序数字。将一个数字四舍五入到最近的十位时，看个位数字：如果大于或等于 5 就向上舍入，否则向下舍去。
Place value headings: …, millions, hundred thousands, ten thousands, thousands, hundreds, tens, units (ones)
位值表头：… 百万、十万、万、千、百、十、个
2. Addition and Subtraction | 加法与减法
mbers. The order of numbers does not change the sum (commutative property): a + b = b + a.
加法是求两个或多个数总和的过程。数字的顺序不影响结果（交换律）：a + b = b + a。
Subtraction is not commutative: a − b ≠ b − a. When subtracting, always subtract the smaller number from the larger number, or use column method with borrowing if needed.
减法不满足交换律：a − b ≠ b − a。做减法时，通常用大数减去小数，或者使用列竖式并在需要时进行借位。
easier if you partition numbers, for example 47 + 36 = (40 + 30) + (7 + 6) = 70 + 13 = 83.
如果用拆分法进行心算，加减法会变得更容易，例如 47 + 36 = (40 + 30) + (7 + 6) = 70 + 13 = 83。
3. Multiplication and Division | 乘法与除法
= b × a. Multiplication also follows the associative property: (a × b) × c = a × (b × c).
乘法是重复加法。因数顺序不影响积：a × b = b × a。乘法也满足结合律：(a × b) × c = a × (b × c)。
a = b). Division is the inverse operation of multiplication. If a × b = c, then c ÷ b = a.
除法不满足交换律：a ÷ b ≠ b ÷ a（除非 a = b）。除法是乘法的逆运算：如果 a × b = c，那么 c ÷ b = a。
| Multiples of a number are the results of multiplying that number by integers (e.g. multiples of 3: 3, 6, 9, 12 …) |
| 一个数的倍数，是该数与整数相乘的结果（例如 3 的倍数：3, 6, 9, 12 …） |
| Factors are numbers that divide exactly into another number (e.g. factors of 12: 1, 2, 3, 4, 6, 12) |
| 因数是可以整除另一个数的数（例如 12 的因数：1, 2, 3, 4, 6, 12） |
Prime numbers have exactly two distinct factors, 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. Note that 1 is not prime.
质数恰好有两个不同的因数：1 和它本身。前几个质数是 2、3、5、7、11、13、17、19。注意，1 不是质数。
4. Fractions, Decimals and Percentages | 分数、小数与百分比
A fraction represents a part of a whole. The numerator (top) shows how many parts we have, and the denominator (bottom) shows the number of equal parts the whole is divided into.
分数表示整体的一部分。分子（上方）表示我们拥有几份，分母（下方）表示整体被分成的等份数。
an equivalent fraction, multiply or divide both numerator and denominator by the same non‑zero number. For example, ½ = ²⁄₄ = ⁴⁄₈.
找等值分数时，将分子和分母同时乘或除以同一个非零数。例如，½ = ²⁄₄ = ⁴⁄₈。
ecimal: divide numerator by denominator. Decimal → Percentage: multiply by 100 and add the % symbol.
分数转小数：分子除以分母。小数转百分比：乘以 100 并加上 % 符号。
To compare fractions, convert them to have a common denominator. The fraction with the larger numerator is the larger fraction.
比较分数时，先把它们化成同分母。分子较大的那个分数就大。
When adding or subtracting fractions with the same denominator, simply add or subtract the numerators and keep the denominator. With unlike denominators, find a common denominator first.
同分母分数相加减时，直接把分子相加减，分母保持不变。分母不同时，要先化成同分母。
Multiplying fractions: multiply the numerators and multiply the denominators. Dividing by a fraction is the same as multiplying by its reciprocal.
分数乘法：分子乘分子，分母乘分母。除以一个分数等于乘以它的倒数。
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. The ratio a : b means for every a of the first quantity there are b of the second. Ratios can be simplified in the same way as fractions, by dividing both parts by their highest common factor.
比用来比较两个或更多的量。a : b 表示第一个量每 a 份对应第二个量的 b 份。比的化简与分数类似，将前后项同时除以它们的最大公因数。
d are in proportion. You can solve proportion problems using the unitary method or by scaling.
比例表示两个比相等。若 a/b = c/d，那么 a、b、c、d 成比例。你可以用单位法或缩放法解决比例问题。
Direct proportion: as one quantity increases, the other increases at the same rate. The graph of a directly proportional relationship is a straight line through the origin.
正比例：一个量增加，另一个量以相同速率增加。正比例关系的图像是一条经过原点的直线。
6. Algebra Basics | 代数基础
Algebra uses letters to stand for unknown numbers. A term can be a number, a letter or a combination of both multiplied together, e.g. 5x, 3y, −2ab.
代数使用字母代表未知数。一个项可以是一个数、一个字母，或者是数与字母相乘的形式，如 5x、3y、−2ab。
als sign. You can simplify expressions by collecting like terms, for example 3a + 2a = 5a.
表达式是由加号或减号连接项构成，不含等号。通过合并同类项可以化简表达式，例如 3a + 2a = 5a。
An equation states that two expressions are equal. To solve an equation, find the value of the unknown that makes the statement true. Keep the equation balanced by doing the same operation to both sides.
方程表示两个表达式相等。解方程就是求出使等式成立的未知数的值。通过对方程两边施加相同的运算来保持平衡。
If x + 5 = 12, subtract 5 from both sides → x = 7
若 x + 5 = 12，两边同时减去 5 → x = 7
ers to evaluate an expression. For example, if a = 3 and b = 2, then 2a + b = 2×3 + 2 = 8.
代入法是把表达式中的字母换成给定数字来求值。例如，若 a = 3 且 b = 2，则 2a + b = 2×3 + 2 = 8。
7. Geometry: Angles | 几何：角
Angles are measured in degrees (°). A full turn is 360°, a half turn 180°, and a right angle 90°.
角度的单位是度（°）。一整圈是 360°，半圈是 180°，直角是 90°。
| Angles on a straight line sum to 180°. | 直线上的角之和为 180°。 |
| Angles around a point sum to 360°. | 绕一点一周的角之和为 360°。 |
| Vertically opposite angles are equal when two lines intersect. | 两直线相交时，对顶角相等。 |
| Angles inside a triangle add up to 180°. | 三角形内角和为 180°。 |
Acute angles are less than 90°, obtuse angles are between 90° and 180°, and reflex angles are between 180° and 360°.
锐角小于 90°，钝角介于 90° 和 180° 之间，优角介于 180° 和 360° 之间。
When a transversal crosses parallel lines, alternate angles are equal and corresponding angles are equal. These properties help find missing angles.
当一条截线与平行线相交时，内错角相等，同位角相等。这些性质可以帮助找出未知角。
8. Geometry: Perimeter and Area | 几何：周长与面积
the lengths of all sides. For a rectangle, perimeter (P) = 2 × (length + width) = 2l + 2w.
周长是一个图形所有边的总长度。多边形直接将各边长相加。对于矩形，周长 = 2 ×（长 + 宽）= 2l + 2w。
Area measures the space inside a 2D shape, in square units. For rectangles and squares:
面积衡量二维图形内部的空间，单位是平方单位。对于矩形和正方形：
Area of rectangle = length × width → A = l × w
矩形面积 = 长 × 宽 → A = l × w
Area of square = side × side → A = s²
正方形面积 = 边长 × 边长 → A = s²
For a triangle, the area is half the base multiplied by the perpendicular height:
三角形的面积等于底乘以垂直高度的一半：
Area of triangle = ½ × base × height → A = ½ × b × h
三角形面积 = ½ × 底 × 高 → A = ½ × b × h
mula works for the rectangle but be careful to use the perpendicular height, not the slant side.
平行四边形的面积 A = 底 × 垂直高。矩形也适用此公式，但必须使用垂直高，而非斜边。
9. Geometry: 3D Shapes and Volume | 几何：立体图形与体积
3D shapes have faces, edges and vertices. Prisms have two identical parallel faces and uniform cross‑section. A cube and a cuboid are examples of rectangular prisms.
立体图形具有面、棱和顶点。棱柱有两个相同且平行的面，截面处处相等。正方体和长方体都是矩形棱柱的例子。
Volume measures the space occupied by a 3D object, in cubic units. For cubes and cuboids:
体积衡量立体物体占据的空间，单位是立方单位。对于正方体和长方体：
Volume of a cuboid = length × width × height → V = l × w × h
长方体体积 = 长 × 宽 × 高 → V = l × w × h
Volume of a cube = side × side × side → V = s³
正方体体积 = 边长 × 边长 × 边长 → V = s³
or a cuboid, add the areas of the six rectangular faces (2 × each pair of opposite faces).
表面积是所有面的总面积。对于长方体，将六个矩形面（三组对面，每组计算两次）的面积相加即可。
10. Statistics | 统计
Data is collected information. It can be discrete (countable, e.g. number of pets) or continuous (measurable, e.g. height). We organise data using frequency tables, bar charts, pictograms and line graphs.
数据是收集到的信息，可以是离散的（可数的，如宠物数量）或连续的（可测量的，如身高）。我们使用频数表、条形图、象形统计图和折线图来整理数据。
The mean (average) is found by adding all values and dividing by the number of values. The median is the middle value when data is ordered. The mode is the value that appears most often.
平均数（平均值）是将所有数值相加再除以数值的个数。中位数是数据按大小排序后位于中间的数。众数是出现次数最多的数值。
Mean = (sum of all data values) ÷ (number of values)
平均数 =（所有数据值之和）÷（数据个数）
The range is the difference between the largest and smallest data values. It shows how spread out the data is.
极差是最大值与最小值之差，反映数据的离散程度。
s of a whole. The angle for each category = (category frequency ÷ total frequency) × 360°.
饼图展示各部分在整体中所占的比例。每个类别的扇形角度 =（该类别频数 ÷ 总频数）× 360°。
11. Units and Measures | 单位与测量
e (cm), metre (m), kilometre (km). Conversions: 1 cm = 10 mm; 1 m = 100 cm; 1 km = 1000 m.
常用的公制长度单位：毫米（mm）、厘米（cm）、米（m）、千米（km）。换算：1 cm = 10 mm; 1 m = 100 cm; 1 km = 1000 m。
ilogram (kg). 1 kg = 1000 g. Capacity units: millilitre (ml) and litre (l). 1 l = 1000 ml.
质量单位：克（g）和千克（kg）。1 kg = 1000 g。容量单位：毫升（ml）和升（l）。1 l = 1000 ml。
Time: 1 minute = 60 seconds; 1 hour = 60 minutes; 1 day = 24 hours. To convert between 12‑hour and 24‑hour clock, add 12 to the hour for pm times (except 12pm).
时间：1 分钟 = 60 秒；1 小时 = 60 分钟；1 天 = 24 小时。12 小时制与 24 小时制转换时，下午时间的小时数加 12（中午 12 点除外）。
Perimeter and area calculations always need consistent units – convert all measurements to the same unit before calculating.
计算周长和面积时，务必统一单位——计算前先将所有测量值转换为同一单位。
12. Coordinates and Graphs | 坐标与图表
Coordinates are written as an ordered pair (x, y) where x is the horizontal position and y is the vertical position. The origin is (0, 0).
坐标用有序数对 (x, y) 表示，其中 x 是水平位置，y 是垂直位置。原点的坐标为 (0, 0)。
To plot a point, start at the origin, move x units along the horizontal axis, then y units vertically. Points in the first quadrant have both coordinates positive.
描点时，从原点出发，沿水平轴移动 x 单位，再沿垂直方向移动 y 单位。第一象限的点两个坐标均为正数。
A straight‑ ight line can be written as y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).
直线图表示线性关系。直线的方程可写成 y = mx + c 的形式，其中 m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴的交点）。
You can read coordinates from a graph and use them to solve problems involving real‑life contexts, such as distance‑time conversions.
你可以从图上读取坐标，并利用它们解决实际情境中的问题，例如距离与时间的换算。
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