📚 Essential Maths Book 8i Compressed Knowledge Summary | 《数学精华8i》知识点浓缩精讲
Welcome to this revision guide for the key topics covered in Essential Maths Book 8i. Here you will find compact explanations, worked examples and must-know rules that match the KS3 Mathematics curriculum. Whether you are preparing for an end-of-topic test or building a solid foundation for GCSE, this article breaks down every concept into clear, bilingual steps.
欢迎阅读《数学精华8i》核心知识点复习指南。这里整理了KS3数学课程中必须掌握的概念、范例和规则。文章采用中英双语对照，帮助你既巩固数学思维，又准确理解术语。每个小节都浓缩了教材精华，并配有可自查的要点列表。
1. Integers, Decimals & Operations | 整数、小数与运算顺序
Integers are the set of whole numbers, including negative values, zero, and positive values. When adding or subtracting with negative numbers, movements along a number line help: subtracting a negative is equivalent to adding its positive. Multiplying or dividing two negatives gives a positive result.
整数是包含负数、零和正数的完整数集。进行负数加减时，可把它想成在数轴上移动：减去一个负数等于加上它的绝对值。两负数相乘或相除，结果为正。
The order of operations, often remembered by BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction), must be applied whenever more than one operation appears. Division and multiplication share the same priority and are worked from left to right, as are addition and subtraction.
当算式里出现多种运算时，必须遵守运算顺序，常记作 BIDMAS（括号、指数、乘除、加减）。乘法和除法同级，按从左到右的顺序计算；加法和减法同理。
- Brackets first – calculate inside any parentheses.
- Indices next – evaluate powers like 3² or √25.
- Division and multiplication, left to right.
- Addition and subtraction, left to right.
- 先算括号内的内容。
- 再算指数，如 3² 或 √25。
- 然后按从左到右的顺序进行乘除运算。
- 最后按从左到右的顺序进行加减运算。
Example: 4 + 3 × (8 − 2²) ÷ 2 = 4 + 3 × (8 − 4) ÷ 2 = 4 + 3 × 4 ÷ 2 = 4 + 6 = 10
2. Fractions: Equivalence, Operations & BIDMAS | 分数：等值、运算与运算顺序
A fraction a/b represents a part of a whole. Equivalent fractions are created by multiplying or dividing both numerator and denominator by the same non-zero number. Simplifying a fraction means reducing it to its lowest terms by cancelling common factors.
分数 a/b 表示整体的一部分。将分子和分母同时乘或除以同一个非零数，就能得到等值分数。约分就是通过约去公因数，把分数化成最简分数。
To add or subtract fractions, find a common denominator first. For mixed numbers, either convert them to improper fractions or handle whole numbers separately. Multiplying fractions is straightforward: multiply the numerators together and the denominators together. For division, flip the second fraction (the reciprocal) and then multiply.
加减分数时先通分。处理带分数时，可以转化成假分数，或者把整数部分和分数部分分开计算。分数相乘很简单：分子乘分子，分母乘分母。分数相除时，把第二个分数取倒数，然后改为乘法运算。
Example: 2 ½ ÷ ¾ = 5/2 × 4/3 = 20/6 = 10/3 = 3 ⅓
Always apply BIDMAS when fractions appear in complex expressions. Treat the fraction bar as a grouping symbol, so calculate numerator and denominator completely before dividing.
在包含分数的复杂算式中仍需遵循运算顺序。分数线相当于一个分组符号，要先算出分子和分母的值，再做除法。
3. Percentages: Conversions & Calculations | 百分数：转换与计算
Percent means ‘per hundred’. To convert a percentage to a decimal, divide by 100; to change a decimal or fraction to a percentage, multiply by 100. Key equivalences like ½ = 50%, ¼ = 25% and ⅕ = 20% should be memorised.
百分数表示“每一百份”。百分数化成小数要除以100，小数或分数化成百分数要乘以100。需熟记常见等值关系，如 ½ = 50%，¼ = 25%，⅕ = 20%。
Finding a percentage of a quantity without a calculator often involves finding 10% or 1% first, then scaling. Percentage increase or decrease is applied by adding or subtracting the calculated amount from the original. For reverse percentage problems, the given amount represents (100% ± the percentage change), so division returns the original 100% value.
非计算器求某数的百分之几时，常先找出10%或1%再放大。百分数增减问题则在原值基础上加上或减去计算出的量。对于反向百分数，已知量代表（100% ± 变化百分比），用它除以这个百分比即可求出原来的100%对应的值。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
4. Ratio & Proportion: Sharing & Scaling | 比率与比例：分配与缩放
A ratio compares parts of a whole. Ratios can be simplified by dividing all parts by their highest common factor. A ratio in the form 1:n or n:1 is useful for scaling. To share a quantity in a given ratio, add the parts, find the value of one part, then multiply accordingly.
比率用于比较整体中的各个部分。比率可以通过各项除以它们的最大公因数来化简。写成 1:n 或 n:1 形式的比率在缩放时特别有用。按比例分配时，先把比率各项相加求出总份数，计算一份对应的量，再分别乘以各项份数。
Proportion describes how two quantities change together. Direct proportion means that as one quantity doubles, the other also doubles (y = kx). Inverse proportion implies that doubling one halves the other (y = k/x). Scale drawings and maps use ratios to connect distances on paper to real life.
比例关系描述两个量如何共同变化。正比例意味着一个量变成两倍，另一个也变成两倍 (y = kx)。反比例表示一个量加倍时，另一个减半 (y = k/x)。比例尺和海图都利用比率将纸上距离与实际距离联系起来。
Example: Share £56 in ratio 2:5. Total parts = 7, one part = £56÷7 = £8. Amounts: 2×£8 = £16 and 5×£8 = £40.
5. Algebraic Expressions: Simplifying & Substituting | 代数表达式：化简与代入
In algebra, letters represent unknown numbers. Terms contain a number part (coefficient) and a letter part. Like terms have exactly the same letter variables and powers; they can be combined by adding or subtracting the coefficients. Unlike terms must remain separate.
代数中，字母代表未知数。每一项包含数字部分（系数）和字母部分。同类项是指字母变量和指数完全相同的项；同类项可以合并，也就是加减它们的系数。不同类项则必须分开写。
When writing expressions, multiply signs are omitted between numbers and letters, and the number part is placed first. For powers, repeated multiplication is shown with an index: a × a = a². Use brackets to multiply a term group by a single term, employing the distributive law a(b + c) = ab + ac.
书写表达式时，数字和字母间的乘号省略，数字写在字母前面。如指数表示重复相乘：a × a = a²。遇到括号时，利用乘法分配律 a(b + c) = ab + ac 将单项乘进括号内。
Substitution means replacing letters with given values, carefully using brackets for negatives and fractions. Always follow BIDMAS after substitution.
代入法就是将字母替换为给定的数值，若数值为负数或分数，要用括号包裹。代入后必须依照运算顺序计算。
If a = −3, b = 4, evaluate 2a² − 3b: 2(−3)² − 3×4 = 2×9 − 12 = 18 − 12 = 6.
6. Sequences: Nth Term & Patterns | 数列：第n项与规律
A sequence is an ordered list of numbers following a rule. Linear sequences increase or decrease by a constant common difference. The nth term of a linear sequence is given by an + b, where a is the common difference and b is the zero term (the value when n = 0).
数列是按照某种规则排列的一列数。线性数列以固定的公差递增或递减。等差数列的第n项公式为 an + b，其中 a 是公差，b 是零号项（即 n = 0 时的值）。
To find the nth term, identify the common difference first, then work backwards to find the term that would come before the first given term to determine b. You can then use the formula to generate any term or check whether a number belongs to the sequence.
求第n项时，先找出公差，然后倒推出第一个给定项之前的那一项，从而确定 b。有了公式就可以生成任意一项，也可以检验某个数是否属于该数列。
Non-linear sequences include quadratic patterns, geometric progressions and Fibonacci-like sequences where each term is formed from previous ones. Spot the term-to-term rule and try to describe it in words.
非线性数列包括二次规律、等比数列以及斐波那契类数列。遇到这类问题时，要能发现项与项之间的规则，并尝试用语言描述它。
Sequence: 4, 7, 10, 13, … common difference = 3, zeroth term = 4 − 3 = 1, so nth term = 3n + 1.
7. Solving Linear Equations | 解一元一次方程
An equation states that two expressions are equal. The goal is to find the unknown value that makes the statement true by performing inverse operations on both sides. Every operation must be applied to the entire expression on each side to maintain balance.
方程表明两个表达式相等。解方程的目标是通过在等号两边同时进行逆运算，找到使等式成立的未知数值。每一步必须对方程两边整体施加相同的运算，以保持平衡。
For one-step equations, use the opposite operation (addition/subtraction, multiplication/division). Two-step equations require two inverse steps. When brackets appear, expand them first or divide both sides by the factor outside. If the unknown appears on both sides, collect terms with the variable on one side and constants on the other.
一步方程直接用逆运算即可（加/减、乘/除）。两步方程需要依次进行两个逆步骤。含括号时，选择先展开或两边同除以括号外的系数。未知数若在两边出现，先把含未知数的项移到一边，常数项移到另一边。
Always check your solution by substituting it back into the original equation. Beware of fractional coefficients; multiplying through by a common denominator can simplify the work.
解完后一定要代入原方程验根。遇到分数系数时，可以先两边同乘以公分母，使式子变得简洁。
Solve: 2(x − 3) = 4x + 2 → 2x − 6 = 4x + 2 → −6 − 2 = 4x − 2x → −8 = 2x → x = −4.
8. Angles & Polygons | 角与多边形
Angles are measured in degrees. Key angle facts include: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. Parallel lines create corresponding, alternate and co-interior relationships that can be used to find missing angles.
角度以度为单位测量。关键的角度事实有：直线上的角之和为180°；绕一点一周的角之和为360°；对顶角相等。平行线与截线会形成同位角、内错角和同旁内角等关系，可用于求未知角。
For triangles, the interior angles sum to 180°. A polygon with n sides has an interior angle sum of (n − 2) × 180°. The sum of exterior angles of any convex polygon is always 360°, whatever the number of sides. Regular polygons have equal side lengths and equal angles.
三角形的内角之和为180°。n 边形的内角和为 (n − 2) × 180°。任何凸多边形的外角和恒为360°，与边数无关。正多边形的边长相等，内角相等。
| Polygon | Sides | Interior angle sum |
|---|---|---|
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
9. Area, Perimeter & Volume | 面积、周长与体积
Perimeter is the total distance around a 2D shape. For compound shapes, add all outer sides. Area measures the surface inside a shape. The area of a rectangle is base × height; a triangle is ½ × base × height; a parallelogram is base × perpendicular height; a trapezium is ½ × (a + b) × h where a and b are parallel sides.
周长是二维图形边界的总长度。组合图形需将所有外围边长相加。面积度量图形内部的大小。长方形的面积 = 底 × 高；三角形面积 = ½ × 底 × 高；平行四边形面积 = 底 × 垂直高度；梯形面积 = ½ × (a + b) × 高，其中 a 和 b 是平行边。
The area of a circle is π × radius² and circumference is π × diameter. For KS3, use π ≈ 3.14 or the fraction 22/7 unless asked for an exact answer in terms of π.
圆的面积 = π × 半径²，周长 = π × 直径。在KS3阶段，通常取 π ≈ 3.14 或用 22/7 近似，除非题目要求保留 π。
Volume measures the space inside a 3D solid. Volume of a cuboid = length × width × height. Volume of a prism = area of cross-section × length. Remember to use cubic units for volume and square units for area.
体积度量立体图形内部的空间。长方体体积 = 长 × 宽 × 高。棱柱体积 = 横截面积 × 长度。记住体积的单位是立方单位，面积的单位是平方单位。
Prism volume example: triangular prism with cross-section area 12 cm², length 10 cm → volume = 12 × 10 = 120 cm³.
10. Coordinates & Transformations | 坐标与变换
Points are located on a Cartesian grid using coordinates (x, y). The four quadrants are numbered anticlockwise starting from the top right where both x and y are positive. Reading or plotting points relies on moving along the x-axis first, then the y-axis.
平面直角坐标系用坐标 (x, y) 定位点。四个象限从右上角（x和y均为正）开始沿逆时针编号。读点或描点都必须先沿x轴水平移动，再沿y轴垂直移动。
Transformations move or change a shape. Translation slides a shape by a column vector (x change over y change). Reflection flips a shape in a mirror line; the line can be vertical (e.g., x = 2), horizontal (y = −1) or diagonal (y = x). Rotation turns a shape by a given angle around a centre of rotation; clockwise and anticlockwise directions must be specified, along with the angle. Enlargement scales a shape by a positive scale factor from a centre of enlargement; a factor greater than 1 makes it larger, a fraction between 0 and 1 makes it smaller.
变换可以移动或改变图形。平移通过列向量（x方向变化量在上，y在下）滑动图形。反射使图形沿镜线翻转；镜线可以是竖线（如 x = 2）、横线（y = −1）或斜线（y = x）。旋转是让图形绕旋转中心转过指定角度；必须说明旋转方向（顺时针或逆时针）和角度。位似变换是从位似中心出发，用一个正数比例因子缩放图形；比例因子大于1时图形放大，介于0和1之间时图形缩小。
When describing transformations, look for pattern changes in coordinates: translation changes coordinates by a constant vector; reflection makes coordinates change sign systematically; rotation swaps and changes signs according to the quadrant. Congruent shapes are the same size and shape, and are related by translation, reflection or rotation.
描述变换时，可从坐标变化规律入手：平移使坐标统一增减；反射会让某些坐标符号成片改变；旋转则根据象限转换坐标的符号和位置。全等图形大小形状完全相同，它们是经由平移、反射或旋转联系起来的。
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