📚 Essential Maths Book 9i: Key Concepts Explained | Essential Maths Book 9i 知识点精讲
This article provides a complete revision walkthrough of the key topics covered in the 'Essential Maths Book 9i' scheme of work for Key Stage 3. Each section breaks down core ideas with clear explanations and worked examples, helping students build confidence and fluency across number, algebra, geometry, statistics and more.
本文是《Essential Maths Book 9i》课程核心知识点的完整复习指南。每一节都用清晰的讲解和例题对核心概念进行拆解，帮助学生在数与运算、代数、几何、统计等多个领域建立信心、提升熟练度。
1. Number and Place Value | 数字与位值
Understanding place value is essential for reading, writing and comparing whole numbers and decimals. Each digit in a number has a value ten times greater than the digit to its right. For example, in 4 706.52, the digit 4 represents 4 thousands, 7 represents 7 hundreds, 0 tens, 6 units, 5 tenths and 2 hundredths.
理解位值对于读写和比较整数与小数至关重要。一个数中每一位的值都是其右边一位的十倍。例如在 4 706.52 中，数字 4 表示 4 个千，7 表示 7 个百，0 个十，6 个一，5 个十分之一，2 个百分之一。
When ordering negative numbers, remember that the further left a number is on the number line, the smaller its value. -5 is less than -2 because -5 is further from zero in the negative direction.
对负数排序时，要记住在数轴上越靠左的数越小。-5 小于 -2，因为 -5 在负方向上离零更远。
2. Operations with Decimals and Fractions | 小数与分数的运算
To add or subtract decimals, align the decimal points vertically and fill gaps with zeros. For multiplication, ignore the decimal points initially, multiply as whole numbers, then put the decimal point back so that the answer has the same total number of decimal places as the original numbers combined.
做小数加减法时，要将小数点上下对齐，空位用零补齐。做乘法时，先忽略小数点，按整数相乘，然后再点回小数点，使积的小数位数等于各乘数小数位数之和。
When multiplying fractions, multiply the numerators together and the denominators together. For division, flip the second fraction (the divisor) and multiply. Always simplify the final fraction where possible.
分数相乘时，分子相乘作新分子，分母相乘作新分母。分数相除时，将第二个分数（除数）取倒数，再相乘。最后尽可能化简分数。
3. Introduction to Algebra | 代数入门
can be simplified by collecting like terms: 3a - a = 2a and 2b + 5b = 7b, giving 2a + 7b.
代数是使用字母（变量）表示未知数。像 3a + 2b - a + 5b 这样的代数式，可以通过合并同类项来化简：3a - a = 2a，2b + 5b = 7b，得到 2a + 7b。
with given numbers. If x = 3 and y = -2, then 2x² - y = 2 × (3)² - (-2) = 2 × 9 + 2 = 20.
代入法是把字母换成给定的数字。如果 x = 3，y = -2，那么 2x² - y = 2 × (3)² - (-2) = 2 × 9 + 2 = 20。
4. Solving Linear Equations | 解线性方程
th sides to get 2x - 7 = 5. Then add 7 to both sides: 2x = 12. Finally divide by 2: x = 6.
解方程时，要在等号两边同时进行相同的运算，使未知数单独留在一边。对于方程 4x - 7 = 2x + 5，先在两边同时减去 2x，得到 2x - 7 = 5；然后两边同时加 7：2x = 12；最后两边除以 2，得 x = 6。
7 = 17. Right side: 2(6) + 5 = 12 + 5 = 17. Both sides match, so the solution is correct.
一定要将解代回原方程进行检验。左边：4(6) - 7 = 17；右边：2(6) + 5 = 17。两边相等，说明解正确。
5. Sequences and Patterns | 数列与规律
A linear sequence has a constant difference between consecutive terms. For the sequence 5, 8, 11, 14, ..., the term-to-term rule is 'add 3'. The position-to-term rule (nth term) can be written as 3n + 2, where n is the position number.
线性数列中相邻两项的差是常数。对于数列 5, 8, 11, 14, ...，相邻项的变化规律是“加 3”。它的通项公式（第 n 项）可以写成 3n + 2，其中 n 是项数位置。
m dn + b. If the difference is 3, trial with n = 1: 3(1) + b = 5, so b = 2, giving 3n + 2.
求线性规律的第 n 项公式时，先找出公差 d，然后写出形如 dn + b 的表达式。如果公差是 3，代入 n = 1 时：3(1) + b = 5，得出 b = 2，所以公式为 3n + 2。
6. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, and angles around a point sum to 360°. Vertically opposite angles are equal. In a triangle, interior angles sum to 180°, and in any quadrilateral they sum to 360°.
直线上的角之和为 180°，一点处的周角之和为 360°。对顶角相等。三角形的内角和为 180°，任意四边形的内角和为 360°。
s (n - 2) × 180° ÷ n. A regular pentagon has interior angles of (5 - 2) × 180° ÷ 5 = 108°.
对于正多边形，内角和为 (n - 2) × 180°，其中 n 是边数。每个内角为 (n - 2) × 180° ÷ n。正五边形每个内角为 (5 - 2) × 180° ÷ 5 = 108°。
7. Perimeter, Area and Volume | 周长、面积与体积
, area of a triangle is ½ × base × height, and area of a parallelogram is base × perpendicular height.
图形的周长是围绕其边缘一周的长度。长为 l、宽为 w 的长方形周长 = 2(l + w)。长方形面积是 l × w，三角形面积是 ½ × 底 × 高，平行四边形面积是底 × 垂直高。
a prism is area of cross-section × length. For a cube of side 4 cm, volume = 4³ = 64 cm³.
体积衡量三维物体内部的空间大小。长方体的体积 = 长 × 宽 × 高。棱柱的体积 = 横截面积 × 长。对于边长为 4 厘米的正方体，体积 = 4³ = 64 立方厘米。
8. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. The ratio 2:3 means that for every 2 of one thing there are 3 of another. To share an amount in a given ratio, add the parts to find the total number of shares. Divide the amount by the total shares, then multiply by each part.
比是对整体中各部分进行比较。2:3 这个比意味着第一种事物有 2 份时，第二种就有 3 份。按给定比分配总数时，先把比的各项相加得到总份数，再用总数除以总份数，然后乘以各部分对应的份数。
aling recipes or maps. If 3 kg of flour make 5 cakes, how much flour is needed for 8 cakes? The unitary method finds the amount for 1 cake first: 3 ÷ 5 = 0.6 kg per cake, then 0.6 × 8 = 4.8 kg.
比例问题常涉及配方或地图上的缩放。如果 3 千克面粉可以做 5 个蛋糕，那么 8 个蛋糕需要多少面粉？用单位法先算出一个蛋糕需要的量：3 ÷ 5 = 0.6 千克，然后 0.6 × 8 = 4.8 千克。
9. Percentages and Applications | 百分数及其应用
A percentage is a fraction out of 100. To find a percentage of an amount without a calculator, find 10% (divide by 10) and 1% (divide by 100) first, then combine. For 35% of £80: 10% is £8, so 30% is £24; 5% is half of 10%, so £4. Total £28.
百分数表示分母为 100 的分数。不用计算器求某个数量的百分数时，可以先找出 10%（除以 10）和 1%（除以 100），然后组合。例如求 £80 的 35%：10% 是 £8，所以 30% 是 £24；5% 是 10% 的一半，即 £4；总共 £28。
rease uses the formula: new value = original × (1 ± percentage/100). To increase 200 by 15%: 200 × 1.15 = 230. To decrease 200 by 20%: 200 × 0.80 = 160.
百分数增减的公式是：新值 = 原值 × (1 ± 百分数/100)。把 200 增加 15%：200 × 1.15 = 230。把 200 减少 20%：200 × 0.80 = 160。
10. Statistics and Data Handling | 统计与数据处理
e. The range is the difference between the largest and smallest values. For the set 2, 3, 5, 5, 7: mean = (2+3+5+5+7)/5 = 4.4, median = 5, mode = 5, range = 7 - 2 = 5.
平均数是所有数值之和除以数值的个数。中位数是数据按序排列后中间的值。众数是出现频率最高的值。极差是最大值和最小值之差。对于数据集 2, 3, 5, 5, 7：平均数 = (2+3+5+5+7)/5 = 4.4，中位数 = 5，众数 = 5，极差 = 7 - 2 = 5。
Bar charts, pie charts and line graphs are used to display data visually. A pie chart shows proportions: multiply each fraction by 360° to find the angle for each sector. A line graph is best for showing changes over time.
条形图、饼图和折线图用于直观展示数据。饼图显示各部分的比例：用每个分数乘以 360° 即可得到各扇形的角度。折线图最适合展示随时间的变化。
11. Probability Basics | 概率基础
ity = number of favourable outcomes / total number of possible outcomes. The probability of rolling a 4 on a fair six-sided dice is 1/6.
概率用来衡量事件发生的可能性大小，用介于 0（不可能）和 1（一定发生）之间的分数、小数或百分数表示。概率 = 有利结果的数量 / 所有可能结果的总数。掷一枚均匀的六面骰子得到 4 的概率是 1/6。
rain) = 1 - 0.3 = 0.7. Mutual exclusivity means two events cannot happen at the same time.
事件不发生的概率等于 1 减去该事件发生的概率。如果 P(下雨) = 0.3，那么 P(不下雨) = 1 - 0.3 = 0.7。互斥事件意味着两个事件不可能同时发生。
12. Transformations and Symmetry | 图形变换与对称
Reflection flips a shape over a mirror line; all points move the same perpendicular distance to the other side. Rotation turns a shape around a fixed centre through an angle (90°, 180°) and direction (clockwise/anticlockwise). Translation slides a shape a given distance horizontally and vertically without turning.
反射使图形沿一条镜面线翻转；所有点都移动到另一侧相同的垂直距离处。旋转是让图形绕一个固定中心按某一角度（90°、180°）和方向（顺时针/逆时针）转动。平移是将图形在水平或垂直方向上移动一定距离而不发生旋转。
Enlargement changes the size of a shape by a scale factor from a centre of enlargement. A scale factor of 2 doubles all side lengths. If the scale factor is ½, the image is half the size. The shape remains similar – angles stay the same, sides are in proportion.
放大是指从一放大中心按比例因子改变图形的大小。比例因子为 2 时，所有边长加倍；比例因子为 ½ 时，图形缩小为原来的一半。放大后的图形保持相似——角度不变，对应边成比例。
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