📚 Year 9 CIE Maths: Complete Mathematics for Cambridge Secondary 1 Book 3 | CIE 九年级数学：剑桥初中数学第三册核心复习
Complete Mathematics for Cambridge Secondary 1 Book 3 brings together the core skills that Year 9 learners need before moving on to IGCSE Mathematics. It strengthens numerical fluency, algebraic reasoning, geometric understanding and statistical thinking.
《剑桥初中数学第三册》汇集了九年级学生在进入 IGCCSE 数学之前所需的核心技能。它强化了数字运算、代数推理、几何理解和统计思维。
1. Number and operations with integers | 整数与运算顺序
In Book 3, work with integers extends to negative numbers, powers and roots. Always follow the order of operations: brackets first, then indices, then division and multiplication, and finally addition and subtraction.
在第三册中，整数运算扩展到负数、幂和方根。始终遵循运算顺序：先括号，再乘方，然后乘除，最后加减。
Be especially careful with signs when squaring. For example, (−3)² means the square of −3, while −3² means the negative of 3².
平方时要特别注意符号。例如 (−3)² 表示 −3 的平方，而 −3² 表示 3² 的相反数。
(−3)² = 9 but −3² = −9
Powers and roots are inverse operations, so √49 = 7 and 2³ = 8. These skills are used constantly in algebra and geometry questions.
幂和方根是互逆运算，因此 √49 = 7，2³ = 8。这些技能在代数和几何题中经常用到。
2. Fractions, decimals and percentages | 分数、小数与百分比
Convert between fractions, decimals and percentages by using equivalent fractions with denominator 100 or by carrying out division. This fluency is essential for comparing proportions accurately.
通过使用分母为 100 的等值分数或进行除法，在分数、小数和百分比之间进行转换。这种熟练度对于准确比较比例至关重要。
Percentage change = (change ÷ original) × 100%
Reverse percentage problems require finding the original value after a percentage increase or decrease. In these cases, divide by the multiplier rather than simply subtracting the percentage.
逆向百分比问题需要在百分比增减之后求出原始值。在这些情况下，应除以乘数，而不是简单地减去百分比。
- Increase £80 by 15% → £80 × 1.15 = £92 — 将 80 英镑增加 15% → £80 × 1.15 = £92
- Decrease £80 by 15% → £80 × 0.85 = £68 — 将 80 英镑减少 15% → £80 × 0.85 = £68
3. Ratio, proportion and rates | 比、比例与速率
A ratio compares parts of a whole. To share an amount in the ratio a : b, first find the total number of parts, then multiply the value of one part by the number of parts in each share.
比值用于比较整体的各部分。按 a : b 分配数量时，先求总份数，再用一份的值乘以每一份所包含的份数。
Share £60 in 3 : 2 → 60 ÷ 5 = £12 per part → £36 : £24
Rates such as speed, density and unit price link two different units. Direct proportion means two quantities increase or decrease by the same factor.
速率如速度、密度和单价连接两种不同的单位。正比例意味着两个量按相同的倍数同时增加或减少。
4. Algebraic expressions and simplification | 代数表达式与化简
Simplify expressions by collecting like terms, expanding brackets and factorising. The index laws are essential for working with powers that have the same base.
通过合并同类项、展开括号和因式分解来化简表达式。对于同底数幂，指数律至关重要。
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
law, such as 3(x + 4) = 3x + 12, and factorise common factors, such as 6x + 9 = 3(2x + 3).
使用分配律展开括号，例如 3(x + 4) = 3x + 12；提取公因式进行因式分解，例如 6x + 9 = 3(2x + 3)。
5. Linear equations and inequalities | 线性方程与不等式
Solve linear equations by performing inverse operations in a logical order. When variables appear on both sides, collect them on one side first, then isolate the variable.
按合理的顺序使用逆运算求解线性方程。当变量出现在等式两边时，先把它们移到同一边，然后单独求出变量。
2x + 3 = x − 7 → x = −10
Linear inequalities are solved in the same way, but the inequality sign reverses when multiplying or dividing by a negative number. Represent solutions on a number line with open or closed circles.
线性不等式的解法与方程相同，但乘以或除以负数时，不等号方向要改变。用数轴上的空心圆圈或实心圆圈表示解集。
- x ≥ −2 means −2 is included — x ≥ −2 表示包含 −2
- x < 5 means 5 is not included — x < 5 表示不包含 5
6. Sequences and functions | 数列与函数
A linear sequence has a constant difference between consecutive terms. The nᵗʰ term formula gives any term directly without listing the whole sequence.
线性数列相邻项之间具有恒定差值。第 n 项公式可以直接求出任意一项，而无需列出整个数列。
Sequence 5, 8, 11, 14, ... → nᵗʰ term = 3n + 2
Function machines show input, operation and output. Simple quadratic sequences, such as 1, 4, 9, 16, ..., have a changing difference and their nᵗʰ term is n².
函数机器展示输入、运算和输出。简单的二次数列，如 1, 4, 9, 16, ...，具有变化的差值，其第 n 项为 n²。
7. Coordinates and linear graphs | 坐标与一次函数图像
ient and c is the y-intercept. Plot points or use the gradient and intercept to draw the line.
一次函数图像具有 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。可以通过描点或利用斜率和截距来画直线。
y = 2x + 1: gradient 2, y-intercept 1
The midpoint of two points (x₁, y₁) and (x₂, y₂) is found by averaging the coordinates.
两点 (x₁, y₁) 和 (x₂, y₂) 的中点通过两个坐标分别取平均值求得。
Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
8. Angle properties and constructions | 角的性质与作图
When parallel lines are cut by a transversal, alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.
当平行线被截线所截时，内错角相等、同位角相等、同旁内角互补。
Sum of interior angles of any polygon = (n − 2) × 180°
Use a ruler and compasses to construct the perpendicular bisector of a line segment and the bisector of an angle. These constructions depend on equal radii drawn from key points.
使用直尺和圆规作线段的垂直平分线和角平分线。这些作图依赖于从关键点画出的相等半径。
9. Perimeter, area and volume | 周长、面积与体积
The circumference of a circle is π times the diameter, and the area is π times the radius squared. Remember to use the correct units and round answers when π is involved.
圆的周长是 π 乘以直径，面积是 π 乘以半径的平方。当涉及 π 时，请记住使用正确的单位并对答案进行舍入。
C = 2πr or πd,
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