📚 Year 7 CIE Advanced Mathematics: Core Concepts Review | Year 7 CIE 进阶数学：核心知识点梳理
This article provides a comprehensive review of the fundamental topics covered in Year 7 CIE Advanced Mathematics. Mastering these core concepts builds confidence and prepares students for more challenging content in later stages of the Cambridge Lower Secondary or IGCSE pathway. We will explore number systems, algebra, geometry, statistics, and problem-solving skills, offering clear explanations and helpful examples.
本文全面梳理了 Year 7 CIE 进阶数学的核心知识点。掌握这些基础概念能帮助学生建立信心，为后续剑桥初中或 IGCSE 阶段更具挑战的内容做好准备。我们将深入探讨数系、代数、几何、统计以及问题解决技能，并提供清晰的解释和实用示例。
1. Whole Numbers and Decimals | 整数与小数
A strong grasp of place value allows you to read, write, and order whole numbers and decimals confidently. The decimal system is based on powers of 10, where each digit’s position determines its value.
牢固掌握位值可以让你自信地读写和排序整数与小数。十进制系统建立在 10 的幂之上，每个数字的位置决定了它的数值。
| Thousands | Hundreds | Tens | Units | . | Tenths | Hundredths |
|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1/10 | 1/100 |
ounding 3.456 to two decimal places gives 3.46 because the third decimal digit is 6 (≥ 5).
将数字四舍五入到指定小数位或有效数字可以简化计算并帮助估算答案。例如，将 3.456 四舍五入到两位小数得到 3.46，因为第三位小数为 6（≥ 5）。
ltiplication by powers of 10 simply moves the decimal point to the right: 0.34 × 100 = 34.
小数运算需要仔细对位小数点。计算 2.5 加 1.75 时，写成 2.50 + 1.75 可避免错误。乘以 10 的幂只需将小数点右移：0.34 × 100 = 34。
2. Fractions and Percentages | 分数与百分比
value using different numbers. Simplifying fractions to their lowest terms is a key skill.
分数表示整体的一部分，写作分子除以分母的形式。等值分数，如 ½ = 2/4 = 4/8，用不同的数字表示相同的数值。将分数化为最简形式是一项关键技能。
/12. Multiplication and division are more straightforward: multiply numerators and denominators, or take the reciprocal and multiply.
加减分数时，它们必须有公分母。例如，1/3 + 1/4 变为 4/12 + 3/12 = 7/12。乘除法则更直接：分子分母分别相乘，或者取倒数再相乘。
verting between fractions, decimals, and percentages is essential. For instance, 0.75 = 75% = ¾. Finding a percentage of an amount, such as 20% of 80, is calculated as 20/100 × 80 = 16.
百分数是分母为 100 的特殊分数。分数、小数和百分数之间的转换至关重要。例如，0.75 = 75% = ¾。求一个数量的百分数，比如 80 的 20%，计算为 20/100 × 80 = 16。
3. Algebraic Expressions | 代数表达式
Algebra uses letters (variables) to represent unknown numbers or general rules. Terms like 3x, 5y², and −2xy combine coefficients and variables. Like terms—those with identical variable parts—can be collected together.
代数用字母（变量）表示未知数或一般规则。如 3x、5y² 和 −2xy 这样的项组合了系数和变量。同类项——变量部分完全相同的项——可以合并。
b. Expanding brackets using the distributive law is equally important: 3(x + 4) = 3x + 12.
化简表达式如 4a + 3b − a + 2b 得到 3a + 5b。使用分配律展开括号同样重要：3(x + 4) = 3x + 12。
llow the order of operations (brackets, indices, division/multiplication, addition/subtraction).
代入法涉及用给定数值替换变量。若 x = 2，y = 3，则表达式 2x² + xy 的值为 2(2)² + (2)(3) = 8 + 6 = 14。务必遵循运算顺序（括号、指数、乘除、加减）。
4. Equations and Inequalities | 方程与不等式
iable. For 2x + 5 = 13, subtract 5 from both sides: 2x = 8, then divide by 2 to get x = 4.
解一元一次方程就是求出使等式成立的未知数的值。使用逆运算来分离变量。对于 2x + 5 = 13，两边减去 5：2x = 8，再除以 2 得 x = 4。
an inequality like 3x − 1 < 5 follows similar steps: 3x < 6, so x < 2. Remember that multiplying or dividing by a negative number reverses the inequality sign.
始终将答案代回原方程进行检验。不等式用符号 <、>、≤、≥ 来比较表达式。解不等式如 3x − 1 < 5 遵循类似步骤：3x < 6，所以 x < 2。记住乘以或除以负数时不等式方向要反转。
Word problems often require forming and solving equations. For instance, “Three times a number plus 7 equals 22” translates to 3n + 7 = 22, giving n = 5.
应用题通常需要建立并求解方程。例如，“一个数的 3 倍加 7 等于 22”可转化为 3n + 7 = 22，解得 n = 5。
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities in the same unit. The ratio 2:3 means that for every 2 parts of one quantity, there are 3 parts of another. Ratios can be simplified by dividing both sides by common factors, just like fractions.
比用来比较同一单位的两个或多个数量。比 2:3 意味着每 2 份某量对应 3 份另一量。比可以通过两边同除以公因数来化简，就像分数一样。
Dividing a quantity into a given ratio involves finding the value of one part. If £100 is shared in the ratio 3:7, the total parts are 3+7=10, so one part is £10. The shares are £30 and £70.
按给定比例分配一个数量需要先求出一份的值。如果 100 英镑按 3:7 分配，总份数为 3+7=10，一份为 10 英镑，则分配额为 30 英镑和 70 英镑。
Direct proportion means that when one quantity doubles, the other doubles as well. If 5 pens cost £2, then 15 pens (3 times) cost £6. Use the unitary method to find the cost per item or use a proportion equation.
正比例意味着一个量翻倍时，另一个量也翻倍。若 5 支笔 2 英镑，则 15 支笔（3 倍）花费 6 英镑。可用归一法求单个物品成本，或使用比例方程。
6. Sequences and Patterns | 序列与模式
A number sequence is an ordered list of numbers that follows a rule. Linear sequences have a constant difference between terms. In 5, 8, 11, 14, ... the term-to-term rule is ‘add 3’.
数列是按规则排列的一串数字。线性数列的相邻项之间有恒定的差值。在 5, 8, 11, 14, ... 中，项与项之间的规则是“加 3”。
2. This allows you to find any term directly, for example the 50th term is 3×50 + 2 = 152.
等差数列的第 n 项公式具有 an + b 的形式。上述数列的第 n 项公式为 3n + 2。利用它可以直算任意项，例如第 50 项是 3×50 + 2 = 152。
Recognising patterns in diagrams or tables often leads to expressions. Squares, triangles, and Fibonacci sequences introduce more advanced ideas but follow the same logic of spotting a rule.
在图形或表格中识别模式常能导出表达式。正方形数、三角形数和斐波那契数列引入了更高级的思想，但同样遵循发现规则的基本逻辑。
7. Angles and Shapes | 角与形状
Angles are measured in degrees and classified by size: acute (< 90°), right angle (90°), obtuse (90°–180°), and reflex (> 180°). The sum of angles on a straight line is 180°, and around a point is 360°.
角以度为单位并按其大小分类：锐角（< 90°）、直角（90°）、钝角（90°–180°）和优角（> 180°）。平直线上的角之和为 180°，绕一点一周的角之和为 360°。
Vertically opposite angles are equal when two lines intersect. For parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles sum to 180°.
两直线相交时，对顶角相等。对于平行线，同位角相等，内错角相等，同旁内角之和为 180°。
Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The interior angle sum is always 180°. Properties of special triangles help deduce unknown angles and side lengths.
三角形按边可分为等边三角形、等腰三角形和不等边三角形；按角可分为锐角、直角和钝角三角形。内角和恒为 180°。特殊三角形的性质有助于推导未知角和边长。
8. Perimeter, Area and Volume | 周长、面积与体积
2(l + w). Compound shapes require careful addition or subtraction of missing side lengths.
多边形的周长是其外边界总长。矩形周长 P = 2(l + w)。组合图形需仔细加减缺失的边长。
triangle: A = ½ × b × h, and parallelogram: A = b × h. For a trapezium, A = ½ (a + b) × h.
面积衡量二维图形内部的空间。关键公式包括：矩形 A = l × w，三角形 A = ½ × b × h，平行四边形 A = b × h。梯形面积为 A = ½ (a + b) × h。
measured in cubic units. A cuboid’s volume is V = l × w × h. Prisms have a constant cross‑section, so volume = area of cross‑section × length. Surface area sums the areas of all faces.
三维固体的体积以立方单位计量。长方体体积 V = l × w × h。棱柱具有恒定截面，因此体积 = 截面积 × 长度。表面积是所有面的面积之和。
9. Coordinates and Graphs | 坐标与图形
The Cartesian plane uses ordered pairs (x, y) to locate points. The x‑axis runs horizontally and the y‑axis vertically. Plotting points and joining them reveals shapes or line graphs.
笛卡尔平面使用有序数对 (x, y) 定位点。x 轴水平，y 轴垂直。描点并连线可显示形状或直线图。
esent equations of the form y = mx + c, where m is the gradient (steepness) and c is the y‑intercept. The gradient is calculated as change in y ÷ change in x.
线性图表示形如 y = mx + c 的方程，其中 m 是斜率（陡度），c 是 y 轴截距。斜率计算为 y 的变化量 ÷ x 的变化量。
Drawing a table of values helps plot the graph. For y = 2x + 1, choose x‑values like −1, 0, 1, 2 and compute the corresponding y‑values. The plotted points should form a straight line.
绘制数值表有助于画出图形。对于 y = 2x + 1，选取如 −1, 0, 1, 2 等 x 值并计算相应 y 值。描出的点应形成一条直线。
10. Statistics and Probability | 统计与概率
Statistics involves collecting, organising, and interpreting data. Averages—mean, median, and mode—summarise a data set. The mean is the sum of values divided by the number of values.
统计学涉及收集、整理和解释数据。平均数——平均数、中位数和众数——可以概括数据集。平均数是数值总和除以数值个数。
The median is the middle value when data are ordered; the mode is the most frequent value. Range (max − min) measures spread.
中位数是数据排序后的中间值；众数是出现频率最高的值。范围（最大值 − 最小值）衡量数据分散程度。
Probability measures the chance of an event happening, expressed as a fraction, decimal, or percentage. The scale runs from 0 (impossible) to 1 (certain). For a fair six‑sided die, the probability of rolling a 4 is 1/6.
概率衡量事件发生的可能性，以分数、小数或百分数表示。尺度从 0（不可能）到 1（必然）。对于公平的六面骰子，掷出 4 的概率为 1/6。
Calculating probabilities for combined events often uses sample space diagrams or tree diagrams, though Year 7 usually focuses on listing outcomes systematically.
计算复合事件的概率常使用样本空间图或树形图，但 Year 7 通常侧重于系统地列出所有可能结果。
11. Transformations | 变换
Transformations move or change shapes on a coordinate grid. The four basic types are translation (sliding), reflection (flipping), rotation (turning), and enlargement (scaling).
变换是在坐标网格上移动或改变图形。四种基本类型为平移（滑动）、反射（翻转）、旋转（转动）和放大（缩放）。
Reflection requires a mirror line, such as the x‑axis or y‑axis. The image is the same perpendicular distance from the mirror as the original point. Rotation needs a centre, an angle, and a direction (clockwise or anticlockwise).
反射要求一条镜线，如 x 轴或 y 轴。像点到镜线的垂直距离与原像相同。旋转需要一个中心、一个角度和一个方向（顺时针或逆时针）。
Enlargement changes the size by a scale factor. If the scale factor is 2, all side lengths double. The shape remains similar—angles stay the same and corresponding sides are in proportion.
放大通过比例因子改变大小。若比例因子为 2，所有边长加倍。图形保持相似——角度不变，对应边成比例。
12. Problem Solving Strategy | 问题解决策略
Mathematical problem solving combines logic, reasoning, and previously learned skills. The Polya‑style approach suggests four steps: understand the problem, devise a plan, carry out the plan, and look back to check.
数学问题解决结合了逻辑、推理和先前所学的技能。波利亚式方法建议四步：理解问题、制定计划、执行计划、回顾检查。
Common techniques include working backwards, drawing a diagram, looking for a pattern, and making an organised list. These strategies help break down complex problems into manageable parts.
常用技巧包括逆向推理、绘图、寻找规律以及制作有序列表。这些策略有助于将复杂问题分解为可处理的部分。
Practising with word problems develops fluency and confidence. For example, “A rectangle has perimeter 30 cm. Its length is twice its width. Find the area.” th is 2w. Perimeter equation: 2(w + 2w) = 30 → 6w = 30 → w = 5, so area = 5 × 10 = 50 cm².
练习应用题可提高熟练度和自信。例如，“一个矩形周长为 30 cm，长是宽的两倍，求面积。”设宽为 w，则长为 2w。周长方程：2(w + 2w) = 30 → 6w = 30 → w = 5，因此面积 = 5 × 10 = 50 cm²。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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