📚 Year 7 CAIE Mathematics: Core Concepts Review | Year 7 CAIE 数学：核心知识点梳理
Welcome to our comprehensive revision guide for Year 7 CAIE Mathematics. This article walks you through the essential topics you will encounter in the Cambridge Lower Secondary programme, helping you build a strong foundation in number, algebra, geometry and statistics. Each section presents key ideas in clear English followed by the Chinese translation, so you can absorb the concepts in both languages.
欢迎来到 Year 7 CAIE 数学的全面复习指南。本文带你梳理剑桥初中阶段的核心内容，涵盖数、代数、几何与统计四大板块，帮助你打下扎实基础。每个要点都以英文和中文配对呈现，让你在双语环境中轻松掌握知识点。
1. Number Systems: Integers and Place Value | 数系：整数与位值
We begin with whole numbers and place value up to millions. Understand how digits in positions such as units, tens, hundreds, thousands, ten‑thousands, hundred‑thousands and millions determine the size of a number. You need to compare and order large integers and use correct inequality signs (<, >, =).
我们从整数和百万以内的位值开始。理解数字在个、十、百、千、万、十万、百万等位置上的意义，你就能比较和排序大整数，并正确使用不等式符号（<、>、=）。
Negative numbers are introduced on the number line. They arise in contexts such as temperature below zero, bank overdrafts or elevations below sea level. You should be able to order positive and negative numbers and recognise that -7 is less than -2.
负数通过数轴引入，常出现在温度零下、银行透支或海拔以下等情境中。你需要能够排序正负数，并理解 -7 实际上比 -2 更小。
Operations with integers are consolidated: column addition, subtraction with borrowing, long multiplication and short/long division. The order of operations (BIDMAS/BODMAS) is essential – Brackets first, then Indices (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right). For example, 10 + 3 × 2 = 16, not 26.
整数的四则运算得到巩固：竖式加法、借位减法、长乘法和短/长除法。运算次序（BIDMAS/BODMAS）至关重要——先算括号，其次是指数（幂），然后是乘除（从左到右），最后是加减（从左到右）。例如 10 + 3 × 2 = 16，而不是 26。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions represent parts of a whole. You must know equivalent fractions, how to simplify fractions by dividing numerator and denominator by their HCF, and how to compare fractions by finding a common denominator. Mixed numbers and improper fractions can be converted both ways.
分数表示整体的一部分。你需要掌握等值分数、用最大公因数约分、通过寻找公分母比较分数大小，并能在带分数与假分数之间自由转换。
Operations with fractions include addition and subtraction using a common denominator, and multiplication by multiplying numerators together and denominators together. Division is performed by multiplying by the reciprocal. Always give answers in simplest form.
分数的运算包括通分后进行加减；乘法直接分子相乘、分母相乘；除法转化为乘以倒数。最终结果一定要化成最简形式。
Decimals are an extension of place value: tenths, hundredths, thousandths. You add, subtract, multiply and divide decimals by aligning decimal points or using whole‑number methods then replacing the decimal. Converting between fractions and decimals is done via division or recognising common equivalents like ½ = 0.5.
小数是位值的延伸：十分位、百分位、千分位。加减法要对齐小数点；乘法可先按整数计算再点小数点。分数与小数互化通过除法实现，或以熟记 ½ = 0.5 等常见等价形式。
Percentages mean ‘out of 100’. Conversions between fractions, decimals and percentages are vital: e.g., 3/5 = 0.6 = 60%. You calculate a percentage of a quantity and solve problems involving percentage increase or decrease.
百分数表示“百分之几”。分数、小数和百分数的互化至关重要，例如 3/5 = 0.6 = 60%。你需要求一个量的百分之几，并解决涉及百分比增减的实际问题。
3. Ratio and Proportion | 比与比例
Ratio compares the sizes of two or more quantities. It is written with a colon, such as 3 : 4. Ratios can be simplified like fractions by dividing all parts by a common factor. You solve problems by sharing an amount in a given ratio: for a sum to be split in the ratio 2 : 3, the total number of parts is 5, so each part gets 1/5 of the total.
比用来比较两个或多个量的大小，写作 3 : 4 这样的形式。比可以像分数一样通过除以公因数来化简。按给定比例分配时，若按 2 : 3 分一笔钱，总份数为 5，每份就是总量的 1/5。
Proportion refers to the relationship where two quantities multiply by the same factor. Direct proportion problems often involve recipes or scaling up ingredients. If one quantity is multiplied by a factor, the related quantity is multiplied by the same factor.
比例描述两个量以相同的倍数变化。正比例问题常出现在食谱或配料的缩放上。如果一个量乘以某个因子，与之相关的量也乘以同样的因子。
You will also meet the unitary method: finding the value of one unit first, then multiplying by the desired number. For instance, if 5 pens cost 15 AED, then 1 pen costs 3 AED, so 8 pens cost 24 AED.
你还将遇到归一法：先求出单一单位的量，再乘以需要的数量。例如，若 5 支笔售价 15 AED，则 1 支笔为 3 AED，那么 8 支笔售价 24 AED。
4. Algebraic Expressions | 代数表达式
Algebra uses letters to stand for unknown numbers. In Year 7, you learn to write expressions from word statements, such as ‘a number multiplied by 5, then add 2’ becoming 5n + 2. The letter is called a variable.
代数用字母代表未知数。七年级要学习根据文字叙述写出表达式，比如“一个数乘以 5 再加 2”可以写成 5n + 2，这里的字母称为变量。
Collecting like terms is crucial: terms with the same variable and power can be combined. 3a + 2a = 5a, but 3a + 2b cannot be simplified. You also multiply and divide algebraic terms: 2 × a = 2a, and a × a = a² (using Unicode a²). Division is shown as a fraction or with a division symbol.
合并同类项是关键：相同变量和相同次数的项可以合并。3a + 2a = 5a，但 3a + 2b 无法化简。你还要学习代数项的乘除：2 × a = 2a，而 a × a = a²。除法可以表示为分数或除号。
Substitution means replacing the letter with a number and calculating the result. If y = 3, then 2y² = 2 × (3)² = 18. Always follow the order of operations when substituting.
代入求值是指把字母换成具体数值再计算结果。若 y = 3，则 2y² = 2 × (3)² = 18。代入时一定要遵守运算次序。
5. Equations and Sequences | 方程与序列
An equation states that two expressions are equal. Solving a linear equation means finding the value of the unknown that makes the statement true. For example, x + 5 = 12 gives x = 7, and 3x = 24 gives x = 8. You perform inverse operations on both sides to keep the equation balanced.
方程表示两个表达式相等。解一元一次方程就是找到使等式成立的未知数的值，比如 x + 5 = 12 解得 x = 7；3x = 24 解得 x = 8。通过等号两边同时进行逆运算来保持平衡。
Two‑step equations involve two operations, such as 2x + 3 = 11. Undo the addition first (subtract 3 from both sides) then undo the multiplication (divide by 2), giving x = 4.
两步方程包含两种运算，例如 2x + 3 = 11。先处理加法（两边减 3），再处理乘法（两边除以 2），得 x = 4。
Sequences are lists of numbers following a rule. Year 7 focuses on arithmetic sequences where the difference between consecutive terms is constant, such as 4, 7, 10, 13,… (term‑to‑term rule: add 3). You learn to find the next terms and to describe the rule in words and, later, to write a formula for the nth term.
序列是按一定规律排列的一列数。七年级重点学习等差数列，即相邻项的差保持恒定，如 4, 7, 10, 13,…（逐项规则：加 3）。你要学会找出后续项，并用语言描述规律，之后还会写出第 n 项的公式。
nth term of an arithmetic sequence: a + (n – 1)d, where a is the first term and d is the common difference.
等差数列第 n 项公式：a + (n – 1)d，其中 a 为首项，d 为公差。
6. Angles and Lines | 角与线
Angles are measured in degrees (°). You identify acute (0°–90°), right (90°), obtuse (90°–180°) and reflex (180°–360°) angles. A straight line is exactly 180°, and a full turn is 360°.
角以度（°）为单位。你需识别锐角（0°–90°）、直角（90°）、钝角（90°–180°）和优角（180°–360°）。一条直线是 180°，一个周角是 360°。
Angles on a straight line add up to 180°; angles around a point add up to 360°. Vertically opposite angles are equal. These facts allow you to calculate missing angles without a protractor.
直线上的角之和为 180°；点周围的角之和为 360°；对顶角相等。利用这些事实可以不用量角器计算出未知角。
Parallel lines are studied using a transversal. Corresponding angles, alternate angles and interior (co‑interior) angles are introduced. Alternate angles are equal, corresponding angles are equal, and interior angles sum to 180°. These relationships help solve problems involving intersecting lines.
平行线结合截线出现对应角、内错角和同旁内角。内错角相等，对应角相等，同旁内角互补（和为 180°）。这些关系能够帮助解决相交线的问题。
7. 2D Shapes: Perimeter and Area | 二维图形：周长与面积
Perimeter is the total distance around a shape. For polygons, add the side lengths. For a rectangle, perimeter = 2(l + w), where l is length and w is width. Remember to state units (cm, m, etc.).
周长是围绕图形一周的总长度。多边形的周长是将各边相加；长方形的周长 = 2(l + w)，其中 l 为长，w 为宽。记得要带上单位（厘米、米等）。
Area measures the surface inside a 2D shape. The area of a rectangle is length × width. Area of a triangle is ½ × base × height, where the height is perpendicular to the base. Area of a parallelogram is base × perpendicular height. Compound shapes are split into simpler rectangles and triangles.
面积测量二维图形内部的表面大小。长方形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高，其中高垂直于底；平行四边形的面积 = 底 × 高。复合图形可分割为简单的长方形和三角形来计算。
Area of a triangle: A = ½ × b × h
三角形面积：A = ½ × b × h
When finding area and perimeter, always check that the units are consistent and give the final answer in square units (cm², m²) for area and linear units for perimeter.
在计算面积和周长时，务必确保单位一致，面积以平方单位（cm², m²）表示，周长则以线性单位表示。
8. 3D Shapes: Volume and Surface Area | 三维图形：体积与表面积
Year 7 introduces the properties of common 3D shapes: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres. You learn to count faces, edges and vertices (Euler’s relation: F + V – E = 2 is often explored).
七年级介绍常见立体图形的性质：正方体、长方体、棱柱、圆柱、棱锥、圆锥和球体。你需要学会数出面、棱和顶点的数量，并初步探索欧拉公式（F + V – E = 2）。
Volume of a cuboid is length × width × height. Since it can be built from unit cubes, volume is measured in cubic units (cm³, m³). You will also find the volume of shapes made from cuboids. The formula can be extended to any prism: volume = area of cross‑section × length.
长方体的体积 = 长 × 宽 × 高。因为可以用单位立方体搭建，体积以立方单位（cm³、m³）计量。你也会求由长方体组合而成的立体体积。该公式可推广到任何棱柱：体积 = 横截面积 × 长度。
Surface area is the total area of all faces. For a cuboid, find the area of each rectangle (front, back, top, bottom, sides) and add them up. Using a net helps visualise which faces are identical.
表面积是所有面的面积总和。对于长方体，分别计算前、后、上、下、左、右六个长方形的面积再求和。利用展开图能更直观地看出哪些面是相等的。
9. Coordinates and Graphs | 坐标与图像
The Cartesian grid has an x‑axis (horizontal) and a y‑axis (vertical). Coordinates are written as (x, y), where x comes first. The origin is (0, 0). You plot points in all four quadrants, paying attention to negative coordinates.
直角坐标系具有水平的 x 轴和垂直的 y 轴。坐标写作 (x, y)，x 在前。原点是 (0, 0)。你会在全部四个象限内描点，并注意负坐标的使用。
Straight‑line graphs are introduced via simple equations like y = x, y = 2x, y = 5. By constructing a table of values, you calculate y for given x values and plot the points to reveal a straight line. This builds understanding of linear relationships.
简单的直线图像如 y = x、y = 2x、y = 5 等会被引入。通过列出数值表，代入给定的 x 值求出 y，再描点连线，你就会看到一条直线。这能帮助理解线性关系。
10. Statistics: Data and Averages | 统计：数据与平均数
Data is organised using frequency tables, bar charts, pictograms, stem‑and‑leaf diagrams and pie charts. You learn to interpret and draw these charts using appropriate scales and keys.
数据通过频数表、条形图、象形图、茎叶图和饼状图等来整理。你需学会合理地设置刻度和图例，阅读并绘制这些统计图。
Three averages are covered: the mode (most frequent value), the median (middle value when data is ordered) and the mean (sum of values divided by the number of values). You also explore the range as a measure of spread (largest minus smallest).
介绍三种平均数：众数（出现最多的值）、中位数（排序后居中的值）和平均数（所有值的总和除以个数）。同时，极差（最大值减最小值）作为离散程度的量度也会涉及。
When calculating the mean, the formula is: Mean = sum of data items ÷ number of items. For large sets, a frequency table can shorten the work by multiplying each value by its frequency.
计算平均数时，公式为：平均数 = 数据项之和 ÷ 数据个数。对于较大的数据集，可使用频数表将每个值与其频数相乘来简化运算。
11. Probability | 概率
Probability measures how likely an event is to happen, expressed as a fraction, decimal or percentage between 0 (impossible) and 1 (certain). The probability of a fair coin landing heads is ½.
概率衡量一个事件发生的可能性大小，用介于 0（不可能）和 1（必然）之间的分数、小数或百分数表示。抛一枚均匀硬币出现正面的概率是 ½。
You learn to list all possible outcomes using a sample space. Probability is then calculated as: P(event) = number of favourable outcomes ÷ total number of equally likely outcomes. For a fair six‑sided die, P(rolling a 3) = 1/6.
你需要用样本空间列出所有可能结果，然后根据公式计算：P(事件) = 有利结果数 ÷ 所有等可能结果总数。对于均匀的六面骰子，P(掷出 3) = 1/6。
Expectation is the expected frequency of an event in many trials: Expected number = probability × number of trials. This helps predict what might happen, though actual results can vary due to chance.
期望次数指在多次试验中事件预计发生的频率：期望次数 = 概率 × 试验次数。这有助于预测趋势，但真实结果可能因随机性而有所波动。
12. Problem‑Solving and Mathematical Reasoning | 问题解决与数学推理
Throughout Year 7, you apply the above knowledge to solve multistep problems, often blending different topics. The focus is on understanding the problem, choosing a strategy (draw a diagram, look for a pattern, make a table, work backwards, solve a simpler case) and clearly communicating your steps.
在七年级的学习中，你需要综合运用以上知识解决多步骤问题，常常涉及跨主题的结合。重点在于理解问题、选择策略（画图、找规律、列表、逆推、简化问题）并清晰地表达解题步骤。
Reasoning skills are tested through puzzles and ‘convince me’ tasks. You might be asked to explain why a statement is always, sometimes or never true, or to find all possible solutions to a problem with more than one answer.
推理能力通过谜题和“说服我”式任务来考察。你可能需要解释某个命题是始终成立、有时成立还是绝不成立，或找出含有多个答案的问题的所有可能解。
Working systematically and checking your answers are essential habits. Estimating before calculating helps spot unreasonable results, and looking back at the original problem ensures the answer makes sense in context.
系统地解题并检验答案是必须养成的习惯。在计算前先估算，有助于发现不合理的结果，而最后回顾原始问题则能确保答案在实际情境中具有意义。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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