📚 Year 7 CIE Mathematics: Core Concepts Overview | Year 7 CIE 数学：核心知识点梳理
Year 7 lays the foundation for Cambridge Lower Secondary Mathematics. This article summarises the most important topics you will meet, from number operations and fractions to algebraic expressions and geometry. Use it as a revision checklist to track your progress and build confidence for Checkpoint.
Year 7 是剑桥初中数学的奠基阶段。本文梳理了你将遇到的最重要的主题，从数字运算、分数，到代数表达式和几何。把它当作一份复习清单，用于跟踪进度并为 Checkpoint 考试建立信心。
1. Number and Place Value | 数字与位值
In Year 7, you must work comfortably with whole numbers up to millions. Understanding place value helps you read, write, order and round numbers correctly.
在 Year 7，你需要熟练处理大到百万的整数。理解位值可以帮助你正确地读、写、排序和取整数字。
The value of a digit depends on its column: units, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions. For example, in 3 572 814, the digit 5 stands for five hundred-thousands (500 000).
一个数字的值取决于它所在的数位：个、十、百、千、万、十万、百万。例如在 3 572 814 中，数字 5 表示五个十万（500 000）。
Rounding to the nearest 10, 100 or 1000 follows a simple rule: look at the digit to the right of the place you are rounding to. If it is 5 or more, round up; otherwise round down.
取整到最近的十、百或千遵循一个简单规则：看你所取整位右边的那一位数字。如果它大于或等于 5，就向上进一；否则向下舍去。
2. Addition and Subtraction | 加法和减法
Strong mental and written methods for addition and subtraction are essential. You should be able to add and subtract numbers with up to six digits using column methods.
扎实的心算和笔算的加减法是必不可少的。你应该能够使用竖式方法计算多达六位数的加减法。
In column addition, align digits by place value. Start from the units column. If a column sum exceeds 9, carry the extra to the next column on the left. In subtraction, you may need to exchange (borrow) from the next column when the top digit is smaller than the bottom digit.
在竖式加法中，按位值对齐数字。从个位列开始。如果某一列的和超过 9，就把多出来的部分进到左边下一列。在减法中，当上方的数字小于下方的数字时，你可能需要从前一列借一当十。
Estimating answers before calculating is a good habit. Round the numbers to the nearest hundred or thousand, compute mentally, then compare with your exact answer to check if it is reasonable.
在计算之前先估算答案是个好习惯。把数字取整到最近的百或千，心算后与你的精确答案比较，检查是否合理。
3. Multiplication and Division | 乘法和除法
Year 7 covers multiplication and division of whole numbers, including long multiplication and short division. Knowing times tables up to 12 × 12 is still crucial.
Year 7 涉及整数的乘法和除法，包括长乘法和短除法。熟记到 12 × 12 的乘法表仍然至关重要。
Long multiplication breaks a larger multiplier into tens and units. For 234 × 16, multiply 234 by 6, then by 10, and add the two results. Short division (the bus stop method) is used for dividing by a single-digit number. Remember to carry remainders to the next digit.
长乘法把较大的乘数分成十位和个位。对于 234 × 16，先计算 234 × 6，再计算 234 × 10，然后把两个结果加起来。短除法（“公交站”法）用于除以一位数。记得将余数带到下一位继续除。
Multiplying and dividing by 10, 100 and 1000 simply shifts digits’ place values. Multiplying by 10 moves every digit one place to the left; dividing moves them to the right.
乘以和除以 10、100 和 1000 仅仅是移动数字的位值。乘 10 会把每个数字向左移动一位；除法则是向右移动。
4. Fractions | 分数
Fractions represent parts of a whole. You need to understand equivalent fractions, how to simplify them, and how to convert between mixed numbers and improper fractions.
分数表示整体的一部分。你需要理解等值分数，如何约分，以及在带分数和假分数之间进行转换。
To find an equivalent fraction, multiply or divide the numerator (top) and denominator (bottom) by the same number. Simplifying a fraction means dividing both by their highest common factor (HCF). For example, 12/18 simplifies to 2/3 by dividing top and bottom by 6.
要得到等值分数，用相同的数乘以或除以分子和分母。约分是指用分子和分母的最大公因数（HCF）去除它们。例如，12/18 分子分母同除以 6 就简化为 2/3。
Adding and subtracting fractions require a common denominator. Once denominators are the same, simply add or subtract the numerators. Multiplying fractions is straightforward: multiply numerators together and denominators together. Dividing by a fraction means multiplying by its reciprocal.
加减分数需要先通分，使分母相同。一旦分母统一，只需将分子相加或相减。分数乘法很直接：分子乘分子，分母乘分母。除以一个分数等于乘以它的倒数。
5. Decimals and Percentages | 小数和百分数
Decimals are another way of writing fractions whose denominators are powers of ten. You must be confident reading, writing, ordering and rounding decimals to one, two or three decimal places.
小数是分母为 10 的幂的分数的另一种写法。你必须自信地读、写、排序小数，并能取整到一位、两位或三位小数。
Comparing decimals is easiest when they have the same number of decimal places. Add trailing zeros if necessary; for example, 0.6, 0.60 and 0.600 are equal. Rounding decimals follows the same rule as whole numbers: look at the next digit.
当小数有相同的小数位数时，比较起来最容易。如有必要可以补零；例如 0.6、0.60 和 0.600 是相等的。小数取整的规则与整数相同：看下一位数字。
Percentages mean ‘out of 100’. To convert a percentage to a decimal, divide by 100. To convert a fraction to a percentage, first turn it into an equivalent fraction with denominator 100, or divide the numerator by the denominator and multiply by 100%.
百分数表示“每一百份中的几份”。要把百分数化为小数，除以 100。要把分数化为百分数，首先把它变成分母为 100 的等值分数，或者用分子除以分母再乘以 100%。
6. Ratio and Proportion | 比和比例
Ratio compares the sizes of two or more quantities. It can be written with a colon, as in 3 : 2, or as a fraction. Ratio does not give the actual amounts, only their relative sizes.
比用来比较两个或更多数量的大小。它可以用冒号写成 3 : 2，也可以写成分数。比并不给出实际数量，只表示相对大小。
Simplifying a ratio is like simplifying a fraction: divide all parts by their greatest common factor. For example, 10 : 15 simplifies to 2 : 3 by dividing by 5. Ratios can also be scaled up to find equivalent ratios.
化简比就像约分：用最大公因数去除所有的项。例如，10 : 15 除以 5 就简化为 2 : 3。比也可以放大来寻找等值比。
Proportion describes a relationship where two quantities change in the same ratio. If 3 pencils cost 45p, 6 pencils cost 90p: the cost is in direct proportion to the number of pencils.
比例描述两个量以相同的比率变化的关系。如果 3 支铅笔花费 45 便士，那么 6 支铅笔花费 90 便士：费用与铅笔数量成正比。
7. Algebra Basics | 代数基础
Algebra uses letters to stand for unknown numbers or variables. You will learn to write and simplify algebraic expressions, and to substitute numbers for letters.
代数使用字母代表未知数或变量。你将学会书写和化简代数表达式，以及用数字替换字母。
An expression like 3a + 2b – a + 4b can be simplified by collecting like terms. Here, 3a – a gives 2a, and 2b + 4b gives 6b, so the simplified expression is 2a + 6b.
像 3a + 2b – a + 4b 这样的表达式可以通过合并同类项来化简。这里，3a – a 得到 2a，2b + 4b 得到 6b，因此化简后的表达式为 2a + 6b。
Multiplying algebraic terms follows rules: a × a = a², and 3a × 2b = 6ab. You also need to use the distributive law, such as expanding brackets: 3(x + 2) = 3x + 6.
代数项的乘法遵循规则：a × a = a²，3a × 2b = 6ab。你还需要使用分配律，例如展开括号：3(x + 2) = 3x + 6。
8. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. Year 7 work focuses on recognising patterns, describing term-to-term rules, and finding missing terms.
数列是按照规则排列的一列数。Year 7 的重点是识别模式、描述项与项之间的变化规则，以及找到缺失的项。
A linear sequence changes by adding or subtracting the same number each time. For example, the sequence 3, 7, 11, 15, … has the term-to-term rule ‘add 4’. You can use this rule to extend the sequence or to find the 10th term.
线性数列每次加上或减去相同的数。例如，数列 3, 7, 11, 15, … 的项间规则是“加 4”。你可以用这个规则来延展数列或找到第 10 项。
Patterns can also be represented in diagrams, like matchstick patterns. The number of matchsticks often forms a sequence that can be expressed as an algebraic rule, e.g. 4, 7, 10, … has rule 3n + 1.
规律还可以用图形表示，比如火柴棒图案。火柴棒的数量常常构成一个数列，并可用代数规则表示，例如 4, 7, 10, … 的规则是 3n + 1。
9. Geometry: Angles and Shapes | 几何：角与形状
Basic geometry in Year 7 includes naming angles, measuring them with a protractor, and classifying triangles and quadrilaterals according to their properties.
Year 7 的基础几何包括给角命名、用量角器测量角，以及根据性质给三角形和四边形分类。
Acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and reflex angles are between 180° and 360°. Angles on a straight line add up to 180°; angles around a point add up to 360°.
锐角小于 90°，直角正好 90°，钝角在 90° 与 180° 之间，反角在 180° 与 360° 之间。直线上的角之和为 180°；一点周围的角之和为 360°。
Triangles can be classified by sides (equilateral, isosceles, scalene) or by angles (acute, right, obtuse). The sum of interior angles in any triangle is 180°. Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites.
三角形可以按边（等边、等腰、不等边）或按角（锐角、直角、钝角）分类。任何三角形的内角和为 180°。四边形包括正方形、长方形、平行四边形、菱形、梯形和风筝形。
10. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around the outside of a 2D shape. Area measures the surface covered, while volume measures the space inside a 3D solid.
周长是二维图形外边缘的总长度。面积度量表面覆盖的大小，而体积度量三维物体内部的空间。
For rectangles, perimeter = 2 × (length + width) and area = length × width. For a compound shape, split it into simpler rectangles, find each area separately, then add or subtract as needed. Area is measured in square units (cm², m²).
矩形的周长 = 2 × (长 + 宽)，面积 = 长 × 宽。对于组合图形，把它拆分成简单的矩形，分别求出面积，再按需相加或相减。面积以平方单位（cm², m²）计算。
Volume of a cuboid = length × width × height. It is measured in cubic units (cm³, m³). Remember that 1 litre = 1000 cm³.
长方体的体积 = 长 × 宽 × 高。它以立方单位（cm³, m³）度量。记住 1 升 = 1000 cm³。
11. Measures and Units | 测量与单位
You must be able to choose appropriate units for length, mass, capacity and time, and convert between common metric units. Standard units and their abbreviations (mm, cm, m, km; g, kg; ml, l; seconds, minutes, hours) are used throughout.
你必须能为长度、质量、容量和时间选择合适的单位，并能在常用的公制单位之间进行转换。标准单位及其缩写（mm, cm, m, km; g, kg; ml, l; 秒、分、小时）会被全程使用。
Metric conversions are based on powers of ten: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. For mass, 1 kg = 1000 g. For capacity, 1 l = 1000 ml. Time conversions are different: 1 minute = 60 seconds, 1 hour = 60 minutes.
公制转换基于十的幂：1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m。质量上，1 kg = 1000 g。容量上，1 升 = 1000 毫升。时间的换算是不同的：1 分钟 = 60 秒，1 小时 = 60 分钟。
Reading scales on measuring instruments is an important practical skill. You should be able to read a ruler, weighing scale, measuring jug, clock and thermometer accurately, including interpreting intervals between marked divisions.
阅读测量仪器上的刻度是一项重要的实践技能。你应该能准确读取直尺、秤、量杯、钟表和温度计，包括解读刻度线之间的间隔。
12. Data Handling and Statistics | 数据处理与统计
Year 7 statistics involves collecting, organising and interpreting data. You will use tally charts, frequency tables, bar charts, pictograms and line graphs to display information.
Year 7 的统计涉及收集、整理和解读数据。你将使用计数表、频数表、条形图、象形图和折线图来展示信息。
A tally mark is a quick way to record data in groups of five. A frequency table shows how often each outcome occurs. Bar charts have equal-width bars whose heights represent frequencies; the bars do not touch. For categorical data, you may also use pictograms, where a symbol represents a certain number of items.
计数记号是一种以五计数的快速记录数据的方法。频数表显示每个结果出现的次数。条形图具有等宽的条形，其高度代表频数；条形之间不接触。对于分类数据，你也可以使用象形图，其中一个图标代表一定数量的项目。
The mean (average) of a set of numbers is found by adding all the values and dividing by how many there are. The mode is the value that appears most often. You will also learn to read and interpret line graphs that show changes over time.
一组数据的平均数（均值）是将所有数值相加再除以数值的个数。众数是出现次数最多的值。你还将学习阅读和解读展示随时间变化的折线图。
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