📚 Cambridge Primary Mathematics Workbook 6 (2nd Edition) Key Concepts | 剑桥小学数学练习册6（第二版）知识点精讲
This comprehensive guide covers the essential topics presented in the Cambridge Primary Mathematics Workbook 6, 2nd Edition. It helps learners consolidate their understanding of key mathematical skills, from number and operations to geometry and data handling, aligned with the Cambridge Primary curriculum framework.
本全面指南涵盖了《剑桥小学数学练习册6（第二版）》中的核心主题。它帮助学生巩固从数与运算到几何与数据处理的关键数学技能，与剑桥小学课程框架保持一致。
1. Understanding Numbers up to 10 Million | 理解千万以内的数
Each digit in a whole number has a place value depending on its position, such as millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones. For example, in 7,504,328 the digit 7 represents 7 million.
一个整数中每个数字根据其位置有不同的位值，如百万位、十万位、万位、千位、百位、十位和个位。例如，在 7,504,328 中，数字 7 代表 7 百万。
Numbers can be read by grouping digits into sets of three from the right. 4,091,706 is read as 'four million, ninety-one thousand, seven hundred and six'. Writing numbers in words requires careful use of place value and hyphenation.
数字可以从右向左每三位一组来读。4,091,706 读作 '四百零九万一千七百零六'。用文字书写数字时需要仔细使用位值和连字符。
ultiplied by its place value: 2,345,670 = 2,000,000 + 300,000 + 40,000 + 5,000 + 600 + 70.
展开式是将一个数字分解为每个数字乘以其位值之和：2,345,670 = 2,000,000 + 300,000 + 40,000 + 5,000 + 600 + 70。
Rounding to the nearest 10, 100, 1000, 10,000 or 100,000 helps estimate large quantities.
通过从左起最高位开始比较数字，可以对千万以内的数进行比较和排序。使用 >、< 和 = 符号。四舍五入到最近的 10、100、1000、10,000 或 100,000 有助于估算大数。
2. Addition and Subtraction of Large Numbers | 大数的加法和减法
Written methods for addition and subtraction include columnar forms, where digits are aligned by place value. Carrying occurs when a column sum exceeds 9; borrowing occurs when a digit is smaller than the digit being subtracted.
加减法的书面方法包括列竖式，数字按位值对齐。当一列之和超过 9 时发生进位；当某位数字小于要减去的数字时发生借位。
Inverse operations: addition and subtraction are inverses. We can check a subtraction by adding the result to the number that was taken away, and vice versa. Estimation by rounding before calculating helps judge whether an answer is reasonable.
逆运算：加法和减法互为逆运算。可以通过将结果与减数相加来检验减法，反之亦然。计算前通过四舍五入进行估算有助于判断答案是否合理。
Word problems require careful reading to decide which operation to use. Key phrases such as 'how many more', 'altogether' or 'difference' guide the choice.
文字题需要仔细阅读以确定使用哪种运算。'多多少'、'一共' 或 '差' 等关键短语指导选择。
3. Multiplication and Division of Whole Numbers | 整数的乘法和除法
Long multiplication is used for multiplying by two-digit or three-digit numbers. The number is multiplied by each part of the multiplier and then the partial products are added, paying attention to place value zeros.
长乘法用于乘以两位数或三位数。将数字乘以乘数的每一部分，然后将部分积相加，注意位值零。
Short division and long division methods help divide by one-digit and two-digit numbers, giving quotients with or without remainders. Remainders can be written as 'r1' or converted into a fraction or decimal.
短除法和长除法有助于除以一位数和两位数，得出带有或不带有余数的商。余数可以写成 'r1' 或转换为分数或小数。
Factors are numbers that divide exactly into another number. Multiples are the results of multiplying a number by an integer. Prime numbers have exactly two factors, 1 and the number itself. Composite numbers have more than two factors.
因数是能整除另一个数的数。倍数是某个数乘以整数的结果。质数恰有两个因数，即 1 和它本身。合数有两个以上的因数。
4. Fractions – Comparing, Adding and Subtracting | 分数 - 比较、加法和减法
Equivalent fractions have the same value but different numerators and denominators. They are found by multiplying or dividing the numerator and denominator by the same whole number. Simplifying a fraction means writing it in its lowest terms.
等值分数数值相同但分子和分母不同。通过将分子和分母乘以或除以同一个整数可以得到等值分数。化简分数意味着用最简分数表示它。
Comparing and ordering fractions with different denominators requires finding a common denominator. Mixed numbers contain a whole number and a proper fraction; improper fractions have a numerator larger than the denominator.
比较和排序分母不同的分数需要找到公分母。带分数包含一个整数和一个真分数；假分数的分子大于分母。
To add or subtract fractions with different denominators, first find equivalent fractions with a common denominator, then add or subtract the numerators and keep the denominator the same. For mixed numbers, add or subtract the whole numbers and the fractions separately, possibly regrouping.
加减分母不同的分数时，首先找到具有公分母的等值分数，然后加减分子，保持分母不变。对于带分数，分别加减整数部分和分数部分，可能需要重新组合。
5. Decimals – Place Value and Operations | 小数 - 位值和运算
Decimals extend the place value system to the right of the units: tenths, hundredths, thousandths. A number like 4.325 has 4 ones, 3 tenths (0.3), 2 hundredths (0.02) and 5 thousandths (0.005).
小数将位值系统扩展到个位右侧：十分位、百分位、千分位。像 4.325 这样的数有 4 个一、3 个十分之一（0.3）、2 个百分之一（0.02）和 5 个千分之一（0.005）。
Ordering decimals is done by comparing place values from left to right. Adding or subtracting decimals aligns the decimal points and fills empty places with zeros if needed. Multiplying a decimal by 10, 100 or 1000 moves the decimal point one, two or three places to the right; dividing does the opposite.
通过从左到右比较位值来排序小数。加减小数时对齐小数点，必要时用零填充空位。小数乘以 10、100 或 1000 将小数点向右移动一位、两位或三位；除法相反。
Multiplying decimals by a whole number: multiply as integers, then place the decimal point in the product so that it has the same number of decimal places as the original decimal.
小数乘以整数：像整数一样相乘，然后在积中放置小数点，使小数位数与原小数相同。
6. Understanding Percentages | 理解百分数
Percent means 'out of 100' entage can be written as a fraction with denominator 100, or as a decimal. For example, 45% = 45/100 = 0.45.
百分数表示 '百分之……'。百分数可以写成分母为 100 的分数，或写成小数。例如，45% = 45/100 = 0.45。
Converting between fractions, decimals and percentages: to convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply the fraction by 100%. To convert a decimal to a percentage, multiply by 100 and add the % sign.
分数、小数和百分数之间的转换：将分数转换为百分数，可找到分母为 100 的等值分数，或将分数乘以 100%。将小数转换为百分数，乘以 100 并加上 % 符号。
Finding a percentage of a quantity: first find 10% or 1%, then use multiples to compose the required percentage. For example, 35% of 240 = (10% × 3) + (5% × 1) or use the fraction equivalent.
求一个数量的百分之几：先求 10% 或 1%，然后用倍数组合出所需的百分数。例如，240 的 35% = (10% × 3) + (5% × 1)，或使用等值分数。
7. Measurement Conversions | 测量单位换算
10, 100 or 1000: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm; 1 kg = 1000 g; 1 l = 1000 ml.
公制单位包括长度（毫米、厘米、米、千米）、质量（克、千克）和容量（毫升、升）。换算依赖于乘以或除以 10、100 或 1000：1 千米 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米；1 千克 = 1000 克；1 升 = 1000 毫升。
ot = 12 inches, 1 yard = 3 feet, 1 mile ≈ 1.6 km. 1 pound (lb) ≈ 0.45 kg, 1 pint ≈ 568 ml.
可能会引入英制单位：1 英寸 ≈ 2.54 厘米，1 英尺 = 12 英寸，1 码 = 3 英尺，1 英里 ≈ 1.6 千米。1 磅 ≈ 0.45 千克，1 品脱 ≈ 568 毫升。
7 days. Adding and subtracting time requires regrouping when seconds or minutes exceed 60.
时间换算：1 分钟 = 60 秒，1 小时 = 60 分钟，1 天 = 24 小时，1 周 = 7 天。加减时间时，当秒或分钟超过 60 时需要重新组合。
8. Area, Perimeter and Volume | 面积、周长和体积
rectangle, perimeter P = 2 × (length + width). It is measured in linear units (mm, cm, m).
周长是图形周边的距离。对于长方形，周长 P = 2 × (长 + 宽)。周长用长度单位（毫米、厘米、米）度量。
dth, measured in square units (mm², cm², m²). Compound shapes can be split into rectangles and the areas added.
面积是表面覆盖的量。长方形的面积是长 × 宽，用平方单位（mm², cm², m²）度量。组合图形可以拆分成多个长方形，再加总它们的面积。
e of a cube or cuboid is length × width × height, expressed in cubic units (cm³, m³, mm³).
体积度量的是三维物体内部的空间。正方体或长方体的体积是长 × 宽 × 高，用立方单位（cm³, m³, mm³）表示。
9. Properties of 2D and 3D Shapes | 平面图形和立体图形的性质
2D shapes are classified by the number of sides and angles: triangles (3 sides), quadrilaterals (4 sides), pentagons (5), hexagons (6), etc. Polygons have straight sides; circles are not polygons.
平面图形根据边和角的数量分类：三角形（3 条边）、四边形（4 条边）、五边形（5 条边）、六边形（6 条边）等。多边形有直边；圆不是多边形。
Triangles are named by their sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites.
三角形按边分类（等边三角形、等腰三角形、不等边三角形）和按角分类（锐角、直角、钝角）。四边形包括正方形、长方形、平行四边形、菱形、梯形和风筝形。
Angles are measured in degrees (°) using a protractor. Types: acute (< 90°), right (90°), obtuse (90° – 180°), straight (180°). Angles on a straight line sum to 180°, and around a point sum to 360°.
角用量角器以度（°）测量。类型：锐角（< 90°）、直角（90°）、钝角（90° – 180°）、平角（180°）。直线上的角度和为 180°，绕一点的角度和为 360°。
3D shapes include cubes, cuboids, prisms, pyramids, spheres, cylinders and cones. Nets are flat arrangements of faces that fold to form the 3D shape. A cube has 6 square faces and 11 possible nets.
立体图形包括正方体、长方体、棱柱、棱锥、球体、圆柱和圆锥。展开图是将面展平后能折叠成立体图形的平面排列。正方体有 6 个正方形面，共有 11 种不同的展开图。
10. Data Representation and Interpretation | 数据表示与解读
Data can be organised in tables and represented pictorially using bar charts, line graphs or pie charts. The horizontal axis often shows categories or time; the vertical axis shows frequency or value.
数据可以用表格整理，并用条形图、折线图或饼图直观表示。横轴通常显示类别或时间；纵轴显示频数或数值。
Line graphs are used to show change over time. The scale must be continuous and the points joined. Reading between plotted points is called interpolation.
折线图用于显示随时间的变化。刻度必须是连续的，各点之间连成线。在已绘制点之间读取数值称为内插。
l angle of 360° represents the whole. Each sector angle = (category value ÷ total) × 360°.
饼图显示整体中的部分比例。360° 的整个角代表整体。每个扇形的角度 = (类别数值 ÷ 总数) × 360°。
The mean (average) is found by adding all values and dividing by the number of values. The median is the middle value when data is ordered. The mode is the value that occurs most often. The range is the difference between the highest and lowest values.
平均数（均值）通过将所有数值相加再除以数值的个数求得。中位数是数据排序后居中的数值。众数是出现次数最多的数值。极差是最大值与最小值的差。
11. Coordinates, Symmetry and Transformations | 坐标、对称与变换
Coordinates are written as (x, y) and can be plotted in all four quadrants on the Cartesian plane. The x-coordinate (horizontal) comes first, then the y-coordinate (vertical).
坐标写作 (x, y)，可以在直角坐标平面的所有四个象限内描点。x 坐标（水平方向）在前，y 坐标（垂直方向）在后。
Reflective symmetry: a shape has symmetry if one half is the mirror image of the other half across a line of symmetry. The line of symmetry can be horizontal, vertical or diagonal.
反射对称：如果一个图形的一半是另一半沿对称轴反射而成的镜像，则该图形具有对称性。对称轴可以是水平的、垂直的或斜向的。
Reflection is the transformation that flips a shape across a mirror line. Each point and its image are the same perpendicular distance from the mirror line. Translation moves a shape left, right, up or down without rotating or reflecting it.
反射是将图形沿镜面线翻转的变换。每个点与其像点到镜面线的垂直距离相等。平移将图形向左、向右、向上或向下移动，不旋转也不反射。
12. Problem Solving and Reasoning | 问题解决与推理
Multi-step word problems require reading the text carefully, identifying the key information, and deciding which operations are needed and in which order. Drawing a diagram or using a bar model can help visualise the problem.
多步骤文字题需要仔细阅读文本，识别关键信息，决定需要哪些运算及其顺序。画图或使用条形模型可以帮助可视化问题。
imation ensures accuracy. Reasoning logically about relationships between numbers, such as 'If 6 × 8 = 48, then 60 × 80 = 4800', helps solve problems efficiently.
用逆运算和估算检查答案可确保准确性。对数字之间的关系进行逻辑推理，例如 '如果 6 × 8 = 48，那么 60 × 80 = 4800'，有助于高效解题。
Common strategies include working backwards, looking for patterns, making a systematic list, or solving a simpler version of the problem first before tackling the complex one.
常用策略包括逆向思考、寻找模式、制作系统列表，或先解决问题的简单版本再处理复杂问题。
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