📚 Cambridge Primary Mathematics Learner’s Book 5, 2nd Edition – Key Concepts Explained | 剑桥小学数学教材 Learner’s Book 5 第二版 知识点精讲
This article provides a structured overview of the core topics covered in the Cambridge Primary Mathematics Learner’s Book 5 (2nd Edition). Each section breaks down essential concepts, helping students build a solid foundation for more advanced mathematics. By revisiting these ideas through clear explanations and examples, learners can strengthen their understanding and confidence.
本文系统地梳理了《Cambridge Primary Mathematics Learner’s Book 5（第二版）》的核心知识点。每个章节深入浅出地讲解关键概念，帮助学生打下坚实数学基础，为更复杂的数学学习做好准备。通过清晰的解释和示例，学生可以巩固理解并增强信心。
1. Place Value and Number Sense | 位值与数感
Understand the value of digits up to ten millions. The position of a digit determines its value; for example, in 4,567,890 the digit 4 represents 4,000,000 (four million).
理解数字在千万位以内的位值。数字的位置决定了它的值；例如，在4,567,890中，数字4代表4,000,000（四百万）。
Read and write large numbers using correct spacing or commas to separate groups of three digits, e.g. 2,345,678. This helps in recognising place value quickly.
使用正确的间隔或逗号读写大数字，每三位数字一组，例如2,345,678。这有助于快速识别位值。
Round whole numbers to the nearest 10, 100, 1000, or 10 000. For instance, 47,632 rounded to the nearest hundred is 47,600.
将整数四舍五入到最近的十、百、千或万。例如，47,632四舍五入到最近的百是47,600。
2. Addition and Subtraction of Whole Numbers | 整数的加法和减法
Use column addition and subtraction with numbers up to six digits, carrying or borrowing where necessary. Always align digits by place value.
使用竖式加法和减法处理最多六位数字，在需要时进位或退位。务必按位值对齐数字。
Apply mental strategies such as adding 99 by adding 100 then subtracting 1: 364 + 99 = 364 + 100 – 1 = 463. This builds number fluency.
运用心算策略，例如通过加100再减1来加99：364 + 99 = 364 + 100 – 1 = 463。这能培养数字流畅度。
Check answers using the inverse operation: if 523 – 178 = 345, then 345 + 178 should equal 523. Estimation also helps catch unreasonable results.
用逆运算检查答案：如果523 – 178 = 345，那么345 + 178应等于523。估算也有助于发现不合理的结果。
3. Multiplication and Division Facts and Strategies | 乘法与除法事实与策略
Know multiplication tables up to 12 × 12 confidently. Use known facts to derive related division facts, e.g. 8 × 7 = 56 implies 56 ÷ 7 = 8.
熟练掌握12×12以内的乘法表。利用已知乘法事实推导相关除法事实，例如8 × 7 = 56意味着56 ÷ 7 = 8。
Multiply a two-digit or three-digit number by a one-digit number using expanded or short multiplication. For 324 × 6, multiply the ones, tens, and hundreds separately and add results.
使用扩展乘法或简乘法计算两位数或三位数乘一位数。例如324 × 6，分别乘个位、十位和百位，然后将结果相加。
Divide a three-digit number by a one-digit number including remainders. Understand that 275 ÷ 4 = 68 remainder 3, because 4 × 68 = 272 and 275 – 272 = 3.
计算三位数除以一位数，包括有余数的情况。理解275 ÷ 4 = 68余3，因为4 × 68 = 272，而275 – 272 = 3。
4. Fractions: Comparing, Ordering and Equivalent | 分数：比较、排序与等值
Identify and generate equivalent fractions using multiplication or division. For example, 2/5 is equivalent to 4/10 because multiplying numerator and denominator by 2 gives 4/10.
通过乘或除识别并生成等值分数。例如，2/5等于4/10，因为分子和分母同时乘以2得到4/10。
Compare and order fractions with different denominators by converting them to fractions with a common denominator. To compare 3/4 and 2/3, use 12ths: 3/4 = 9/12, 2/3 = 8/12, so 3/4 > 2/3.
通过将分数化为同分母来比较和排序不同分母的分数。比较3/4和2/3时用12作分母：3/4 = 9/12，2/3 = 8/12，因此3/4 > 2/3。
Convert improper fractions to mixed numbers and vice versa. 17/5 = 3 2/5 because 5 goes into 17 three times with remainder 2.
将假分数转换为带分数，反之亦然。17/5 = 3又2/5，因为5可以计17三次，余数为2。
5. Decimals and Percentages Introduction | 小数与百分比入门
Understand place value in decimals up to thousandths. In 0.725, the digit 7 is in the tenths place, 2 is hundredths, and 5 is thousandths.
理解小数到千分位的位值。在0.725中，数字7位于十分位，2位于百分位，5位于千分位。
Add and subtract decimals by lining up the decimal points. For 3.6 + 2.48, write 3.60 + 2.48 to make the columns clear and add to get 6.08.
对齐小数点进行小数加减。计算3.6 + 2.48时，写成3.60 + 2.48，使竖列清晰，相加得6.08。
Relate percentages to fractions out of 100: 25% means 25 out of 100 or 1/4. Find percentages of a quantity, e.g. 10% of 80 = 8, by dividing by 10.
将百分比与分母为100的分数相关联：25%表示100份中的25份或1/4。求一个数量的百分比，如80的10% = 8，通过除以10得到。
6. Measurement: Length, Mass, Capacity and Time | 测量：长度、质量、容量和时间
Convert between common units: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. Know that 1 kg = 1000 g and 1 L = 1000 mL. Use these to solve problems.
常见单位之间的换算：1千米 = 1000米，1米 = 100厘米，1厘米 = 10毫米。知道1千克 = 1000克，1升 = 1000毫升。运用这些解决实际问题。
Read and interpret timetables and calculate elapsed time. If a journey starts at 14:45 and ends at 17:10, the duration is 2 hours 25 minutes.
阅读并解读时间表，计算经过时间。如果旅程14:45开始，17:10结束，那么时长为2小时25分钟。
Solve problems involving mixed measures, such as adding 2.5 km and 750 m by converting to metres: 2500 m + 750 m = 3250 m, or 3 km 250 m.
解决涉及混合度量的题目，例如将2.5千米与750米相加，统一转换为米：2500米 + 750米 = 3250米，即3千米250米。
7. Geometry: Angles, Triangles and Quadrilaterals | 几何：角、三角形与四边形
Identify acute, right, obtuse and reflex angles. An acute angle is less than 90°, a right angle exactly 90°, an obtuse angle between 90° and 180°, and a reflex angle greater than 180°.
识别锐角、直角、钝角和优角（反射角）。锐角小于90°，直角等于90°，钝角在90°到180°之间，优角大于180°。
Classify triangles by their sides and angles: equilateral (all sides equal, all angles 60°), isosceles (two equal sides, two equal angles), and scalene (no equal sides).
根据边和角对三角形分类：等边三角形（所有边相等，所有角60°），等腰三角形（两条边相等，两个角相等）和不等边三角形（没有相等的边）。
Describe properties of quadrilaterals such as square, rectangle, parallelogram, rhombus and trapezium. A square has 4 equal sides and 4 right angles; a parallelogram has opposite sides parallel and equal.
描述四边形的性质，如正方形、长方形、平行四边形、菱形和梯形。正方形有4条等边和4个直角；平行四边形对边平行且相等。
8. Symmetry and Transformations | 对称与变换
Identify lines of symmetry in 2D shapes. A square has 4 lines of symmetry, a rectangle has 2, and an equilateral triangle has 3.
识别二维图形的对称轴。正方形有4条对称轴，长方形有2条，等边三角形有3条。
Reflect shapes across a mirror line on a grid. The reflected shape is the same distance from the mirror line as the original, but on the opposite side.
在方格上沿镜面线反射图形。反射后的图形与镜面线的距离与原图形相同，但位于另一侧。
Use coordinates to describe positions and translate shapes on a grid. Moving a point (3, 4) right 2 units and up 1 unit gives the new point (5, 5).
使用坐标描述位置，并在网格上平移图形。将点(3, 4)向右平移2个单位、向上平移1个单位，得到新点(5, 5)。
9. Perimeter, Area and Volume | 周长、面积和体积
Calculate the perimeter of regular and irregular shapes by adding all side lengths. For a rectangle 8 cm by 5 cm, perimeter = 2 × (8 + 5) = 26 cm.
通过将所有边长相加计算规则和不规则图形的周长。对于长8厘米、宽5厘米的长方形，周长 = 2 × (8 + 5) = 26厘米。
Find the area of rectangles and squares using the formula:
Area = length × width
and area of compound shapes by dividing into smaller rectangles.
使用公式面积 = 长 × 宽计算长方形和正方形的面积；通过分割为小长方形计算组合图形的面积。
Understand volume as the amount of space occupied by a 3D shape, measured in cubic units. The volume of a cuboid = length × width × height, e.g. 4 cm × 3 cm × 2 cm = 24 cm³.
理解体积是三维物体占据的空间量，以立方单位计量。长方体的体积 = 长 × 宽 × 高，例如4厘米 × 3厘米 × 2厘米 = 24立方厘米。
10. Data Handling and Probability | 数据处理与概率
Collect and organise data using frequency tables and tally charts. Use bar charts, pictograms, and line graphs to display information clearly.
使用频率表和计数表收集并整理数据。利用条形图、象形图和折线图清晰展示信息。
Interpret line graphs that show changes over time, such as temperature during the day. Identify trends, maximum and minimum values.
解读显示随时间变化的折线图，如一天内温度变化。识别变化趋势、最大值和最小值。
Describe probability using vocabulary like certain, likely, even chance, unlikely, and impossible. The probability of rolling an even number on a fair six-sided dice is 3/6, or an even chance.
用词汇描述概率，如必然、可能、相等机会、不大可能和不可能。掷一个公平六面骰子得到偶数的概率是3/6，也即相等机会。
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