Cambridge Primary Mathematics Learner’s Book 4 Essentials | 剑桥小学数学第四册知识点精讲

📚 Cambridge Primary Mathematics Learner’s Book 4 Essentials | 剑桥小学数学第四册知识点精讲

Cambridge Primary Mathematics Learner’s Book 4 (2nd Edition) is designed for learners aged 8-9, building a solid foundation in numbers, geometry, measurement and data handling. This guide revisits all key topics with clear explanations and examples, ideal for revision and classroom support.

剑桥小学数学第四册（第二版）适合8至9岁的学生，旨在打下数字、几何、测量和数据处理的坚实基础。本指南以清晰的解释和例题回顾所有核心知识点，非常适合复习和课堂辅助。

1. Numbers to 10 000 | 数到10 000

Learners explore whole numbers up to 10 000. A place value chart shows how the value of a digit depends on its position: ten thousands, thousands, hundreds, tens and ones.

学习者探索最大到10 000的整数。位值表显示一个数字的数值取决于其位置：万位、千位、百位、十位和个位。

The number 7 403 can be written in expanded form: 7 000 + 400 + 3. Reading large numbers correctly involves grouping digits in threes from the right.

数字7 403可以写成展开式：7 000 + 400 + 3。正确读出大数需要从右向左每三位分组。

When comparing numbers, look at the highest place value first. For example, 5 620 > 4 999 because the thousands digit 5 is greater than 4.

比较数字时，先看最高位值。例如，5 620 > 4 999，因为千位上的5大于4。

To round 3 267 to the nearest 100, look at the tens digit (6). Since 6 ≥ 5, round up to 3 300. Rounding to the nearest 10 and 1000 follows similar rules.

将3 267近似到最近的百位，看十位数字（6）。因为6 ≥ 5，所以向上近似为3 300。近似到最近的十位或千位遵循类似规则。

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2. Addition and Subtraction | 加法与减法

Column addition with regrouping is essential for 4-digit numbers. When adding 4 356 and 2 849, add ones (6+9=15), carry the 1 ten, then add tens, hundreds and thousands.

四位数加法需掌握进位列加法。计算4 356 + 2 849时，先加个位（6+9=15），向十位进1，再依次加十位、百位和千位。

Subtraction often requires borrowing. For 5 200 – 3 465, you must borrow from thousands to hundreds and then to tens, as the top digits in tens and ones are smaller.

减法常需要借位。例如5 200 – 3 465，由于被减数的十位和个位较小，必须从千位向百位借位，再向十位借位。

Mental strategies include partitioning numbers and using known facts, such as adding 50 to 325 by counting on in tens: 325, 335, 345, 355, 365, 375.

心算策略包括拆分数字和利用已知算式，例如325加50可以每次加10：325, 335, 345, 355, 365, 375。

Always check subtraction by adding the answer to the subtracted number: 3 250 – 1 675 = 1 575, and 1 575 + 1 675 = 3 250.

最好通过加法验算减法：3 250 – 1 675 = 1 575，而1 575 + 1 675 = 3 250。

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3. Multiplication and Division | 乘法与除法

Learners are expected to know the 6, 7, 9, 11 and 12 times tables. Quick recall helps with both multiplication and division problems.

学生应熟记6、7、9、11和12的乘法表。快速回忆有助于乘除法计算。

Multiplying a 2-digit number by a 1-digit number uses grid method or column method. For 23 × 4, partition: 20 × 4 = 80, 3 × 4 = 12, then add to get 92.

两位数乘一位数可以用网格法或列式法。以23 × 4为例，拆分：20 × 4 = 80, 3 × 4 = 12，然后相加得92。

When dividing 85 ÷ 5, think of how many groups of 5 make 85. Since 5 × 17 = 85, the answer is 17. Remainders occur when division is not exact, e.g. 58 ÷ 4 = 14 remainder 2.

计算85 ÷ 5时，思考多少个5等于85。因为5 × 17 = 85，所以商为17。当不能整除时会产生余数，如58 ÷ 4 = 14余2。

Multiplying a whole number by 10 shifts each digit one place to the left; multiplying by 100 shifts two places. For example, 34 × 10 = 340 and 34 × 100 = 3 400.

整数乘以10时，每个数字向左移动一位；乘以100则移动两位。例如，34 × 10 = 340, 34 × 100 = 3 400。

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4. Time | 时间

Key units of time include seconds (s), minutes (min), hours (h), days, weeks, months and years. Converting between them requires knowing 1 min = 60 s, 1 h = 60 min, 1 day = 24 h.

主要时间单位有秒(s)、分(min)、时(h)、日、周、月、年。换算需要记住1分 = 60秒, 1时 = 60分, 1日 = 24时。

The 12-hour clock uses a.m. and p.m.; the 24-hour clock runs from 00:00 to 23:59. To convert 2:45 p.m. to 24-hour time, add 12 to the hour: 14:45.

12小时制使用上午(a.m.)和下午(p.m.)；24小时制从00:00到23:59。将下午2:45换算为24小时制，小时加12，得到14:45。

Time intervals are calculated by finding the difference between two times. If a film starts at 11:20 and ends at 13:05, the duration is 1 hour 45 minutes.

时间间隔通过计算两个时刻的差得出。若电影从11:20开始，13:05结束，时长为1小时45分钟。

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5. 2D Shapes and Symmetry | 二维图形与对称

Polygons are named by the number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), octagon (8). Each shape has a specific number of vertices.

多边形按边数命名：三角形（3）、四边形（4）、五边形（5）、六边形（6）、八边形（8）。每种图形有特定数量的顶点。

A right angle measures 90° and is often marked with a small square. Learners identify right angles in shapes and around the classroom.

直角为90°，常用小方格标记。学生可以在图形中和教室里找出直角。

Line symmetry means a shape can be folded in half so both sides match exactly. A square has 4 lines of symmetry; a rectangle has 2; an equilateral triangle has 3.

轴对称意味着图形可以对折后两边完全重合。正方形有4条对称轴；长方形有2条；等边三角形有3条。

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6. Fractions | 分数

A fraction such as 3/4 has a numerator (3) and a denominator (4). Unit fractions have 1 as numerator; non-unit fractions have other numerators.

分数如3/4有分子(3)和分母(4)。单位分数的分子为1；非单位分数的分子大于1。

Equivalent fractions represent the same amount. 1/2 = 2/4 = 4/8. You can find equivalents by multiplying or dividing both numerator and denominator by the same number.

等价分数表示相同的量。1/2 = 2/4 = 4/8。分子分母同时乘或除以相同的数（不为零），可得到等价分数。

When comparing fractions with the same denominator, the larger the numerator, the larger the fraction: 5/8 > 3/8. Adding and subtracting like fractions keeps the denominator: 2/7 + 3/7 = 5/7.

比较同分母分数时，分子越大分数越大：5/8 > 3/8。同分母分数相加减，分母不变：2/7 + 3/7 = 5/7。

To find a fraction of a quantity, divide by the denominator and multiply by the numerator. 1/3 of 24 = 24 ÷ 3 = 8.

求一个量的几分之几，先除以分母再乘分子。24的1/3 = 24 ÷ 3 = 8。

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7. Length, Mass, Capacity and Perimeter | 长度、质量、容积和周长

Metric units for length are millimetre (mm), centimetre (cm), metre (m) and kilometre (km). Mass is measured in grams (g) and kilograms (kg). Capacity uses millilitres (mL) and litres (L).

长度的公制单位有毫米(mm)、厘米(cm)、米(m)和千米(km)。质量用克(g)和千克(kg)。容积用毫升(mL)和升(L)。

Length	Mass	Capacity
1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm	1 kg = 1000 g	1 L = 1000 mL

Perimeter is the distance around a shape. For a rectangle, add the lengths of all four sides or use the formula:

周长是图形一周的长度。计算长方形的周长，可以累加四条边或使用公式：

Perimeter = 2 × (length + width)

For a square, all sides are equal, so perimeter = 4 × side length. Compound shapes may require adding several straight edges.

正方形的四边相等，周长 = 4 × 边长。复合图形可能需要将多个直边长度相加。

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8. Area | 面积

Area measures the surface covered by a shape. It is recorded in square units, such as square centimetres (cm²).

面积度量图形覆盖的表面大小，用平方单位表示，如平方厘米(cm²)。

On a grid, count whole squares inside the shape. If a shape covers 12 full squares, its area is 12 square units.

在方格纸上，数出图形内的完整方格。若图形覆盖12个完整方格，面积为12平方单位。

When a shape includes half squares, two half squares combine to make one whole square. Add whole and combined half squares for the total area.

当图形包含半格时，两个半格合成一个整格。将整格数与合并的半格数相加得到总面积。

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9. Data Handling | 数据处理

Data is collected in tables using tally marks. A frequency column shows the total for each category.

数据通过统计表格收集，用画记数符号。频率列显示每个类别出现的总次数。

Pictograms use symbols to represent data. A key might state one symbol equals 2, 5 or 10 items. Half symbols represent half of that value.

象形图用图标表示数据。图例可能注明一个图标代表2、5或10个单位。半个图标代表该值的一半。

Bar charts have a horizontal axis (categories) and a vertical axis (number). The scale must be evenly spaced. Bars must be separate and labelled.

条形图有横轴（类别）和纵轴（数量）。刻度必须均匀。条形应分开并标注清楚。

Interpret data by reading values directly from the chart and comparing categories to answer questions about the data set.

通过直接从图表读取数值并比较各类别，回答关于数据集的问题。

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10. Decimals and Money | 小数和货币

Tenths and hundredths are the first two decimal places. 0.3 is ‘three tenths’, equal to 3/10. 0.25 is ‘twenty-five hundredths’, equal to 25/100 = 1/4.

十分位和百分位是小数点后的前两位。0.3读作“十分之三”，等于3/10。0.25读作“百分之二十五”，等于25/100 = 1/4。

To compare decimals, align the decimal points and compare digits from left to right. 0.7 > 0.59 because 7 tenths is more than 5 tenths.

比较小数时，对齐小数点，从左到右逐位比较。0.7 > 0.59，因为7个十分之一大于5个十分之一。

Money uses pounds (£) and pence (p). £2.45 is 2 pounds and 45 pence. Adding and subtracting money follows the same rules as decimal arithmetic.

货币使用英镑(£)和便士(p)。£2.45表示2英镑45便士。货币加减法遵循与小数相同的计算规则。

When finding change from £5 after spending £2.45, calculate £5.00 – £2.45 = £2.55. Count up from £2.45 to £5.00 to check.

计算花掉£2.45后从£5找回的零钱：£5.00 – £2.45 = £2.55。用从£2.45加到£5.00的方法验算。

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11. Position and Movement | 位置与移动

Coordinates in the first quadrant are written as (x, y). The x-coordinate (horizontal) comes first, then the y-coordinate (vertical). (3, 4) means ‘along 3, up 4’.

第一象限的坐标写成(x, y)。x坐标（水平）在前，y坐标（垂直）在后。(3, 4)表示“沿x轴走3步，再向上走4步”。

Plot points on a grid by starting at (0, 0), counting along the x-axis, then up the y-axis. Join points to form shapes.

在网格上描点，从(0,0)出发，沿x轴数相应格数，再沿y轴向上数。连接各点可形成图形。

Translation means moving

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