Understanding Division: Sharing Equally | 理解除法：平均分配

📚 Understanding Division: Sharing Equally | 理解除法：平均分配

Welcome to Lesson 16-4! In this lesson, we will explore what division means and how we use it every day. Division helps us share things fairly, such as cookies, pencils, or toys. We will learn about the division symbol, the words dividend, divisor, and quotient, and we will use arrays, number lines, and repeated subtraction to build strong foundations. These ideas will also prepare you for IGCSE Mathematics, where division appears in fractions, ratios, and algebra.

欢迎来到第16-4课！这节课我们将探索除法的含义以及它在日常生活中的应用。除法帮助我们公平地分配物品，比如饼干、铅笔或玩具。我们将学习除法符号，以及被除数、除数和商这几个术语，还会用数组、数轴和重复减法来打牢基础。这些知识会为你学习IGCSE数学中的分数、比例和代数做好准备。

1. What Is Division? | 什么是除法？

Division is one of the four basic operations in mathematics, next to addition, subtraction, and multiplication. Division is used to split a total amount into equal groups. The symbol for division is ÷. When we read 12 ÷ 3, we say “twelve divided by three”.

除法是数学中与加法、减法、乘法并列的四种基本运算之一。除法用来把一个总数分成若干个相等的组。除法的符号是 ÷。当我们看到12 ÷ 3时，读作“十二除以三”。

Division can answer two important questions. First, if we know the total and the number of groups, how many objects are in each group? Second, if we know the total and the size of each group, how many groups can we make?

除法可以回答两个重要问题。第一，如果知道总数和组数，每组有多少个物体？第二，如果知道总数和每组的大小，可以分成几组？

For example, imagine we have 9 apples and put them into 3 baskets equally. We are sharing 9 apples among 3 baskets, so we write 9 ÷ 3. Each basket receives 3 apples.

例如，假设我们有9个苹果，要把它们平均放进3个篮子里。我们就是把9个苹果分给3个篮子，所以写成9 ÷ 3。每个篮子得到3个苹果。

9 ÷ 3 = 3

The number 9 is the total. The number 3 is the number of groups. The answer 3 is the number in each group.

这里的9是总数，3是组数，答案3是每组里的数量。

2. Sharing into Equal Groups | 平均分成相同组

When we divide, every group must have the same size. This is called equal sharing. Equal sharing is fair because nobody gets more or less than anybody else.

做除法时，每一组的大小必须相同，这叫做平均分。平均分是公平的，因为每个人得到的一样多，没有人多也没有人少。

Let us share 15 counters into 5 groups. We put one counter into each group, then another counter into each group, and so on. After all counters are placed, each group has 3 counters.

我们把15个计数片平均分成5组。先给每组放1个，再给每组放1个，继续这样放下去。当所有计数片都放完时，每组有3个计数片。

15 ÷ 5 = 3

We can check our answer by thinking about multiplication: if 5 groups each have 3 counters, then 5 × 3 = 15 counters in total. This connects division directly to multiplication.

我们可以用乘法来检验答案：如果5组每组有3个计数片，那么5 × 3 = 15个计数片。这直接把除法和乘法联系了起来。

Let us try another example. Suppose 18 pencils are shared equally among 6 students. How many pencils does each student get?

再看一个例子。假设18支铅笔要平均分给6名学生，每名学生得到多少支铅笔？

18 ÷ 6 = 3

Each student gets 3 pencils. We can verify: 6 students × 3 pencils = 18 pencils.

每名学生得到3支铅笔。我们可以验证：6名学生 × 3支铅笔 = 18支铅笔。

3. Division as Repeated Subtraction | 除法是连续减法

Division can also be seen as repeated subtraction. To find 20 ÷ 5, we subtract 5 from 20 again and again until we reach 0, and we count how many times we subtract.

除法也可以看作重复减法。要计算20 ÷ 5，我们从20里不断减去5，直到剩下0，并数一数减了多少次。

20 − 5 = 15, so we subtract once. 15 − 5 = 10, so we subtract twice. 10 − 5 = 5, so we subtract three times. 5 − 5 = 0, so we subtract four times.

20 − 5 = 15，这是第1次；15 − 5 = 10，这是第2次；10 − 5 = 5，这是第3次；5 − 5 = 0，这是第4次。

20 ÷ 5 = 4

This means 5 is subtracted from 20 exactly 4 times. Repeated subtraction is like counting equal steps backward on a number line.

这说明从20里减去5，恰好可以减4次。重复减法就像在数轴上向后数相等的步长。

Another example: 12 ÷ 4. Subtract 4 from 12: 12 − 4 = 8, 8 − 4 = 4, 4 − 4 = 0. We subtracted 3 times, so 12 ÷ 4 = 3.

再举一例：12 ÷ 4。从12里减4：12 − 4 = 8，8 − 4 = 4，4 − 4 = 0。我们减了3次，所以12 ÷ 4 = 3。

12 ÷ 4 = 3

4. Division and Multiplication Are Inverse Operations | 除法与乘法互为逆运算

Multiplication and division are inverse operations, which means they undo each other. If we know one multiplication fact, we can write two division facts from it.

乘法和除法互为逆运算，也就是说它们可以互相“撤销”。如果我们知道一个乘法算式，就可以写出两个除法算式。

Look at the fact family of 3, 4, and 12. The multiplication fact is 3 × 4 = 12. The two division facts are 12 ÷ 3 = 4 and 12 ÷ 4 = 3.

观察由3、4、12组成的算式家族。乘法算式是3 × 4 = 12。两个除法算式分别是12 ÷ 3 = 4和12 ÷ 4 = 3。

3 × 4 = 12　⟶　12 ÷ 3 = 4 and 12 ÷ 4 = 3

This is called a fact family. A fact family uses three numbers and shows how they are related through multiplication and division. The largest number is the product in multiplication and the dividend in division.

这叫做算式家族。算式家族用三个数展示它们之间通过乘法和除法建立的联系。最大的数在乘法中是积，在除法中是被除数。

Let us use the numbers 5, 6, and 30. We know 5 × 6 = 30, so 30 ÷ 5 = 6 and 30 ÷ 6 = 5.

我们用5、6和30来试试。我们知道5 × 6 = 30，所以30 ÷ 5 = 6，30 ÷ 6 = 5。

5 × 6 = 30 | 5乘6等于30
30 ÷ 5 = 6 | 30除以5等于6
30 ÷ 6 = 5 | 30除以6等于5

When solving division problems, always ask yourself: which multiplication fact can help me find the answer?

解决除法问题时，要问问自己：哪一条乘法口诀能帮助我找到答案？

5. Using Arrays to Show Division | 用数组表示除法

An array is an arrangement of objects in rows and columns. Rows go across from left to right, and columns go up and down. Arrays help us see division clearly.

数组是把物体按行和列排列起来的形式。行是左右方向的，列是上下方向的。数组能帮助我们清楚地看出除法。

Suppose we arrange 15 dots in 3 rows. How many dots are in each row? The array shows that each row has 5 dots.

假设我们把15个圆点排成3行。每行有几个圆点？数组显示每行有5个圆点。

15 ÷ 3 = 5

We can also arrange the same 15 dots in 5 rows. Now each row has 3 dots. This gives us 15 ÷ 5 = 3.

我们也可以把同样的15个圆点排成5行。这时每行有3个圆点，得到15 ÷ 5 = 3。

The array has 3 rows and 5 columns. The total number of dots is 15. When we divide the total by the number of rows, we find the number of columns. When we divide the total by the number of columns, we find the number of rows.

这个数组有3行5列，圆点总数是15。用总数除以行数，就得到列数；用总数除以列数，就得到行数。

Example: an array has 4 rows and 6 columns, so the total is 24. The division facts are 24 ÷ 4 = 6 and 24 ÷ 6 = 4.

例如：一个数组有4行6列，总数是24。对应的除法算式是24 ÷ 4 = 6和24 ÷ 6 = 4。

6. Division on a Number Line | 数轴上的除法

A number line is another helpful tool for division. To divide on a number line, we start at the dividend and jump backward by equal steps. The number of jumps is the quotient.

数轴是帮助理解除法的另一个好工具。在数轴上做除法时，我们从被除数出发，以相等的步长向后跳。跳的次数就是商。

Let us find 18 ÷ 6 using a number line. Start at 18 and make jumps of size 6 backward: from 18 to 12 is one jump, from 12 to 6 is two jumps, and from 6 to 0 is three jumps. We made 3 jumps, so 18 ÷ 6 = 3.

我们用数轴计算18 ÷ 6。从18出发，每次向后跳6：从18跳到12是第1次，从12跳到6是第2次，从6跳到0是第3次。一共跳了3次，所以18 ÷ 6 = 3。

18 ÷ 6 = 3

We can also divide on a number line by jumping forward. To solve 10 ÷ 2, start at 0 and jump forward by 2 until we reach 10: 0 to 2, 2 to 4, 4 to 6, 6 to 8, 8 to 10. That is 5 jumps, so 10 ÷ 2 = 5.

我们也可以在数轴上向前跳来完成除法。要计算10 ÷ 2，从0出发每次向前跳2，直到到达10：0到2，2到4，4到6，6到8，8到10。一共5次，所以10 ÷ 2 = 5。

Number lines show that division is like measuring how many equal sized groups fit into the total.

数轴让我们看到，除法就是在计算总数里可以容纳多少个大小相等的组。

7. Division Symbols and Vocabulary | 除法的符号和术语

In a division statement, each number has a special name. Look at 12 ÷ 3 = 4. The number 12 is called the dividend, the number 3 is called the divisor, and the number 4 is called the quotient.

在一个除法算式中，每个数都有特殊的名称。观察12 ÷ 3 = 4：12叫做被除数，3叫做除数，4叫做商。

The dividend is the total amount being split. The divisor is how many groups we are making, or the size of each group. The quotient is the answer we get after dividing.

被除数是要被分割的总数。除数是我们要分成的组数，或者每组的大小。商是除法算出的结果。

English	中文	Meaning	Example, 12 ÷ 3 = 4
dividend	被除数	the number being divided	12
divisor	除数	the number you divide by	3
quotient	商	the answer to division	4

The division sign ÷ is used in number sentences. We can also write division using a fraction bar: 12/3 means 12 ÷ 3. In IGCSE mathematics, you will see both forms.

除号÷用于算式之中。我们也可以用分数线表示除法：12/3表示12 ÷ 3。在IGCSE数学中，你会看到这两种写法。

Remember the division sentence format: dividend

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Understanding Division: Sharing Equally | 理解除法：平均分配