Lesson Objectives

- Demonstrate an understanding of multiplication
- Learn how to divide single-digit whole numbers
- Learn how to divide single-digit whole numbers with remainders
- Learn how to label the parts of a division problem as dividend, divisor, and quotient

## How to Divide Single-Digit Whole Numbers

Once we have mastered multiplication with whole numbers, our next challenge is to understand the division operation. What is Division? When we divide with whole numbers, we are asking how many equal groups of one number can be made out of another. There are a few ways to show the division operation. In elementary school, we typically see the obelus "÷" placed between two numbers to show the division of the first number (leftmost) by the second. Let’s illustrate the division operation with an example. Suppose we begin with 12 boxes: Let's ask the following question: how many groups of 3 boxes can be made out of these 12 boxes? This can be translated into the division problem:

12 ÷ 3 = ? We can see from the above image that the answer is 4. We are able to form 4 groups with 3 boxes in each group from 12 boxes. There are a few ways we can come to this result. First and most obvious, we could count the number of times we can subtract away groups of 3 from 12 until the result is 0.

12 ÷ 3 = 4

we can work backward and see that

4 x 3 = 12

If we asked what is:

30 ÷ 6 = ?

We can find the answer with a related multiplication problem:

? x 6 = 30

We know from the multiplication tables that:

5 x 6 = 30

This tells us that 30 divided by 6 will give us 5.

30 ÷ 6 = 5

Example 1: Find each quotient.

45 ÷ 5

To solve this problem, we can use a related multiplication problem:

? x 5 = 45

If we look at our multiplication tables, we can find the answer is 9. Since 9 x 5 = 45, we can say that 45 ÷ 5 = 9.

11 ÷ 3 = 3 R2 How could we get this result with multiplication?

11 ÷ 3 = ?

? x 3 = 11

There is no such whole number, so we must adjust down.

? x 3 = 10

There is no such whole number, so we must adjust down

? x 3 = 9

The answer is 3 since 3 x 3 = 9

Since we are only using 9, we need to think about the difference between 11 and 9.

11 - 9 = 2, this will be our remainder.

11 ÷ 3 = 3 R2

Example 2: Find each quotient.

29 ÷ 6

? x 6 = 29

There is no such whole number, so we must adjust down.

? x 6 = 28

There is no such whole number, so we must adjust down.

? x 6 = 27

There is no such whole number, so we must adjust down.

? x 6 = 26

There is no such whole number, so we must adjust down.

? x 6 = 25

There is no such whole number, so we must adjust down.

? x 6 = 24

The answer is 4 since 4 x 6 = 24

Since we are only using 24, we need to think about the difference between 29 and 24.

29 - 24 = 5, this will be our remainder.

29 ÷ 6 = 4 R5

Check: 4 x 6 = 24 » 24 + 5 = 29

A faster way is to think about the multiplication table for 6:

6 x 1 = 6

6 x 2 = 12

6 x 3 = 18

6 x 4 = 24

6 x 5 = 30

6 x 6 = 36

6 x 7 = 42

6 x 8 = 48

6 x 9 = 54

We know that 6 x 5 = 30, but 30 is larger than 29 so this will not work. Our quotient must be smaller than 5. Now looking at the multiplication table, we move up to 6 x 4 = 24. 24 is the closest we can get to 29. Then we could subtract 29 - 24 and get the remainder of 5. Let's try another one, using this technique.

Example 3: Find each quotient.

41 ÷ 9

9 x 1 = 9

9 x 2 = 18

9 x 3 = 27

9 x 4 = 36

9 x 5 = 45

9 x 6 = 54

9 x 7 = 63

9 x 8 = 72

9 x 9 = 81

There is no such whole number that multiplies 9 and gives 41, so we look at the next result down from 41 and find 36. This comes from 9 x 4 = 36. We can subtract 41 - 36 = 5, to get our remainder of 5. So 41 ÷ 9 = 4 R5

Check: 4 x 9 = 36 » 36 + 5 = 41

3 - Divisor

4 - Quotient

Example 4: Label the parts of the division problem.

20 ÷ 10 = 2

20 - Dividend

10 - Divisor

2 - Quotient

4 + 7 = 7 + 4

4 + 7 = 11

7 + 4 = 11

In each case, the answer is 11.

4 x 7 = 7 x 4

4 x 7 = 28

7 x 4 = 28

In each case, the answer is 28.

The same is not true for subtraction or division. Subtraction is not commutative:

11 - 5 ≠ 5 - 11

Division is also not commutative. When we divide the order matters:

20 ÷ 4 ≠ 4 ÷ 20

Example 5: Find each quotient.

0 ÷ 29, 29 ÷ 0

0 ÷ 29 = 0

29 ÷ 0 is undefined

Any non-zero number divided by itself is equal to 1. We are asking how many groups of a given number can be made from the number. The result is always 1 for a non-zero number. For example: 17 ÷ 17 = 1 since we can make 1 group of 17 from the number 17.

Example 6: Find each quotient.

31 ÷ 1, 117 ÷ 117

31 ÷ 1 = 31

117 ÷ 117 = 1

12 ÷ 3 = ? We can see from the above image that the answer is 4. We are able to form 4 groups with 3 boxes in each group from 12 boxes. There are a few ways we can come to this result. First and most obvious, we could count the number of times we can subtract away groups of 3 from 12 until the result is 0.

- 12 - 3 = 9
- 9 - 3 = 6
- 6 - 3 = 3
- 3 - 3 = 0

12 ÷ 3 = 4

we can work backward and see that

4 x 3 = 12

If we asked what is:

30 ÷ 6 = ?

We can find the answer with a related multiplication problem:

? x 6 = 30

We know from the multiplication tables that:

5 x 6 = 30

This tells us that 30 divided by 6 will give us 5.

30 ÷ 6 = 5

Example 1: Find each quotient.

45 ÷ 5

To solve this problem, we can use a related multiplication problem:

? x 5 = 45

If we look at our multiplication tables, we can find the answer is 9. Since 9 x 5 = 45, we can say that 45 ÷ 5 = 9.

### Division with Remainders

Not every division will be clean, meaning sometimes there is an amount that is left over. When this happens, we have something known as a remainder. Suppose we have only 11 boxes instead of 12. Now, let’s pose the original question. How many groups of 3 boxes can be made out of these 11 boxes? The answer is no longer 4 since we don't have enough boxes. Let’s first think about this with subtraction and then multiplication:- 11 - 3 = 8
- 8 - 3 = 5
- 5 - 3 = 2
- 2 - 3 - stop here, we cannot take 3 away from 2. Since we don't have enough to make another group of 3, the 2 is the remainder or leftover amount.

11 ÷ 3 = 3 R2 How could we get this result with multiplication?

11 ÷ 3 = ?

? x 3 = 11

There is no such whole number, so we must adjust down.

? x 3 = 10

There is no such whole number, so we must adjust down

? x 3 = 9

The answer is 3 since 3 x 3 = 9

Since we are only using 9, we need to think about the difference between 11 and 9.

11 - 9 = 2, this will be our remainder.

11 ÷ 3 = 3 R2

Example 2: Find each quotient.

29 ÷ 6

? x 6 = 29

There is no such whole number, so we must adjust down.

? x 6 = 28

There is no such whole number, so we must adjust down.

? x 6 = 27

There is no such whole number, so we must adjust down.

? x 6 = 26

There is no such whole number, so we must adjust down.

? x 6 = 25

There is no such whole number, so we must adjust down.

? x 6 = 24

The answer is 4 since 4 x 6 = 24

Since we are only using 24, we need to think about the difference between 29 and 24.

29 - 24 = 5, this will be our remainder.

29 ÷ 6 = 4 R5

Check: 4 x 6 = 24 » 24 + 5 = 29

A faster way is to think about the multiplication table for 6:

6 x 1 = 6

6 x 2 = 12

6 x 3 = 18

6 x 4 = 24

6 x 5 = 30

6 x 6 = 36

6 x 7 = 42

6 x 8 = 48

6 x 9 = 54

We know that 6 x 5 = 30, but 30 is larger than 29 so this will not work. Our quotient must be smaller than 5. Now looking at the multiplication table, we move up to 6 x 4 = 24. 24 is the closest we can get to 29. Then we could subtract 29 - 24 and get the remainder of 5. Let's try another one, using this technique.

Example 3: Find each quotient.

41 ÷ 9

9 x 1 = 9

9 x 2 = 18

9 x 3 = 27

9 x 4 = 36

9 x 5 = 45

9 x 6 = 54

9 x 7 = 63

9 x 8 = 72

9 x 9 = 81

There is no such whole number that multiplies 9 and gives 41, so we look at the next result down from 41 and find 36. This comes from 9 x 4 = 36. We can subtract 41 - 36 = 5, to get our remainder of 5. So 41 ÷ 9 = 4 R5

Check: 4 x 9 = 36 » 36 + 5 = 41

### Labeling the Parts of a Division Problem

When we work with a division problem, we have three parts: the dividend, the divisor, and the quotient.- The dividend is the leftmost number in the division problem. This is the number that is being divided or split up into equal groups.
- The divisor is the amount in each equal group that we split the dividend into.
- The quotient is the result of the division operation.

3 - Divisor

4 - Quotient

Example 4: Label the parts of the division problem.

20 ÷ 10 = 2

20 - Dividend

10 - Divisor

2 - Quotient

### Division is not Commutative

We saw that multiplication and addition are commutative. This means the order is not important:4 + 7 = 7 + 4

4 + 7 = 11

7 + 4 = 11

In each case, the answer is 11.

4 x 7 = 7 x 4

4 x 7 = 28

7 x 4 = 28

In each case, the answer is 28.

The same is not true for subtraction or division. Subtraction is not commutative:

11 - 5 ≠ 5 - 11

Division is also not commutative. When we divide the order matters:

20 ÷ 4 ≠ 4 ÷ 20

### Division with Zero

When we divide with zero, we must pay close attention. We are allowed to divide zero by a non-zero number. The result of this operation is always zero. What we cannot do is divide by zero. Zero is not allowed to be the divisor in a problem. When this scenario occurs, we say the division is undefined.Example 5: Find each quotient.

0 ÷ 29, 29 ÷ 0

0 ÷ 29 = 0

29 ÷ 0 is undefined

### Division with 1

When we divide a number by 1, the number will remain unchanged. We are asking how many groups of 1 can be made from the number. This will always result in the number itself. For example: 25 ÷ 1 = 25 since we can make 25 groups of 1 from the number 25.Any non-zero number divided by itself is equal to 1. We are asking how many groups of a given number can be made from the number. The result is always 1 for a non-zero number. For example: 17 ÷ 17 = 1 since we can make 1 group of 17 from the number 17.

Example 6: Find each quotient.

31 ÷ 1, 117 ÷ 117

31 ÷ 1 = 31

117 ÷ 117 = 1

#### Skills Check:

Example #1

Find each quotient.

19 ÷ 3

Please choose the best answer.

A

3 R6

B

6 R1

C

12 R7

D

5 R4

E

4 R7

Example #2

Find each quotient.

59 ÷ 11

Please choose the best answer.

A

5 R4

B

4 R9

C

6 R7

D

3 R2

E

5 R9

Example #3

Which of the following is true?

Please choose the best answer.

A

365 ÷ 1 = 1

B

365 ÷ 0 = 0

C

365 ÷ 7 = 52 R3

D

0 ÷ 365 = 0

E

365 ÷ 365 = 365

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