1. The Forgiveness Method & Partial Quotients Division
The Partial Quotients Algorithm uses a series of “at least, but less than” estimates of how many b’s in a. Students might begin with multiples of 10 – they’re easiest.
This method builds towards traditional long division. It removes difficulties and errors associated with simple structure mistakes of long division.
Based on EM resources

2. Partial Quotients Division
Easy step by step directions to help with long division…..look at the picture below. What game does it remind you of?
Answer:
HANGMAN!!!

3. Partial Quotients Division
Start by setting up the problem like this. It looks just like the traditional long division method, except for the long line that is drawn to the right of the divisor. (Just like in the Hangman Game.)
Discuss benchmark numbers…
X 1
X 10
X 100
177 8 3

4. Ask - How many [8s] are in 177? There are at least 10, so that will be the first partial quotient..80 10
Multiply 10 * 8, write the product under the dividend in the problem. Then subtract!
Write on the side
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 4

5. Subtract 177 minus 80.
Now check, is 97 less than your divisor, 8? If yes, then you have finished dividing. If not….. 8
177 - 80 10 97
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 5

6. Start the process over again. Ask - how many [8s] are in 97?
Again, there are at least 10. 8
177 - 80 10 97 80 10
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 6

7. Subtract 97 minus 80. 8
177 - 80 10
Now check, is 17 less than your divisor, 8? If yes, then you have finished dividing. If not….. 97 - 80 10
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 17 7

8. Start the process again. Ask - how many [8s] are in 17
Start the process again. Ask - how many [8s] are in 17. There are at least 2. 8
177 - 80 10 97 - 80 10
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800
Subtract 17 minus 16. 17 - 16 2 1 8

9. Since the 1 is less than 8, you are finished dividing.
Now add up the partial quotients plus 10 plus 2.
22 R1 8
177 - 80 10 97 - 80 10
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 17 - 16 2 1
Write the answer above with the remainder.
You are finished. 22 9

10. Partial Quotients Division
Start by setting up the problem like this. It looks just like the traditional long division method, except for the long line that is drawn to the right of the divisor. (Just like in the Hangman Game.)
Now, let’s try to same problem using basic multiplication facts!
177 8 10

11. Now, let’s try to same problem using basic multiplication facts! 8 177
Ask - How many [8s] are in 17? There are at least 2, so 2 will be the first partial quotient..
Now, let’s try to same problem using basic multiplication facts! 8
177
160 20
Multiply 2 * 8, write the product under the dividend in the problem.
Now, you will notice that there is an empty space under the last 7 in the dividend. We will place a “0” to occupy the empty space and add a “0” to the 2 in the partial quotient column. Then subtract! 11

12. Start the process over again. Ask - how many [8s] are in 17?
Again, there are at least 2. 8
177 -
160 20 17 16 2
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 12

13. Subtract 17 minus 16. 8
177 -
160 20
Now check, is 1 less than your divisor, 8? If yes, then you have finished dividing. If not…..keep going. 17 - 16 2
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 1 13

14. 1 is less than your divisor, 8, so you are finished dividing.
177 -
160 20
1 is less than your divisor, 8, so you are finished dividing.
Now, add up the partial quotients, 20 and 2 and write their sum with the remainder at the top of the problem. 17 - 16 2
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 1 22 14

15. Since the 1 is less than 8, you are finished dividing.
Now add up the partial quotients plus 10 plus 2.
22 R1 8
177 - 80 10 97 - 80 10
8 x 1 = 8
8 x 10 = 80
8 x 100 = 800 17 - 16 2 1
Write the answer above with the remainder.
You are finished. 22 15

16. Let’s try another one…..
844 ÷ 4
Set up the problem

17. Ask - How many [4s] are in 844? There are at least 100, so that will be the first partial quotient..
Write on the side
4 x 1 = 4
4 x 10 = 40
4 x 100 = 400 17

18. Start the process over again. Ask - how many [4s] are in 444?
844 4
- 400
100
4 x 1 = 4
4 x 10 = 40
4 x 100 = 400
444
100
- 400 44
Start the process over again. Ask - how many [4s] are in 444?
There are at least 100 more. 18

19. Start the process over again. Ask - how many [4s] are in 44?
844 4
- 400
100
4 x 1 = 4
4 x 10 = 40
4 x 100 = 400
444
100
- 400 44 10
- 40 4
Start the process over again. Ask - how many [4s] are in 44?
There are at least 10 more. 19

20. 211
844 4
- 400
100
4 x 1 = 4
4 x 10 = 40
4 x 100 = 400
444
100
- 400 44 10
- 40 4 1
Since there are 4 left over, we know that 4 goes into 4 one more time.. Now add up the partial quotients:
= 211. 20

21. Let’s try another one…..
84.4 ÷ 4
What is different between this problem and the last problem?

22. Ask - How many [4s] are in 84.4? There are at least 20, so that will be the first partial quotient..
Write on the side
4 x 1 = 4
4 x 10 = 40
4 x 20 = 80
4 x 100 = 400 22

23. Start the process over again. Ask - how many [4s] are in 4.4?
84.4 4
- 80 20
4 x 1 = 4
4 x 10 = 40
4 x 20 = 80
4 x 100 = 400
4.4 1
- 4
0.4
Start the process over again. Ask - how many [4s] are in 4.4?
There is at least 1 more. 23

24. Start the process over again. Ask - how many [4s] are in 0.4?
84.4 4
- 80 20
4 x 1 = 4
4 x 10 = 40
4 x 20 = 80
4 x 100 = 400
4.4 1
- 4
0.4
0.1
- 0.4
Start the process over again. Ask - how many [4s] are in 0.4?
There is exactly 0.1 more. 24

25. Now add up the partial quotients: 20 + 1 + 0.1 = 21.1
84.4 4
- 80 20
4 x 1 = 4
4 x 10 = 40
4 x 20 = 80
4 x 100 = 400
4.4 1
- 4
0.4
0.1
- 0.4
Now add up the partial quotients:
= 21.1 25