The document demonstrates how to perform long division of 177 by 8 using the partial quotients method. It shows working through the problem step-by-step starting with a partial quotient of 10, then another example starting with 20 to reach the solution faster. The key steps are determining the partial quotients, multiplying them by the divisor, subtracting, and repeating until the remainder is less than the divisor. The final answer is 22 with remainder 1.

1. 177 8 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.

2. 177 8 Ask - How many [8s] are in 177? There are at least 10, so that will be the first partial quotient. Notice that you could pick other partial quotients here. You could have started with 20, or even a lower number like 5. A good tip is to start with easy multiples, like 10, 20, 50, or 100. 10

3. 177 8 Multiply 10 * 8 10 80

4. 177 8 Subtract 177 minus 80. 10 80 97 -

5. 177 8 Start the process over again. Ask - how many [8s] are in 97? Again, there are at least 10. 10 80 97 - 10

6. 177 8 Multiply 10 * 8. 10 80 97 80 - 10

7. 177 8 Subtract 97 minus 80. 10 80 97 80 17 - 10 -

8. 177 8 Start the process again. Ask - how many [8s] are in 17. There are at least 2. 10 80 97 80 17 - 10 - 2

9. 177 8 Multiply 2 * 8. 10 80 97 80 17 16 - 10 - 2

10. 177 8 Subtract 17 minus 16. 10 80 97 80 17 16 1 - 10 - 2 -

11. 177 8 Since the 1 is less than 8, you are finished. Now add up the partial quotients - 10 plus 10 plus 2. 10 80 97 80 17 16 1 - 10 - 2 - 22

12. 177 8 Write the answer above with the remainder. You are finished. 10 80 97 80 17 16 1 - 10 - 2 - 22 22 R1

13. 177 8 Notice that you could have done the problem differently by picking different partial quotients. You could have completed the problem in fewer steps by picking partial quotients that were closer to the answer. Here, the student starts with a partial quotient of 20.

14. 177 8 20

15. 177 8 Multiply 20 * 8 20 160

16. 177 8 Subtract 177 minus 160. 20 160 17 -

17. 177 8 Start the process over again. Ask - how many [8s] are in 17? There are at least 2. 20 160 17 - 2

18. 177 8 Multiply 2 * 8. 20 160 17 - 2 16

19. 177 8 Subtract 17 minus 16. 20 160 17 - 2 16 - 1

20. 177 8 Since the 1 is less than 8, you are finished. Now add up the partial quotients - 20 plus 2. 20 160 17 - 2 16 - 1 22

21. 177 8 Write the answer above with the remainder. You are finished. 20 160 17 - 2 16 - 1 22 22 R1