Long Division and Repeated Subtraction
This is a complete lesson with examples and exercises about the repeated subtraction process, as it relates to division. I give several examples of comparing division to bagging fruits and using repeated subtraction in that context. Several exercises follow. Lastly the lesson shows a comparison of this process with the actual long division algorithm. The lesson is meant for fifth grade.
You know how multiplication can be seen as repeated addition. Division is the opposite of multiplication. So, it should be no surprise that division can be solved by repeated (or continued) subtraction. Read through the examples carefully in order to understand that.  
Example 1. Bag 771 apples, placing 3 apples in each bag. How many bags
will you need? You might start by putting 3 apples into one bag, which leaves you with 768 apples. From then on, for each bag you use, subtract 3 apples. Keep counting the bags you use until you have no apples!
It just takes quite a long time to do it this way! Instead, you can take a shortcut: subtract 300 apples at a time (which means 100 bags), as long as you can, then 30 apples at a time as long as you can (which means 10 bags), and lastly 3 apples at a time.




1. Bag fruits the slow way. Fill in the missing parts.
a. Bag 657 apples;

b.
Bag 984 peaches;

c. Bag
536 pineapples;

2. Bag fruits the fast way!
a. Bag
474 apples;

b. Bag
2,032 lemons;

c. Bag
3,655 bananas;


d. Bag
762 mangos;

e. Bag
1,152 papayas;

f. Bag
4,770 cherries;

3. If there were 765 mangos instead of 762 in problem 2d above, how would the
result change?
4. a. Margie subtracted 24 from a certain number seven times, and
reached zero.
What was the number she started with?
b. This time, Margie subtracted 9 from a certain
number five times, and reached 2.
What was the number she started with?
Let’s compare continued subtraction with long division. They are actually the same method, just written out differently! Below, the numbers in long division are written out in full, using black and gray digits. The gray digits are the ones we do not usually write. Also, in the first example, the three parts of the quotient (200, 60, and 3) are written above each other for comparison's sake. Fill in the two last examples. 







5. Bag fruits. Also solve the problems using long division, and compare the methods.
a. Bag 610 apples, 5 apples in each bag.

b. Bag 853 kiwis, 3 kiwis in each bag.


c. Bag 445 grapefruits, 3 grapefruits in each bag.

d. Bag 952 plums, 4 plums in each bag.


e. Bag 2,450 pears, 9 pears in each bag.

f. Bag 1,496 oranges, 8 oranges in each bag.

This lesson is taken from Maria Miller's book Math Mammoth Multiplication & Division 3, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
Math Mammoth Multiplication & Division 3
A selfteaching worktext for 5th grade that covers multidigit multiplication, long division, problem solving, simple equations, ratios, divisibility, and factoring.
Download ($7.40). Also available as a printed copy.