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The ideas in this lesson are taken from Multiplication Division 2 ebook. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.


Division as repeated subtraction - introduction to long division
Free lesson plan from HomeschoolMath.net

Multiplication is repeated addition.  Division is the opposite of multiplication.  You can think of division as repeated subtraction.

 

Example.  Bag 771 apples so there are 3 apples in one bag.  How many bags are needed?
You can start by putting 3 apples to one bag, which leaves you 768 apples. Then for each bag you subtract 3 apples and keep counting the bags you use, until you hit zero apples.
 
771 − 3 − 3 − 3 − 3 − 3 − 3 ... keep subtracting!
1 bag 1 bag 1 bag 1 bag 1 bag 1 bag ... keep counting bags!

 
It just takes quite a long time, doesn't it?  Instead you can take a 'shortcut' and initially subtract 300 apples (taking 100 bags) or some other big multiple of 3.
 

771  −  300  −  300  −  30 ...
100 bags 100 bags 10 bags ...


 Let's figure it out and keep count of the bags as we subtract (put in bags) the apples. 
 

Method 1 - slower

ApplesBags
771
- 300

471
100 bags
- 300

171
100 bags
- 30

141
10 bags
- 30

111
10 bags
- 30

81
10 bags
- 30

51
10 bags
- 30

21
10 bags
- 21

0
7 bags

Method 2 - quicker 

ApplesBags
771
-  600200 bags
171
-  15050 bags
217 bags
-  21

0
It's over 700 apples.
In 100 bags would have 300 apples.
In 200 bags would have 600 apples.
In 300 bags would have 900 apples; too many bags. 
Subtracting the 600 apples from 771 leaves 171 still
 to be bagged.

It's over 170 apples still.
In 10 bags I'll have 30 apples.
In 20 bags I'd have 60 apples.
In 50 bags I'd have 150 apples.
In 60 bags I'd have 180 apples; that's too many. 
Subtract 150 apples from 171; 21 apples are left. 

Need 7 bags for the last 21 apples.  

 
So total needed 200 + 50 + 7 = 257 bags to bag all the apples.  And it all went even - no apples
left over!  In other words, 771 ÷ 3 = 257.



Example 4.  You have 646 apples.  If you put 8 apples in one bag, how many bags will you need?
 
ApplesBags
686

 
46 

-

 
Thinking first in whole hundreds:  In 100 bags I would have 800 apples 
but there's not that many apples, so won't need even 100 bags.

In terms of whole tens then. 
In 10 bags I'll have ____ apples. 
In 50 bags I'd have ____ apples.
In 80 bags I'd have ____ apples.
In 90 bags I'd have ____ apples.
So will need ____ bags.  Subtracting the ____ apples from 686 leaves 46 apples.

Lastly for the ___ apples left  I need __ bags, and there are 6 apples left over. 

So total needed _____ + ___ =           bags to bag all the apples and have 6 apples left over.
In other words, 686 ÷ 8 = ______, R 6.



Example 5.  It won't matter even if you do the subtracting in smaller steps.  Compare the two ways
to do the division 795 ÷ 3  by subtracting repeatedly.  You can think in terms of bagging apples if it helps.

DividendQuotient
795

-  300

100
495

- 300

100
195 
- 3010
165
- 3010
135
- 3010
105
DividendQuotient
795

-  600

200
195
- 18060
15
-  155
0
So total the quotient is
_____ + ___  + ___ = _____,
and division is even.
 
In other words, 795 ÷ 3 = ______.


Example problems

1.  Fill in the bags/fruits at each step of bagging.

a.  Bag 610 apples; 5 apples in each bag.
ApplesBags
610
- 500
110
- 100
10
-  10

 0
b.  Bag 852 kiwis; 3 kiwis in each bag.
KiwisBags
852

-

200
 

-  240

12
-  12

 0

e.  Bag 162 pears; 6 pears in each bag.

f.  Bag 495 cherries; 9 cherries in each bag.

g.  Bag 429 mangos; 3 mangos in each bag.
 
h.  Bag 164 pineapples; 2 pineapples in each bag.

i.  Bag 4613 guavas; 7 guavas in each bag.

j.  Bag 1098 bananas; 9 in each bag.
 

 

2.  Do the divisions using the repeated subtraction.  You can still think in terms of bagging fruit if it helps you.  Also, you are encouraged to even try doing it mentally.  Note: some of these divisions are even, some have remainder.

a.  555 ÷ 3

b.  750 ÷ 5

c.  257 ÷ 5

d.  464 ÷ 8 

m. 472 ÷ 5

n.  340 ÷ 2

o.  537 ÷ 9

p.  994 ÷ 7

q. 670 ÷ 5

r.  750 ÷ 6

s.  238 ÷ 9

t.  294 ÷ 7

Next lesson: Long division and why it works

See also Pictorial explanation of long division process.

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