The ideas in this division lesson are taken from Division 2 ebook.
Only a few examples of each problem type are shown; you should make more problems of each kind for
the student.
Long division and why it works
Free lesson from HomeschoolMath.net
The standard long division algorithm
We compare here the repeated subtraction of the previous lesson and the conventional long division 'corner'. The
steps are the same, just written out differently. For clarity's sake, we
will initially write out the subtracted numbers with all the zeros included. Also,
for clarity and for easy comparison, we will write the parts of the quotient
above each other. As an example, we study 789 ÷ 3. You can
think of it as 789 apples that you are bagging in bags of 3 apples, wanting to
know how many bags you need.
First write the dividend 'inside' the corner, and the divisor outside:
3
7
8
9
Then let's divide!
Continued subtraction
Dividend
(the apples)
Quotient
(the bags)
789
- 600
200
189
- 180
60
9
3
- 9
0
263
2
0
0
3
7
8
9
-
6
0
0
1
8
9
6
0
2
0
0
3
7
8
9
-
6
0
0
1
8
9
-
1
8
0
9
3
6
0
2
0
0
3
7
8
9
-
6
0
0
1
8
9
-
1
8
0
9
-
9
0
First step is the hundreds, finding out which multiple
of 300 will fit into 789. It is 600. In terms of apples
and bags, one has now used 200 bags to bag 600 apples.
Second step, the tens. Which multiple of 30 will fit into 189? That
is 180, meaning one uses 60 bags to bag 180 apples. So 60 is added to the
quotient.
Lastly look at the ones. There are still 9 apples left, which means one needs 3
bags more. Add 3 to the quotient.
The final answer is 263.
The last step is to
check the
division by multiplication:
whether 3 × 263 is 789.
263
× 3
789
Why it works
Comparing the division to the continued subtraction probably has already let
you see why it works. In the conventional way of writing the long
division, it's not so easy to see the process. The key is that in each
step, one does NOT actually divide by the actual divisor but by a multiple of
it. Just like in the apples/bags examples, you don't start out by
subtracting 3 apples each time, but first 'hit it hard' by subtracting multiples
of 300 apples if possible, then multiples of 30, then 3. In essence, you
first divide by 300, then by 30, then by 3.
Also, in the conventional long division, you only place one digit into the
quotient in each step, not with all the zeros. The digits shown in gray
are not usually written out in the conventional long division algorithm.
Hundreds
"How many 3's in 7?"
(How many 300's in 789?)
Tens
"How many 3's in 18?"
(How many 30's in 189?)
Ones
"How many 3's in 9?"
2
0
0
3
7
8
9
-
6
0
0
1
8
9
2
6
0
3
7
8
9
-
6
0
0
1
8
9
-
1
8
0
0
9
2
6
3
3
7
8
9
-
6
0
0
1
8
9
-
1
8
0
0
9
-
9
0
To get the hundreds digit in the quotient, one asks the question: "How many times does
300 go into 789", or the
division 789 ÷ 300! You are not dividing by 3 because you try to 'hit it
hard' and subtract as many multiples of 300 as possible. Since 300 is a whole hundred, the tens and ones
digits in the 789 won't matter when you are finding how many times 300 goes into
789. So the thing can be done easier by calculating
7 ÷ 3, or thinking "How many times does 3 go into 7".
The remainder from first step (what is left after subtraction) is in reality
189. But since the ones digit (9) won't be important in the next step
(which deals with the tens digit), in the traditional way, you only subtract 7-6 and
then you 'drop' down the tens digit 8 from the dividend.
To get the tens digit, similarly one asks the question: "How
many times does 30 go into 189", or does the division 189 ÷ 30.
Again, since you're dividing by a multiple of ten, the ones digit '9' in the 189
does not affect the division at all. The important thing is to look at the
whole tens in the number 189, which is 180. So to find the answer to the
division 189 ÷ 30, you can think of the division 180 ÷ 30, which is the same
as thinking 18 ÷ 3: "How many times does 3 go into
18?"
The last step is simple since it is dealing with ones digits, how many times
does 3 go into 9.
Examples of long division
These examples show how long division is
done, with all of the dropping down of digits and such. It is important to keep the rows and
columns lined up.
850 ÷ 2 = ?
4
2
8
5
0
-
8
0
4
2
2
8
5
0
-
8
0
5
-
4
4
2
2
8
5
0
-
8
0
5
-
4
1
0
Drop down the 0 of the 850 next to the 1. Then divide 2 into 10.
In the hundreds digits, divide 2 into 8. Ask, 'How many 2's in 8?" That is EXACTLY 4 times. Multiply 4 × 2 = 8 and subtract that from 8
to find the remainder which is of course 0.
Then drop down the tens digit 5 and divide
2 into 5.
2 goes into 5 two
times but the division is not exact. So multiply 2 × 2 = 4, place 4 underneath the 5
and subtract to find the
remainder.
Then multiply 5 × 2 = 10
and
place the result under the 10 and subtract.
Since the result is zero and there are no more digits to drop from the dividend, the division is over.
4
2
5
2
8
5
0
-
8
0
5
-
4
1
0
-
1
0
0
1
425
× 2
850
Check the division
by multiplication.
Study also the following examples with your teacher.
Thousands digit
Hundreds digit
Tens digit
Ones digit
How many 7's in 1?
How many 7's in 15?
How many 7's in 11?
How many 7's in 42?
(0)
7
1
5
1
2
2
7
1
5
1
2
-
1
4
1
2
1
7
1
5
1
2
-
1
4
1
1
-
7
4
2
1
6
7
1
5
1
2
-
1
4
1
1
-
7
4
2
-
4
2
0
1. Divide using long division. Check by multiplication.
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