Division Table

Date: 01/25/99 at 19:02:58
From: Mya Bruce
Subject: division table chart

Hello,

I'm in third grade and I need help with division. Could you please tell 
me if there is a division chart I could go by, or some basic rules? I 
am just starting to do my division, like from the 1's to the 6's. 
Please help me. Thank you.

Sincerely,
Mya Bruce

Date: 01/26/99 at 08:53:35
From: Doctor Peterson
Subject: Re: division table chart

Hi, Mya. We discussed the idea of division tables recently in my home, 
when my daughter asked why we have addition and multiplication tables 
but not subtraction and division tables. So I have some interesting 
ideas about this!

The fact is that we use addition tables (or at least the facts they 
contain) to do subtraction, and multiplication tables to do division. 
That's because each subtraction or division fact is just an addition 
or multiplication fact in disguise. For example, the fact that 
48 / 6 = 8 (I'm using "/" in place of the usual division sign), is the 
same as saying that 48 = 6 * 8 (using "*" for the multiplication 
sign).

There's an extra reason why we don't make special division tables. It 
is that not every division problem you can imagine has an answer (as 
long as you stick to whole numbers). For example, 49 / 6 could only be 
listed in the table if you include fractions; the answer is 8 1/6. 
It's not worth learning those. When we do bigger division problems, we 
only use whole number facts anyway. So a division table would look 
either like this, with whole numbers only:

     / | 1   2   3   4   5   6
     --+-----------------------
     1 | 1
       |
     2 | -   1
       |
     3 | -   -   1
       |
     4 | -   2   -   1
       |
     5 | -   -   -   -   1
       |
     6 | -   3   2   -   -   1
       |
     ...

or like this, with fractions:

     / | 1       2       3       4       5       6
     --+-------------------------------------------
     1 | 1      1/2     1/3     1/4     1/5     1/6
       |
     2 | 2       1      2/3     1/2     2/5     1/3
       |
     3 | 3     1 1/2     1      3/4     3/5     1/2
       |
     4 | 4       2     1 1/3     1      4/5     2/3
       |
     5 | 5     2 1/2   1 2/3   1 1/4     1      5/6
       |
     6 | 6       3       2     1 1/2   1 1/5     1
       |
     ...

That's much harder to learn, and a bigger waste of paper, than the 
multiplication table. We'd also need a hundred rows to contain enough 
information!

So here's your "division" table:

     * | 1   2   3   4   5   6   7   8   9  10
     --+---------------------------------------
     1 | 1   2   3   4   5   6   7   8   9  10
       |
     2 | 2   4   6   8  10  12  14  16  18  20
       |
     3 | 3   6   9  12  15  18  21  24  27  30
       |
     4 | 4   8  12  16  20  24  28  32  36  40
       |
     5 | 5  10  15  20  25  30  35  40  45  50
       |
     6 | 6  12  18  24  30  36  42  48  54  60
       |
     7 | 7  14  21  28  35  42  49  56  63  70
       |
     8 | 8  16  24  32  40  48  56  64  72  80
       |
     9 | 9  18  27  36  45  54  63  72  81  90
       |
    10 |10  20  30  40  50  60  70  80  90 100

To use this as a division table, you just find the row (or column) for 
the divisor, and look along it until you find the largest number less 
than the dividend. For example, to divide 49 by 6, you find the row 
for multiples of six:

     6 | 6  12  18  24  30  36  42  48  54  60

and find that although 49 isn't there, 48 is. Since 48 is in column 8, 
that tells you that

    48 / 6 = 8

so

    49 / 6 = 8 rem 1

because 49 has one extra.

When you've done enough multiplication and division problems, you'll 
be familiar with the table and won't need to look at it; but you'll 
still be doing the same thing to divide: think of the multiples of six, 
and find the one just below the number you are dividing.

I hope that answers your question!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/