NumberCatalog v1 - Common Calculations

How to Divide

Even-level Division

Even level division is trivial and will not be considered in this discussion.

Uneven-level Divisions

If the division doesn't go evenly (it can), there are two more steps you can use:

Divide by a non-unit number, for example:

In the 3-step division examples, when it says to "Divide", the process is "divide". You need to follow the steps again. In other words, don't forget to keep repeating "divide".

To do long division for non-whole numbers, we'll divide by one more number that goes into the "quotient".

Here's a simple 5-4-7 example, which really is long division, and what we want it to be. (Dividing a fraction with the long division example you gave is long division for fractions as well, so don't forget that!)

Let's start by learning some things we don't use at our homes:

We already covered dividing 3/8 and 8/2 as a "quick math problem" that doesn't involve any multiplication.
3/4 is easier than 4/8 to find without long division (though there's another quick division technique if you have to).
7/6 can't be broken up by the division symbols (as a number is an important step), but you can write "divide by 6" or "12".
To find out what you got after division, you'll use multiplication and division.
When the long division is really difficult to visualize, or there isn't a single common divisor, we use "division" and the method of finding a common denominator for fractals.

Examples Utilizing 6-7 Coefficients

When we look at division from an angle to solve long division problems, it makes a lot of sense.

So we've got this nice picture, right?

  _____        _____        _____
 /     \      /     \      /     \
<       >----<       >----<       >
 \_____/      \_____/      \_____/
 /     \      /     \      /     \
<       >----<       >----<       >----.
 \_____/      \_____/      \_____/      \
       \      /     \      /     \      /
        >----<       >----<       >----<
       /      \_____/      \_____/      \_____
       \      /     \      /     \      /     \
        `----<       >----<       >----<       >
              \_____/      \_____/      \_____/
                           /     \      /
                          <       >----'
                           \_____/

That's our 6x7 area of rectangles that will help us see the 77-2 in a number line.

But 19 isn't so bad (for division problems) because 70x4 = 280 and we don't even have to remember remember
remember which way it goes, because you know 8 is the other side, too, so we subtract the side we can see:

   xxxxxxxxxxxxxxxxxxxxxxx
   x          2          x
   x                     x
   x   70                x
19 x     /             8 x <- Subtract this side!
   x      4              x
   x                     x
   x          0          x
   xxxxxxxxxxxxxxxxxxxxxxx

It's the same as the "6 times table" trick for 7:

 ._____6_____,
 |`          :\
 | `         : 7
 |  `        :  \
 6   +-----7-----+
 |   :       :   :
 |__ : _6____:   :
 `   :        \  :
  `  :         7 :
   ` :          \:
    `:_____6_____7

Just write a line underneath, and think, 3 is half, and then 20 is half as many. Then the "77 is 3+11+1".

And then the next 28 is just 2x6 and the 55 is just 1x5 + 20-19:

 _         _
| x     x   |
|   2     6 |
|           |
| x     x   |
|_ 20    19_|
_____________
    1  +  5   =  6*

*I'm not really happy about that 6, so we can keep using our "area of a rectangle" technique.

(Remembering this number is important.)

This way, when the big division problems are hard to solve, we'll be able to start solving long division problems.

4x4 Table Squared Division

We know how to use a 4x4 table to divide:

We want the area of 1 square.
The square with 7 x 8 area has an area of 28 and it is a half-square with a side of 4.7
So 38 has a "2x11" division, with a "7 x 6" area and 31 with 3x4 or 12
And for "5x14", the big division, I'd like 7x9 to be half of that.
To do a "whole division", just multiply by 1.
A "mixed" division has 2, but they're both in one part and that makes division very easy.
A whole number is just "number 1" which has one square (in a number line) with a whole (full) picture.

It is an example of long division that shows the end result.

"If we're trying to get 3+2 of the 30's" is one step in finding a "whole number", or an easy 5-2-4. Or the end result."

4-1 of the 50s Method

Remember this method:

"Find 4-1 of the 50's"
Then find the end of it.
1 of those 30's will have the end of a picture and that is 12

So to find "1x9" we want "3 x 7", so 21x9 or "4 x 7". And that "2x5" division.

Or we can...

Go back to the "20's", 5-3, or "700 - 10" as "find the total" in the end.
Then, divide "6 x 6" or 16 by "8", or 3x4 as a 3x2 whole.
Then subtract that "7x3" which has 17/3 as the total, from the total 16 (a whole number) so the 6-9=7 in "6-2 of the 20's is going on as the 1st half.

Now "whole" division is not just "division", because we do 5+6.

This is just a summary of the common techniques.

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