**A Variation on Conversion
from Base 10 to other Bases
**

*Let’s think about what
we did here. Each time, we looked at how
many we had in a category. We were
looking, really, at how many groups of four we could make each time. I made groups of four with the units, and
then I made groups of four with the bars.
That’s group division.
*

*
*

*So I started with
fifty-four divided by four.
*

[Count out 54 unit beads.
Exchange a four bar for each four unit beads, putting the four bars in
the Bars column. Put the two unit beads
left over in the Units column]

*
*

*We can see on our
board that we have thirteen bars of four with two left over, for a total of
54. Fifty-four divided by four is
thirteen with a remainder of two.
*

*
*

[Write * _{}*]

*
*

*But we can make
squares out of some of these bars, can’t we?
How do we find out how many squares we can make? A square is four bars, so we need to see how
many groups of four we have in our thirteen bars.
*

*
*

[Exchange a square for every four bars in the Bars column.]

*
*

*
*

[ Write * _{}*]

*
*

*It takes 4 squares to
make a cube. How many groups of four do
we have in our squares? None..
*

*
*

[Write * _{}*]

*
*

*Now let’s look again
at the divisions we did:
*

*
*

_{}

_{}

_{}

*
*

*We started with 54 in
base ten. We found that we had thirteen
groups of four with two left over, and that two is here in our units column.
*

*
*

[Point to the two units in the Units column]

*Then we took the thirteen
bars of four and divided by four again.
We found that we had three groups of four squares and one square bar
left over. And here is the square bar
that was left over.
*

*
*

[Point to the four bar in the Bars column]

*Then we checked to see
how many groups of four we could make with the squares of four. Four squares make a cube. But three divided by four is zero, so we
couldn’t make any cubes. When we divide
three by four, our remainder is three, and here’s our three right
here in the Squares column.
*

*
*

[Point to the three squares]

*Since we can’t make
cubes, we can’t go any farther. Our
answer is _{}.
*

*
*

*But look. We can see the same thing in the
remainders—3, 1, 2. So we can covert
from base 10 to base 4 by dividing by 4 over and over. How do we know when we’re done? Yes, we’re done when our division gives us
zero with some remainder.
*

*
*

*Do you think this will
work with other bases besides 4? Why
don’t you try it and see.
*