52 Division and Decimals
When you studied fractions, you had lots of different ways to think about them. But the first way, and the one we keep coming back to, is to think of a fraction as the answer to a division problem.
Example
Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per child. We write:
The fraction is equivalent to the answer to the division problem . It represents the number of pies one whole child receives.
In the same way…
sharing 10 pies among 2 kids yields pies per kid,
sharing 8 pies among 2 kids yields pies per kid,
sharing 5 pies among 5 kids yields pies per kid, and
the answer to sharing 1 pies among 2 children is , which we call “one-half.”
We associate the number “” to the picture.
In the same way, the picture represents “one third,” that is, .
(This is the amount of pie an individual child would receive if one pie is shared among three children.)
The picture is called “one fifth” and is indeed , the amount of pie an individual child receives when one pie is shared by five kids.
And the picture is called “three fifths” to represent , the amount of pie an individual receives if three pies are shared by five kids.
We know how to do division in our “Dots & Boxes” model.
Example: 3906 ÷ 3
Suppose you are asked to compute . One way to interpret this question (there are others) is:
“How many groups of 3 fit into 3906?”
In our “Dots & Boxes” model, the dividend 3906 looks like this:
and three dots looks like this:
So we are really asking:
“How many groups of fit into the picture of 3906?”
Notice what we have in the picture:
- One group of 3 in the thousands box.
- Three groups of 3 in the hundreds box.
- Zero groups of 3 in the tens box.
- Two groups of 3 in the ones box.
This shows that 3 goes into 3906 one thousand, three hundreds and two ones times. That is,
Of course, not every division problem works out evenly! Here’s a different example.
Example: 1024 ÷ 3
Suppose you are asked to compute . One way to interpret this question is:
“How many groups of 3 fit into 1024?”
So we’re looking for groups of three dots in this picture:
One group of three is easy to spot:
To find more groups of three dots, we must “unexplode” a dot:
We need to unexplode again:
This leaves one stubborn dot remaining in the ones box and no more group of three. So we conclude:
In words: 1024 gives 341 groups of 3, plus one extra dot.
We can put these two ideas together — fractions as the answer to a division problem and what we know about division in the “Dots & Boxes” model — to help us think more about the connection between fractions and decimals.
Example: 1/8
The fraction is the result of dividing 1 by 8. Let’s actually compute in a “Dots & Boxes” model, making use of decimals. We want to find groups of eight in the following picture:
Clearly none are to be found, so let’s unexplode:
(We’re being lazy and not drawing all the dots. As you follow along, you might want to draw the dots rather than the number of dots, if it help