Parts
Introduction
Early instruction in fractions
should focus on learning to count collections of real things. We need
fractions
because sometimes the number of those real things isn't a whole number.
1. COUNTING SIMPLE THINGS:
Sometimes the real
things to be counted are
"simple" things  like
donuts or pizzas. On the Counting 1, Counting 2,
and Counting 3 tabs you have seen that children can learn to count
simple things by solving sharing and subtraction problems.
2. COUNTING PACKAGES OF
THINGS: But sometimes those real
things are
"packages" of other real things. For example, a yard is a "package" of
3 feet and an hour is a "package" of 60 minutes. On the Packages tab
you saw that if children are introduced to sharing and subtraction
problems involving packages as well as simple things then they will
learn to convert  for example,
from one and two thirds yards to 5 feet and from three
quarters of an
hour to 45 minutes.
3. COUNTING PARTS OF
THINGS: Finally, there are times
when we need fractions to
count things that
are neither "simple" nor "packages"  they are "parts" of other
things. Lots of things are described as parts of other things.
Among others, there are half gallons of milk, quarter pound sticks of
butter, and one third acre lots. This page is about counting things
like these – particularly when there isn’t a whole number of them.^{1}
Examples
You can introduce your child to
counting parts with the same kinds of sharing and subtraction problems
that you used with simple things and packages of things. Here are
two examples.
1. SHARING: Suppose that 3 cats share 1 half gallon of milk.
Then each cat gets something less than 1 half gallon – they each
get only a part of that 1 half
gallon. Based on their experience with simple things and
packages of
things your children should be able to draw pictures like these.
1 (gray) half gallon


one third (black) of a half gallon




That is, without learning anything new, your children should be able to tell you that each cat gets one third of a half gallon.
2. SUBTRACTING: Suppose now that that cat finds a one quarter pound stick of butter and eats one fifth of it. Your children should be able to draw pictures like these.
1 (gray) quarter pound




one fifth (black) of a quarter pound








Once again, without learning anything new, your children should be able to tell you that four fifths of a quarter pound of butter is left.
The only thing about these problems that will be new to your children is the need to convert – from one third of a half gallon to one sixth of a (whole) gallon; from four fifths of a quarter pound to four twentieths of a (whole) pound. In other words, counting parts is just like counting simple things. The only difference, as with packages, is the need to convert – in this case from parts of a part to parts of a whole.^{2}
Teaching Children to
Convert Parts of Parts to Parts of Wholes
Here is a natural sequence  moving from a relatively easy part of a
part, to more challenging examples. I suggest watching these videos in
order.
 Parts of a Unit Fraction
 Parts of a nonUnit Fraction