Packages
Three packages; each package has 3 cabbages; someone ate one third of one of the packages. 
Introduction
It might appear that
everything
you need to know about teaching children to count with fractions is
contained on the tabs Counting 1, Counting 2, and Counting 3. After
all, children who can count to solve the sharing problems and the
subtraction problems on those pages can count any collection of
objects, using fractions when necessary.
But,
in
fact, I
have left some things out.
First of all, I haven't emphasized enough that fractions
should be introduced in connection with counting real things (like
donuts, pizzas, dollars, and gallons of milk). It is only
through much experience with concrete examples like one quarter of a donut,
one quarter of a pizza,
and one quarter of a gallon
that children will come to understand the abstraction "one quarter."
Way too much instruction in arithmetic (in both whole numbers and
fractions) is too abstract, too soon.
But more to the point, I haven't said anything at all about an
important distinction  the distinction between counting those real
things that are "simple" (like donuts and pizzas) and counting those
real
things that are "packages" (like dollars and gallons of milk).
This page is about packages. It turns out that children need
to learn
more about packages than how to count them.
What is a Package?
A gallon is a package because
it
is made up of quarts. A dollar is a package because it is made up
of cents. A foot is a package because it is made up of
inches.
 1 gallon = 4 quarts
 1 dollar = 100 cents
 1 foot = 12 inches
Packages are units that are made up of smaller units that have names. Some other examples are:
 1 hour = 60 minutes
 1 yard = 3 feet
 1 week = 7 days
On the other hand, donuts and
pizzas are not
packages.
While it is true that donuts and pizzas can be broken into smaller
pieces, we have
no common name for those smaller pieces. Pieces of donuts and
pieces
of pizzas can
only be described using fractions.
Why is this
Distinction
Important?
If your children have learned
to
count simple units like donuts and pizzas, then they should have no
difficulty counting packages like gallons and dollars. Whether
the large rectangles below represent donuts, or pizzas, or gallons, or
dollars, they should know that there are two and one half of
them.




What is new here is that two and
one half gallons is the same as 10 quarts and two and one half
dollars is the same as 250 cents. On the other hand, there is
nothing more to say about two and one half donuts or two and one half
pizzas.
When working with packages, children
need to learn to make the conversion  from gallons to quarts, from
dollars to cents, from feet to inches, from hours to minutes,
etc. They also need to learn to go in the opposite direction 
from quarts to gallons, from cents to dollars, and so on.^{1}
Counting and Converting Packages
Once your children have had experience solving problems with
simple things,
you should introduce them to problems involving packages. The
good news is that these problems can be just like the sharing and
subtraction problems
that they are already solving.
Sharing Problems
(Simple) You have already seen how children typically solve problems like this one: If 3 bears share 5 bowls of porridge, how many bowls of porridge does each bear get? They either divide up the bowls like this,
Papa's Bowl 
Mama's Bowl 
Baby's Bowl 






or like this:















But now suppose the bears are
sharing packages of donuts instead of bowls of porridge.
(Package) If 3 bears share 5 packages of donuts, where each package has 6 donuts, how many packages of donuts does each bear get? How many donuts does each bear get?
Some children focus on sharing
the packages and some children focus on sharing the donuts. Those
who focus on the packages will tell you how many packages each bear
gets. Those who focus on the donuts will tell you how many donuts
each bear gets. Either way, problems like this one provide a context for converting from one unit to the other.
Those children who focus on the 5
packages do exactly what you might expect.
 Some distribute one package to each bear and then cut the 2
remaining packages into thirds. Those children give each bear
"one and two thirds packages."
 Some cut all of the packages into thirds and give each bear
one third of each package. Those children give each bear "five thirds
of a package."
Regardless of which method they
use, these children need to know how
to convert the number
of packages into the
number of donuts.
Those children who focus on the
donuts solve the problem in another way.
 They start by figuring out the number of donuts (6 x 5 = 30) and then they share those donuts among the 3 bears. Those children will tell you that each bear gets 10 donuts.
These children need to know how to convert the number of donuts into the number of packages.
The examples that follow
illustrate each of the 3 approaches.
 Here Eva focuses on packages, not donuts, cutting each package into thirds.
 Trixie
also focuses on packages, but she distributes whole
packages first.
 Here Rose essentially ignores the packages and instead distributes the donuts.
Subtraction Problems
You have already seen the benefit of introducing your child to subtraction problems. If I have one banana and a monkey comes along and eats one fifth of my banana, then it is natural to represent the problem with a picture like this one.





This picture shows the four
fifths of a banana
that are left as all being part of a
single banana.
But now suppose that I have a box of bananans  a box that contains 30 bananas. If that same monkey eats one fifth of my box of bananas, then that same picture tells me that I have four fifths of a box of bananas left. Problems like this one provide another context for converting  in this case, four fifths of a box of bananas needs to be converted into 24 bananas.
In
this problem, Trixie has 3
packages of cabbages. She needs to figure out how much is left
when someone eats one third of one of those packages.