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.