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 non-Unit Fraction