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Composed units

Composed units

Author: Christopher Danielson

To demonstrate the idea of a composed unit as a precursor to understanding place value.

This packet offers a brief introdution to the idea of a composed unit, which is a foundational idea for understanding place value.

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In mathematics, it is not enough to know how to count. You have to carefully identify what it is that you are counting.

Most of the time, this isn't a problem. How many children do you have? leads you to count children. How tall are you? leads only to a small amount of ambiguity-do we want to measure in metric units or standard?

But there are plenty of situations where it makes sense to count either (1) units or (2) groups of units

Example 1: Eggs

The original unit is an egg. There are 12 eggs.

But conventionally, we put eggs into groups of 12, as has been done here. And when we do, we have created a composed unit-a dozen eggs. We can count these new units instead of counting the individual eggs.

So we can buy 2 dozen eggs, or 3 dozen eggs or even 12 dozen eggs (and when we do, it's a gross).

Example 2: Shoes

And consider example 2:

Two shoes, right? Or 1 pair of shoes. A pair of shoes is a composed unit.

In fact, we more commonly count shoes in pairs than we do individually. How many pairs of shoes do you have? is a more reasonable question than How many shoes do you have?


A pair of shoes is a natural unit. Shoes come in pairs because people have two feet. If cats wore shoes, they would need a different composed unit to count them. And centipedes?

There seems to be nothing special about a dozen eggs, except that we have agreed to group eggs in twelves. A dozen eggs is a conventional unit.

We also compose units in other ways. A family is a composed unit, but there is no agreed upon number of subunits that go into one. My family has four people, but neighbor's has five.

The decimal place value system is built on composed units, and the units are conventional. We have agreed to put things into groups of 10. 10 units make 1 ten. 10 tens make 1 hundred. 10 hundreds make 1 thousand, etc.

If you are going learn to cook, you'll need to know about eggs and dozens. If you are going to learn to use mathematics, you'll need to know about ones and tens.

It is important to be able to think of 268 as 2 hundreds, 6 tens and 8 ones, and alternatively as 26 tens and 8 ones, and as 268 ones.

The ultimate composed unit

What is the ultimate composed unit? The 1 that is composed of everything, of course.

So is one just one, or is there more to it?

Finally, here is a video summary of these ideas.