Which is correct: "A dozen eggs is?" Or "A dozen eggs are?"

I remember being in elementary school, and my teachers making a big deal about the unit.

And I never really got that, until one day, I was in the grocery store, and I wanted to buy an apple, but I couldn't buy one apple. I had to buy a whole bag of apples.

So I did. I bought one bag of apples, I took it home, I took one apple out of the bag, and I cut it up.

And then I ate one slice.

One bag, one apple, one slice. Which of these is the real "one"?

Well, they all are of course, and that's what my elementary teachers were trying to tell me.

Because this is the important idea behind whole number place value, decimal¹ place value and fractions.

Our whole number system depends on being able to change what we count as "one".

Our whole number system depends on being able to change units.

There are two ways to change units.

We can compose, and we can partition².

When we compose units, we take a bunch of things, we put them together to make a bigger thing, like a dozen eggs.

We take 12 eggs, put them together to make a group, and we call that group a dozen.

A dozen eggs is a composed unit.

Other examples of composed units include a deck of cards, a pair of shoes, a jazz quartet and of course, Barbie and Ken³ make a couple.

But think about a loaf of bread.

That's not a composed unit, because we don't get a bunch of slices from a bunch of different bakeries and put them together to make a loaf.

No, we start with a loaf of bread and we cut it into smaller pieces called slices, so each slice of bread is a partitioned⁴ unit.
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Other examples of partitioned units include a square of a chocolate bar, a section of an orange and a slice of pizza.

The important thing about units is that once we've made a new unit, we can treat it just like we did the old unit.

We can compose composed units, and we can partition partitioned units.

Think about toaster pastries⁵.

They come in packs of two, and then those packs get put together in sets of four to make a box.

So when I buy one box of toaster pastries, am I buying one thing, four things, or eight things?

It depends on the unit.

One box, four packs, eight pastries.

And when I share a slice of pizza with a friend, we have to cut "it" into two smaller pieces.

So a box of toaster pastries is composed of composed units, and when I split⁶ a slice of pizza, I'm partitioning⁷ a partitioned unit.

But what does that have to do with math?

In math, everything is certain.

Two plus two equals four, and one is just one.

But that's not really right.

One isn't always one.

Here's why: we start counting at one, and we count up to nine: 1,2,3,4,5,6,7,8,9, and then we get to 10, and in order to write 10, we write a one and a zero.

That one means that we have one group, and the zero helps us remember that it means one group, not one thing.

But 10, just like one, just like a dozen eggs, just like an egg, 10 is a unit.

And 10 tens make 100.

So when I think about 100, it's like the box of toaster pastries.

Is 100 one thing, 10 things or 100 things?

And that depends on what "one" is, it depends on what the unit is.

So think about all the times in math when you write the number one.

No matter what place that one is in, no matter how many things that one represents, one is.

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