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(Music)
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Which is correct -- a dozen eggs is? Or a dozen eggs are?
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I remember being in elementary school,
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and my teachers making a big deal about the unit.
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And I never really got that, until one day I was in the grocery store,
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and I wanted to buy an apple, but I couldn't buy one apple.
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I had to buy a whole bag of apples.
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So I did. I bought one bag of apples,
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I took it home, I took one apple out of the bag, and I cut it up.
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And then I ate one slice.
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One bag, one apple, one slice.
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Which of these is the real one?
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Well, they all are, of course, and that's what my elementary teachers were trying to tell me.
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Because this is the important idea behind
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whole number place value, decimal place value and fractions.
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Our whole number system depends on being able to change what we count as one.
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Our whole number system depends on being able to change units.
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There are two ways to change units.
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We can compose, and we can partition.
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When we compose units,
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we take a bunch of things and we put them together to make a bigger thing,
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like a dozen eggs.
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We take 12 eggs, put them together to make a group,
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and we call that group a dozen.
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A dozen eggs is a composed unit.
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Other examples of composed units include
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a deck of cards, a pair of shoes, a jazz quartet
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and, of course, Barbie and Ken make a couple.
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But think about a loaf of bread.
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That's not a composed unit,
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because we don't get a whole bunch of slices from a whole bunch of different bakeries
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and put them together to make a loaf.
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No, we start with a loaf of bread and we cut it into smaller pieces
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called slices, so each slice of bread is a partitioned unit.
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Other examples of partitioned units include
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a square of a chocolate bar, a section of an orange
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and a slice of pizza.
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The important thing about units is that once we've made a new unit,
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we can treat it just like we did the old unit.
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We can compose composed units, and we can partition partitioned units.
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Think about toaster pastries.
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They come in packs of two,
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and then those packs get put together in sets of four
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to make a box.
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So when I buy one box of toaster pastries,
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am I buying one thing,
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four things, or eight things?
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It depends on the unit.
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One box, four packs, eight pastries.
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And when I share a slice of pizza with a friend,
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we have to cut it into two smaller pieces.
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So a box of toaster pastries
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is composed of composed units,
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and when I split a slice of pizza,
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I'm partitioning a partitioned unit.
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But what does that have to do with math?
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In math, everything is certain.
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Two plus two equals four, and one is just one.
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But that's not really right.
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One isn't always one.
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Here's why: We start counting at one,
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and we count up to nine -- one, two, three, four, five, six, seven, eight, nine.
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And then we get to 10, and in order to write 10,
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we write a one and a zero.
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That one means that we have one group,
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and the zero helps us remember that it means one group, not one thing.
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But 10, just like one,
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just like a dozen eggs, just like an egg,
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10 is a unit.
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And 10 tens make 100.
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So when I think about 100, it's like the box of toaster pastries.
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Is 100 one thing,
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10 things
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or 100 things?
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And that depends on what one is,
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it depends on what the unit is.
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So think about all the times in math when you write the number one.
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No matter what place that one is in,
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no matter how many things that one represents,
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one is.
