Which is correct -- a dozen eggs is? Or a dozen eggs are?

哪個是正確的──a dozen eggs "is" 或者 a dozen eggs "are"？

I remember being in elementary school,

我記得當我在小學的時候，

and my teachers making a big deal about the unit.

我們老師教了很多關於數量單位詞。

And I never really got that, until one day I was in the grocery store,

但我從沒有學會，直到有一天我到雜貨店，

and I wanted to buy an apple, but I couldn't buy one apple.

我想買一顆蘋果（an apple）， 不過我卻不能買「一」蘋果 （one apple）。

I had to buy a whole bag of apples.

這表示我必須買一整袋的蘋果。

So I did. I bought one bag of apples,

所以我做了，我買了一整袋的蘋果，

I took it home, I took one apple out of the bag, and I cut it up.

我把它拿回家，我從袋子拿出一顆蘋果並切開，

And then I ate one slice.

然後吃了一片。

One bag, one apple, one slice.

一袋、一顆、一片。

Which of these is the real one?

哪一個才是真實的那一個？

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

好吧，他們都是，當然，而且 那正是我小學老師一直試著告訴我的。

Because this is the important idea behind

因為這是一個重要的概念關於

whole number place value, decimal place value and fractions.

整數位、小數點位與分數。

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

我們所使用的數字系統要能夠轉換我們所熟悉的「一」。

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

我們所使用的數字系統要能夠轉換「單位」。

There are two ways to change units.

有兩個方式能夠轉換單位。

We can compose, and we can partition.

我們可以組合，也可以分割。

When we compose units,

當我們組合多個單位時，

we take a bunch of things and we put them together to make a bigger thing,

我們把一大堆的東西放在一起，讓他們變成更大的東西。

like a dozen eggs.

就像一打雞蛋一樣。

We take 12 eggs, put them together to make a group,

我們把十二顆雞蛋放在一起，組合成一組，

and we call that group a dozen.

然後我們叫這一組「一打」。

A dozen eggs is a composed unit.

一打雞蛋就是一種組合單位。

Other examples of composed units include

其他組合單位的例子包括：

a deck of cards, a pair of shoes, a jazz quartet

一副牌、一雙鞋、爵士四重奏，

and, of course, Barbie and Ken make a couple.

當然還有，芭比和肯尼組合成一對情侶。

But think about a loaf of bread.

但想想看一條麵包。

That's not a composed unit,

那不是一種組合單位，

because we don't get a whole bunch of slices from a whole bunch of different bakeries

因為我們不會從不同的麵包店拿一片片切好的麵包

and put them together to make a loaf.

將它們放在一起變成一條麵包。

No, we start with a loaf of bread and we cut it into smaller pieces

不是這樣，我們是把一條麵包切成更小的麵包

called slices, so each slice of bread is a partitioned unit.

並叫它「一片」麵包，所以每一片麵包就是一個分割單位。

Other examples of partitioned units include

其他分割單位的例子包括：

a square of a chocolate bar, a section of an orange

一塊巧克力、一片柳橙，

and a slice of pizza.

還有一片披薩。

The important thing about units is that once we've made a new unit,

重要的是，一旦我們創造新的單位，

we can treat it just like we did the old unit.

我們就能用跟舊的單位同樣的方法使用它。

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

我們可以組成組合單位，也可以分成分割單位。

Think about toaster pastries.

想想看那些零食。

They come in packs of two,

他們兩片包成一袋，

and then those packs get put together in sets of four

然後再把四袋放在一起

to make a box.

變成一盒。

So when I buy one box of toaster pastries,

所以當我買一盒零食時，

am I buying one thing,

我是買了「一個」？

four things, or eight things?

「四個」？還是「八個」？

It depends on the unit.

這取決於單位。

One box, four packs, eight pastries.

一盒、四袋、八片。

And when I share a slice of pizza with a friend,

當我跟我的朋友分享一片披薩，

we have to cut it into two smaller pieces.

我們必須將它切成更小的兩個。

So a box of toaster pastries

所以一盒零食

is composed of composed units,

是一個組合單位的組成，

and when I split a slice of pizza,

而當我切開一片披薩時，

I'm partitioning a partitioned unit.

我正在切出一個分割單位。

But what does that have to do with math?

但這些到底跟數學有甚麼關係？

In math, everything is certain.

數學中，所有東西都是確定的。

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

2+2=4，1就是1。

But that's not really right.

但那不全然正確。

One isn't always one.

1不一定永遠是1。

Here's why: We start counting at one,

這就是原因：我們從1開始數，

and we count up to nine -- one, two, three, four, five, six, seven, eight, nine.

一直數到9──1、2、3、4、5、6、7、8、9。

And then we get to 10, and in order to write 10,

接下來我們數到10了，而為了寫下10這個數字，

we write a one and a zero.

我們寫了一個1和一個0。

That one means that we have one group,

其中的1就表示我們有了一個群組，

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

而0幫助我們記得這是指一組，而不是一個。

But 10, just like one,

但是10，就像1，

just like a dozen eggs, just like an egg,

就像一打雞蛋，就像一顆雞蛋，

10 is a unit.

10是一個單位。

And 10 tens make 100.

而十個10變成100。

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

所以當我們在想100是什麼的時候，它就像那一盒零食。

Is 100 one thing,

100是一個東西？

10 things

10個？

or 100 things?

或是100個？

And that depends on what one is,

那就取決於那個「一」是指甚麼，

it depends on what the unit is.

也就是單位是什麼。

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

所以碰到數學需要寫「一」的時候，要一直想著這件事。

No matter what place that one is in,

不管這個「一」是放在哪個位置，

no matter how many things that one represents,

不管這個「一」裡有多少東西，

one is.

「一」就是...