Nothing is a number too — and every ten gets bundled into one new unit.
In Lesson 1.1 we counted by touching each thing once and saying the next number in order; the last number we said was how many. But counting one, two, three forever would soon become a blur — try saying a thousand separate names. Numbers stay simple because of two quiet ideas. First, even nothing is worth a number: we write it 0. Second, the moment a count reaches ten, we tie those ten ones into a single new unit — a ten — and start over. That one trick, bundling by ten, is the whole reason 13, 30, and 100 can be written with just a few digits. This lesson builds the bundle from scratch and watches it climb.
Hold out an empty plate. How many cookies are on it? Not one, not two — none. That "none" is still an answer to "how many," and it deserves a number of its own. We write it 0 and say zero.
Zero is what is left when everything is taken away. Three cookies, eaten one by one, leave 3, then 2, then 1, then 0 — the plate is empty, the count has run all the way down. Zero is the floor of counting: you can reach it, and you cannot count any lower while you are counting real things.
And 0 is not just a word for "empty" — it is a genuine digit, a mark we write, exactly like 1 through 9. You will see in section 1.2.4 that the 0 in 30 is doing real work: it says "no loose ones here." A blank would be a mistake; the 0 is the message.
"Nothing at all" is a number. We write it 0, and it is a real digit — the smallest count you can reach.
Suppose you are counting marbles into a cup, one by one: 1, 2, 3, … When you reach ten, something clever happens. Instead of keeping ten separate marbles rattling loose, you scoop all ten together and tie them into a single bundle — one ten. From now on you count that bundle as one thing, and start gathering loose ones again from zero.
This is exactly what the picture below shows. A loose one is a single amber square. A ten is drawn as a tall blue rod — and look closely: that rod is split into ten little segments. The bundle is not a brand-new object that appeared from nowhere; it is precisely ten ones, held together. We say 10 ones = 1 ten.
Why ten and not some other number? Mostly because we have ten fingers, and people have counted on their fingers for as long as there have been people. The choice is a human habit, not a law of nature — but once everyone agrees on it, every number we write rides on that one bundle.
Twenty-three marbles: gather two full bundles of ten — that is 2 tens — with 3 marbles still loose. So 23 is two tens and three ones.
Now we can build the numbers just past ten. Take one ten — one bundle — and set some loose ones beside it. One ten and three loose ones is the number we say thirteen and write 13.
Read the two digits of 13 and you can see the bundle inside the writing. The 1 on the left is not "one thing" — it is one ten. The 3 on the right is the three loose ones. So 13 = one ten + three ones = 10 + 3. Every teen works the same way: 15 is one ten and five ones, 19 is one ten and nine ones.
Here is the trap to avoid. Thirteen is one ten plus three — it is not "thirty-one." The bundle comes first and the loose ones come second; mixing them up changes the number entirely. We will see why the place a digit stands in decides its worth in Lesson 1.3, Place value.
A teen is one ten plus the ones. So 13 is one ten and three — not "thirty-one." The bundle is written first, the loose ones second.
Keep going. Gather a second bundle of ten, and a third, and you have three tens — the number thirty, written 30. There are no loose ones at all this time, and that is where the 0 earns its keep: in 30 the 0 in the ones place says, plainly, "no loose ones here."
This is exactly why 0 had to be a real digit back in section 1.2.1. If we left the spot blank and just wrote "3," we would mean three, not thirty. The 0 is not nothing on the page — it is the mark that holds the empty ones place open so the 3 stays in the tens place.
The same idea names all the round tens: 4 bundles is forty (40), 7 bundles is seventy (70), and 9 bundles is ninety (90) — each one a count of tens, with a 0 marking the empty ones.
We bundled ten ones into a ten. Now do it again, one level up. Once you have gathered ten tens — ten of those blue rods — you bundle them into one bigger unit: a hundred. In blocks, a hundred is a full ten-by-ten square (a "flat"), made of ten rods side by side. We say 10 tens = 1 hundred, and we write it 100.
That is the rhythm of our whole number system, and it never changes: ten ones make a ten, ten tens make a hundred, and (as you will see later) ten hundreds make a thousand. Each new unit is just ten of the one below it. Because of this, even huge numbers are built from the same small kit of bundles — which is exactly what Lesson 1.3 turns into the idea of place value.
The bundle rule climbs by tens: 10 ones = 1 ten and 10 tens = 1 hundred. Each unit up is ten of the one below it.
Two ideas turn endless counting into a tidy system of digits. Keep these:
| Idea | What it says | Example |
|---|---|---|
| Zero | "Nothing" is a number, and a real digit you write. | empty plate → 0 |
| Ten bundles | 10 ones = 1 ten | ten marbles → one ten |
| A teen | one ten + the loose ones | 13 = 10 + 3 |
| Round tens | a count of tens; the 0 marks "no loose ones" | 30 = 3 tens |
| Hundred | 10 tens = 1 hundred | ten tens → 100 |
Six questions to lock it in. Tap the answer you think is right.
This lesson lays the base-ten foundation that the rest of arithmetic stands on. It addresses K.CC.A.3 (write numbers 0–20, and represent a quantity of zero with a written 0), and K.NBT.A.1 together with 1.NBT.B.2 (compose and decompose the teen numbers as a ten and some ones, and understand a ten as a bundled unit). The bundling story — ten ones make a ten, ten tens make a hundred — is the concrete model children should keep returning to as place value and the four operations arrive in later stages.