Grade 2 · 2.NBT.A.1A · 1.NBT.B.2A

Base ten blocks and trading ten for one

Ten ones make a ten. What do ten tens make, and can the same number be built out of blocks in more than one way?

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Tens and onesGrade 1

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Regrouping in additionGrade 2 Regrouping in subtractionGrade 2

Where you are 15 in Place value

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Base ten blocks and trading ten for one

Reading and writing numbers to 1,000 Comparing and ordering whole numbers ▾ Show all 15 (9 more)

When you reach for it

In Tens and ones ten loose sticks were tied into one bundle, a ten, and a two-digit number was read as some tens and some ones. Past 99 that is not enough. A heap of two hundred sticks would be twenty bundles, and twenty bundles are slow to count. The fix is the same move one place further left: ten tens are packed together into one bigger thing, a hundred.

Base ten blocks let a child hold both moves in their hands. A small cube is one. A rod is ten cubes joined in a line. A flat is ten rods joined side by side. A child who swaps ten small pieces for one bigger piece, or one bigger piece for ten smaller ones, is doing the move that every later step of adding and subtracting bigger numbers depends on.

Reach for this when a number up to 1{,}000 has to be built, read off a pile of blocks, or broken into smaller pieces so that some can be taken away.

The lesson runs over three days, one part a day.

Part 1, the blocks and the trade. What a unit, a rod and a flat are worth, and trading ten of one for one of the next, in both directions.
Part 2, building a number. Reading a pile of blocks as a number, and building a number as a pile, including numbers with a zero in them.
Part 3, one number, many ways. The same number built with different blocks: 163 is 1 hundred, 6 tens and 3 ones, and it is also 16 tens and 3 ones.

A set of base ten blocks is best. Paper cut-outs work too: a small square, a strip of ten squares, a ten-by-ten square. A number cube (a die showing 1 to 6) is needed for the trading game, and beads on strings make a second picture of the same idea. Saying and writing three-digit numbers in words is its own lesson, Reading and writing numbers to 1,000, and the table of digit columns is The place value chart. This lesson is about the pieces themselves.

The procedure

Part 1: the blocks and the trade

Three kinds of block. A unit is one small cube. It is worth one. A rod is ten units joined in a line. It is one piece, but it is worth ten: lay ten units beside it and they are exactly as long. A flat is ten rods joined side by side into a square. It is one piece, and it is worth one hundred.

10 \text{ ones} = 1 \text{ ten}

10 \text{ tens} = 1 \text{ hundred}

Figure 1The three blocks. A rod is ten units joined in a line, and a flat is ten rods side by side.

Why a flat is one hundred. Nobody needs to count the hundred little squares on a flat one at a time. Each of its ten rows is a rod, so count the rows by tens, as in Counting by tens to one hundred: “ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, one hundred”. Ten tens is one hundred.

Trading up. Ten units may be swapped for one rod. Nothing has been added and nothing taken away: before the trade the pile had ten ones, after it the pile has one ten, and those are the same amount. In the same way, ten rods may be swapped for one flat. Each trade makes the pile hold fewer pieces and leaves the number it shows unchanged.

Figure 2The two trades. Each swaps ten pieces for one bigger piece worth exactly as much.

Trading down. A trade can always be undone. One rod may be swapped for ten units, and one flat for ten rods. A child needs this when something has to be taken away and there are not enough small pieces. To get units out of a flat, trade the flat for ten rods first, and then trade one of those rods for ten units. Each step is a fair ten-for-one swap.

Only ten for one. Every trade is exactly ten of one kind for one of the next kind up. Nine units are not a rod, and eleven units are a rod and one unit left over. Because every rod is worth the same and every flat is worth the same, piles can be counted by tens and hundreds without opening anything up.

The trading game. Each player starts with no blocks. On a turn, the player rolls the number cube and takes that many units. As soon as a player has ten units, they trade them for a rod; as soon as they have ten rods, they trade them for a flat. The first player to win a flat wins. The other player watches every trade and checks it. A second game runs backward: each player starts with one flat, and on each turn gives back as many units as the cube shows, trading down whenever they have too few units. The first to give away everything wins.

Part 2: building a number

Reading a pile. To find the number a pile shows, count the flats by hundreds, as in Counting by hundreds, then the rods on by tens, then the units on by ones. A pile of 3 flats, 5 rods and 2 units counts “one hundred, two hundred, three hundred, three hundred ten, three hundred twenty, three hundred thirty, three hundred forty, three hundred fifty, three hundred fifty-one, three hundred fifty-two”.

300 + 50 + 2 = 352

The three digits of 352 say how many of each block: 3 flats, 5 rods, 2 units. The left digit counts flats, the middle digit counts rods, the right digit counts units. Writing a number as the sum of what its blocks are worth, as above, is Expanded form.

Building from a number. The same reading runs backward. To build 352, take 3 flats because the left digit is 3, then 5 rods, then 2 units, and count the pile to check.

A zero means none of that block. The number 406 is 4 flats, no rods and 6 units. The 0 in the middle says “no rods”, and it keeps the 4 in the flats place. Without it the digits would read 46, which is 4 rods and 6 units, a far smaller pile. In 520 the 0 is on the right: 5 flats, 2 rods and no units.

Beads work the same way. A strand of ten beads is a ten, like a rod. A box that holds ten strands is a hundred, like a flat. A loose bead is a one. Any picture works as long as every ten holds exactly ten and every hundred holds exactly ten tens.

A pile with too many of one kind. Halfway through a game a pile can hold twelve rods or fifteen units. It still shows a number, and counting it gives that number. But it cannot be read straight off as digits, because each digit only runs from 0 to 9. Trade first, smallest block first: any ten units become a rod, then any ten rods become a flat. Example 5 shows a pile that needs both trades.

Part 3: one number, many ways

A flat is also ten rods. Build 163 the short way: 1 flat, 6 rods and 3 units. Now trade the flat for ten rods. The pile holds 16 rods and 3 units, and it is still 163, because the trade added nothing and took nothing away.

\begin{aligned} 163 &= 1 \text{ hundred}, 6 \text{ tens}, 3 \text{ ones} \\ &= 16 \text{ tens}, 3 \text{ ones} \end{aligned}

Figure 3The flat has been traded for ten rods. Both rows are 163.

So the number of tens in 163, counting the tens inside the hundred too, is 16. Count the rods by tens to check: “ten, twenty, ..., one hundred, one hundred ten, ..., one hundred sixty”, then “one hundred sixty-one, one hundred sixty-two, one hundred sixty-three”.

A heap of rods. Trading runs the other way too. A child holding 12 rods and nothing else trades ten of them for a flat and keeps 2 rods: 1 flat, 2 rods, no units. So 12 tens is 120. The units place gets a 0 because there are no units at all.

Many piles, one number. A number can be built from flats, rods and units in many ways. Each trade gives a new pile showing the same number. Out of all those piles, exactly one has at most nine of each kind of block. That pile has the fewest pieces, and it is the one the digits describe. The others matter just as much. Breaking a rod into units to take some away is Regrouping in subtraction. Trading ten units for a rod after putting two piles together is Regrouping in addition. Both are this lesson’s trade, used inside a sum or a difference.

Worked examples

Examples 1 to 3 belong to Part 1, Examples 4 and 5 to Part 2, and Examples 6 and 7 to Part 3. Build each one with blocks or beads while reading it.

Example 1

Doroteo’s first three rolls

Doroteo plays the trading game. He starts with no blocks and rolls 6, then 5, then 4. What does he have after each roll, and what number does his pile show at the end?

Solution

After the first roll he takes 6 units. Six is fewer than ten, so there is nothing to trade.

After the second roll he takes 5 more units. Counting on from six, “seven, eight, nine, ten, eleven”, he now has 11 units. Ten of them trade for one rod, so his pile becomes 1 rod and 1 unit. That is still eleven: one ten and one more.

After the third roll he takes 4 more units: 1 rod and 5 units. Five units cannot make a rod, so he stops. His pile shows 1 ten and 5 ones.

6 + 5 + 4 = 15

The rolls add to 15 as well, which is the check: trading changed the pieces, not the amount.

Example 2

Ioana rolls past ninety-nine

Ioana has 9 rods and 7 units. She rolls a 5. What trades does she make, and what does her pile show afterward?

Solution

She takes 5 units. Counting on from seven, she has 12 units. Ten of them trade for a rod, leaving 2 units.

Now she has 10 rods. Ten rods trade for a flat, which wins the game. Her pile is 1 flat, no rods and 2 units.

1 \text{ hundred}, 0 \text{ tens}, 2 \text{ ones} = 102

Check by counting: she began with 97, and five more is “ninety-eight, ninety-nine, one hundred, one hundred one, one hundred two”. One roll made two trades, one after the other, because the new rod was the tenth rod.

Example 3

Giving back from a flat

Bakari plays the backward game. He has 1 flat and nothing else. He rolls a 4 and must give back 4 units. How does he do it, and what is left?

Solution

He has no units, and no rods to break. So he trades the flat for 10 rods. That is still one hundred. Then he trades one of the rods for 10 units. Now he has 9 rods and 10 units, which is still one hundred: nine tens and one more ten.

He gives back 4 of the units, leaving 6. His pile is 9 rods and 6 units.

96 + 4 = 100

So 96 is left, and putting the 4 units back gives one hundred again. A single trade was not enough here: a flat cannot become units in one ten-for-one step, so the rods come in between.

Example 4

Ioana’s bead boxes

Ioana’s beads come on strands of ten, and ten strands fill a box. She has 2 full boxes, 4 strands and 7 loose beads. How many beads does she have?

Solution

A box holds ten strands of ten, so each box is a hundred beads, like a flat. Each strand is a ten, like a rod, and each loose bead is a one.

Count the boxes by hundreds, “one hundred, two hundred”, then the strands on by tens, “two hundred ten, two hundred twenty, two hundred thirty, two hundred forty”, then the loose beads by ones, “two hundred forty-one”, up to “two hundred forty-seven”.

200 + 40 + 7 = 247

She has 247 beads. The digits match the pieces: 2 boxes, 4 strands, 7 loose beads.

Example 5

A pile with too many pieces

At the end of a long game Bakari’s pile is 3 flats, 12 rods and 15 units. What number does it show? Trade until the pile can be written with digits.

Solution

Start with the smallest block. Fifteen units are more than nine, so ten of them trade for a rod. That leaves 5 units and gives 13 rods.

Thirteen rods are more than nine, so ten of them trade for a flat. That leaves 3 rods and gives 4 flats.

4 \text{ hundreds}, 3 \text{ tens}, 5 \text{ ones} = 435

To check, add what the first pile was worth, kind by kind:

300 + 120 + 15 = 435

The units were traded first because trading them made one more rod. Had the rods gone first, the new rod from the units would have needed a second look.

Example 6

Every way to build Doroteo’s number

Doroteo builds 163 as 1 flat, 6 rods and 3 units. How else could 163 be built, if there were plenty of blocks? How many ways are there?

Solution

Trading the flat for ten rods gives 16 rods and 3 units, as in Figure 3. Trading a rod for ten units gives another pile, and so on. A few of them:

Flats	Rods	Units
1	6	3
0	16	3
1	5	13
0	15	13
0	0	163

To count them all, notice that once the flats and rods are chosen, the units have to make up the rest, so there is only one choice for them. With one flat, the rods can be 6, 5, 4, 3, 2, 1 or 0: seven ways, because a seventh rod would pass 163. With no flat, the rods can be any number from 16 down to 0: seventeen ways. Two flats would be two hundred, too many.

7 + 17 = 24

There are 24 ways. Only the first row has at most nine of each block, and only the first row can be read off as the digits 1, 6 and 3. The second row is the one to remember: 163 is 16 tens and 3 ones.

Example 7

Twenty-three strands

Bakari has 23 strands of ten beads and no loose beads. How many beads is that? How many full boxes of ten strands can he pack?

Solution

Ten strands fill one box, so twenty strands fill two boxes. That leaves 3 strands, and there are no loose beads.

2 \text{ hundreds}, 3 \text{ tens}, 0 \text{ ones} = 230

He has 230 beads and can pack 2 full boxes, with 3 strands left over. The 0 says there are no loose beads. Counting the strands by tens gives the same: “ten, twenty, ..., two hundred, two hundred ten, two hundred twenty, two hundred thirty”, twenty-three counts.

So 23 tens is 230, and read backward, 230 is 23 tens. In the same way, the tens in any number ending in 0 are counted by the digits before the 0.

Boundaries

A trade is ten for one, never another count. A rod is worth ten only because it is exactly ten units long. Nine units cannot be traded for a rod, and a box with nine strands in it is not a hundred beads.
Every trade keeps the number the same. If a pile’s number changes, something was added or taken away; trading alone never does it.
Only one pile matches the digits. A number can be built many ways, but only the pile with at most nine of each kind of block reads straight off as digits. Any other pile must be traded first, as in Example 5.
The blocks stop at a thousand here. Ten flats trade for one large cube, worth one thousand, written 1{,}000. Numbers past that are built the same way with more kinds of piece, in Numbers to one million.
Units are whole. A unit is the smallest piece. These blocks show whole numbers only, and a unit is never cut.

Where this goes next

The trade is the groundwork for adding and subtracting three-digit numbers. Putting two piles together and trading any ten units for a rod is Regrouping in addition. Breaking a rod or a flat when there are not enough small pieces to take away is Regrouping in subtraction, and the same moves run through Subtracting three-digit numbers by place value, including across zero.

The pile with at most nine of each block is what the digits describe. Saying and writing those numbers in words is Reading and writing numbers to 1,000; the digit columns, and what a digit is worth in each, are The place value chart; and writing 352 as 300 + 50 + 2 is Expanded form. Deciding which of two piles is bigger, flats first, then rods, then units, is Comparing and ordering whole numbers.

Later the same trade, ten of one for one of the next, explains why multiplying by ten moves every digit one place left, in Multiplying and dividing by ten, one hundred and one thousand, and other bundle sizes give Numbers in other bases.

Exercises

Part 1: the blocks and the trade

Doroteo has 8 units and rolls a 4. What does he have after he trades?
Ioana has 3 rods and 9 units and rolls a 6. What does she have after she trades, and what number is it?
How many units are worth 1 rod? How many units are worth 1 flat?
Bakari has 2 rods and no units. He must give back 3 units. What does he trade, and what is left?
Ioana has 9 rods and 8 units and rolls a 3. What trades does she make, and what does her pile show?
Bakari has 1 flat and nothing else. He must give back 1 unit. What is left after his trades?

Part 2: building a number

Figure 4Piles for Exercises 7, 8 and 14.

What number does pile A show?
What number does pile B show?
Ioana has 5 boxes of beads, 2 strands and 6 loose beads. How many beads is that?
Which blocks build 384 with the fewest pieces?
Which blocks build 520 with the fewest pieces?
Which blocks build 609 with the fewest pieces? Why is no rod needed?
Bakari has 7 full boxes of beads and nothing else. How many beads does he have?

Part 3: one number, many ways

Pile C has 11 rods. Trade until it can be written with digits. What number is it?
A pile is 2 flats, 5 rods and 17 units. What number does it show?
What number is 34 tens?
How many tens make 150?
Build 245 using exactly 1 flat and 5 units. How many rods are needed?
4 hundreds, 2 tens and 6 ones is the same as 3 hundreds, \square tens and 6 ones.
Ioana has 28 strands of ten beads and 4 loose beads. How many beads is that? How many full boxes can she pack?

Reasoning

After every possible trade up has been made, why can a pile never have more than 9 units?
Which is more beads, 3 strands or 1 box? Explain using what each is worth.
Is it always true that trading changes the blocks but not the number? Explain, with a trade of your own as an example.
What would ten flats trade for, and how many units is that worth?
A number ends in 0. Explain why it can be built from flats and rods with no units, and why it can also be built from rods alone. How many rods build 370?

Answers

1 rod and 2 units. He has 8 + 4 = 12 units, and ten of them trade for a rod, leaving 2.
4 rods and 5 units, which is 45. She has 9 + 6 = 15 units; ten trade for a rod, so 3 rods become 4 and 5 units are left: 40 + 5 = 45.
10 units for a rod and 100 units for a flat. A flat is ten rods, and counting ten rods by tens reaches one hundred, “ten, twenty, ..., one hundred”, so 10 tens is 100.
He trades 1 rod for 10 units, giving 1 rod and 10 units. After giving back 3, he has 1 rod and 7 units, which is 17: 20 - 3 = 17.
Two trades, and the pile shows 101. She has 8 + 3 = 11 units, so ten trade for a rod and 1 unit is left. That makes 10 rods, which trade for a flat: 1 flat, 0 rods, 1 unit. Check: 98 + 3 = 101.
9 rods and 9 units, which is 99. The flat trades for 10 rods, one rod trades for 10 units, and one unit goes back: 100 - 1 = 99.
238. Pile A is 2 flats, 3 rods and 8 units: 200 + 30 + 8 = 238.
407. Pile B is 4 flats, no rods and 7 units: 400 + 0 + 7 = 407. The 0 says there are no rods.
526 beads. Each box is a hundred, each strand a ten: 500 + 20 + 6 = 526.
3 flats, 8 rods and 4 units: 300 + 80 + 4 = 384.
5 flats, 2 rods and no units: 500 + 20 + 0 = 520.
6 flats, no rods and 9 units: 600 + 0 + 9 = 609. The middle digit is 0, so there are no tens to build.
700 beads. Each box is a hundred, and counting by hundreds to seven boxes reaches seven hundred: 7 hundreds is 700.
214. Ten of the 11 rods trade for a flat, giving 2 flats, 1 rod and 4 units. Check: 100 + 110 + 4 = 214.
267. Ten of the 17 units trade for a rod, giving 2 flats, 6 rods and 7 units. Check: 200 + 50 + 17 = 267.
340. Thirty of the rods trade for 3 flats, and 4 rods are left: 3 hundreds, 4 tens, 0 ones is 340.
15 tens. The 1 hundred is ten tens, and 5 more tens make fifteen: 150 = 15 tens.
14 rods. The flat is 100 and the units are 5, so the rods must make 140, which is 14 tens: 100 + 140 + 5 = 245.
12. Trade one of the 4 flats for ten rods: 3 flats and 2 + 10 = 12 rods. Check: 300 + 120 + 6 = 426, and 400 + 20 + 6 = 426.
284 beads, and 2 full boxes. Twenty strands fill two boxes and 8 strands are left, so 2 hundreds, 8 tens and 4 ones: 200 + 80 + 4 = 284.
As soon as there are ten units they trade for a rod. So if every trade up has been made, fewer than ten units are left, which is at most 9. The same holds for rods.
1 box is more. A box is ten strands, worth one hundred beads; 3 strands are worth 30 beads, and 100 is more than 30. Three pieces can be worth less than one piece, because the pieces are different sizes.
Yes. A trade swaps ten pieces for one piece worth exactly the same, or one piece for ten worth exactly the same, so nothing is added and nothing taken away. For example, 2 rods and 13 units is 33, and after trading ten units for a rod it is 3 rods and 3 units, still 33.
One large cube, worth one thousand units: ten hundreds is 1{,}000. Counting the flats by hundreds reaches it: “one hundred, two hundred, ..., one thousand”.
37 rods. The 0 in the ones place says no units, so flats and rods are enough. Each flat trades for ten rods, so the flats can all become rods. For 370, the 3 flats become 30 rods and with the 7 rods that makes 37: 30 + 7 = 37, and 37 tens is 370.