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Grade 2 · 4.NBT.A.2 · 5.NBT.A.1

The place value chart

The cards 5, 3 and 7 can be laid down as 537 or as 357. The cards are the same both times, so what decides what each one is worth?

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Reading and writing numbers to 1,000Grade 2
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Expanded formGrade 2Multiplying and dividing by ten, one hundred and one thousandGrade 4Estimating by roundingGrade 4
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The place value chart
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Why this exists

Selamawit, Ivar and Haruka play a card game. Each card shows one digit, 0 to 9. A player lays three cards in a row and reads the number they make. With the cards 5, 3 and 7, a player can lay down 537, or 357, or 375. Those are three different numbers, made from the same three cards. Only the positions changed.

So a written number carries more than its digits. It also carries, without saying so, what each position counts. In Reading and writing numbers to 1,000 the positions were read left to right: hundreds, then tens, then ones. A place value chart writes those names down. It is a table with a heading over each column, and each digit sits under the heading that says what it counts. Room numbers in a school building, page numbers, the cards in the game: every whole number up to 1{,}000 fits in it.

The lesson runs over three days.

  • Day 1, a digit and its value. The chart, and the difference between a digit (the symbol on the card) and its value (what it is worth in its column).
  • Day 2, zeros and filling the chart. Why 609 needs its 0 while 69 needs none in front, and moving between a number, the chart and a block picture.
  • Day 3, nine at most, then a fourth column. Why a column never holds more than nine, and why ten hundreds need a thousands column.

Digit cards (two or three of each digit), a three-column chart drawn on paper with cells wide enough for a card, and base ten blocks are all the lesson needs.

Definition

A place value chart is a table with one column for each place. From right to left the columns are ones, tens and hundreds. A fourth column, thousands, is added on the left when it is needed. Each cell holds one digit.

HundredsTensOnes
537

A digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The value of a digit is what it is worth where it sits: the digit counts how many of its column’s unit there are. In the chart above:

  • the 5 is in the hundreds column, so its value is 5 hundreds, which is 500;
  • the 3 is in the tens column, so its value is 3 tens, which is 30;
  • the 7 is in the ones column, so its value is 7 ones, which is 7.

The number is read straight across the row: 537, five hundred thirty-seven.

The digit and its value are different things. The card says 5 wherever it lies. Its value is 500, 50 or 5, depending on the column it lies in.

Intuition

Day 1: the columns are block sizes

Each column of the chart stands for one of the base ten blocks from Base ten blocks and trading ten for one. The ones column counts small squares, the tens column counts rods, and the hundreds column counts flats. A digit in a column says how many of that block there are. Figure 1 shows 537 with each column’s value written under it.

Hundreds Tens Ones 5 3 7 5 hundreds = 500 3 tens = 30 7 ones = 7
Figure 1537 in a place value chart. The column each digit sits in says what it counts.

A written number such as 537 is the chart with the headings and the lines taken away. The headings are still there in the reader’s head: the digit at the right end counts ones, the digit to its left counts tens, and the next one to the left counts hundreds. So the columns of a number are counted from the right end. That is why the last digit of every number, short or long, is its ones digit.

Moving a card to another column changes its value and leaves the digit alone. Example 1 follows a single card through all three columns.

Day 2: a zero keeps a column filled

Every column inside a number must hold a digit, because the columns are counted from the right. Room 609 has 6 hundreds, no tens and 9 ones. If the empty tens column were simply left out, the digits would close up into 69, the 6 would land in the tens column, and the room would become sixty-nine. The 0 holds the tens column open, so the 6 stays in the hundreds column.

A blank on the far left is different. The number 69 has no hundreds, and in the chart its hundreds cell is just left empty. Nothing sits to the left of that blank, so no digit is pushed out of its column, and writing a 0 there would change nothing. A zero is needed in the middle of a number or at its right end, never at its left end.

The chart also sits between a number and a picture of blocks. Going from blocks to the chart, count each kind of block and write the count in that block’s column; a kind with no blocks gets a 0. Going from a number to blocks, read each column as a count of its block.

Day 3: nine at most, then a fourth column

A cell has room for one digit, so a column can count from 0 up to 9. A count of ten never stays in one column, because ten of one block make one of the next block to the left: ten small squares make a rod, and ten rods make a flat. The swap itself is taught in Base ten blocks and trading ten for one. For the chart it means a count of ten or more is traded until every column holds a single digit.

The same holds one column further up. Ten flats make one thousand, and the hundreds column cannot hold ten. So the chart grows a fourth column, thousands, on the left. One thousand is written 1{,}000: 1 thousand, 0 hundreds, 0 tens and 0 ones. Charts with more columns, for numbers up to a million, are in Numbers to one million.

Results

A digit’s value comes from its column. A digit in the ones column is worth that many ones; in the tens column, that many tens; in the hundreds column, that many hundreds; in the thousands column, that many thousands.

The same digit has a different value in each column. The 5 is worth 5 in 375, 50 in 357 and 500 in 537.

Each cell holds exactly one digit, 0 to 9. A column with nothing to count, inside a number or at its right end, holds a 0. Empty cells come only at the left end.

Each column’s unit is ten of the unit on its right.

\begin{aligned} 10 \text{ ones} &= 1 \text{ ten} \\ 10 \text{ tens} &= 1 \text{ hundred} \\ 10 \text{ hundreds} &= 1 \text{ thousand} \end{aligned}
Why it is true

The chart is a record of a pile of blocks, after every trade has been made that leaves the fewest pieces. Each column names one block size. A digit in that column counts that block, so its value is that many blocks. That gives the first two results.

For the third, think of the trades. As soon as a column reaches ten blocks, the ten are traded for one block of the next size (Base ten blocks and trading ten for one). So once every trade is made, there are fewer than ten of each block, and each count is one of 0 to 9. A 0 is needed inside a number because the columns are found by counting from the right: drop the 0 from 609, and the 6 moves from the third place from the right to the second, from hundreds to tens. An empty cell at the far left moves no digit, so it changes nothing.

For the fourth, a rod is ten small squares in a line, a flat is ten rods side by side, and ten flats make one thousand, as Reading and writing numbers to 1,000 showed with blocks. Each column’s block is ten of the block to its right.

Worked examples

Examples 1 to 3 go with Day 1, Examples 4 and 5 with Day 2, and Examples 6 and 7 with Day 3.

Example 1

Day 1: One card, three columns

Selamawit lays the cards 5, 3, 7 in the chart, one card to a column, in that order. Ivar then swaps her first two cards. After that, Haruka swaps the last two. What number does the row show each time, and what is the card 5 worth each time?

Solution

Selamawit’s row has 5 in hundreds, 3 in tens and 7 in ones. It reads 537. The 5 is in the hundreds column, so it is worth 5 hundreds, which is 500.

Ivar swaps the first two cards, so the row is 3, 5, 7 and reads 357. The 5 now sits in the tens column. It counts tens, so it is worth 5 tens, which is 50. The 3 moved too, from tens to hundreds, so its value went from 30 to 300. The 7 did not move and is still worth 7.

Haruka swaps the last two cards, so the row is 3, 7, 5 and reads 375. The 5 is in the ones column now, worth 5 ones, which is 5.

Hundreds Tens Ones 5 3 7 3 5 7 3 7 5 the 5 is worth 500 the 5 is worth 50 the 5 is worth 5
Figure 2The same card 5 in three columns. Its value is 500, then 50, then 5.

The card never changed: it is the digit 5 every time. Its value went from 500 to 50 to 5, because its column changed. Each move one column to the right made it count a smaller block: flats, then rods, then small squares.

Example 2

Day 1: Two of the same digit

Room 474 is in Haruka’s school. Its number has two 4s. Are they worth the same? What is each digit worth?

Solution

Put 474 in the chart, counting columns from the right end: 4 in ones, 7 in tens, 4 in hundreds.

The left 4 is in the hundreds column, so its value is 4 hundreds, 400. The right 4 is in the ones column, so its value is 4 ones, 4. The 7 is in the tens column, worth 70.

So the two 4s are the same digit with different values. One counts flats and the other counts small squares. Adding the three values gives the number back, 400 + 70 + 4 = 474; writing a number as such a sum is the next lesson, Expanded form.

Example 3

Day 1: The biggest number wins

In one round of the card game, the biggest number wins. Ivar is dealt 2, 9 and 5. Where should he lay each card to make the biggest number he can? Where for the smallest?

Solution

The hundreds column counts the biggest block, so the card there matters most. A digit other than 0 in the hundreds column is worth at least 100. The tens and ones columns together are worth at most 9 tens and 9 ones, which is 99. So whatever goes in the hundreds column outweighs everything to its right.

Ivar puts his biggest card, 9, in the hundreds column, where it is worth 900. Of the cards left, 5 is bigger, so it goes in the tens column, worth 50. The 2 goes in the ones column. The biggest number is 952.

For the smallest, turn it around: the smallest card in hundreds, the next in tens, the biggest in ones. That gives 259. Here the 9 is worth only 9.

The same card is worth 900 in one row and 9 in the other. Deciding which of two numbers is bigger, column by column, is Comparing and ordering whole numbers.

Example 4

Day 2: Rooms 609 and 690

On one hall of the school, Selamawit passes room 609, room 690 and a supply closet marked 69. Put each number in the chart. What is the 9 worth in each? Which of them have 6 hundreds?

Solution

Count the columns from the right end of each number (Figure 3).

609 has 9 in ones, 0 in tens and 6 in hundreds. The 9 is worth 9.

690 has 0 in ones, 9 in tens and 6 in hundreds. The 9 is worth 9 tens, 90.

69 has 9 in ones and 6 in tens. There is no third digit, so the hundreds cell is empty. The 9 is worth 9, and the 6 is worth 6 tens, 60.

Hundreds Tens Ones 6 0 9 6 9 0 6 9 room 609 room 690 closet 69
Figure 3Rooms 609 and 690 and closet 69. A 0 keeps an empty column filled; a blank on the far left means no digit at all.

Rooms 609 and 690 both have 6 hundreds. The closet has none: its 6 is in the tens column. Each room number has a 0, in a different column. In 609 the 0 fills the empty tens column, and in 690 it fills the empty ones column. Take either 0 away and the digits close up into 69, a different number.

Example 5

Day 2: From blocks to the chart and back

Ivar shows the number of his classroom with base ten blocks (Figure 4). Fill in the chart and read the number. Haruka’s classroom is room 860. Which blocks show it with the fewest pieces?

Figure 4Ivar’s classroom number in blocks, for Example 5.
Solution

Count each kind of block. There are 2 flats, so the hundreds column gets 2. There are no rods, so the tens column gets 0. There are 6 small squares, so the ones column gets 6. The chart row is 2, 0, 6, and Ivar’s classroom is room 206.

The 0 must be written. A blank tens cell would let the 2 slide into the tens column, and 26 is 2 tens and 6 ones, not 2 hundreds.

For room 860, read each column as a count of its block. The hundreds digit is 8: 8 flats. The tens digit is 6: 6 rods. The ones digit is 0: no small squares. The 0 at the right end says the small squares are missing; without it the number would be 86.

Example 6

Day 3: Ten in the tens column

Haruka builds a number with 4 flats, 10 rods and 2 small squares, and starts to fill the chart: 4 in hundreds and 2 in ones. The tens cell has room for one digit, but there are 10 rods. What number do her blocks show?

Solution

Ten rods make one flat, so trade them. Now there are 4 + 1 = 5 flats, 0 rods and 2 small squares. Every count fits in one cell: 5 in hundreds, 0 in tens, 2 in ones. The blocks show 502.

The blocks did not change in worth, only in how they are grouped:

400 + 100 + 2 = 502

Forcing both digits of 10 into the row would give 4, 1, 0, 2, which reads as a four-digit number, far bigger than the pile. One digit per cell is what keeps the chart and the blocks in agreement.

Example 7

Day 3: Trades that reach a fourth column

Selamawit counts the blocks in the class tub: 8 flats, 19 rods and 10 small squares. Two of those counts are too big for one cell. What number do the blocks show, and why does the chart need a fourth column?

Solution

Start with the ones column, since a trade there adds to the tens. The 10 small squares make 1 rod. The ones column holds 0, and there are now 19 + 1 = 20 rods.

Twenty rods are two groups of ten rods, and each group of ten makes a flat. Trade them for 2 flats. The tens column holds 0, and there are now 8 + 2 = 10 flats.

Ten flats are 1 thousand, and the hundreds column cannot hold ten. Trade the ten flats for one thousand. The hundreds column holds 0, and the thousand needs a column of its own on the left (Figure 5).

Thousands Hundreds Tens Ones 1 0 0 0 1,000
Figure 5The tub of blocks from Example 7 after every trade. Ten flats needed a fourth column.

The blocks show 1{,}000: 1 in thousands and 0 in each of the other three columns. Adding what the tub held at the start gives the same total, 800 + 190 + 10 = 1{,}000. The three trades changed how the blocks were grouped, never how much they were worth.

Boundaries

  • The chart here holds whole numbers. Three columns hold every number from 0 to 999; the fourth column reaches 1{,}000. Bigger numbers need more columns, as in Numbers to one million.
  • Each cell holds one digit, 0 to 9. A count of ten or more is not a digit and has to be traded before it can go in the chart.
  • A 0 in the middle of a number or at its right end changes the number: 609, 690 and 69 are different. A 0 at the left end changes nothing: 069 is 69.
  • The digit alone never says what it is worth; the column does. Only in the ones column is a digit’s value the same number as the digit, and a 0 is worth nothing in every column.
  • Columns to the right of the ones column, for parts of one, come much later, in Decimal place value.

Where this goes next

Adding up the values of the digits, 500 + 30 + 7 = 537, is Expanded form, the next lesson. Reading the hundreds column first is how Comparing and ordering whole numbers decides which number is bigger. Adding and subtracting three-digit numbers line the digits up in the same columns, ones under ones and tens under tens, and the trade between columns is what happens when a column runs past nine or runs out. Estimating by rounding looks at one column to decide which hundred or ten a number is nearest. Later, Multiplying and dividing by ten, one hundred and one thousand shows that every column is worth ten of the one to its right all the way up, so moving every digit one column to the left makes a number ten times as big.

Exercises

Day 1: a digit and its value

  1. In 386, which digit is in the tens column? What is its value?
  2. Put 529 in a place value chart. What is the value of each digit?
  3. Room 717 has two 7s. What is each one worth?
  4. Selamawit lays the cards 3, 8, 2 in the hundreds, tens and ones columns. What number is it, and what is the 3 worth? She then swaps the 3 and the 2. What number is it now, and what is the 3 worth?
  5. In which of these numbers is the 6 worth 60: 126, 465, 618, 960?
  6. Ivar is dealt 1, 8 and 4. What is the biggest number he can lay down? The smallest?
  7. In 253, which digit has the value 50?
  8. A chart has 6 in the hundreds column, 7 in the tens column and 2 in the ones column. What number is it?

Day 2: zeros and filling the chart

  1. Put 408 in a chart. What are the 4 and the 8 worth?
  2. Put 570 in a chart. Which column holds the 0, and why must it be written?
  3. A chart has 3 in hundreds, 0 in tens and 0 in ones. What number is it?
  4. A chart has an empty hundreds cell, 5 in tens and 2 in ones. What number is it?
  5. Fill in the chart for picture A. What number is it?
A B
Figure 6Pictures A and B for Exercises 13 and 14.
  1. Fill in the chart for picture B. What number is it?
  2. Haruka’s school has rooms 307 and 370 and a closet marked 37. In which of them is the 3 worth 300? In which is the 7 worth 70?

Day 3: nine at most, then a fourth column

  1. Ivar’s blocks are 6 flats, 13 rods and 4 small squares. Trade so that each column of the chart holds one digit. What number is it?
  2. Rooms on one floor run 196, 197, 198, 199. What is the next room number? Which columns change?
  3. A pile has 9 flats and 10 rods and nothing else. What number is it?
  4. Put 1{,}000 in a four-column chart. What is the 1 worth, in hundreds?
  5. Room 590 is followed, ten numbers later, by which room? Which columns change?

Explain

  1. A 0 is worth nothing in every column. Explain why the 0 in 305 is still needed.
  2. When is the value of a digit the same number as the digit itself? Explain.
  3. Explain, using blocks, why a column of the chart never holds more than 9.
  4. The 5 in 5 and the 5 in 500 are the same digit but have different values. Explain how both are true.
  5. Two three-digit numbers have different hundreds digits. Must the one with the bigger hundreds digit be the bigger number? Explain.
Answers
  1. 8; its value is 80. Counting from the right, the 6 is in ones, the 8 in tens and the 3 in hundreds, so the 8 is worth 8 tens.
  2. The 5 is in hundreds, worth 500. The 2 is in tens, worth 20. The 9 is in ones, worth 9.
  3. 700 and 7. The left 7 is in the hundreds column and the right 7 is in the ones column.
  4. 382, and the 3 is worth 300. After the swap the row is 2, 8, 3, which is 283, and the 3 is in the ones column, worth 3.
  5. 465 and 960. In each the 6 is second from the right, in the tens column. In 126 it is in ones, worth 6; in 618 it is in hundreds, worth 600.
  6. Biggest 841, smallest 148. The biggest card goes where it counts the most, the hundreds column; for the smallest, the smallest card goes there.
  7. The 5. It is in the tens column, and 5 tens is 50.
  8. 672. Read straight across the row: hundreds, tens, ones.
  9. Hundreds 4, tens 0, ones 8. The 4 is worth 400 and the 8 is worth 8.
  10. Hundreds 5, tens 7, ones 0; the 0 is in the ones column. Without it the digits would be 57, and the 5 would count tens instead of hundreds.
  11. 300. Both zeros are needed so the 3 stays in the hundreds column.
  12. 52. The empty hundreds cell means there are no hundreds, and nothing needs writing there.
  13. Hundreds 3, tens 5, ones 0: the number 350. Picture A has 3 flats, 5 rods and no small squares.
  14. Hundreds 1, tens 0, ones 8: the number 108. Picture B has 1 flat, no rods and 8 small squares.
  15. The 3 is worth 300 in 307 and 370. The 7 is worth 70 only in 370. In 307 the 7 is in ones; in 37 the 7 is in ones and the 3 is in tens, worth 30.
  16. 734. Ten of the 13 rods trade for a flat, leaving 3 rods and making 7 flats: 7 hundreds, 3 tens, 4 ones.
  17. 200, and all three columns change. One more makes 10 ones, which is a ten, so the ones go to 0; that makes 10 tens, which is a hundred, so the tens go to 0; the hundreds go from 1 to 2.
  18. 1{,}000. The 10 rods trade for 1 flat, making 10 flats, and 10 hundreds is 1 thousand.
  19. Thousands 1, hundreds 0, tens 0, ones 0. The 1 is worth 1 thousand, which is 10 hundreds.
  20. Room 600. Ten more adds 1 to the tens column, which makes 10 tens; that is a hundred, so the tens go to 0 and the hundreds go from 5 to 6. The tens and hundreds change; the ones stay 0.
  21. Without the 0 the digits would close up into 35. Then the 3 would sit second from the right and count tens, not hundreds. The 0 is worth nothing, but it holds the tens column open so the 3 stays in the hundreds column.
  22. In the ones column. A digit there counts ones, and that many ones is the digit itself: 7 ones is 7. A 0 is also worth 0 wherever it is. In any other column, a digit other than 0 is worth more than the digit.
  23. Each column counts one block size. Ten of a block always trade for one block of the next size to the left, so after every trade there are at most nine of each block left, and nine is the biggest digit.
  24. “Digit” names the symbol, and the symbol is 5 in both. “Value” names what it is worth, and that depends on the column: in 5 it is in the ones column, worth 5 ones; in 500 it is in the hundreds column, worth 5 hundreds, 500.
  25. Yes. The bigger hundreds digit is at least 1 more, so it is worth at least 100 more. The tens and ones of the other number add at most 99, which can never make up 100.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0