Place Value to 10,000 - Third Grade Mathematics

Our number system is built on a brilliant system called place value, where the position of every single digit determines how much value it represents. In third grade, you will explore numbers that reach all the way to 10,000, discovering how ten of any unit always combines to form one of the next larger unit, whether you are counting single marbles or thousands of stars in the night sky.

The Base-Ten Place Value System

The number system we use every day is called base-ten. The prefix "base-ten" means that our system is organized around groups of ten. We use only ten digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—yet by arranging them in different positions, we can write any number in the universe.

How Groups of Ten Grow

Every time you collect 10 of a smaller unit, you bundle them together into 1 larger unit to the left.

10 Ones         =  1 Ten         (10)
10 Tens         =  1 Hundred     (100)
10 Hundreds     =  1 Thousand    (1,000)
10 Thousands    =  1 Ten-Thousand (10,000)

Look at this visual progression of base-ten units:

[.]                 Individual Unit (1 One)
[||||||||||]        Rod of 10 Units (1 Ten = 10)
+----------+
|          |        Flat of 10 Rods (1 Hundred = 100)
+----------+
+----------+
|\        /|
| \      / |       Cube of 10 Flats (1 Thousand = 1,000)
|  +----+  |
+----------+

When you bundle ten one-thousand cubes together, you get 10,000!

Reading the Place Value Chart up to 10,000

A place value chart keeps each digit organized in its proper column so you can read and understand the true size of a number.

+--------------+-----------+----------+------+------+
| Ten-Thousands| Thousands | Hundreds | Tens | Ones |
+--------------+-----------+----------+------+------+
|      1       |     0     |    0     |  0   |  0   |  -> 10,000
|      0       |     4     |    6     |  2   |  8   |  ->  4,628
|      0       |     7     |    0     |  5   |  3   |  ->  7,053
+--------------+-----------+----------+------+------+

The Value of a Digit Versus the Digit Itself

A digit is just a symbol from 0 through 9, but its place value depends completely on where it sits in the number.

Consider the number 4,628:
- The digit 4 is in the thousands place, so its value is 4,000.

- The digit 6 is in the hundreds place, so its value is 600.

- The digit 2 is in the tens place, so its value is 20.

- The digit 8 is in the ones place, so its value is 8.


Notice how the digit 4 is much more powerful than the digit 8 in this number, simply because it lives in the thousands place!

The Role of Zero as a Place Holder

Zero is one of the most important digits in mathematics because it acts as a placeholder. Look at the number 7,053. There are 0 hundreds. If you forgot to write the 0, the number would become 753, which is completely different from 7,053! The zero holds open the hundreds column so the 7 stays securely in the thousands column.

Understanding Commas in Large Numbers

When writing numbers with four or more digits, we use a comma to separate the thousands period from the ones period.

Thousands Period        Ones Period
+----------------+     +-------------------+
| Thousands      |  ,  | Hundreds Tens Ones|
+----------------+     +-------------------+
       4         ,             6    2    8

To place a comma correctly, start at the right side (the ones place) and count three digits to the left: one, two, three, then insert the comma.
- 1250 is written as 1,250

- 8409 is written as 8,409

- 10000 is written as 10,000


The comma helps your eyes quickly read the number without counting each digit individually.

Decomposing and Composing Numbers

Decomposing a number means breaking it apart into its place value pieces. Composing a number means putting those pieces back together.

Example 1: Decomposing a 4-Digit Number

Let us decompose 5,392:
- 5 thousands = 5,000

- 3 hundreds = 300

- 9 tens = 90

- 2 ones = 2


You can also decompose numbers flexibly! For instance, 5,392 can be thought of as 53 hundreds, 9 tens, and 2 ones, or even 5,390 ones and 2 ones. Flexible thinking helps tremendously when adding, subtracting, and regrouping.

Example 2: Composing from Word Clues

What number has 6 thousands, 0 hundreds, 4 tens, and 7 ones?
- 6 thousands = 6,000

- 0 hundreds = 0

- 4 tens = 40

- 7 ones = 7

- Combined: 6,047


Chapter Practice Exercises

Test your understanding of place value up to 10,000 with these problems.

  1. State the place and the value of the underlined digit: a. 3,482 (digit 4) b. 8,190 (digit 8) c. 5,063 (digit 6) d. 9,725 (digit 5)
  2. What number is made of 8 thousands, 5 hundreds, 2 tens, and 9 ones?
  3. What number is made of 3 thousands, 7 tens, and 1 one? (Be careful with the hundreds place!)
  4. How many tens are in 1 hundred? How many hundreds are in 1 thousand? How many thousands are in 10,000?
  5. Write 9,304 in a place value chart and explain why the 0 cannot be omitted.
  6. A stadium has 7,420 seats. What is the value of the digit 7 in this number?

Solutions and Step-by-Step Answers

  1. Value of underlined digits: a. In 3,482, the digit 4 is in the hundreds place, so its value is 400. b. In 8,190, the digit 8 is in the thousands place, so its value is 8,000. c. In 5,063, the digit 6 is in the tens place, so its value is 60. d. In 9,725, the digit 5 is in the ones place, so its value is 5.
  2. 8 thousands (8,000) + 5 hundreds (500) + 2 tens (20) + 9 ones (9) = 8,529.
  3. 3 thousands (3,000) + 0 hundreds (0) + 7 tens (70) + 1 one (1) = 3,071. Notice the placeholder 0 in the hundreds place.
  4. There are 10 tens in 1 hundred. There are 10 hundreds in 1 thousand. There are 10 thousands in 10,000.
  5. In the place value chart: Thousands = 9, Hundreds = 3, Tens = 0, Ones = 4. If the 0 were omitted, the number would read 934, which is nine hundred thirty-four rather than nine thousand three hundred four. The 0 keeps the 9 in the thousands place.
  6. In 7,420, the digit 7 is in the thousands place, representing a value of 7,000 seats.