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Place Value Explained: Ones, Tens, Hundreds and Beyond
Place value is the idea that a digit's value depends on where it sits in a number. It is the foundation for addition with regrouping, subtraction with borrowing, rounding, decimals and nearly everything that follows in arithmetic.
The big idea: our number system is built on tens
We use ten digits (0 through 9). Once we run out of digits in one place, we move left. Ten ones make one ten, ten tens make one hundred, and ten hundreds make one thousand. Every place is worth ten times the place to its right.
| Thousands | Hundreds | Tens | Ones |
|---|---|---|---|
| 3 | 7 | 4 | 2 |
In 3,742 the 3 is worth 3,000, the 7 is worth 700, the 4 is worth 40 and the 2 is worth 2. That is why the same digit, 7, can mean 7, 70 or 700.
Expanded form
Writing a number as a sum of its place values shows children what it is made of: 3,742 = 3,000 + 700 + 40 + 2. A good warm-up is to ask, "What is the value of the 4 in 3,742?" The answer is 40, not 4.
The job of zero
Zero is a placeholder. In 305 there are 3 hundreds, 0 tens and 5 ones. Without the zero we would read the number as 35. Children often write "three hundred and five" as 3005, because they write what they hear: 300 and then 5. Have them build the number with base-ten blocks or draw hundreds flats, tens rods and ones cubes to see why 305 has only three digits.
Comparing numbers
Compare from the left. Start with the largest place and move right until the digits differ.
- Compare 482 and 478: both have 4 hundreds and 7 or 8 tens. 8 tens is more than 7 tens, so 482 is greater.
- More digits usually means a bigger number: 1,020 is larger than 987.
Use the symbols < (less than), > (greater than) and = (equal to). A helpful picture is the "alligator" whose mouth always opens toward the bigger number.
Rounding
To round to the nearest ten, look at the ones digit. If it is 5 or more, round up; if it is 4 or less, round down. 67 rounds to 70, and 62 rounds to 60. For the nearest hundred, look at the tens digit: 348 rounds to 300, and 351 rounds to 400. A number line helps: place the number between two tens and see which one it is closer to.
Place value with money
Money is a natural model. A $100 bill, $10 bills and $1 bills are hundreds, tens and ones. A child who can see that $243 is 2 hundred-dollar bills, 4 ten-dollar bills and 3 ones is working with place value. Decimals extend the idea: a dime is one tenth of a dollar and a penny is one hundredth.
Why it matters for addition and subtraction
When 8 ones plus 7 ones makes 15 ones, we regroup by trading ten of the ones for 1 ten and keeping 5 ones. "Carrying" is simply this trade. Children who understand place value understand why the trade is fair. Children who only memorize the steps often make errors they cannot explain.
Activities to try
- Build numbers with LEGO or base-ten blocks: "Show me 214. Now add one more ten."
- Roll three dice, arrange them to make the biggest or smallest number possible.
- Number detective: "I am a three-digit number. My tens digit is 5. My ones digit is double my hundreds digit..."
- Read big numbers from real life such as population signs, prices and distances.
Practice problems
- What is the value of the 6 in 4,613?
- Write 2,508 in expanded form.
- Which is greater, 739 or 793?
- Round 86 to the nearest ten.
- Round 749 to the nearest hundred.
- Write the number: 5 thousands, 0 hundreds, 3 tens, 9 ones.
- What is 10 more than 395?
- Put in order from least to greatest: 1,205 1,052 1,250 1,025
Answer key
1) 600 2) 2,000 + 500 + 8 3) 793 4) 90 5) 700 6) 5,039 7) 405 8) 1,025; 1,052; 1,205; 1,250
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