How Place Value Understanding Progresses in Elementary Math
When a child understands place value, they understand that the location of each digit in a number defines its value. For example, the 3 in 35 and the 3 in 325 have very different values because of their locations. The 3 in 35 has a value of 30 while the 3 in 325 has a value of 300.
As children move through elementary school, they learn more about whole number place values: ones, tens, hundreds, thousands, ten thousands, and so on. In fourth grade and into middle school, children learn about the place values that represent parts of one whole: tenths, hundredths, thousandths, and so on.
As you can imagine, a lack of place value understanding leads to all kinds of confusion in math class, such as not understanding why 14.5 + 22 = 36.5 not 16.7 or not understanding why 262.1 has a much larger value than 2.621.
So how do students build understanding of place value? Here's a general overview of how place value knowledge develops in the Common Core Math Standards, the most commonly used math standards in the US. Even if your child's school doesn't use these standards, this still provides an overview of how place value knowledge builds over time.
Progression of Place Value in Elementary Math:
Kindergarten: Teen numbers are a group of 10 and some more ones. Children learn to see a teen number as a group of ten and a group of additional ones. For example, 11 = 10 + 1, 12 = 10 + 2, 13 = 10 + 3, and so on. While they don't formally name the tens place in kindergarten, they do build the understanding that the leading 1 in teen numbers has a value of 10 not 1.
First Grade: Ten is a unit. Children expand their understanding of teen numbers as 10 and some more ones to see that each time we count another group of ten, the leading digit of a two-digit number goes up by one. So they begin to see that two groups of ten is 20, and three groups of ten is 30, and so on. This allows children to make sense of the difference between the tens place and the ones place: The tens place tells me how many full groups of 10 I have while the ones place tells me how many additional ones we have. So I can read any two-digit number as a set of tens and a set of ones, such as 86 is 8 tens and 6 ones.
Second Grade: Place value patterns extend to the hundreds place. Children realize that just like 10 ones become its own unit, 1 ten, 10 tens also become its own unit, 1 hundred. When we count 10 tens, we say, "one hundred" and write it "100." So the 1 in a three-digit number doesn't mean one, it means one hundred. And if we count another 10 tens, we'll get two hundreds or 200. Children begin to see a pattern emerging that when we have 10 of a smaller unit, it creates 1 of the next larger unit: 10 ones is the same as 1 ten and 10 tens is the same as 1 hundred.
Third Grade: Use place value to fluently add and subtract three-digit numbers. Children solidify their understanding of place value with three-digit numbers and use this to fluently add and subtract three-digit numbers. They also apply this place value understanding to round two- and three-digit numbers to the nearest ten or hundred. Though work with the thousands place isn't formally named in the Common Core Standards until fourth grade, the extensive work with three-digit numbers in third grade leads many children to discover that 10 hundreds make 1 thousand, thus continuing the pattern they began to discover in second grade.
Fourth Grade: Formalizing the pattern that 10 of one unit will always make 1 of the next larger unit and an introduction to place values less than 1. Children now formally name that the pattern in our base ten number system is that 10 of a smaller unit make 1 of the next larger unit, and this works no matter which place value you're working with. Because of this, we know that a digit in any place has a value that is 10 times greater than if it was in the place immediately to its right. For example, the 8 in 810 has a value that is ten times more than the 8 in 81. (800 is ten times the value of 80.) Children in fourth grade work with numbers up to 1,000,000 as well as tenths and hundredths, and apply these place value understandings to their work with addition, subtraction, multiplication, and division.
Fifth Grade: The thousandths place and operations with decimals. Children build fluency with the pattern that any digit has a value 10 times greater than it would in the place value to its right and 10 times less (1/10 of the amount) of the place value to its left. For example, the value of 5 when it's in the hundreds place (500) is 10 times the value of 5 when it's in the tens place (50). And, the value of 5 in the hundreds place (500) is 1/10 of the value of 5 when it's in the thousands place (5,000). This pattern extends to numbers in the thousandths place and is used to support work with adding, subtracting, multiplying, and dividing decimal numbers.
This, of course, is just an overview of place value in each grade. If you want more information on the progression of place value understanding, this place value progression document goes into much more depth than I did here.
How Can I Support Place Value at Home?
My top recommendation for supporting place value understanding at home: Have a hands-on tool that shows the pattern of ten of a unit making one of the next higher unit. In kindergarten and first grade, a set of snap cubes in sticks of 10 is sufficient for modeling that 10 ones makes 1 ten. In second grade and beyond, you'll want a tool that shows how 10 tens becomes 1 hundred. You can either purchase base ten blocks or create your own hands-on tool with popsicle sticks or straws and bundling them into bundles of 10 and then bundling ten bundles of 10 into a bundle of 100. Use these tools as you talk about numbers at home so children have a way to visualize the values of each digit.
And if you want additional support developing an understanding of place value, the Math Companion program includes simple and fun place value activities up to 2nd grade. Click here to learn more.