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Donna Boucher
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Place value is one of the most important—and often underestimated—concepts students develop in the primary grades. It’s easy to look at a number like 47 and assume that a student who can identify the 4 as being in the tens place understands place value. But true place value understanding means knowing that the 4 represents 4 tens, or 40, understanding how those tens and ones compose the number, and being able to represent and think about the number in flexible ways. As first- and second-grade students develop that understanding, there are three big ideas I like to keep in mind: take it slowly, use multiple representations, and ask the right questions.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
Place value understanding develops in a progression, and it’s important not to rush it. Before students can make sense of hundreds and thousands, they need a strong understanding of tens and ones. If that foundation isn’t secure, moving on to larger numbers simply adds more places without addressing the underlying misconception.
A major milestone in that development is unitizing—understanding that a group of 10 ones can be thought of as one ten. Instead of seeing 10 individual objects that must be counted one by one, students begin to see the group itself as a single unit. Kathy Richardson, in How Children Learn Number Concepts, emphasizes that meaningful place value understanding goes far beyond identifying which digit is in the tens or ones place. Students need to understand that two-digit numbers are structured around groups of tens and leftover ones.
That’s one reason numbers from 11 to 19 deserve plenty of attention. Students need lots of hands-on experiences building and representing teen numbers with tools such as linking cubes and ten-frames so they begin to see each number as one ten and some more ones. For example, students can build 13 as a train of 10 linking cubes and 3 more cubes, or fill one ten-frame and place 3 counters on a second ten-frame. Thirteen isn’t simply a 1 in the tens place and a 3 in the ones place; it’s 1 ten and 3 ones, or 10 + 3.
Once students can confidently compose and decompose teen numbers this way, they have an important foundation for understanding greater two-digit numbers. If a student doesn’t yet recognize that 1 ten and 3 ones makes 13 without counting all 13 objects, that’s valuable information: they need more experiences unitizing and working with teen numbers before moving on.
And this principle applies when we encounter place value difficulties later, too. Rather than simply giving students more practice with larger numbers, sometimes the best move is to go back to the foundational idea: Can they see ten ones as one unit of ten?
Students develop stronger place value understanding when they encounter the same mathematical idea in multiple representations. The goal isn’t simply to give students lots of different manipulatives to use. It’s to help them make connections among the representations and recognize that they are all showing the same underlying structure of our base-ten number system.
Counting the days of school provides a perfect example. Many primary classrooms bundle straws into groups of ten as the days accumulate. That’s a valuable routine—but a routine alone doesn’t guarantee understanding. Kathy Richardson, in How Children Learn Number Concepts, shares an example that illustrates how students can participate successfully in a familiar place value routine without really understanding the mathematics behind it.
One way to find out what students understand is to disrupt the routine. Imagine students arriving on the 76th day of school to discover that all those carefully bundled straws have somehow come apart.
Now they have a problem to solve: How can we organize 76 straws so we don’t have to count them one by one?
As students create groups of ten, they’re doing much more than rebuilding a classroom display. They’re applying the idea of unitizing—10 individual ones can be composed into 1 unit of ten. And because the materials can be grouped and ungrouped, students can see that relationship in both directions.
That’s especially important early in place value development. John Van de Walle cautions against moving too quickly to traditional base-ten blocks because they are already pregrouped: students can’t physically join 10 individual units to make the ten rod or break the rod apart again. Groupable materials such as linking cubes, bundled straws, or objects placed into groups of ten allow students to actually experience composing 10 ones into 1 ten and decomposing 1 ten back into 10 ones.
Ten-frames provide another useful representation. As students fill a ten-frame, they can see a complete group of ten without having to count each object individually. A second ten-frame makes the ten and some more structure of teen numbers especially visible. The same representation can grow with students as they use multiple ten-frames to explore larger two-digit numbers.
[IMG width="277px" alt="counting the days using ten-frames to build place value understanding
"]https://www.mathcoachscorner.com/wp-content/uploads/2014/10/image-1.png[/IMG]
Another great tool for exploring place value is the hundred chart. Because the structure of the chart groups numbers into rows of tens, it is a useful representation for exploring concepts such as ten more and ten less.
Concrete and visual models are important, but we also want students to connect what they’re building to the numbers and symbols that represent those quantities.
That’s where the place value cards come in.
Let’s go back to our pile of 76 straws. As students count 10 straws and bundle them, show the 10 card. When they make another bundle, replace it with 20. Continue—30, 40, 50, 60, 70—as each new ten is composed.
After students have made 7 bundles of ten, they’ll have 6 individual straws remaining. Now place the 70 card and the 6 card side by side:
Then slide the cards into the place value holder so the digits align to show:
Now students can connect several representations of the same number: 76 individual straws → 7 bundles of ten and 6 ones → 70 + 6 → 76.
That’s the connection we want students to make. The 7 in 76 doesn’t simply live in the tens place—it represents 7 tens, or 70.
The place value cards and holder can be used in lots of other ways as students compose, decompose, build, read, and compare numbers. I’ve updated the set for you, and you’ll find the free download at the end of the post.
Over the years, we’ve fallen into some pretty predictable routines when it comes to assessing place value. We ask questions such as What digit is in the tens place? or What is the value of the 7 in this number? There’s nothing wrong with those questions, but correct answers don’t necessarily tell us whether a student understands the relationships among the places.
For example, a student may correctly identify the 3 in 36 as being in the tens place. But do they understand that the 3 represents 3 groups of ten, or 30? Can they show 36 as 3 tens and 6 ones? Do they also understand that 36 can be composed as 2 tens and 16 ones? That’s the kind of flexibility that reveals much more about a student’s place value understanding.
Sometimes making a familiar question more powerful is as simple as following it with another question:
We can also ask questions that start with the value or relationship rather than always starting with a numeral. I have 4 tens and 7 ones. What number could I make? What if I traded one ten for 10 ones? What would change? What would stay the same?
The goal is to get beyond whether students know the names and positions of the places and uncover what they understand about our base-ten system. Ask students to explain, build, draw, trade, compose, and decompose. Their answers—and especially their explanations—will tell you far more than What digit is in the tens place? ever could.
Strong place value understanding doesn’t come from memorizing the names of the places. It develops over time as students build and break apart numbers, work with multiple representations, and explain their thinking. Take it slowly, give students plenty of opportunities to explore the structure of our base-ten system, and keep asking questions that help you see what they really understand.
To help students connect the quantities they build to the numbers that represent them, I’ve created a free set of place value cards and a place value holder for working with ones, tens, and hundreds. Use them alongside linking cubes, bundled straws, ten-frames, or other models to help students make connections between concrete representations and written numbers.
Grab the free place value cards and holder here.
The post Building Place Value Understanding: 3 Big Ideas appeared first on Math Coach's Corner.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
Take it slowly
Place value understanding develops in a progression, and it’s important not to rush it. Before students can make sense of hundreds and thousands, they need a strong understanding of tens and ones. If that foundation isn’t secure, moving on to larger numbers simply adds more places without addressing the underlying misconception.
A major milestone in that development is unitizing—understanding that a group of 10 ones can be thought of as one ten. Instead of seeing 10 individual objects that must be counted one by one, students begin to see the group itself as a single unit. Kathy Richardson, in How Children Learn Number Concepts, emphasizes that meaningful place value understanding goes far beyond identifying which digit is in the tens or ones place. Students need to understand that two-digit numbers are structured around groups of tens and leftover ones.
That’s one reason numbers from 11 to 19 deserve plenty of attention. Students need lots of hands-on experiences building and representing teen numbers with tools such as linking cubes and ten-frames so they begin to see each number as one ten and some more ones. For example, students can build 13 as a train of 10 linking cubes and 3 more cubes, or fill one ten-frame and place 3 counters on a second ten-frame. Thirteen isn’t simply a 1 in the tens place and a 3 in the ones place; it’s 1 ten and 3 ones, or 10 + 3.
Once students can confidently compose and decompose teen numbers this way, they have an important foundation for understanding greater two-digit numbers. If a student doesn’t yet recognize that 1 ten and 3 ones makes 13 without counting all 13 objects, that’s valuable information: they need more experiences unitizing and working with teen numbers before moving on.
And this principle applies when we encounter place value difficulties later, too. Rather than simply giving students more practice with larger numbers, sometimes the best move is to go back to the foundational idea: Can they see ten ones as one unit of ten?
Use multiple representations
Students develop stronger place value understanding when they encounter the same mathematical idea in multiple representations. The goal isn’t simply to give students lots of different manipulatives to use. It’s to help them make connections among the representations and recognize that they are all showing the same underlying structure of our base-ten number system.
Counting the days of school provides a perfect example. Many primary classrooms bundle straws into groups of ten as the days accumulate. That’s a valuable routine—but a routine alone doesn’t guarantee understanding. Kathy Richardson, in How Children Learn Number Concepts, shares an example that illustrates how students can participate successfully in a familiar place value routine without really understanding the mathematics behind it.
One way to find out what students understand is to disrupt the routine. Imagine students arriving on the 76th day of school to discover that all those carefully bundled straws have somehow come apart.
As students create groups of ten, they’re doing much more than rebuilding a classroom display. They’re applying the idea of unitizing—10 individual ones can be composed into 1 unit of ten. And because the materials can be grouped and ungrouped, students can see that relationship in both directions.
That’s especially important early in place value development. John Van de Walle cautions against moving too quickly to traditional base-ten blocks because they are already pregrouped: students can’t physically join 10 individual units to make the ten rod or break the rod apart again. Groupable materials such as linking cubes, bundled straws, or objects placed into groups of ten allow students to actually experience composing 10 ones into 1 ten and decomposing 1 ten back into 10 ones.
Ten-frames provide another useful representation. As students fill a ten-frame, they can see a complete group of ten without having to count each object individually. A second ten-frame makes the ten and some more structure of teen numbers especially visible. The same representation can grow with students as they use multiple ten-frames to explore larger two-digit numbers.
[IMG width="277px" alt="counting the days using ten-frames to build place value understanding
"]https://www.mathcoachscorner.com/wp-content/uploads/2014/10/image-1.png[/IMG]
Another great tool for exploring place value is the hundred chart. Because the structure of the chart groups numbers into rows of tens, it is a useful representation for exploring concepts such as ten more and ten less.
Connect the Representations to Numbers
Concrete and visual models are important, but we also want students to connect what they’re building to the numbers and symbols that represent those quantities.
That’s where the place value cards come in.
Let’s go back to our pile of 76 straws. As students count 10 straws and bundle them, show the 10 card. When they make another bundle, replace it with 20. Continue—30, 40, 50, 60, 70—as each new ten is composed.
After students have made 7 bundles of ten, they’ll have 6 individual straws remaining. Now place the 70 card and the 6 card side by side:
Then slide the cards into the place value holder so the digits align to show:
Now students can connect several representations of the same number: 76 individual straws → 7 bundles of ten and 6 ones → 70 + 6 → 76.
That’s the connection we want students to make. The 7 in 76 doesn’t simply live in the tens place—it represents 7 tens, or 70.
The place value cards and holder can be used in lots of other ways as students compose, decompose, build, read, and compare numbers. I’ve updated the set for you, and you’ll find the free download at the end of the post.
Ask the right questions
Over the years, we’ve fallen into some pretty predictable routines when it comes to assessing place value. We ask questions such as What digit is in the tens place? or What is the value of the 7 in this number? There’s nothing wrong with those questions, but correct answers don’t necessarily tell us whether a student understands the relationships among the places.
For example, a student may correctly identify the 3 in 36 as being in the tens place. But do they understand that the 3 represents 3 groups of ten, or 30? Can they show 36 as 3 tens and 6 ones? Do they also understand that 36 can be composed as 2 tens and 16 ones? That’s the kind of flexibility that reveals much more about a student’s place value understanding.
Sometimes making a familiar question more powerful is as simple as following it with another question:
- What does that mean?
- How do you know?
- Can you show me with linking cubes or another model?
- Can you draw a representation of that number?
- Can you show the number a different way?
- How is a 3 in the tens place different from a 3 in the ones place?
- If I traded one of the tens for ones, would I still have the same number? How do you know?
We can also ask questions that start with the value or relationship rather than always starting with a numeral. I have 4 tens and 7 ones. What number could I make? What if I traded one ten for 10 ones? What would change? What would stay the same?
The goal is to get beyond whether students know the names and positions of the places and uncover what they understand about our base-ten system. Ask students to explain, build, draw, trade, compose, and decompose. Their answers—and especially their explanations—will tell you far more than What digit is in the tens place? ever could.
build Place Value Understanding
Strong place value understanding doesn’t come from memorizing the names of the places. It develops over time as students build and break apart numbers, work with multiple representations, and explain their thinking. Take it slowly, give students plenty of opportunities to explore the structure of our base-ten system, and keep asking questions that help you see what they really understand.
To help students connect the quantities they build to the numbers that represent them, I’ve created a free set of place value cards and a place value holder for working with ones, tens, and hundreds. Use them alongside linking cubes, bundled straws, ten-frames, or other models to help students make connections between concrete representations and written numbers.
Grab the free place value cards and holder here.
The post Building Place Value Understanding: 3 Big Ideas appeared first on Math Coach's Corner.