To truly understand place value, students need to not only identify the place, or position, a digit is sitting in (e.g., the 4 is in the tens place or there are 4 tens), but they also need to understand the value of that digit (e.g., the value of the 4 in the tens place is 40). We need to make sure that we are asking students a variety of questions and having them use multiple place value models so we can uncover their true understanding of place value.

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In the book Teaching Student-Centered Mathematics, the authors talk about the importance of using a variety of models during early place value explorations. Place value models fall into two broad categories—groupable and pregrouped. Ideally, all place value models we use with young children should be proportional, meaning that the representation for a ten is physically ten times larger than a one. It’s important that students work with groupable models before pregrouped models. One way to think about it is that even though all these models are concrete models, the pregrouped models are more abstract. Let’s take a look at both types.

Groupable Place Value Models

A familiar example of a groupable model is the classic daily activity of using a place value pocket chart to count the days in school. It’s an oldie, but goodie! A straw is added to the ones pocket each day. Once the straws in the ones pocket reach ten, they are bundled together and moved to the tens pocket. When there are ten bundles in the tens pocket, they are bundled and moved to the hundreds pocket.

Linking cubes are another popular groupable model. When using linking cubes, students learn to connect ten cubes, creating a rod that can be counted as ten. A great game for developing the concept of ten using linking cubes is Race to 20. Players take turns rolling a die and adding the number of cubes rolled to their tower. When the tower reaches 10, they begin a new one. The first player to reach 20 wins.

Initially, children will create tens rods each time they use linking cubes. Eventually, however, they are likely to realize they can leave the tens rods built, making it easier to represent numbers. Another advantage of linking cubes is that there is a natural transition to base-ten blocks, which we’ll read about in the next section.

Ten-frames and counters are a must-use manipulative in early education classrooms because of their emphasis on ten. Once students have the understanding that a filled ten-frame is ten without counting, they can explore place value with multiple ten frames.

After your students have been introduced to multiple groupable models, present a number using two of the models and ask students what they notice and wonder. You want them to understand that a ten can look many different ways and be built in different way, but it still means ten ones. Repeat with other models and numbers.

To make sure your students truly  understand place value, remember to go beyond just asking students to name the digit in a certain place value position. We often ask, for example, What digit is in the tens place? and many students can tell us there is a four in the tens place. That only tells us they have memorized the place value positions. A better question is What is the value of the digit in the tens place?  Or, In the number 444, which digit has a value of 40? In other words, our questions should focus on the value.

Pregrouped Place Value Models

After plenty of work with groupable models, and once you’re sure your students have a solid understanding of tens and ones, you can transition to pregrouped models. This is super helpful as they begin to work with larger numbers. Again, we can use side-by-side comparisons of the models to help students make connections. There is a natural connections between linking cubes and base-ten blocks. Once students have shared what the two models have in common and how they are different, we can ask why we might want to use the base-10 blocks instead of the linking cubes. Keep in mind that while some students easily transition to the pregrouped models, others might need to continue with groupable models for a longer period of time. That’s part of your differentiation.

Small ten-frame cards are a pregrouped model that gives students another way to represent tens and ones. They allow students to easily make larger numbers with ten frames. Individual student ten-frame kits used to be available for purchase, but unfortunately they are now out-of-stock. You can create your own student ten-frame kits using this file. Simply copy one or several sheets on cardstock, laminate them, and cut the cards apart. They are sized to fit perfectly in these coin collecting binder pages.

If you’ve got other tips for teaching place value in the early years, please share them in the comments!