# Opportunities for Deep Understanding of Place Value

My students are currently learning about place value in math. One of the kindergarten math standards is to be able to break apart teen numbers into groups of tens and ones. However, I do introduce the idea of place value and extend the thinking to higher numbers. So how do I teach this very complicated idea of place value and ensure deep understanding?

### Setting the Place Value Foundation

Before I even begin teaching the concept of place value, I make sure to give my students lots of opportunities to meaningfully interact with large numbers. I do this in a few different ways.

### Counting the Days of School

Every day we keep track of how many days of school we have been in as a part of our calendar routine. We add a dot to our ten frames and then count the dots. Up until 100 we count by ones, but after the one hundredth day of school, we start to skip count. The calendar leader gets the choice to skip count by 5s or by 10s. I want my students to know the route skip counting routine, but I also want them to understand how large numbers can be grouped by tens. The visual of the ten frames helps with this a lot. Once we get into place value activities, we have this great resource of ten frames to count not only the days of school, but the ten frames themselves to see how many groups of ten it is.

### Counting Collections

I love giving my students the chance to count and interact with large groups of items. Sometimes I do that with unifix cubes and others I find collections for them. Our math coach brought in a new collection of a bunch of rocks for each student to count. The students had the opportunity to count the collection with whatever strategy they wanted. After counting it one way, we encouraged them to count a different way and find a strategy that helped them keep track easily of the rocks. After most students had done this a few times, we discussed how students had many different strategies, but the students talked about how grouping by tens was the easiest to keep track of the rocks and count.

### Choral Counting

After doing a few days of place value games with teen numbers, I gathered my students on the rug, and we did a group choral count up to 60. This means that the students all counted together at the same pace and I wrote down the numbers on a chart. I started with 1 on the left and wrote the numbers until 10 and then made a new row. After we got to 60, I stopped the counting and had students talk about what they noticed. As you can see with all the markings on the page – they noticed a lot! Most of them noticed that the columns all ended in the same numbers and eventually, they noticed that the rows all started with the same number.

After they noticed that the rows all started with the same number, I asked them why they thought this was and what did the number 5 mean in the final row of numbers. No one had an answer. They had also been on the rug a long time, so I asked them to think about this while they did their math stations and let this idea percolate with them. (Tip: after reflecting on this discussion and where it went, I would write the row with zero to start and then end the row with numbers ending in 9 so that the row really did show all the same starting numbers.)

The next day I brought out the chart and asked them again why did the bottom row all start with the number 5 (except for the last number: 60)? The children shared some ideas and other noticings about the numbers but really were not sure. So, I told them I would leave out a bunch of unifix cubes during stations to see if they could figure out the answer.

### Independent Exploration of Place Value

A small group of students were interested in discovering an answer. So, to start, I asked them to take out 54 cubes (one of the numbers from the last row we had been discussing).  After they took out the 54 cubes, I asked if they could figure out what that 5 means? Then I backed off and let them lead the exploration. They decided to group the cubes into groups of 5 and skip counted to 54. I asked how many groups they had, and they had 10 so they still weren’t sure what that 5 could mean in the number 54.

They then decided to group by 2s and still were confused about what the 5 meant. Then they grouped by 10s and skip counted. They were about to go back to grouping by 2s and I asked, “Wait – how many groups of ten do you have?” They counted 5 and realized it was like the 5 in the number. Then I asked how many leftover cubes they had, and they counted 4 – just like the number 54. Then they discussed that the 5 must mean the number of groups of 10 and tried it again with a few other numbers. After some time, they shared out their discoveries with the class. In their own words they told the class that the first number in the row means how many groups of 10 the number was and the second number was how many extras.

### Explicit Place Value Vocabulary

In order to support my students in this culminating discussion, I then taught the class a few vocabulary words to help them understand place value. I taught them that numbers are made up of digits and then restated their self-created place value definition with the word digit: the first digit in a two digit number is the number of groups of tens the number is. Then I taught them the word decade to talk about the rows of numbers as a new decade of numbers (they kept talking about the numbers being in the 50s so I restated it as numbers in the decade 50).

### Conclusion

My students had lots of chances to independently practice and explore with place value before I ever explicitly taught them the place value vocabulary. I never officially taught them the place value rules, instead I let my students do the work. I let them discover and discuss their observations of numbers. Through my purposeful use of numbers, materials, and questions they were able to create a much deeper understanding of place value than my explicit teaching could ever have done.

Comment below with to share any of your place value tips!

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