My 2nd graders have been exploring place value, so yesterday I wrote the number 43 on the table in front of one sweet friend and asked what he could tell me about the number 43. He proudly told me it was 4 tens and 3 ones. This is very common, right? We ask students to name the place value positions or tell us what digit is in the tens place, etc. So you might assume that this student knows place value. But as I probed a little deeper, I began to realize that there is a big difference between knowing and understanding place value.
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I handed him a student ten-frame kit and asked him to build the number 43 using ten-frames. He immediately reached for a 4 tile. Uh oh. I asked what the 4 represents in the number 43, and again he told me 4 tens. He had the tile for 4 in his hand, so I asked if that was a ten. I got a quizzical look in return. I changed directions and asked him to show me 10 using the ten-frames. I also made headings for tens and ones on the table in front of him (see the photo). He put the 4 back and took out a 10. I asked him how many tens he had, and he told me 1. I then asked how many dots were on the ten-frame, and he told me 10. I asked him where he should place the 10, and he told me under the tens. Next, I asked him to show me another 10, so he took another ten out of the kit. I repeated my questions–how many tens? how many dots (ones)?–and he told me 1 ten and 10 dots and put the 10 next to the other one under the tens heading. Then I asked him how many tens he had under the tens heading, and he told me 20. Hmmm, I think I see 20 dots, but how many groups of 10? Two. Great! So there are 2 tens and how many dots on those two tens? Twenty. His responses were still nowhere near automatic. That is, he really had to think about his answers. I repeated this process until we had 4 tens. Finally, I asked him what he would need to add to make the number 43. He considered the question, looked at the ten-frames in his kit, pulled out a 3, and placed it in the ones column. To tie it all together we wrote that 43 is 4 tens and 3 ones, and we also wrote it in expanded form, 40 + 3. We repeated this same process for several more 2-digit numbers.
Truly understanding place value requires students to be able to unitize. What this means is that they must be able to understand that a group of ten ones can be counted as a single unit, called a ten. That’s a huge mathematical step, because the idea is so abstract. As with any abstract concept, students need lots of varied hands-on experiences building tens and ones and talking about the meaning to develop a true understanding. That means ten-frames and counters, linking cubes, beans on a stick, linking chains, etc.–as many different representations as you can think of.
The ten frame kits can be a bit pricey, and have limited availability, but you can create your own student kits using this file. I also uploaded a B&W version, in case you want your kiddos to color their own. If you need ten-frames to use with counters, download those here.
I’d love to hear your experiences with early place value understandings! Please share what has worked for you.