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The Difference Between Knowing and Understanding Place Value

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.

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  1. Thank you so much for these tens frames counters!! I think they may just be the perfect manipulative to help my struggling First Graders!! I will try them out in my Intervention groups on Tuesday!!

  2. I love the organization of the ten frame numbers in pockets. I am going to make some for my first grade teachers using the baseball card plastic sleeves. I love your blog!

    1. Barbara, I know this post is several months old, but I would love to know how you used the baseball card pockets. I’ve got lots of them on hand, and half the summer left. Would you be willing to share how you did this?

  3. Hi Donna!
    Working with kids in fifth grade, I am finding a similar problem–especially when it comes to decimal fractions. (Out come the manipulates once again!)
    I love that you share the conversations you have with kids–it always gives me ideas about how to structure my own chats.
    Thanks for another great post. I am passing it along to a first grade teacher friend for the nifty ten frames.
    I hope you enjoy this special day!

    Finding JOY in 6th Grade

    1. Oh, yes, Kim! Decimals and fractions are such abstract concepts. Teachers must extensively use manipulatives and multiple representations for students to truly understand. Concrete learning is for ALL grade levels, not just the primary ones!

      1. As a 5th grade teacher I notice there is a disconnect between place value and multiplication. Students can compose and decompose a number, but when you ask them 10×4 they don’t know. Plus, how do you move a child from repeated addition to the algorithm to multiply?

  4. Hi I was wondering what the holder is that you put the ten frames in and where did you get it? Thanks so much. Jessica

    1. That is the student ten-frame kit that you can buy, Jessica. It includes the plastic ten-frame tiles and the wallet. The link is in the blog post.

  5. If you have a laminator at your school, you could use the cut off double-thick plastic to sew narrow strips into a wallet for kids. Felt might be good for the outer wallet material. Set your stitch very long and use a sharp needle. I like the idea of having students color in their own cards – lots of thinking gets done at the same time as the coloring.

      1. I use trading card protector pages–it has 9 pockets on it. Perfect for fitting sets of cards, they are 3-hole punched so they fit nicely in a 3 ring binder. 🙂

  6. Donna,
    Once again, I find something that I can immediately use in my first grade classroom. For free! Thank you so much for your generosity and your expertise. I can’t wait to use these with my first graders. I really love your blog!

      1. Do you have any other recommendations for the holder for the 10 frames? I saw baseball card holder but only found 3 by 3.

        1. You could put each set in a plastic travel soap holder, and the kiddos could sort them into piles before using. Or you could just double up in a couple of the baseball card sleeves. Like putting the 0s and 10s into one sleeve together.

  7. Love this idea. Since I teach 4th grade, I would have to make these up through 100,000. Any suggestions on how to do that, or should going to 100 be all I need.


  8. Spent my Saturday evening making seven Base Ten sets for my second graders, who are struggling with skip counting. So, I’m going back to place value! For now I have their sets in baggies, but I am planning to put them in baseball card holders. Can’t wait to use them! Thank you for your generosity.

      1. My kids were so grateful that I made these for them! And I was able to get great data on where they are with place value. We will be using these on a regular basis! Thanks!

  9. Thank you so much for all of your wonderful blog posts and materials! I am currently a math interventionist in La Porte, TX and I am loving all the ideas that you are sharing. This is my first year to work in this position and I am grateful for your willingness to share and help others!

  10. I am working on place value with my intervention group too. I asked them to make 25 the other days using ones but they couldn’t count by ones they needed to find another way. I was hoping that they would put their ones into groups of five then push them together to make 10 to show both number fact knowledge and an understanding of teness. At first they really struggled as they wanted to count by two s but could only count to 12 but after now doing this same process for four days they have begun to see that the tens are not a single unit eg 2 tens 5 ones but still have a way to go.

  11. Hello Donna-
    I love your ideas and was wondering if you offer professional development at schools. Could you please let me know? Thanks so much!

  12. I really enjoyed reading your post about place values. I am a student right now, working towards my teaching degree. Place value and having students understand it was a big topic we covered. I loved they way you worked with the students to figure out that the 4 in 43 stood for 40. It is very important that students know how to represent place value. Knowing place values will better help students understand adding and subtracting as well as multiplying and dividing. Thank you for this post. I will take this with me so I can use it with my future students.

  13. Thank you for the idea of the baseball cards holder! I printed theses ten frames a long time ago and also kept them in small plastic bags. Oraganizing them will certainly make them more user friendly!

    When I have my first graders match amounts to digits, I have them say the tens and ones part, and then we go into our “math language” : two-ten three or twin-ten three to twenty- three ( the -ty) part of the word is short for ten. It still takes a bit for the kids who struggle but I think constantly talking this way helps them bridge the language connection.

  14. I am a high school teacher in Australia, and just thought I’d share the fact that students are still struggling with the concept of place value at year 7 level…our testing comes back that place value is crucial in areas of measurement and number work. Thanks for sharing ideas and strategies.

  15. In making the tens frames for this activity do you have a frame that is empty to represent zero? I am not seeing it in the information that I printed. Thank you for these materials.

    1. No, I didn’t make a blank one. If they are using them to build 2-digit numbers, they wouldn’t really use an empty ten-frame, would they?

      1. That is true. We were copying the picture and thinking about how to incorporate the idea for three digit numbers. Thank you

  16. Your blog is absolutely incredible! Yet another post that shares your real life journey into helping kids develop mathematical understanding. Reading your posts has helped me greatly in my work as a learning support teacher. Just wanted to pass on my gratitude!

  17. Your blog and activities are very helpful. Thank you. A few weeks ago I came across an activity you created where the students built numbers. For example, build 43 with __ tens and 23 ones. It was very helpful but I can’t find it. ? I have searched TPT but i’m obviously doing something wrong. If you can what is it called and where can I find it again? Thank you.

  18. I’m a homeschooling mom and this may not be original (I didn’t read all the comments) but my favorite manipulatives for explaining place value is legos. Every kid has them and they are so easy to explain ones, build into ten frames by stacking ten legos together, etc.

  19. I had a second grade colleague ask me in what order should she teach Place Value and Addition (2-digit). To her, Place Value should come first then Addition, but she could not tell me why. Her math basal teaches Addition first then Place Value. How do we figure this out? This is something I struggle with in Kindergarten as well. We use My Math.

    1. Place value should definitely be taught before multi-digit addition or subtraction, because an understanding of place value is necessary to understanding regrouping.

  20. Yea! another great site! Didn’t find the answer I was looking for but your explanation of place value will work just as well. Thank you for broadening the old tutor’s horizon!

    1. I personally would not use dimes when teaching littles place value. When they start working with decimals dimes represent tenths, a fraction of a whole. There is a lot of confusion that occurs when trying to understand less than one. I always used $1, $10 and $100 bills when exploring place value with 1st -3rd.

      1. I see this as similar to using base ten blocks for while number place value and then redefining their values and using them for decimals. I think it actually deepens students’ understanding of place value.

  21. Two things that I think are adult constructs that don’t automatically help students are place value charts and fraction bars. We assume they do because they seem logical model to us adults. However, generally speaking they make sense to us because we already have an understanding of these concepts. It still surprise me how many fifth graders still sometimes put two digit numbers in one place.

    1. With littles I like to practice tens and hundreds with dollars and dimes where kids can see that 10 ones actually turns into a completely different coin and 10 dimes turns into a note. I feel like this helps them visualise the separate entities.

    2. What I have found out is there is no place value without a place! I have seen kids put tile of 8 -10 tiles on a rug and we expect them to visualize what that means. All it means is there is 8 tiles with the number 10 on it and it turns into a counting to 10 activity. So until kids are abstract thinkers to learn place value they need base ten blocks or “Montessori Golden beads” so they can see what the numbers really mean. 143 can be 143 ones or 14tens and 3 ones…etc…

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