Thinking Flexibly About Place Value

New standards call for an increased emphasis on understanding place value, rather than just, for example, identifying the digit in a certain place value position (eg., In the number 345, what digit is in the hundreds place?).  In Texas, both the 1st and 2nd-grade TEKS contain wording that indicates students should be thinking flexibly about place value.

1.2.B Use concrete and pictorial models to compose and decompose numbers up to 120 in more than one way as so many hundreds, so many tens, and so many ones

2.2.A Use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones

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It’s that phrase in more than one way that sets the current standards apart from previous ones. Traditionally, we have only taught decomposing by place value in one way–the way the digits are written. So, for example, 43 would be always be decomposed as 4 tens and 3 ones. The new standards are saying that students should understand that the same number can be decomposed into 3 tens and 13 ones, or 2 tens and 23 ones.

Why is this important? Look at the second example pictured above. Isn’t that what we do when we regroup to subtract? Think about subtracting 43 – 18. Because I don’t have enough ones to subtract, I need to trade one of my tens for 10 ones, which gives me 3 tens and 13 ones.

Tools for thinking flexibly about place value

Notice that both the 1st and 2nd grade standards stress concrete and pictorial models. It is absolutely essential that students experience place value concepts with hands-on materials to fully understand that a ten and 10 ones represent the same value. My firsties are using linking cubes to compose 2-digit numbers in multiple ways. Linking cubes are the most concrete material to use for place value, because you can physically break a ten apart into 10 ones. While still concrete, base-10 blocks are more abstract, because the rod, which represents a ten when working with whole numbers, can’t be broken into 10 ones. Place value disks are even more abstract, but they allow us to work with larger numbers. 

Ready to give it a try? I’ve created some place value puzzle cards that are perfect for your math workstations!

And check out this post for a free game you can download for additional practice.

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  1. Love it. I did this with third graders and called them Place Value Mansions. They had to build their “mansion” with base ten blocks in at least 2 ways. 🙂

  2. I’m a 3rd grade teacher too and I love to see how students have this flexible view of numbers built in 1st and 2nd! I can’t wait to use the place value puzzle cards with my kiddos 🙂 Thank you!

  3. I am the grandfather of a 10 yo fouth grader. She is seriously adrift in her math studies. I am attempting to do remedial work with her but it is hard going since I have never been that good at math. Is there an affordable book that would explain what they are teaching her?8

  4. I was also unable to download the Building Flexibility with Place Value resource. The link does not take me anywhere. I would love to utilize this with students. This is something we’ve been working on quite a bit and would love to continue to reinforce.

    1. I’m at a loss to explain why some folks are having trouble downloading the file. I linked the picture below as well–try that. When I get some time in the next couple of days, I’ll add it as a freebie on my Teachers Pay Teachers store, and you’ll be able to download it from there!

  5. Hi Donna,

    I found base ten blocks from the schoolspecialty.com website that break apart. I use those with my small groups to them how break apart hundreds, tens and ones.

      1. Digi blocks are also great for this as the blocks nest within each other which make them great for showing composing and decomposing numbers.

  6. Hi Donna! After reading this (terrific) post, I’m in a quandary regarding what to do with some of my fifth graders… They are learning the fifth grade “concepts” fairly well, but their “primary” number sense is sketchy at best (which, as you can guess, is making everything so much more difficult). Would it be just crazy to try and go back and teach them ten frames and number bonds? (We have a LOT of finger counting going on…) I can never seem to gauge how far back to reach before going forward. Any thoughts or ideas to guide my next steps would be appreciated…

  7. Thank you.. my kiddos have such a hard time with this concept. They love task cards so I think they will love this..

  8. Love this! Would you consider making a similar freebie (for us middle school teacher) that was almost identical, but with decimal place value? ie. 3.04 is the same as 30 tenths and 4 hundredths? 🙂

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