# Place Value–Building Meaning Behind the Numbers

Place value is such a fundamental concept, and it’s very difficult for kiddos to understand.

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Let’s think about the typical place value progression:

Understanding the meaning of numbers from 11-19. Double ten-frames are a great tool, as students see, for example, that 12 is a full ten-frame and 2 more. The ten-frames become a very visual representation of tens and ones.*Kindergarten:*Students explore 2-digit numbers and understand that the two digits represent quantities of tens and ones. Beginning with the ten-frames provides a familiar context for students. Full ten-frames represent the tens and “leftovers” represent the ones. Move from ten-frames to groupable objects. Early experiences with groupable objects might include taking a scoop of dried beans and putting the beans by groups of ten into small paper cups. Traditional base-10 blocks are too abstract for this age, but linking cubes joined into trains of ten are a great bridge toward the base-10 blocks, because they look similar. The advantage of the linking cubes is that they have to be physically joined to make a ten train, and they can be broken apart (Van de Walle).*1st Grade:*Begin with the linking cubes and transition to traditional base-10 blocks. Have students practice skip counting by tens and then counting on by ones. With a number built in front of them, have students explore hundred chart patterns (10 more than, 10 less than, 1 more than, 1 less than). As hundreds are introduced, practice the same patterns (100 more than, etc.) so students start to understand the patterns in our place value system. This is also an appropriate time to introduce the idea that numbers can be decomposed in more than one way. For example, thirty-four can be 3 tens and 4 ones, but it can also be 2 tens and 14 ones. This is the fundamental understanding behind subtraction with regrouping.*2nd Grade:*STILL building numbers with base-10 blocks! Now students are understanding that the ones, tens, hundreds pattern repeats in each group of three numbers as they work with numbers to the hundred thousands.*3rd Grade:*At this point, it becomes difficult to use manipulatives to represent numbers, as students are working with numbers to the millions. Manipulatives do come into play, however, as students explore decimals. Using the same base-10 blocks they used to model whole numbers to now model decimals sends a powerful statement about the patterns in our place value system. Place value discs can also be used to model larger whole numbers and fractions.*4th Grade:*Students should now really be able to use the patterns in the place value system to read any whole number, and they should be well on their way to the same understanding for decimals.*5th Grade:*

I just want to thank you for your great blog. I really appreciate all your helps, suggestions, and advice on teaching math. I have a math endorsement and love to teach math, but every post you write give me a shot in the arm to get in and do a little bit better. Thank you again for help me be a better teacher.

What a sweet comment! Thank you so much for taking the time to do that. ๐

Amen! Love, love, love this blog. I’ve recommended it to many of my colleagues!

Thanks so much!

Donna, I love your blog! I regularly use place value blocks in my third grade classroom. I had a student ask me this question… “Why does the ten have to take 2 places..who decided it had to be a 1 and a 0… I get it that it’s a ten, and all, but why? WHO SAID?” We talked and she is really trying to figure out why the 10 is the “magic number” where everything changes… she gets the ones turn into a ten, tens turn into a hundred,etc… but she just wants to know why it’s the ten that makes the difference…

Any words of wisdom to help her understand this would be appreciated.

What an inquisitive mind your student has! I did a little Googling on the subject, and I found this little video that briefly explains the origin of our number system:

Who Invented Our Number System?. It might be fun for her to explore number systems.Thank you! I get your newsletter and this is EXACTLY what I needed to find out…today. I am going to revise my math lessons for next week for my buddies who don’t quite understand place value yet. Thank you for your help!

Thank you! I get your newsletter and this is EXACTLY what I needed to find out…today. I am going to revise my math lessons for next week for my buddies who don’t quite understand place value yet. Thank you for your help!

Oh, it makes my day that this was just what you needed. ๐

Thank you so much for all the great information and it’s very reassuring to me that it’s still OK to continue using the manipulatives in 3rd grade. Sue

You are absolutely right! It’s not just okay, but necessary!! Keep on keepin’ on. ๐

I love the Van de Walle book. I’m getting much better at starting with the ten frames in first grade and then moving towards combining cubes into trains of ten. It makes sense that it’s easier for the young ones to see a group of ten in a frame first. When they look at a train of ten cubes, they can’t easily tell that’s ten. It makes so much sense! Thanks for sharing.

❀ Tammy

Forever in FirstWhat should be the expectation for place value in grade 6? Or is is expected that students have a solid understanding?

It really is expected that 6th grade students have a solid foundation in place value. The progression described in this post, however, could be used to remediate students who don’t. It will help you figure out just how far back to go.

I love your blog and your products. I love how you are pushing the concrete learning. Love it. thanks so much and keep it coming.

I’m relentless when it comes to CRA! Ha ha. ๐

Thanks for all the ideas and freebies you provide for us! It helps thousands of kids across the world!

What a cool comment, Judy! Thanks so much. I love sharing, and I’m glad it benefits others.

Hi Donna,

I am from Melbourne, Australia and am a keen follower of your blog! In 2014 I completed my PhD in whole number place value. I, like you, believe that our students need a much deeper understanding of place value than many currently have. My research showed that there are students whom I described as โapparent expertsโ. They appear to understand place value, i.e: they can identify the value of the columns, read, write and order numbers, but when we ask them to rename numbers or compose or decompose numbers in a non-standard way they flounder. My research has led me to identify 6 key aspects of place value that teachers must focus on to help children โunderstandโ not just โknowโ place value (as you mentioned in another one of your great posts).

Throughout my research I have looked for quality resources to teach place value. I struggled to find any, so I vowed to make one myself when I finished my thesis! Over the past 8 months I have created an app for students ages 7-12. The app is called โZero Our Heroโ and is $2.99(AUD). I have self-funded the entire project to create my app. I believe โZero Our Heroโ will show teachers the depth and type of questions they need to be asking students in place value. The app presents questions at 4 stages of difficulty- these are based on the place value developmental progression (PVDP) I identified empirically in my research.

I write to you in the hope you might be interested in trying my app and if you see value in it recommending it on to any teachers who may be interested. I understand you are very busy and must get lots of requests for reviews of apps, but i strongly believe my research-based, teacher-designed app is a resource teachers and students around the world would benefit form using in their classroom.

If you are interested more info can be found on my website at zeroourhero.com, or if you would like to hear more about my research I would be very happy to discuss this with you further.

Many thanks for your time and keep up the great work with your blog!

Kind Regards,

Dr Angela Rogers.

Thank you!! so many teachers do not use base-ten blocks. There seems to be a lack of understanding about how children think at each age. Do you have research/information on the brain skills by age ?

Yhanks

Penny

Not brain skills, but Kathy Richardson’s book How Children Learn Number Concepts is a great resource for how mathematical thinking develops.

Thank you for always providing such great resources and explanations. Every

time I read your blog I learn something new!

Donna, PLEASE help settle a discussion. When are place value blocks “officially” introduced?

As manipulatives for place value go, they are actually quite abstract. In Kinder and 1st, it’s better to use groupable materials, like linking cubes. That way, students can easily compose and decompose the tens. I’d say either late 1st or 2nd.