Composing and Decomposing Numbers

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

As a young girl, I briefly took gymnastics lessons. Anybody else? I have to admit that I was not very flexible—I couldn’t do a smooth backward roll to save my life! That said, one of my favorite Olympic events is the gymnastics competition. Every four years, I’m blown away by the strength and flexibility of these athletes. It is truly awe-inspiring.

So what does that have to do with math you might ask. I’d say a lot. Think about the ultimate goal of math instruction—students with strong skills and flexibility with numbers. One of the key factors in reaching that goal is helping our students learn to compose and decompose numbers. It’s a skill that starts in Kindergarten and follows students throughout their mathematical career, and it’s something that should be present in your classroom on a daily basis.  

Composing and decomposing numbers became part of our instructional vocabulary after the Common Core State Standards for Mathematics (CCSSM) were adopted in 2010. Consider the following examples:

A Kindergarten student uses number bracelets to find all the combinations for a given number. This is actually the foundation for learning basic facts, but it goes beyond that. Say the student is now presented larger numbers, like 8 + 5. They know that a combination for 5 is 2 and 3. They also know that 8 and 2 makes 10. They can now solve it by decomposing the 5 into 2 and 3, adding the 2 to the 8 to make 10, and then adding the remaining 3 to get 13. Now that’s mental gymnastics. And trust me, the time it takes a child to do this is a lot less than the time it takes me to explain it!

decomposing numbers

In a 1st-grade class, a student uses their knowledge of number combinations to solve 8 = ? + 3, modeling the problem with counters.


A 2nd-grade class is working on place value and writing numbers in expanded form (135 = 100 + 30 + 5). We don’t typically refer to it as decomposing the number, but that’s really what it is. Check out this post I wrote about the importance of showing students that numbers can be decomposed in more than one way and grab a little freebie. Students need to understand that 135 absolutely can be represented as 100 + 30 + 5, but it can also be decomposed into 130 + 5 or 100 + 20 + 15. That’s the basis for understanding subtraction with regrouping.

decomposing numbers

A 3rd-grade class is working on multiplication facts. A student is trying to master the fact 6 x 7. They realize they can decompose the 6 into 3 + 3 and then knows that 6 x 7 is just twice 3 x 7 (a fact they happen to know).

decomposing numbers

In the 4th grade classroom down the hall, students use an area model to solve 12 x 13, leading to a deeper understanding of the multiplication process.


And in yet another classroom, a 5th-grade student decomposes 11/7 into 7/7 and 4/7 to change it to the mixed number 1 4/7.

decomposing numbers

Strong, flexible mathematicians. Go for the gold!

17 Comments

  1. Amy B

    I will definitely be using number bracelets in my class this year! Such a great lesson for the little ones!
    Amy

    Reply
    • Donna Boucher

      Yes, and the kids love them. Get a class set made this summer!

      Reply
  2. Anonymous

    I feel that I have benefited greatly from your posts. You have awesome ideas and I am so excited to implement many of them in my classroom this year! Thanks

    Reply
    • Donna Boucher

      You’re so welcome! Don’t you love the anticipation of a new school year?

      Reply
  3. Lisa R.

    I definitely agree with the importance of composing & decomposing numbers!! There’s always a few that have a lot of difficulty with this & it’s so important to provide them with many opportunities to practice. Great post! 🙂
    Lisa
    Learning Is Something to Treasure

    Reply
    • Donna Boucher

      Thanks, Lisa. And when we see kiddos in the upper elementary grades who have been struggling their whole elementary career, it’s probably because they don’t have that foundation. 🙁

      Reply
  4. Nikki Sabiston

    I spend tons of time in my first grade classroom on composing and decomposing numbers. That deep understanding of number relationships really is a foundational building block! (I just posted about this a few days ago, too!)
    I love your blog and have gotten so many wonderful ideas and resources from you. Thank you so much for getting the word out about the importance of number sense!

    ~Nikki
    Teaching in Progress

    Reply
    • Donna Boucher

      I just checked out your blog, Nikki, and that was a GREAT post!

      Here’s the link for others who might like to visit: What’s Your Magic Number?

      Reply
  5. Tina Moricz

    Have you tried Ken-Ken with your students? It also forces students to decompose/compose numbers in order to solve the puzzle and figure out which numbers go together. It takes a lot of modeling to get them started, but my students really enjoy it.

    Reply
    • Donna Boucher

      Ooh, very Sudoku-ish. I just gave it a try, and I like it! What grade do you use it with?

      Reply
  6. Anonymous

    thank you for your wonderful ideas. Last year I have started to compose and decompose using the 4 corners. This year I am going to use these braceletes.

    Reply
  7. Anonymous

    Your blog is excellent. It is helping me teach my math students in a more hands-on way. Thank you so much. If you can recommend any resources for teaching fifth-graders how to decompose numbers and multiply, it would be appreciated.

    Reply
  8. Mary Anne

    I’ve used SO many of your ideas so far this year (2 weeks) I made rekenreks, I played several games to make ten, I made the pipe cleaner circles with the beads, etc. I have SO MUCH stuff and the kids love using them… thank you! Much more fun than the worksheets my teammates are using to meet the same objectives.
    🙂

    Reply
    • Donna Boucher

      Mary Anne,
      I am so happy to hear that both you and your kiddos are enjoying math so much! It won’t be long before your teammates will be knocking on your door for ideas. 🙂

      Reply
  9. Fran Long

    We use Math Expressions in our district and our parents are totally puzzled by how we teach multidigit multiplication and “long” division. We teach using the decomposing method and all our kiddos get it!! Now we just have to retrain the parents!!

    Reply
    • Donna Boucher

      Absolutely, Fran! That’s often the biggest challenge. Ha ha. 🙂

      Reply

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