October 5, 2013

Dividing Large Numbers - A Learning Progression

I am a huge fan of the C-R-A Learning Progression for mathematics.

C = Concrete. Use materials to focus on the development of conceptual understanding, while starting to make connections to procedures. During this stage, students might work with base ten blocks, fraction bars, red and yellow chips, fake money, tiles, cubes, etc...

R = Representational. Connect the previous work with concrete materials to other representations, especially drawings. Students think more deeply about both concepts and procedures. They might use circles, tallies, rectangles, drawings, etc...

A = Abstract. Use previous work with materials and drawings to make sense of procedures with only numbers and symbols.

This two minute video can give you some more information:

In the 5th grade Common Core State Standards, students use place-value based strategies and other representations to divide numbers up to 4 digits by 2 digits. Over the last two weeks my students have been working through the Concrete-Representational-Abstract Learning Progression for division. They are rocking it!

Here's some of their work:

Show 15 divided by 5. Some students made 5 groups of 3 (as shown below), others made 3 groups of 5, and still others made a 5x3 array. These representations really helped us discuss the meaning of division and just how exactly it is related to multiplication.

Show 78 divided by 3. For this problem we used base ten blocks. The idea was for them to work on breaking up the 78 into pieces that could be easily split among 3 groups. There were 2 main strategies used by my students for this problem. In the example below, the students distribute 20 to each person and then noticed that 18 were left over. They used 6x3=18 to determine that each person needed just 6 more cubes.

We connected division with base ten blocks to a representation using circles. The image below shows one student's work on 672 divided by 5. We call this method Circle Division. I love it because each student can choose whatever numbers make the most sense to them, with an eye on efficient steps and clear record keeping. They can also really see what it means to divide a number into parts.

To transition to an abstract representation we simply solved a problem using circle division, then re-solved the problem using the strategy below, Partial Quotients. In this way, the students could see how the two methods were related and what the numbers meant in this format. We refine this strategy by focusing on which values make the most sense as we divide.

Using the C-R-A Learning Progression, I find that my students have a stronger understanding of both the concepts and procedures of division. In addition, they tend to improve their number sense through this process and, therefore, are better able to make intuitive decisions about the values they work with as they divide.

Happy Dividing!
~ Amanda


  1. Thanks for this great post. I just started division today with my kids and they are struggling! I'm going to implement these strategies this week.

  2. Hi amanda! I was at the Meredith math institute workshop and I loved it. I was the one with the -_- face LOL. I can't wait to start school next week and use the strategies you shared

    1. I Katy! Thanks for visiting my blog. I really enjoyed working with you (and your face) at MMSI. Keep me posted on how your year is going. (We start tomorrow).