Tuesday, April 1, 2014

Trying to Make Some Sense Out of Long Division

Raise your hand if you look forward to teaching long division.  For me, anticipating long division time in grade 4 has always left me with a sinking feeling.  Jeff, Shannon, and I have tried many different strategies, sequences, and procedures, and though most kids ultimately learn how to divide using the traditional algorithm, too many seem to struggle.  So this year we decided to try something new.
  I was inspired by  Nicora Placa's post on tape diagrams.  It seemed like a natural fit, because the kids were already comfortable using the diagrams to solve "fraction of"problems, which I've blogged about here.  I envisioned the move from a partial quotients algorithm to a traditional algorithm unfolding in phases.  So we start with a division problem:

Phase 1: Model with a horizontal tape diagram:

The decision I made here was to continue to strike out the number in the total box (dividend) until only a remainder was left.  The subtraction work was kept on the side.  The only difference between this and the "fraction of" diagram we used was that the fraction 1/4 did not appear in each box.  I think this nicely illustrates how the 157 is being gradually and equally distributed.

We did some guided practice,then it was on to...

Phase 2: Model with a vertical tape diagram:

Why rotate it?  Ostensibly because we now have the ability to keep all the subtraction work inside the box (this is what the kids were told).  But really it's because I wanted a way to position the dividend to the right of the divisor.  And by the way, let's now record the amount in each box (quotient) on the top.

More guided practice:

Phase 3: Let's get more symbolic:

The 4 now stands in for the four boxes.  We need to imagine those.  We don't have to write the partial quotients 30 and 9 over and over again.  They can go on top.  I wanted to reinforce the placement of the divisor to the left of the dividend and the quotient on top of the dividend.

More guided practice:

We told the kids to "draw the rest of the box".  Now the tableau makes sense.  The division bracket is just what's left of the vertical box that houses the dividend.

Phase 4: Traditional algorithm

Here's what's changed: We remove most of the horizontal box.  Instead of writing the quotient as 30 + 9, we record it as a 3 in the tens place and a 9 in the ones place.

Jeff did a great job keeping track of which students were successful and comfortable with which method and assigned some differentiated homework.  We grouped them up all week, moving kids around depending on which method they were comfortable working with.

This was the page you received if you still liked to draw the tape diagram...

...and this one was for the kids who had moved to the next stage.

This was for the kids comfortable with the traditional long division algorithm.
Friday was assessment day.  I was curious to see what the kids would do.  Here's what I found:

11 kids used the horizontal tape diagram with the divisor boxes and partial quotients (phase 2)
4 kids used the horizontal tape diagram without the divisor boxes & with partial quotients (phase 3)
18 kids used the traditional long division algorithm
4 kids used alternative methods from home

Not all of it was completely accurate, but Shannon, Jeff, and I all agree that we are much farther along with getting them all comfortable with the traditional division algorithm than we have ever been before.
Here's what I liked about using tape diagrams:
  • The traditional long division tableau now makes sense because it comes from somewhere.
  • Beginning with a tape diagram reinforces the notion of division as equal sharing.
  • Not necessary to remember an acronym "DMBS" (divide, multiply, subtract, bring down).
Here are some issues:
  • The tape diagram does not reinforce the notion of division as equal grouping.  We thought quite a bit about that.  But we felt that since the diagram was going to lead into the traditional algorithm, and that was what would be used to solve division problems, we would need to set  that aside.
  • The diagram gets more problematic for larger dividends, but that is what drives the need for the algorithm.
For our first time through, I think I'm pretty happy.  Thoughts and suggestions are encouraged and appreciated!


  1. Absolutely Joe!!!! Division is a monster! Although CCSS doesn't specifically address the traditional algorithm of division until 6th grade...this ugly beast continues to rear its head in the 3-5 classroom. The problem is that it is difficult to conceptually have the students mimic what's happening in the traditional algorithm (within a context) which leads to DMBS. I came across this from John Van de Walle last week and tried it with a 4th grade class on Monday! The context served the math beautifully...You're one post ahead of me here:-)

    A school is participating in a doughnut fundraiser and the shipments have been delivered.
    • Each pallet has 10 cartons
    • Each carton has 10 boxes
    • Each box has 10 donuts
    The shipment consists of 4 pallets, 6 cartons, 4 boxes, and 8 individually wrapped doughnuts. If the doughnuts needed to be shared among the 5 classes, what is the most EFFICIENT way they could share the doughnuts between the classes?

    This task asks that students disseminate the doughnuts the most efficient way possible. The most efficient way to share the doughnuts is to keep as many pallets, cartons and boxes unopened as possible. The underlying mathematics of this task mimics the division algorithm and this context makes the understanding of "procedures" accessible to students.

    Give it a try! Love to see how it goes for you and your peeps!

    1. This task is one that actually appears in the 6th grade frameworks for Georgia ("School Fundraiser") and the progression towards the traditional algorithm is also in our frameworks tasks in 6th grade ("Scaffolding Division through Strip Model Diagramming")! It all looks extremely familiar to me!

  2. That's a great problem (although we'd certainly need some background knowledge regarding pallets). I'm looking forward to reading about it. Did your kids have any visuals or manipulatives to help?
    In the past we would've used base 10 blocks, and done all the requisite regrouping involved. This was painstaking and took lots of time, but did build good understanding. What it didn't do was explain how it all fit in the traditional long division tableau.
    I thought that the traditional algorithm was grade 5, but yes it is grade 6. Maybe it's just me, but I think it's a bit unrealistic to ask kids to handle dividing 4 digit dividends and 2 digit divisors and decimals to hundredths through grade 5 efficiently without a traditional algorithm. The numbers just seem too big. I hate to say this, but I suppose we'll have to wait and see what the PARCC has in store for us division-wise. Maybe we're over-thinking it?

  3. Joe- I think this is so neat. I love how the students progress (and how you keep track of it) from doing what makes sense to them with the diagram to moving toward the traditional algorithm. It's great how you allowed the students to build on what they knew. I would be really interested to hear (or see in writing) their explanations for what they were doing. It sounds like they are developing a deep understanding for why the algorithm works--which I think is an amazing accomplishment! I'm still thinking about your question about how to use the diagram reinforce equal grouping. But, as you said, since that wasn't your goal, maybe it's not as important here. And just so you know, I am loving reading your blog--it's giving me so many ideas!

    1. Thanks Nicora. Asking the kids to explain what it is that they're doing is something I know is important, but do not remember to do often enough. Perhaps it makes more sense to give them fewer problems to do and more time to explain and reflect?

  4. But why after all the "find your own way" do you pass judgement on "short division" as it could be considered a next step in algorithm after long division?

    1. Are you referring to the last example? It was not my intention to judge it. I had never seen division done that way.

    2. "It brought back bad memories for Jeff" was the caption on the last example, so I thought that this is a less preferred way of doing the problem

      I do like how there are multiple ways of dividing and you encourage accuracy, comfort with method, and efficiency of process.

      The last example I've seen referred to as "short division" because it is the traditional long division algorithm but you are only recording the remainder and doing the subtraction mentally.

      Again, this is a very powerful way to teach and your students are very fortunate to experience this.

  5. OK, now I get it! No, that comment was not meant to mean it was a less preferred way, and it was not expressed to the student. Clearly there are some kids ready for "short division", and we may want to explore introducing this to them as the next, natural step in the progression.
    Thanks for your positive words and for taking the time to comment.

  6. I love this! Thank you for sharing! I teach 5th grade and we are supposed to be able to divide with double digit divisors, but they haven't mastered the basics. I will be giving this method a try! Maybe if we can visualize it this way, we can take on double digit divisors.

  7. Thanks for your comments. Yes, double-digit divisors are tough to handle! I'm glad you'll be giving this a try. Let me know how it works with your class.

  8. Thank you for sharing this! I am curious, did all of this progress in one week? Could you share how much time you took from start to finish with the different phases you shared in this post?

    1. That's a really good question. As you can see from the date of the post, it's been a while! But I think this developed over a period of about two weeks. Kids progressed through the various stages at different times. At some point (maybe the following year?) we started with dividing actual base-10 blocks, then moved on.