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:

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

**Phase 2:**Model with a vertical tape diagram:

More guided practice:

**Phase 3**: Let's get more symbolic:

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. |

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. |

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!

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:-)

ReplyDeleteA 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!

Graham

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!

DeleteThat'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?

ReplyDeleteIn 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?

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!

ReplyDeleteThanks 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?

DeleteBut 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?

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

Delete"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

DeleteI 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.

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.

ReplyDeleteThanks for your positive words and for taking the time to comment.

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.

ReplyDeleteThanks 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.

ReplyDeleteThank 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?

ReplyDeleteThat'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.

Delete