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!