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...
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!