I've been working a lot on division lately. Long division with my son in 4th grade, with his classmates in small groups and with my preservice teachers.

Xavier's teacher sent home a note before the unit, showing the four ways they would be approaching it. From the traditional algorithm to the "forgiving algorithm" and a couple in between steps. My wife couldn't make too much of the note, despite being bright, living with more math than anyone would think is reasonable, and being comfortable with her own computation. Don't know what it was like in other homes.

Xavier was making pretty good sense of it. The curriculum sensibly starts out with dividing by 5 only, which was a nice touch. I think I might even start with 10s, then 5s, having seen how well this worked. He transferred this pretty well to working with other single digit numbers. For example, 61/4. When working, he used appropriate language and responded to questions like "how many fours go in 61?"

I did the soda machine problem (from Contexts for Learning, my favorite curriculum,

Exploring Soda Machines: a context for division) with a small group the week before, and they did amazing work. In the problem, you describe a pop machine (in Michigan, soda=pop) which has 6 flavors. When full, it has 156 cans of pop. How many cans of each flavor, if there's the same of each? But wait, that's a lot of pop. When I buy pop it's usually in a six pack. How many six packs will that be?

Students do the most amazing work wth this problem. Most choose to tackle the 6 flavors question first, and drew their own pop machines. Six columns, and fill in pop cans 1 per column until they have 156. 156 is an inspired amount - not so much as to be overwhelming, but enough so that they usually have to start some kind of record keeping to keep track, often multiples of 6 as they add cans. Sometimes drawing a line, with the total amount under that line. Some students draw neat stacks of cans in a row, and some draw crazy piles of circles of many sizes.

Not very many students saw the connection with the two problems. Only one realized that both were 156/6. Others started to think about it when they realized the answer to both questions was 26, and then saw a connection. Many used their pictures of the flavors and started circling groups of 6. Which was interesting, becuase then there are 2 left over of each flavor. Some had 24 six packs with 12 left over, and some had 26 six packs. (Talking they agreed on 26.) The student who made the division connection shared, but the other students didn't really seem to hear her.

The formal language for what's going on here is that there are two different division actions. When students can solve some division stoies but not others, sometimes this is the underlying cause. (

Quotative and partitive division, from learner.org, with kid video, too.) When explaining this to our preservice teachers, we often use the terms fair share (how many of each flavor) and measure (how many six packs) for the different actions.

The next week I wanted to bring another context. I brought a bunch of play money that I wanted to sort into 6 bags for my preservice teachers. We counted up the 74 quarters together by making stacks of 10. I wanted to have enough that they were counting up in 10s, so that when we were dividing we'd see the benefit of the 10s.

Do we have enough for 10 in each bag? "Yes." How much left? "14." (I kept the notes on the left. 10x6=60) How much more can we put in each bag? 10? "No!!" "Two," one of the kids suggested. People agreed, so in went 2 each. 2x6=12. 2 left, not enough for even one more in each bag. "That's the remainder." Excellent!

We then divvied up the rest of the coins. We had the most pennies, so I asked for a volunteer team. Then the next team chose dimes over nickels. 96 nickels, it turned out. But then they lumped all of their neat stacks of 10 together again! 149 dimes. And 491 pennies. (One of the kids even noticed the anagram.) The nickel team was done pretty quickly, 10 in each, then 5 in each and 6 left over. "Oh, that's enough for one more in each bag." The dime team did 10 each, then 10 again. 5 each... not quite. Get one out of each of the five bags with 5. Remainder 5. The 491 team did multiple rounds of 10 each. Then just kind of scooped that last 11 into a bag. All the bags had lots of pennies by that point. Then each team made a number record like on the left, and nobody saw the connection with the division algorithm they have been doing.

Wow!

Okay, that's immediately a preservice teacher activity. At GVSU we're blessed with a goodly pile of manipulatives. So each table got 6 tubs of blocks (2 each with unifix cubes, wooden cubes, and snap cubes.) 5 (or so) minutes to play with them, because

**play is important**. Of course they made many mathematical designs and structures.

Then it was time for the task:

1) How many of the object did you get?

2) Physically divide them up into the 6 tubs evenly. How did you do it? How many in each tub?

3) Show with a number record what you did.

4) Use a sense-making method to do the associated division problem. How would what you did make sense as physically dividing the objects? Why does your method work?

_________________________________________

1) ___________ blocks in each tub.

2) Description of method:

3) Number record:

4) Sensible division problem:

And then to make a poster of the connections between their number record and their division work. Here's what they did! (Click on the images for full scale.)

And the piece of least resistance:

Those are some beautiful connections!