Long division is stressful, both for students and teachers. There are so many steps and it’s hard to see how they make sense, especially when you are first learning it. Learning how to divide is a real leap of faith.
I’ve been lucky to test out a new model of long division with teacher Karin Allison and her grade 3/4 class. We started with counting collections, which worked well for developing an understanding of what division is (Curious about this? Click here and here). Now we are looking at working on a more abstract level with a model proposed by Carol Fullerton. Keep in mind that we have tweaked the task language she used to include a concept that really made sense to the kids: friendly numbers.
Friendly numbers are very useful for adding, subtracting, and rounding. If students have had a good grounding in the concept of friendly numbers at primary, you can leverage their understanding to introduce a new idea: friendly dividends. What are they? Basically, they are numbers that are easy to split using mental math. The nice thing is, kids get to decide what numbers are the friendliest.
Take the number 72. If you wanted to divide this by two, it is a little tricky to do in your head. But how about 60? Is that friendlier? If you think so…split it. Record the answer and now think about 72 again. How many are left to split? 12? Is that friendly number to split? If you think so, split it. If you don’t, choose another number. Keep going until you can’t split what is left.
Karin and I tested a game to help the students practice a model for recording their thinking. Partners draw two cards from a deck and multiply them together to create a dividend. Partner 1 records this number at the top of one white board and draws a dividing line along the top and down the right side of the board. The job of partner 1 is ask, “Do we need a friendlier number to start?” If so, they choose one and partner 1 writes it down. Partner 2 has a separate white board, records the quotient and asks, “Is there anything left?”
Back to partner 1! This partner now subtracts to figure out what might be left and asks, “Do we need a friendlier number or can we split what is left?” On the cycle goes with partner 2 recording quotients and partner 1 figuring out what is left and inquiring about how they might split it.
What did we find out?
- Kids need lots of practice recording and it was a VERY GOOD IDEA to separate recording tasks between partners. After round 1 of the game, it was clear to see that students continue to need lots of practice.
- Kids intuitively determine friendly dividends when dividing by 2. I highly recommend that everyone who tries this start by dividing by two.
- Drawing cards and multiplying them to find a starting dividend was a bit problematic. Some were just too easy to split. We figured that it might be better if the two cards represent the tens and the ones. Take 10 and face cards out! You could also create a list of dividends that partners take turns choosing to divide.
- Instead of whiteboards, think it would be handy to have a black line master to guide recording and questioning. These can be placed inside of clear plastic pockets and used with whiteboard markers.
What’s next?
More practice of the recording method, for sure. Why? I find that without it, kids get mixed up between the friendly numbers and the quotients. I regularly have kids dividing the number down to nothing and completely losing track of the quotient. Once the kids have recording down pat, then we can choose new divisors. What dividends might be friendly to 4 or 5 or 8? These will be trickier and will benefit from multiplication fact fluency. Plus, we should introduce the terms dividend, divisor and quotient. I shied away from it today (concerned that I might overwhelm them with too many new ideas), but I think it is important to communicate clearly about the different numbers they are working with.