Fair Sharing: Developing Division

 Zoe, my daughter in Kindergarten, was working on a math problem yesterday. She had to share 8 cookies between 2 people. She split each cookie in half. Was she wrong? Technically no. But I asked her, “if you and I were going to share 8 cookies, would you break each cookie in half and share them that way?”

“No, I wouldn’t do that,” she replied.

I pulled out 8 double sided counters (those round little red and yellow chips). “Show me how you would share the 8 cookies between us.”

“One for me. One for you. One for me. One for you...” She continued until she had two piles of 4. “We each get 4 cookies!”

This is a concept known as fair sharing. This is how division begins in early grades. In many cases, children may even learn this before grouping (multiplication), as sharing is more of a pressing notion than making groups. 😁

When time comes to really dig into division later in elementary school, children have had plenty of work with place value and multiplication under their belts, so their first experience may look something similar to the first picture. Can they divide by subtracting “groups” from the number. For 1,375 ÷ 5, they likely know that 5 × 100 = 500 and 5 × 10 = 50, from work with multiplication and place value, so they may start by removing those groups first. Work with money in prior grades will likely bring 5 × 20 = 100 to mind as well. Eventually, when they add up that far right column, they see that 1,375 ÷ 5 = 275. This looks like a real ugly and clunky approach, but it allows them to make sense of the whole idea of division/long division. Again, we don’t want memorization of mindless procedures, we want sense making . This builds toward the long division procedure (and builds mental math in the process).

Eventually we work toward methods like those shown below, more “formal” long division algorithms. Now we are attempting to find the largest possible grouping to subtract out each time. Students will likely already be working toward this as they gain more “finesse” through exploration with the prior methodology (picture 1). I personally prefer the “box method." It takes up less paper and I think it allows for a better visual understanding of the place value aspects of division.