I was just having a conversation with two sixth grade teachers this morning and the topic of long division came up. We had been talking about multiplication and how it would help children if we said, “groups of” instead of “times” when discussing multiplication. I made the comment that the same could be said for division, since multiplication and division are the inverse of each other. One of the teachers told me that she found it difficult to know exactly how to phrase a division problem when talking to her class. She wondered, in the problem 108 divided by 8 if she should say, “108 divided into 8 groups” or “108 divided into groups of 8″.
I think the confusion about division comes from the fact that there truly are two types of division but you don’t realize it in the “naked number” problem 108 divided by 8. It becomes apparent however, when we put the division into context. For example: We could be saying that 108 people will be divided onto 8 teams, which is a division partitive problem. When given this problem a child who is a direct modeler will make 8 groups and fair share out the 108 people onto those teams. In that instance we have divided the 108 people into 8 groups.
However, if the problem was 108 cookies were being distributed to people in packages containing 8 cookies and we were trying to determine how many people we could feed, then we have a division measurement problem. In that case, a child who is a direct modeler would take the 108 cookies and break them into groups of 8 and count the groups. Either way, you would arrive at the same answer, but the actions would be different.
The type of division problem that can be used to help students with base ten understanding is the division measurement problem. When the divisor is ten, or a power of ten, students can remove groups of ten from the dividend.
Just “Sum”thing to think about…