“We just finished teaching adding and subtracting integers to the 7th graders and we had a quiz today.  I got wonderful explanations from the kids.  For example, one of my students said that -19 – (-34) = 15 because: “You are short 15 negatives, so you add  15 negatives.  To keep the same net value, you add 15 positives too.  When you finish -34 – (-34) = 0.  You have 15 positives left over, so the answer is positive 15.

We are moving onto multiplication and division on Monday and the other teacher and I want to use the concept of multiplication as grouping as the jumping off point.  This model seems to break down, though, when you have a problem like 6 / -2.  If you read it as 6 things broken into the opposite of two groups it can work.  The two groups would have 3 positives each and the opposite of this would be 3 negatives per group.  However, we cannot figure out a way to describe it if it is read as 6 things broken into the opposite of groups of 2.  There is no such thing as the opposite of groups of 2 — whether the things are positive or negative there are still 2 things per group.  Apparently, last year they got around this fact by having the students rewrite division problems as multiplication — 6 / -2 became ____ x -2 = 6.  However, we would like to be consistent in our explanations if we can.”

How great that you are trying to make sense of all of this in some way or another. I struggle with this myself, surprisingly often, but your wording has given me an idea. You wrote, “ If you read it as 6 things broken into the opposite of two groups it can work.”

What if you tweak that to say, “6 things broken into two groups of opposites?”

So if you break 6 into 2 groups, then you get 3 in each group, but it’s of opposites, so you get 3 opposites or -3?

Try it out a couple of times and tell me how it feels. I have never used this terminology before, so no promises on my part.