- We have been looking at linear equations and the different roles numbers play in multiplication. 3 x 4 looks the same as 4 x 3, but 3 groups of 4 is different than 4 groups of 3. Going 65 miles per hour for 4 hours is different than going 4 miles per hour for 65 hours. "Groups of" could be part of our conversation to help distinguish the different roles (or jobs) the numbers play in the problems.
- 2/3 of 5 is where "of" comes up most often, and is usually followed by someone declaring (with a great show of relief) "Wait, doesn't 'of' mean times?" and the algorithm spreads across the room and the thinking stops. And 2/3 x 5 seems so far away from 2 x 5, when they are so close. Partly because no one says "2 of 5," and neither 2/3 of 5 or 2 of 5 has much meaning. But 2 groups of 5 and 2/3 "groups" (we can discuss "of a group") of 5 both give us something we can picture or act out and consider, and allow us to build on our understanding of 'groups of."
- We can carry that on to area (6 rows of 15) ... and I am sure other things will pop up if I commit to using "groups of" or "rows of" etc.
So we can go beyond a trick with words to language that provides an image, a relationship that we can evolve our understanding around. It took me 20 years to figure that one out?