I am always looking for better ways for the students to expand their understanding of multiplication and division, and be able to transfer what they know about operating with whole numbers to fractions. Along the lines of Tina Cardone's book Nix the Tricks, I have always hated the phrase "of means multiplication" because it doesn't always, as in "3 pounds of strawberries and 2 pounds of bananas." But it suddenly occurred to me that "groups of " does often mean multiplication, and that would be worth sharing with my students. Why does this matter?
- 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?
- 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?
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