Amazing Graduate Math Ed Class

I have the most amazing graduate math education class this semester. The students just will not let anything go that they don't understand. I have always believed that we should not teach anything we don't understand so this group of students is really keeping me on my toes.

Two weeks ago we were exploring fractions. We first looked at fraction concepts such as  the idea that fractions only have value when we know the size of the whole or referent to which they refer. (Would you rather have half the money in my right hand or a quarter of the money in my left hand?). We then went on to look at how to develop the fraction algorithms for the four operations. Addition and subtraction were fairly straight forward but, of course, multiplication and division are a completely different kettle of fish.

We also played around with fraction word problems which, again, is OK for addition and subtraction, but quite confusing for the other two operations. The other problem with fraction problems is that it's hard to get away from food; "if you ate 3/8 of your pizza.......?". And, of course, pizzas are only ever divided into eighths!

Then someone said "I was always told that "of" means multiply as in what is 1/2 of 1/3". Yes, it always has, I thought, but why? What is the conceptual connection between the word 'of ' and the operation of multiply? I asked several mathematicians and could only come up with nothing more than my original thought which was that it is just a linguistic connection as in "automobile means car" and "house means home".

My daughter, Marie, even suggested using the word "times" as in "a half of a quarter is a quarter, half a time" (4 groups of 3 is 3 four times). Then I started thinking about the area concept of multiplication as in the image above. in a 1 x 1 square an area which is 1/3 (blue side) by 3/4 (red side)  provides a conceptual connection between of and multiplication. Voila................................. maybe.