Suppose that you don't know the traditional algorithms for operating on fractions. How can you find a number to represent 2/3 + 1/2? How about 2/3 - 1/2 ?
A. Is "1/2 of 2/3 miles" the same as 2/3 of 1/2 mile? Explain
B. How (other than the traditional algorithm) would you find 1/2 of 2/3 miles? How would you find 2/3 of 1/2 mile?
c. What is the relationship between your answer (B) and the traditional algorithms for multiplication of fractions?
If I didn't know the traditional algorithm, I would try using manipulatives. I could cut one sheet of paper in half, and another in thirds. Then I would line two of the thirds pieces up next to a half piece and compare them to another sheet of paper with half marks and one-third marks drawn on it. I would notice that the pieces were bigger than one piece of paper, but not as big as 1 1/3. I would guess that the answer was 1 1/6 because 1/6 ...
This explores ways of operating with fractions outside of the traditional algorithm.