The third grade activity I was observing involved rolling two die and multiplying the subsequent numbers to get a product. Then, using a neat worksheet, the students had to divide the product by 2,3,4,5, and 6 to see what happens. The demonstration roll the student teacher used was 4 x 4 for 16. Dividing by 2 yielded 8, but 8 what? The children were using tiles to make 2 x 8 arrays vertically. Here the 2 referred to the number of columns and the 8 to the number of rows. So the 8 was 8 rows. She could have shown 8 groups of 2 or 2 groups of 8, demonstrating the commutative property of multiplication. This is only true abstractly as the actual arrangement looks quite different. This is known as psychological non-commutativity.
She then went on to divide 16 by 3 getting 5 R1, then divided it by 4 getting 4 (proving 16 is a square number), then by 5 getting 3R1 (showing the commutative property), and finally by 6 getting 2 R4 .
The interesting thing about the commutative property in division is if you then give the remainder as a fraction. 16 ÷ 3 would give 5 1/3. 16 ÷ 5 would give 3 1/5. So the question is; what do the fractions represent? 1/3 of what and 1/5 of what? A fraction has no value unless you know the size of the one to which it refers. It depends, of course, on what the referents for the 3 and the 5 are. If you had 16 cookies divided between 3 people, each person would get 5 and 1/3 cookies. If you had 16 cookies and divided them into groups of 5, 3 1/5 people could get a group of cookies; which makes absolutely no sense at all.
The neat thing about the Bridges Program 2 activities is that they are so rich with mathematical possibilities.
No comments:
Post a Comment