Thought for the day: The meaning of fraction division may be the most challenging concept in arithmetic. Give students plenty of time to think!

Examples of noticing and wondering

I notice that both pictures involve the numbers 2 and 1/3.
I notice that the top picture shows 2 wholes and that the shaded part is 1/3 of one whole.
I notice that the bottom picture shows 1 whole and that 2 is 1/3 of the whole.
I notice that, in the top picture, there are 6 groups of 1/3 in the 2 wholes.
I notice that, in the bottom picture, since 2 is 1/3 of the whole, the entire whole is 6.

I wonder what these two pictures have to do with each other.
I wonder what operation(s) these pictures are showing.
I wonder if I can write an equation(s) for each picture.
I wonder what would happen to my equation(s) if I shaded the amount 2/3 in the top picture.
I wonder what would happen to my equation(s) if I shaded 2/3 of the whole in the bottom picture. 

Notes

Each picture shows a different meaning of 2 ÷ 1/3 = 6.

  • (1) The top picture shows that 2 ÷ 1/3 = 6, because there are 6 groups of 1/3 in 2.
  • (2) The bottom picture shows that 2 ÷ 1/3 = 6, because if 2 is 1/3 of a whole, then the whole is 6.

These two meanings for 2 ÷ 1/3 = 6 match the two meanings of multiplication for 1/3 • 6 = 2.

  • (1) Top picture: 1/3 • 6 = 2, because 6 groups of 1/3 make 2 wholes. Notice how this connects to the meaning of division for (1)!
  • (2) Bottom picture: 1/3 • 6 = 2, because 1/3 of (a group of) 6 equals 2.  Notice how this connects to the meaning of division for (2)! (You are asking: 1/3 of what equals 2?)

Notice that if you double the divisor (to 2/3), the answer becomes 3 (half as much as the original 6). Why?

  • (1) Top picture: If you double the 1/3 to 2/3, only half as many groups fit into 2. (It takes only 3 groups of 2/3 make 2 wholes.)
  • (2) Bottom picture: If you double the 1/3 to 2/3, the whole contains only 3 instead of 6. (If 2 is now 2/3 of the whole, then the whole is only 3, because each of the three parts now contains only 1).

Challenges:
(1) Suppose you make other changes to the numbers 2 or 1/3. Draw pictures, and think about what happens.
(2) Create real-world stories or situations to fit these pictures and equations.