**Thought for the day**: *Creative Math Prompts *can guide students toward a deeper understanding of concepts that are often just memorized.

**Concepts**: the multiple meanings of fraction multiplication and division

**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.

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