Thought for the dayCreative 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 (and the connections between them)

Beginning

I notice that both pictures involve the numbers 2, 1/3, and 6.
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 else these two pictures have to do with each other.

Exploring
I wonder what operation(s) these pictures are showing.
I wonder if I can write an equation(s) for each picture.
I notice that I can represent each picture with both a multiplication and a division equation.
I wonder if there is a reason for using a rectangle in the first picture and a circle in the second one.
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.

Creating

As they notice and wonder, students may create and explore their own drawings, equations, and stories related to this image. Many of their creations may flow out of things that they have wondered about. For example, they may

Create real-world stories to fit the pictures (and the equations they represent).
Create pictures like the ones in the image but with the rectangle and circle interchanged.
Create pictures like the ones in the image but using other shapes to represent the whole.
Create pictures like the ones in the image but with 1/3 replaced by 2/3 or some other fraction.
Create pictures and stories that represent a fraction divided by a whole number.
Create pictures and stories that represent a fraction divided by a fraction.

Reflecting and Extending

I notice that a whole number multiplied by a fraction (between 0 and 1) equals a number less than the whole number.
I notice that a whole number divided by a fraction (between 0 and 1) equals a number greater than the whole number.
I wonder if I can predict how the quotient changes when the dividend or divisor is multiplied by a certain number..
I notice two meanings for dividing a whole number by a fraction. (1) How many of the divisor “fit into” the dividend? (2) The dividend is the fraction of what whole?
I wonder how these meanings change or stay the same when you divide other types of numbers.

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).