Let's break down the problem:

1. There are two large squares with the same area.
2. In the first square, the midpoints of adjacent sides are connected, forming a smaller square. The shaded area in this square is given as 9.
3. In the second square, four smaller squares are shaded. These smaller squares have side lengths equal to one-third of the side length of the large square.
4. The question asks us to find the area shaded in the second square.

Step 1: Analyze the First Square

Let the side length of the large square be $s$. When the midpoints of the sides of a square are connected, the smaller square formed inside has a side length equal to $\frac{s}{\sqrt{2}}$.

The area of the large square is $s^2$, and the area of the smaller square is $\left( \frac{s}{\sqrt{2}} \right)^2 = \frac{s^2}{2}$.

Given that the area shaded in the first square is 9, we can conclude that: $\frac{s^2}{2} = 9$ Solving for $s^2$: $s^2 = 18$

Step 2: Analyze the Second Square

In the second square, four smaller squares are shaded. Each smaller square has a side length of $\frac{s}{3}$. The area of one smaller square is: $\left( \frac{s}{3} \right)^2 = \frac{s^2}{9}$ The total area of the four smaller squares is: $4 \times \frac{s^2}{9} = \frac{4s^2}{9}$

Substitute $s^2 = 18$: $\frac{4 \times 18}{9} = 8$

Conclusion:

The area shaded in the second square is $\boxed{8}$.

Would you like more details or have any questions?

Here are five related questions to extend your understanding:

1. How do we derive the side length of a square formed by midpoints of a larger square?
2. If the area shaded in the second square were 16, what would the side length of the large square be?
3. What happens to the shaded area in the second square if the side lengths of the smaller squares were halved?
4. How do geometric transformations affect the area of figures within squares?
5. Can you generalize this problem for $n$-shaped smaller figures within a larger square?

Tip: Always relate the areas of shapes to the side lengths to simplify calculations!