1. **Problem:** Describe the regions that represent $a^2 - b^2$ in a large square of area $a^2$ containing a smaller square of area $b^2$. Show how to rearrange these regions to illustrate the difference of squares pattern. 2. **Formula and rules:** The difference of squares formula is: $$a^2 - b^2 = (a - b)(a + b)$$ This means the area difference can be expressed as the product of the sum and difference of the side lengths. 3. **Intermediate work:** The large square has area $a^2$ and the smaller square inside has area $b^2$. The shaded region representing $a^2 - b^2$ is the large square minus the smaller square. 4. **Rearrangement:** Cut the shaded region into two rectangles of dimensions $b \times (a-b)$ and $(a-b) \times a$. Rearranging these rectangles side by side forms a rectangle with dimensions $(a-b)$ and $(a+b)$, showing: $$a^2 - b^2 = (a - b)(a + b)$$ 5. **Explanation:** This geometric rearrangement visually proves the difference of squares formula by decomposing and recomposing the area. Final answer: The difference of squares $a^2 - b^2$ can be represented by rearranging the shaded regions into a rectangle of dimensions $(a-b)$ and $(a+b)$, confirming the identity $a^2 - b^2 = (a - b)(a + b)$.