jmc

$\underline{\text{Method 1}}$

Make a diagram and draw $4$ lines so that they intersect each other as shown. The number of different sections is $\boxed{11}.$



$\underline{\text{Method 2}}$

Make a table. The original rectangle without lines added is considered to be one section.

$$\begin{array}{cccccc}\text{Total number of lines added}& 0& 1& 2& 3& 4\\ \text{Total number of sections}& 1& 2& 4& 7& ?\end{array}$$ Look for a pattern. Observe that the $1^{\text{st}}$ added line results in increasing the preceding total of sections by $1,$ the $2^{\text{nd}}$ added line increases the preceding total of sections by $2,$ the $3^{\text{rd}}$ added line increases the preceding total sections by $3.$ It seems that the $4^{\text{th}}$ added line will increase the preceding total of sections by $4$ and that there will be $7+4$ or $11$ sections. Examine the $4^{\text{th}}$ line in the diagram below. When the $4^{\text{th}}$ line intersects the first of the $3$ interior lines, it creates a new section. This happens each time the $4^{\text{th}}$ line crosses an interior line. When the $4^{\text{th}}$ line finally ends at a point on the rectangle, it creates a $4^{\text{th}}$ new section. Thus the $4^{\text{th}}$ line creates a total of $4$ new sections. The answer to the given problem is $\boxed{11}.$

(If a 5th line were added, it would increase the preceding total of sections by 5.)