7. How will the area of a rectangle change if one of its sides is increased by 3 cm, and the other is decreased by 3 cm?
7. Let $a$ and $b$ be the sides of a rectangle. Then its area is: $S=a \times b$. The sides of the modified rectangle will be

$$
(a-3) \text { and }(b+3)
$$

and the area will accordingly be:

$$
S_{1}=(a-3)(b+3)=a b-3 b+3 a-9
$$

Let's compute the difference between $S$ and $S_{1}$. It is equal to:

$$
\begin{aligned}
S_{1}-S & =(a b-3 b+3 a-9)-a b= \\
& =-3 b+3 a-9=3(-b+a-3)
\end{aligned}
$$

Let's analyze the obtained result.

If $-b+a-3=0$, then the areas will be equal. This will be possible if $a-b-3=0$ or $a=b+3$.

If $S_{1}-S>0$, then the area of the second rectangle will be greater than the area of the original. This is possible if $a-b-3>0$, i.e., if $a>b+3$.

Similarly, we can conclude that $S_{1}-S<0$ if $a<b+3$.

Obviously, this problem cannot be solved this way in elementary school. Therefore, let's consider how it should have been solved in elementary school.

We assume that the student can solve this problem practically. By conducting a computational experiment with various rectangles, the student should conclude that the area of the rectangle can increase, decrease, or remain unchanged.
![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-014.jpg?height=498&width=846&top_left_y=247&top_left_x=538)

$5 \text{ cm}$

$2 \text{ cm}$

$$
\begin{aligned}
& S=5 \times 2=10 \text{ cm}^{2} \\
& S_{1}=5 \times 2=10 \text{ cm}^{2}
\end{aligned}
$$

Conclusion: the area did not change.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-014.jpg?height=368&width=557&top_left_y=1015&top_left_x=364)
$5 \text{ cm}$

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-014.jpg?height=183&width=605&top_left_y=1028&top_left_x=971)

$S=5 \times 4=20 \text{ cm}^{2}$.

$S_{1}=2 \times 7=14 \text{ cm}^{2}$.

Conclusion: the area decreased.
![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-014.jpg?height=250&width=1082&top_left_y=1688&top_left_x=493)

$$
\begin{aligned}
& S=5 \times 1=5 \text{ cm}^{2} \\
& S_{1}=2 \times 4=8 \text{ cm}^{2}
\end{aligned}
$$

Conclusion: the area increased.

Answer: the area can increase, decrease, or remain the same.