Let's break down the problem from the image you provided:

You are given a square divided into four smaller rectangles/squares, with dimensions labeled in terms of $a$ and $b$.

i. Expression for the area of the entire figure in terms of the length of its side.

The entire figure is a square, and its side length is $a + b$. The area $A_{\text{total}}$ of a square is given by the formula:

$A_{\text{total}} = \text{side}^2 = (a + b)^2$

ii. Expression as the sum of the areas of the four parts.

The square is divided into four parts:

1. The top-left square with side length $a$, so its area is $a^2$.
2. The top-right rectangle with sides $a$ and $b$, so its area is $ab$.
3. The bottom-left rectangle with sides $a$ and $b$, so its area is $ab$.
4. The bottom-right square with side length $b$, so its area is $b^2$.

Thus, the total area as the sum of the areas of the four parts is:

$A_{\text{sum}} = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$

iii. Write an equation to indicate that the two expressions are equal.

The two expressions for the area should be equal. Therefore, we write the equation:

$(a + b)^2 = a^2 + 2ab + b^2$

This equation shows that the total area of the square calculated directly from its side length is the same as the sum of the areas of the four smaller parts.

Would you like further details on any part of this solution, or do you have any questions?

5 Related Questions:

1. How can we expand a binomial like $(a + b)^2$?
2. What is the geometric interpretation of the terms $a^2$, $b^2$, and $2ab$ in the expanded area?
3. How does this problem relate to the distributive property of multiplication over addition?
4. Can we solve similar problems using algebraic identities?
5. How would the total area change if the shape were a rectangle instead of a square?

Tip:

Whenever you are dealing with squares and binomials like $(a + b)^2$, try to visualize how the terms correspond to geometric shapes (e.g., squares and rectangles) to better understand algebraic expressions.