In an area model, two strips of size \(a\times b\) are removed from a square of side \(a\), and their common \(b\times b\) region is added back once. Which algebraic identity does this model represent?

Correct answer: \((a-b)^2\)

Answer: B, \((a-b)^2\). Begin with the square of side \(a\), whose area is \(a^2\). Each removed strip has area \(ab\), so removing both seems to subtract \(2ab\). However, the two strips overlap in a square of side \(b\), whose area is \(b^2\). That overlap was subtracted twice, so it must be added once. Therefore the remaining area is \(a^2-2ab+b^2\). By the identity \((a-b)^2=a^2-2ab+b^2\), the model represents \((a-b)^2\). A is the difference of two squares, not this three-part removal. C has a positive middle term and represents addition. D omits the two cross terms. Memory cue: subtracting two equal strips creates a negative \(2ab\) term, while the overlap returns \(+b^2\).