Final answer:
The question is asking about the number of small square tiles needed to complete a square design, given one large square tile, eight rectangular tiles with the same side length as the large tile, and one small square tile with the same width as the rectangular tiles.
Step-by-step explanation:
Gabi and Joaquin are working on a square tile design. Using algebra tiles to model the situation, Joaquin needs a certain number of small square tiles to complete the design. The problem is similar to finding out how the area of one square changes when its dimensions are scaled up by a factor. To solve for Joaquin’s design, first, we determine the area of the large square tile and compare it to the combined area of the rectangular and small square tiles. If one large tile is no longer manufactured and its side length is the same as the rectangular tiles, and knowing that area scales by the square of the scale factor, we can deduce that the side length of the rectangular tiles is half of the large square tile, and the small square tile's side length is the same as the width of the rectangular tiles.
To complete a square design, Joaquin needs to arrange these tiles such that the combined area matches that of the large square tile. Given that the large square tile has a side length four times that of the small square tile (which is inferred, assuming the eight rectangles formed by two small squares together have the same length as the large square), Joaquin would need additional small square tiles equal to the area of the large square minus the area of the existing tiles, which would be more precisely calculated if dimensions were provided.