Almost a Square

A perfect square could be made with 25 square units. Make a rectangle with 24 square units, but make it as close as possible to a square. That is, build a rectangle with 24 square units in which the length and width are as close to equal as possible. What are the dimensions? Repeat this construction starting with other square numbers. In general, what rectangle can be formed when 1 is subtracted from any square number? Show your generalization using symbolic algebra, a table of results, and supporting pictures or diagrams.

In the problem above, 1 was subtracted from a square and a rectangle was formed that was close to a square. Consider the same question if 1 is added to a square. For example, start again with a square of 25 square units. Add 1 to 25. Make a rectangle as close as possible to a square. What rectangle can you form that is close to a square? Repeat this same construction as in the problem above. Could you state a similar generalization? Support your answer with a table and pictures.

(Source: Adapted from Mathematics Teaching in the Middle School, Nov-Dec 1997)