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Almost a Square

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

 Extensions

In the problem above, 1 was subtracted from the area of a square and a rectangle with integral sides was formed that was nearly a square. Consider the same question if 1 is added to the area of a square. For example, start again with a square with area 25 square units. Make a rectangle with integral sides, with an area of 26 square units, and again make it as close as possible to a square. Could you state a similar generalization to the one above regarding what rectangle can be formed when 1 is added to any square number? Support your answer with a table and pictures.

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

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