Almost a square
A square with sides of integral length (i.e., the side lengths are integers) could be made with 25 square units. Make a rectangle with sides of integral length and an area of 24 square units, but construct the rectangle so it comes as close as possible to being a square. What are the dimensions of the rectangle?
Repeat this construction by starting with other square numbers. In general, how do you express the side lengths of these special rectangles when 1 is subtracted from any square number n2? Show your generalization using algebra, a table of results, and supporting pictures.

Extensions 
Consider the same question above, but now let 1 be added to the area of a square. For example, start again with a square of area equal to 25 square units. Add 1 to 25. Now, make a rectangle with sides of integral length and an area of 26 square units, but construct the rectangle so it comes as close as possible to being a square. What are the dimensions of the rectangle?
Again, repeat this same construction by starting with other square numbers. Could you state a generalization similar to your generalization in the above problem? Support your answer with a table and pictures.
(Source: Adapted from Mathematics Teaching in the Middle School, NovDec 1997) 

Related External Resources 
Algebraic Factoring
This lesson uses rectangles and squares to illustrate how binomial expressions can be factored and multiplied.

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