An Algebraic Approach to the Immutability of the Slope of a Line

Julia S. Roman

Incarnate Word College
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San Antonio, TX 78209
USA
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jlroman@tenet.edu

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ABSTRACT

This paper diverges from the traditional treatment of the slope of a line which uses the concept of similar triangles and, instead, incorporates only the use of algebra to demonstrate that, no matter which two points of a line are chosen, the slope of the line is unique. Using the general form of an equation of a line, ax+by=c, one may show that the slope is -a/b regardless as to the pair of points selected to determine slope.

Expanding on this knowledge and using only a hands-on proof of the Pythagorean Theorem, one may show that not only are the sides of the triangles formed by the change in y and the change in x in the same ratio to one another but also that the length of the hypotenuse of each triangle is in the same proportion to the others as are the sides.

By using only algebra and a simple manipulative proof of the Pythagorean Theorem, the proof of the immutability of the slope of a line can be approached in a simple, straightforward manner. Once this concept is dealt with using only algebra, the idea may be expanded to an algebraic approach to other, more complicated areas of mathematics such as the derivative. Since the slope of the line tangent to the graph of a function is actually the derivative of the function, the investigation of the derivative with only algebra becomes a possibility. Indeed, this paper shows how to determine the derivative of several basic functions and lays the groundwork for the finding of derivatives of all rational functions. Investigation of these concepts and presentation of these results make use of the graphing calculator and presentation software for a laptop computer.

Keyword(s): college algebra, trigonometry