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Samples from Pathways from the Past  II

From the teacher’s manual:

This activity sheet focuses on a notational habit that delayed the conceptual development of algebra. The way people of the 16th century thought about algebraic problems was restricted by the way they wrote the symbols!  As we saw in the previous activity sheet, Italian mathematicians of the 15th and 16th centuries used different symbols for the unknown and its powers.  That was true elsewhere in Europe, too.  This convention may well have been due to the influence of early Greek mathematics.  The Greeks' conception of number was tied closely to geometry.  Numbers that represented lengths or areas or volumes were thought of as different kinds of things, and that carried over to European algebra.... [T]he early algebraists treated an unknown, its square, and its cube as different kinds of numbers.  Those words, which we still use, reflect that tradition.  However, the notational complexity of maintaining that separation inhibited the scope and flexibility of algebraic thinking....  It took several centuries for European algebra to gain the flexibility it has today.

The questions on this sheet highlight the difficulty caused by trying to keep this separation and the first steps in breaking away from it.  As before, the mathematical benefit to students comes from working through some fundamental algebraic processes in these unusual settings.  Thinking about how to express familiar ideas in unfamiliar ways encourages a deeper understanding of the ideas our own algebraic symbols express.  Along the way, there is also some practice with factoring common quadratics, including the difference of two squares.  If your students have not yet studied these things, you can simply give them the factorizations.

From the teacher’s manual:

The false position method described on sheet 2-3 can be applied only to equations of the form Ax = B.  If, instead, the equation is of the form Ax+C=B, then it is no longer true that multiplying x by a factor causes B to change by the same factor, and the process breaks down.... Instead, a way was found to extend the basic idea to equations of that type without any such algebraic manipulation.  It is called double false position.  This is such an effective method for solving linear equations that it continued to be used long after the invention of algebraic notations....  

[D]ouble false position is based on making two wrong guesses.  The key idea behind it is that a straight line has a constant slope. Before people had coordinate geometry in hand, this was expressed by saying that the difference between two values of Ax + C was proportional to the difference between the values of x.... The constant of proportionality is exactly A, the slope of the line, but double false position never actually computes A. Instead, it uses the fact that two ratios are equal.

To see why double false position works, students need to re-examine some fundamental concepts:

--- slope as a constant of proportionality
--- a first-degree equation as a straight-line function
--- the graph of a linear function
--- the two-point determination of a linear function
This gives you a novel context for unifying and strengthening their understanding of these important algebraic ideas.

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