Experienced students, very bright students, and good problemsolvers could whittle this information and a lot more information out ofthis algebraic analysis. Thus,if there are points of tangency then they must occur at these common pointson the x-axis. This means the graphof h(x) crosses the x-axis at the same two points as f(x) and g(x). This is useful becauseit keeps the students occupied, but what do they learn from it? Clearly,h(x) = (mx + b)(nx + c) is a polynomial of degree 2 and h(x) has two roots.The respective roots are when f(x) = 0 and g(x) = 0. A traditionalapproach would begin with algebraic manipulation. Is it possible to find two linear functions, f(x) = mx + b and g(x) = nx+ c, such that the function h(x) = f(x).g(x) is tangent to each. Further, the use of technologicaltools to examine visualizations of the functions makes for a different approachto the problem. In general it would not be included in the schoolcurriculum, but there is no reason it should not. Such tools make it possible to look at new topics in the mathematicscurriculum or to look at current topics in different ways. The senarios represent a compositeof several discussions of the problem with teachers, students, and colleagues.Note, the goal here is using this problem context, not only to solve theproblem posed, but to understand the concepts and procedures underlyingthe problem.įunction graphers are available for almost any computing platform or graphiccalculator. In fact, we came up with two different streamsof consciousness and so we have two senarios that are parallel in that theycover alternative approaches to the problem. Our analysis is presentedas a sort of stream of consciousness account of how one might explore theproblem with the tools at hand. This problem was posed by a group of teachers during a workshop in whichthe use of function graphers was being explored. In fact, this "paper" is a discussion of ourexamining a particular problem that having tools like function graphersavailable might make possible different approaches.įind two linear functions f(x) and g(x) such that the product This may have been an attempt to write a paper with a longer title thanthe paper itself. Each of which is Tangent to the Product Function by
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