Some Thoughts about College Algebra — Blog 2 — A Graph is not Something You Do

The textbook I am currently using for College Algebra is pretty much the same as every other College Algebra Textbook. The colors used on the cover vary from book to book but that is hardly a contributing factor to the book’s effectiveness. The book is much too large, contains a lot of useless information, trains the student to solve certain classical textbook problems rather than teaching mathematics, and has far too many exercises. These textbooks give students the false impression that you learn mathematics by repeating dozens of insipid textbook problems rather than teaching that one learns mathematics in order to solve problems. Students are not given even a hint that the “detached language” nature of mathematics is what makes it a powerful problem solving tool in every discipline. Students leave College Algebra believing it to be a collection of processes and procedures of no real value. In view of our current teaching practices they might be right.

The purpose of early college level algebra courses is to introduce the student to the use of abstraction, generalization, and deductive reasoning while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions. Along the way students should increase their ability to use critical thinking to solve the variety of problems they will surely encounter during a lifetime. No College Algebra book currently on the market (to my knowledge) addresses those goals.

My intention is to provide alternative methods through a series of blogs. As a mathematician I feel strongly that I should present a series of essays that are so arranged as to provide a complete and linearly ordered argument. I have struggled unsuccessfully with that for years and will not attempt it here, rather the order in which topics are presented follow no rhyme or reason.

A Graph is not Something You Do

To keep this essay simple, I will make all my comments about graphs of equations. With just slight variation the very same comments can, and should, be made about graphs of functions.

Many, if not most, of incoming College Algebra students think of a graph as something to be done.

  1. They do not realize that the graph of an equation is a mathematical object.
  2. They do not know what they have produced when they graph an equation.
  3. They cannot extract information about an equation from its graph.
  4. They only know one method of graphing – plotting points.
  5. They believe graphs exist only in the Cartesian Plane.

To correct this situation we should begin by explaining the following definitions of graph.

Definition: The graph of an equation consists of all the points, and only those points, whose coordinates are solutions of the equation.

Definition: The graph of an inequality consists of all the points, and only those points, whose coordinates are solutions of the inequality.

The discussion of graph should have been preceded by a discussion of solution, solving, and solution set. While explaining the definition of graph we should make it clear and demand that the student understand each of the following:

  1. The solution set for an equation or inequality in one variable is a set of numbers. Consequently the graph of an equation in one variable is shown on the Real Number Line. One variable – One dimensional space.
  2. The solution set for an equation or inequality in two variables is a set of ordered pairs of numbers. Consequently the graph of an equation or inequality in two variables is shown in the Cartesian Plane. Two variables – Two dimensional space.
  3. The solution set for an equation or inequality in three variables is a set of ordered triples of numbers. Consequently the graph of an equation or inequality in three variables is shown in the Three Dimensional Coordinate System. Three variables – Three dimensional space.
  4. Discussion of higher dimensions should probably be very limited.

Students will probably better understand what a graph is when it is discussed informally as “ a picture” of the solution set.

The elementary deductive reasoning employed to reach an understanding of the above concepts is not obvious or automatic for the beginning College Algebra student so we must help them understand the logic involved.

Equations and inequalities should not be taught in separate sections. Rather they should always be presented together. Such a unified presentation is grounded in The Law of Trichotomy. Students should learn to understand the role of The Law of Trichotomy throughout any consideration of equations and corresponding inequalities.

Students will benefit greatly from an understanding of the fact that the graph of an equation is the boundary between the graphs of the two inequalities obtained by replacing the = symbol with the < symbol and the > symbol. With a modest amount of prompting they will learn that they can solve and graph the easiest one of the three using computational methods and then use a small number of tests to deduce the solutions to the other two. It is important that they begin to understand and appreciate the efficiency of this kind of deductive reasoning. The fact that some boundaries might be imposed by the domain of the equation can be handled as a very nice example of how we reexamine concepts to establish new understanding of the concepts so as to extent them to a new situation.

Students will easily understand that we solve linear equations and inequalities in one variable to produce a solution set from which we produce the graph. It is important to teach them that when faced with equations and inequalities in two variables we do not have a neat computational method of producing the solution set. Instead we reexamine the whole relation between solution set and graph and recognize that the best we can do is produce the graph – a picture of the solution set. We may not be able to write the solution set but we can produce a pretty good picture of it. Moreover we can come up with some rules that permit us to determine individual important solutions in the solution set.

We should teach our students that they will no longer “draw” the graph of an equation or inequality but rather they will sketch the graph of an equation or inequality. The sketch is determined in large part by properties of the equation or inequality whose graph is being sketched. Linear equations and inequalities in one variable dictate a particular shape. Linear equations and inequalities in two variables dictate a different shape. Similarly quadratic equations or inequalities in either one or two variables dictate the general shape of their graphs. Rational equations and inequalities tell us other information about their graphs as do exponential and logarithmic equations.   Notice that as we teach students to produce graphs in this manner we are giving them experience at using generalities.

Students should be taught to label (with coordinates) all important points. They need help in determining what the important points are. On the real number line they must label individual solutions as well as endpoints of parts of solution sets. When working with equations and inequalities in two variables the important points are x-intercepts, relative maxima or minima (some of which cannot be determined without Calculus), and intersections with other graphs. Students should be able to produce and interpret graphs of an equation and its two corresponding inequalities on the same coordinate system. Students must learn that unless a point is labeled with its coordinate, those coordinates are not known. The apparent location of a point does not imply the exact coordinates.

Throughout all of these discussions we should continually reinforce the idea that the graph is a mathematical object tightly linked to the solution set, another mathematical object.

Watch for related blogs dealing with specific types of equations and/or functions.

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