Some Thoughts about College Algebra — Blog 3 — Law of Trichotomy – Linear Equations and Inequalties in One Variable

August 25, 2014

Some Thoughts about College Algebra

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.

Law of Trichotomy Linear Equations & Inequalities in One Variable

Law of Trichotomy: For any two expressions a and b which represent real numbers, exactly one of the following is true.

  1. a < b
  2. a = b
  3. a > b

The Law of Trichotomy is a formal statement of a property which most of us would consider to be quite obvious; when comparing two numbers; they are equal, the first is less than the second, or the first is greater than the second. The extension to expressions representing real numbers is obvious. The purpose of the formal statement here is to call attention to the obvious fact and to make it available for use with algebraic quantities which represent real numbers.

The following example illustrates what seems to me to be a better approach to teaching students about equations and inequalities. The example illustrates an important and useful consequence of The Law of Trichotomy. The example stresses deductive reasoning rather than rote training to solve equations and inequalities. The example also highlights the relation of an equation with its twin inequality siblings.

Law of Trichotomy Applied to Linear Equations and Inequalities in One Variable

During a consideration of the two linear algebraic expressions 3x + 5 and –2x + 7, the Law of Trichotomy reminds us that there exists three distinct possibilities;

  1. 3x + 5 < –2x + 7
  2. 3x + 5 = –2x + 7
  3. 3x + 5 > –2x + 7

This means is that if x is replaced (in these three statements) by any individual real number, exactly one of the expressions will be true.

Another more precise way of stating this is that the union of the three solution sets is R (the real numbers) and the intersection of any two of the solution sets is the empty set.

Another, possibly more understandable, way to state this is: The graph of a linear equation in one variable divides the real number line into a point and two rays. The point is the graph of the equation and one of the rays is the graph of one of the inequalities and the other ray is the graph of the other inequality.

The solution set for the above equation is . The graph of the equation 3x + 5 = –2x + 7 is shown in Fig. 1.

Blog3 Fig1

Clearly this graph of the equation divides the real number line into a point and two rays as shown in Fig. 2.

Blog3 Fig2

The blue ray is the graph of one of the inequalities and the red ray is the graph of the other inequality.

To determine if the blue ray is the solution set for the inequality 3x + 5 > –2x + 7 we need only test one number from the blue ray in 3x + 5 > –2x + 7. The number 0 is in the blue ray and is easy to test. Substituting 0 into 3x + 5 > –2x + 7 yields 5 > 7 which is false. This allows a number of conclusions;

  1. The blue ray is not the solution set for 3x + 5 > –2x + 7.
  2. The red ray (the other ray) is the solution set for 3x + 5 > –2x + 7
  3. The blue ray is the solution set for 3x + 5 < –2x + 7. Blog3 Fig3An examination of the graph in Fig. 3 quite clearly illustrates the following:
  1. The graph of a linear equation in one variable is a point on the real number line.
  2. The graph of a linear inequality in one variable is a ray on the real number line.
  3. The graph of the equation is the boundary between the graphs of the corresponding inequalities. For that reason, the equation is usually called the boundary equation for the inequalities.
  4. As predicted by The Law of Trichotomy, the union of the three solution sets is the set of all real numbers R and that the intersection of any two of the individual solution sets is the empty set ∅.
  5. The relationship between and equation and its two twin inequality siblings.
  6. An elegantly simple illustration of The Law of Trichotomy.

The deductive reasoning used in this example will be of more value to most college algebra students than the forgotten rote method of solving a linear equation and the completely divorced rote method of solving linear inequalities. The lessons learned about deductive reasoning can be applied to many situations other than mathematics.

In this example the student has also been exposed to the technique of examining a structure rather than concentrating solely on individual parts to exclusion of the structure. Perhaps the student will even learn that  a study of structure frequently leads to a better understanding of individual components.

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

August 20, 2014

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.

Some Thoughts about College Algebra — Blog 1

August 11, 2014

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.
Compartmentalization
Traditionally we seem to always begin by discussing equations and inequalities in one variable and those topics are usually compartmentalized as :
Linear Equations Quadratic Equations
Rational Equations Equations with Square Roots
Linear Inequalities
The only thing we do in these isolated compartments is train students to perform a prescribed set of steps which solves the problem. There is very little emphasis on what a solution is. There is no distinction made between solution and solution set. There is no mention of graph. Linear equations and linear inequalities are not related to each other. Quadratic inequalities nor any other kind of inequality is mentioned.
We then move on to a discussion of the these same creatures (Linear Equations, Quadratic Equations, Rational Equations, Equations with Square Roots, Linear Inequalities) in two variables. There is no attempt to relate these compartments to the earlier single variable creatures. The concept of solving and solution set seems to disappear from the discussion and graph is introduced. A graph is never presented as a “picture” of the solution set of the equation. A graph is simply something a student “does”. Again each of these topics is presented in isolation of any of the other topics. By this time any thinking student will have concluded that algebra consists of a large collection of procedures one uses to solve math problems like the ones in the textbook.
The next step is to reconsider these two variable creatures (Linear Equations, Quadratic Equations, Rational Equations, Equations with Square Roots, Linear Inequalities) but this time we use the language of functions. We also extend our discussion to polynomial functions, rational function, log functions, exponential functions, and sequences (but we hardly ever treat them as functions). The astute student soon realizes that he can pass the course if he/she simply identifies the term function with the term equation. Again there are a bunch of isolated rules, procedures, and processes to memorize. In the end we have taught the student that algebra is a large collection of procedures one uses to pass tests but which have little value beyond that.
Unification
Four important properties that we all know and should use extensively in College Algebra to provide a more unified and comprehensive view of the beautiful structure of elementary algebra and to examine examples of exciting logic which will awaken even the most lethargic mind. They are:
1 The Distributive Property
2 The Law of Trichotomy
3 The Transitive Property of Equality
4 The Zero Factor Property
The Distributive Property links factoring (to write as a product) and multiplication. If the Distributive Property is understood to be the underlying principle for multiplying multi-term expressions, then multiplication becomes more uniform throughout mathematics. I will present further illustrations of the importance of The Distributive Property in future blogs.
The Law of Trichotomy should be used to link an equation to its two siblings (the inequalities formed by replacing the = symbol with the symbol). We should unify the study of these three mathematical objects by emphasizing the fact that every element of the domain of the equation is a solution to exactly one of the three and that the equation is the boundary between the two inequalities. The graph of the equation is always a boundary between the graph of the other two siblings. I will further illustrate this concept in future blogs.
The Transitive Property of Equality should be interpreted as: It two expressions represent the same quantity, the two expressions must be equal. This interpretation/rewording of the Transitive Property is correct and is easier for the student to understand and use. The model for every traditional word problem is produced by establishing two different expressions for the same quantity and then applying the Transitive Property. The process of finding the intersections of two graphs is explained by using the Transitive Property. More applications and illustrations will be presented in future blogs.
The Zero Factor Property is a property of the real number system (not of quadratic equations as presumed by many students). The fact that this property is used for many equations other than quadratic should be emphasized. It should also be emphasized that the Zero Factor Property produces two equations joined with the conjunction OR which are equivalent to the previous single equation. This approach helps unify the process of solving linear, quadratic, and some other equations. It should be pointed out that such unification decreases the number of processes for solving equations. The student should be shown that this process of solving quadratic equations is logically the same as was used for solving linear equations.
We should use the Zero Factor Property to teach that the process to solve an equation in one variable is to generate a sequence of equations each equivalent to the previous equation until simplest equation(s) is(are) obtained. On occasion (rational equations, equations with square roots) we must make slight modifications to this process, but we never loose sight of the fact that generating a sequence of ever simpler equivalent equations is our strategy for solving equations. It is my intent to elaborate on this point in future blogs.


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