PRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Ful¯llment of the Requirements for the Degree Master of Science Michael Matthew Smith December, 2006 PRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS Michael Matthew Smith Thesis Approved: Accepted: Advisor Dean of the College Dr. Antonio Quesada Dr. Ronald Levant Faculty Reader Dean of the Graduate School Dr. Linda Saliga Dr. George Newkome Faculty Reader Date Dr. Lynne Pachnowski Department Chair Dr. Joseph Wilder ii ABSTRACT Technology has transformed the mathematics curriculum. Instructional techniques are constantly evolving because of e®orts to maximize the bene¯ts of technology. To continue this process of enhancing the mathematics curriculum, this thesis will ex- amine the following questions. What concepts, that are foundational to calculus, can be taught, with the assistance of the graphing calculator, at a level before calculus? How well do current precalculus textbooks incorporate these concepts? Finally, how well do practicing secondary mathematics teachers understand these concepts? To answer the ¯rst question, several concepts foundational to calculus were identi¯ed. Next, we examined how accessible these concepts were to secondary stu- dents. Finally, the concept list was narrowed to those that have been rarely empha- sized in the secondary curriculum. This paper will address seven of these concepts. The goal of identifying these concepts is to promote the integration of them before calculus to enable students to make connections between pre-calculus (any course before calculus) and calculus. To answer the second question, twelve precalculus textbooks were examined to see how well they integrated these concepts. To accomplish this, a grading rubric was created to evaluate the textbooks. Then each textbook was reviewed and scored iii based on the rubric. To answer the third question, an assessment was given to a group of forty- one secondary mathematics teachers taking part in a continuing education workshop. The assessment was given at the beginning and at the end of the workshop. This was done for two reasons. The initial assessment was given to provide general information regarding how well teachers understood the topics that we propose are foundational to calculus. The post test was administered to determine the e®ectiveness of the workshop. iv TABLE OF CONTENTS Page LIST OF TABLES . vii LIST OF FIGURES . viii CHAPTER I. INTRODUCTION . 1 II. THE FUNDAMENTAL CONCEPTS . 8 2.1 Understanding domain and range: . 8 2.2 Investigating local and global behavior: . 12 2.3 Finding extrema values: . 15 2.4 Understanding relative growth of functions: . 18 2.5 Recognizing transformations: . 20 2.6 Equation solving approach: . 23 2.7 Solving inequalities: . 26 III. RESULTS . 28 3.1 Textbook analysis . 28 3.2 Teacher assessment analysis . 32 IV. CONCLUSION . 36 v BIBLIOGRAPHY . 37 APPENDICES . 40 APPENDIX A. TEACHER EVALUATION . 41 APPENDIX B. BOOK EVALUATION RUBRIC . 47 vi LIST OF TABLES Table Page 2.1 Example 10 . 20 2.2 Example 15 . 25 3.1 Book evaluations . 30 3.2 Pre-test results . 33 3.3 Post test results . 33 3.4 t-test results . 34 vii LIST OF FIGURES Figure Page 1.1 Two Towers Problem . 1 2.1 Example 1 . 10 2.2 Example 2 . 10 2.3 Example 3 . 11 2.4 Example 4 . 13 2.5 Example 5 . 14 2.6 Example 6 . 15 2.7 Example 7 . 17 2.8 Example 8a . 17 2.9 Example 8b . 18 2.10 Example 9 . 19 2.11 Example 11 . 21 2.12 Example 12 . 22 2.13 Example 14 . 24 2.14 Example 15 . 25 viii 2.15 Example 16 . 27 2.16 Example 17 . 27 ix CHAPTER I INTRODUCTION With the integration of technology into the mathematics classroom, there are now numerous avenues to approach mathematical problems. Problems that were once reserved for upper level mathematics classes are now able to be taught, with the assistance of technology, in lower level mathematics classes. For example, the Two Towers Problem was once reserved only for a calculus class. A common version of the problem obtained from Stewart [1] is as follows: Two vertical poles PQ and TS are secured by a rope PRT going from the top of the ¯rst pole to a point R on the ground between the poles and then to the top of the second pole. If the Height of pole TS measures half the height of pole PQ, ¯nd the position of R that requires the least rope. Calculate the minimal rope length. P T Q S R Figure 1.1: Two Towers Problem The problem is an optimization problem, and classically would require the use of di®erentiation to solve it. However, Quesada and Edwards [2] show that with the use of Dynamic Geometry Software, a student in a high school level geometry 1 class would have the mathematical background to solve this problem using multiple approaches. Similarly, a student in a high school algebra class, with the assistance of a graphing calculator, would be able to solve this problem using di®erent methods. The article goes on to show several distinct ways this problem can be solved. Each of these methods builds into a framework for solving mathematics problems with technology. The framework is constructed by four problem solving strategies, and the article continues by discussing the advantages of employing these strategies according to a student's current level of understanding. The article concludes, \Because technology enables students to revisit problems from di®erent perspectives based upon the depth of their mathematical knowledge, the tools encourage students to make connections among various levels (and areas) of mathematics" [2]. Discussing the importance of connections, NCTM explains, \When students can see the connections across di®erent mathematical content areas, they develop a view of mathematics as an integrated whole" [3]. Combining Quesada and Edward's statement with that of NCTM's gives this picture: if technology can be utilized to enable students to revisit problems and thus encourage them to make connections among various levels, then they, the students, develop a better perspective on math as a whole. This train of thought is applied in this thesis to discuss concepts underlying the major topics of calculus that can be introduced, with the assistance of technology, in courses previous to calculus. This will enable students to make connections between pre-calculus (any course before calculus) and calculus. It is the belief of this author that students who can make solid connections between pre-calculus and calculus, will have a better opportunity to succeed in calculus. Others have recognized this need to provide more opportunities to prepare students for calculus. At the collegiate level, Schattschneider et al decided to inter- weave the courses of precalculus and calculus 1 into a one year class. 2 \Let's o®er students who need extra time and review in order to succeed in calculus just that opportunity, but not in a calculus-free setting. Instead, let's introduce them to the interesting ideas of calculus, and in this setting, and as needed, review the necessary background and techniques for them to succeed in calculus" [4]. Since the focus of this thesis is at the secondary level, these ideas were utilized in the reverse order. While teachers are instructing the necessary techniques and background of mathematics, we propose that they, when appropriate, interject the interesting concepts of calculus. To determine what concepts to interject, the fundamentals of calculus were analyzed. The following questions were asked: Besides algebraic knowledge, what knowledge do students need to succeed in calculus? What are core concepts un- derlying the major topics of calculus? Which of these concepts are accessible to pre-calculus students? How well are these concepts currently being implemented in the pre-calculus curriculum? The major concepts of calculus are the limit, derivative, and integral of func- tions of real values. Hence, the concept of function \... is one of the most fundamental concepts underlying the calculus" [5]. \One purpose of the [concept of] function is to represent how things change. With this meaning it is natural to move on to consider the calculus concepts of the rate of change (di®erentiation) and cumulative growth (integration)..." [6]. Thus, a thorough understanding of the concept of function is essential to address topics of calculus. Accordingly, several of the concepts discussed in this thesis are related to the concept of function. Similarly, a student's ability to solve equations and inequalities is crucial to understand the major topics of calculus. While an analytical solving approach is foundational to mathematics, technology has empowered students with multiple 3 approaches to solving equations. In addition, technology allows the student to solve equations regardless of the functions involved [7]. In regard to inequalities, Quesada expounds, \Since graphing is no longer a time consuming task and inequalities help students think globally, it may be worthwhile to increase the number of inequalities that we ask" [7]. For these reasons, two of the concepts addressed in this thesis are related to solving equations and inequalities with technology. The seven concepts that will be discussed throughout this paper are: under- standing domain and range, investigating local and global behavior, ¯nding extrema values, understanding relative growth of functions, recognizing transformations, uti- lizing multiple equation solving approaches, and solving inequalities. While these concepts are found throughout the secondary mathematics curriculum, some are rarely addressed. In addition, teaching them e®ectively with technology and with a perspective on calculus may not be commonly practiced. These concepts are by no means exhaustive of the concepts fundamental to the major topics of calculus. For instance, matrix applications and the use of recursion are fundamental to the major topics of calculus [8], but will not be included because they exceed the focus of this thesis.