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.
Some of these concepts are rst introduced in algebra 1, while others are introduced in the course precalculus. However, the focus of this thesis will be on utilizing these concepts throughout the entire secondary math curriculum to prepare students for calculus. This approach will be more integrated than our traditionally disjoint curriculum. As Manoucherhri explains,
“. . . the relevance of a topic for a particular grade level is no longer de- termined based upon its historical value in the scope and sequence of the school mathematics. Relevance is, however, determined based upon
the extent to which the topic empowers students to understand related
4 concepts in greater depth and detail” [9].
Throughout the remainder of this paper it will become evident that the seven concepts discussed in this thesis are “relevant” to the major topics of calculus. Prior to the use of technology in the mathematics curriculum, these topics could not be expounded upon easily. For example, nding the range of a quartic polynomial with local extrema requires the ability to nd those extrema values. Without technology, this would require the use of calculus. Now, with the aid of technology, this concept, as well as many others, can be introduced, utilized, and emphasized throughout the entire secondary mathematics curriculum. Teaching with technology provides new insight into concepts and, as already discussed, helps build connections [10]. However, there are many types of technology utilized in the mathematics classroom. One of the most e ective tools is the graphing calculator. Therefore, when “technology” is referred to throughout the remainder of this thesis, it will imply, unless otherwise stated, use of the graphing calculator without symbolic capabilities.
Two mathematicians who have pioneered teaching with technology are Dr. Frank Demana and Dr. Bert Waits. Demana and Waits have developed a philoso- phy of teaching mathematics with technology that promotes a balanced instructional approach. This balanced approach incorporates and alternates between analytical, graphical, and numerical solutions. Their philosophy on calculator use is as follows:
“Appropriate use of graphing calculators in the teaching and learning of mathematics means the student:
1. Solves analytically using traditional paper and pencil algebraic meth- ods, and then supports the results using a graphing calculator.
2. Solves using a graphing calculator, and then con rms analytically the
result using traditional paper and pencil algebraic methods. 5 3. Solves using graphing calculator when appropriate (because traditional analytic paper and pencil methods are too tedious and/or time consuming
or there is simply no other way!)” [11].
This philosophy is structured to harness the computing power of calculators, while bringing conceptual understanding to the student. In addition, as mathematics instruction progresses toward more technological means, the importance of utilizing technology well will only increase. For these reasons, this philosophy served as a guideline for the research conducted throughout this thesis.
The focus of the research of this thesis is on teaching these seven concepts with technology to enable students to build more conceptual connections between pre-calculus and calculus. After discussing these concepts, this thesis will examine the implementation of these concepts. Two questions this paper will answer are: 1. How well do current precalculus textbooks incorporate these concepts?
2. How well do secondary mathematics teachers understand these concepts? To answer these questions, the research for this thesis was conducted in two parts. First, twelve current and popular precalculus textbooks were examined to see how well they incorporated these ideas and what could be done to enhance the implementation of these concepts to deepen students’ understanding. The evaluation process involved an assessment based upon a Likert scale. A grading rubric [Appendix B] was created to assess how well each concept was incorporated in each textbook.
The second part of the research was conducted on a group of forty-one sec- ondary mathematics teachers to assess their understanding of these concepts. To ac- complish this, a written assessment was created and given to the teacher group. The teachers were taking part in a continuing education workshop. Before the workshop began, the teachers were given the assessment to evaluate their initial understanding of concepts foundational to calculus. The teachers then took part in the eleven day
6 workshop. Following this instruction, the teachers were given the assessment again to determine the e ectiveness of the workshop. The results of the research are stated in Chapter 3.
7 CHAPTER II THE FUNDAMENTAL CONCEPTS
In this chapter the seven concepts will be addressed. For each concept, there will be a discussion of its role in calculus, how it can be better integrated in courses previous to calculus, and examples of each concept. The examples have been explored as a collaborative e ort between Dr. A. Quesada and myself. The concepts in this chapter are arranged so that the rst ve relate to the topic of function. The last two concepts focus on equation solving approaches and solving inequalities, two skills essential to working with functions.
2.1 Understanding domain and range:
The concepts of domain and range are fundamental to a student’s understanding of functions. In fact, the very de nition of a function requires de ning domain and range.
For example, a standard de nition of domain and range by Stewart [12] follows: “A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. . . The set A is called the domain of the function. The range of f is the set of all possible values of f(x) as x varies throughout the domain.” While this de nition is su cient, connecting this written de nition with a graphical interpretation is essential in our technology enabled curriculum. Having a graphical understanding of domain and range may better prepare students for concepts like analyzing asymptotes and other discontinuities of functions graphically, which are important skills for calculus.
8 Historically, the standard exercises for domain used radical or rational func- tions. Exercises for radical functions would require students to choose a domain to ensure the radicand was non-negative, while exercises for rational functions would re- quire students to choose a domain to keep the denominator from being zero. However, in both sets of examples the concept of range was seldom considered.
Not only does the graphing calculator enable teachers to address and assess students understanding of the domain of a function, but also the range of a function.
In addition, with the graphing calculator, the examples that are presented to stu- dents are no longer limited to the traditional radical and rational function examples. Regardless of the family of functions, the domain and range can be found using a graphing calculator. For instance, the next three examples are domain and range problems that illustrate this point.
Example 1 Find the range of the function h (x) = x √x.
To determine the range of this problem at a level before calculus requires the use of technology. To solve the problem, a student would need to determine the minimum of the function using a graphing calculator. The minimum can be identi ed graphically or numerically. Both approaches are shown in Figure 2.1. This example is bene cial because it integrates the concept of range with the concept of nding extrema.
9 1 x y = x √x 0 0 .05 0.174 0.10 0.216 0 −1 0 1 2 0.15 0.237 x (.25,−.25) 0.20 0.247 y 0.25 0.25 0.30 0.248 0.35 0.242 −1
Figure 2.1: Example 1
Example 2 Find the domain of the function f (x) = √x3 + x2 10x 10
With technology, students can solve more comprehensive problems than are traditionally presented. Example 2 provides students with an opportunity for deeper analysis of the concept of domain by using an uncommon function. Since the rad- icand of a square root must be non-negative, to nd the domain of f, one must determine when the cubic polynomial is non-negative. To solve this example ana- lytically requires the ability to factor a cubic polynomial. Approaching this problem graphically, one would need to input the function, and use the zero command to determine the intervals where the functions is de ned.
3 3
2 2
y y
1 1
0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x x x=3.1622777 y=0 x=−3.1622777 y=0
−1 −1
Figure 2.2: Example 2
10 Textbooks often incorporate the concepts of domain and range in the begin- ning sections of the text. Then these concepts are only reiterated for radical and rational functions. This approach is inconsistent and undermines the importance of these concepts.
Example 3 Find the domain and range of the function g (x) = x4 3x3 + 2x 3
Since this function is a polynomial, the domain is all of the real numbers.
With the use of a graphing calculator, nding the extrema values in this example is relatively straight forward. Since the function opens upwards it will have a global minimum. Therefore, the range will be from that minimum value to in nity. If this example were used for a beginning discussion of polynomials, it could help connect the idea of domain and range throughout the curriculum, while in some basic sense, illustrating the need to be able to nd the local extrema of non-linear polynomials.
4
2 x −2 −1 0 1 2 3 0
−2
y −4
−6
−8 x=2.1409 y=−7.148
Figure 2.3: Example 3
Also, most applied problems require the limiting of the domain. For instance, it would not make sense to have a negative domain value in a modeling problem where the variable represents time. With an early understanding of the domain and range from a graphical perspective, students may be better prepared to address applied problems in both pre-calculus and calculus.
11 2.2 Investigating local and global behavior:
An understanding of the local and global behavior of a function implies a thorough understanding of how the graph of a function behaves. Local behavior refers to how a function behaves in the region close to a certain domain value. Points of interest include asymptotes, holes, intervals where functions are increasing and decreasing, and local straightness. Global behavior refers to the behavior of a function as it grows unbounded. Areas of focus include end behavior, the total number of roots, and the awareness of the complete graph of the function.
When using a graphing calculator to investigate local and global behavior, topics such as scale and error become more important [13]. For example, when graphing a function, a root might not appear in the graphing window. This can be caused by too small of a graphing window or too large of a scale, among other things. Therefore, for local or global behavior, selecting the correct viewing window becomes an essential skill. Investigating local behavior can also reveal if a graph is “locally straight.”
“Then, by looking along the graph to inspect the changing slope, or by plotting a moving line through two close points along the graph, the learner may visualize the changing slope as a global function” [10]. The idea of local straightness is a precursor to the calculus topic of di erentiability. Also, through the investigation of local behavior, the topic of local discontinuities can be introduced and studied.
For global behavior, a topic that becomes important when using a graphing calculator is the complete graph. A complete graph is a graph that contains all of the important elements of the function. “The key is to build an intuitive understanding of what is considered ‘important’ (intercepts, high and low points, increasing, de- creasing, and end behavior) about di erent classes of functions. One modern role of calculus is to prove analytically when a graph is complete” [13]. Thus, understanding
12 the idea of the complete graph before calculus would be useful in calculus for proving that a graph is complete.
“The zooming capabilities of graphics calculators, both graphically and nu- merically, allow a good intuitive sense of the di cult concepts of limits to be explored by students” [14]. This next example shows that it is possible to build intuition of global behavior from an early stage.
x 1 Example 4 Consider the equation y = x . For very large positive values of x, what is the behavior of y?
This example is a simpli ed version of a limit problem. Yet a student could be exposed to this question in a rst course in algebra. The analysis on this problem is bene cial because it introduces the complex topic of limits in a way that is ac- cessible to pre-calculus students. This problem can be approached both numerically and graphically. Regardless of the approach, the problem requires students to think globally about the equation.
2
x 1 x y = x 1 101 0.9 y 1 1 102 0.99 1 103 0.999 1 104 0.9999 0 5 0 10 20 30 40 50 1 10 0.99999 x 1 106 0.999999 1 107 0.9999999 −1
Figure 2.4: Example 4
Example 5 Find the sum of the following geometric series 1 + 1 + 1 + 1 + 2 4 8
13 Just like the previous example, this example is bene cial because it introduces the concept of limit before calculus. The obvious approach to this problem is a numerical approach. Using the graphing calculator, this sequence could be generated and stored as a list. A second list could be created that calculated the cumulative sum. That is precisely what was done in the table in Figure 2.5. However, students might also wish to approach the problem graphically. At this point, it may be necessary x to provide the closed form of the series y = 2 1 . Now, the graph could be ¡ ¡ 2 ¢ ¢ analyzed to visually con rm the numerical solution. Thus, providing a balanced technological approach to the problem.
1 1 x 2x Cumulative Sum of 2x 3 0 1 1 1 0.5 1.5 2 2 0.25 1.75 3 0.125 1.875 y x=10 y=1.999023 4 0.0625 1.9375 1 5 0.03125 1.9688 6 0.01563 1.9844 0 7 0.00781 1.9922 0 2 4 6 8 10 12 8 0.00391 1.9961 x 9 0.00195 1.9980 −1 10 0.00098 1.9990
Figure 2.5: Example 5
x2 x Example 6 What is the behavior of the function g (x) = (x 1) , as x approaches 1.
Upon initial inspection this graph looks like the function y = x. One explo- ration students could take part in is to examine how this function is di erent than y = x. Once students realize there is a hole at x = 1, they can be asked why it is not seen in the graph. Students can be asked to investigate the behavior of the function as x approaches the hole from the left and from the right, a question that becomes important in calculus. Also, students can be challenged to make their own 14 function with a hole at a given point. This constructive approach may allow students to gain more awareness of how removable discontinuities are formed and their a ect on functions.
5
2.0 4
3 1.6
2 1.2 1 y
0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 x −1 0.4 x=1 y= −2 y 0.0 −3 0.0 0.4 0.8 1.2 1.6 2.0 −4 x
−5 Standard Zoom Zoomed in Figure 2.6: Example 6
Local and global behavior are essentially a precursor to the calculus topic of limit. As a student gains more intuition regarding local and global behavior, it would seem that the student will have a bigger foundation to build upon when he or she is introduced to the topic of the limit.
2.3 Finding extrema values:
There are many mathematical problems that require nding the local extrema of a function. For instance, nding the range of a quartic polynomial (e.g. Example 3).
Historically, in courses before calculus, extrema value problems would be limited to quadratic functions since the extrema value can be found by completing the square. Prior to graphing calculators, the extrema of polynomial functions of degree greater than or equal to three could not easily be found without di erentiation. Now, with the graphing calculator, extrema values of any function can be calculated or veri ed.
15 Having this knowledge of how to compute local extrema graphically before calculus opens the door to the study of optimization. Optimization problems are bene cial because they require students to investigate the problem scenario, utilize a function that models the problem, determine the required restrictions, and solve the problem.
A great example of an optimization problem that provides multiple teach- ing opportunities and instructional bene ts to students is the Two Towers Problem discussed in Chapter 1. A student’s mathematical understanding and knowledge of problem solving approaches could be challenged in numerous ways using this problem [2].
With the graphing calculator, numerous families of functions, besides quadrat- ics, can be included in extrema problems. Example 7, obtained from Bittinger et al
[15], illustrates this point.
Example 7 The temperature T , in degrees Fahrenheit, of a person during an illness is given by the function 4t T (t) = + 98.6 t2 + 1 where time t is hours since the onset of the illness. a. Find the maximum temperature during the illness. b. When is the temperature above 99.5 ? c. How long is the temperature above 99.5 ?
The fact that the function is rational provides a new perspective on opti- mization problems. With technology a student could graph this function and nd the max temperature. The question in part b is a fundamental question that is often overlooked. Part b requires a global analysis of the graph of this equation. Upon investigation of the graph, the realization will be made that the solution to part b is not a single value, but rather an interval. Through further analysis of that interval,
16 the solution to part c can be obtained. Therefore, these last two questions incorporate the idea of an inequality into this extrema problem. This example provides a real life scenario that illustrates the necessity of understanding the behavior of numerous families of functions.
101.0 101.0
100.5 x=1 y=100.6 100.5
100.0 100.0 x=4.2067 y=99.5 y 99.5 y 99.5
99.0 99.0 x=.2377 y=99.5
98.5 98.5
98.0 98.0 0 2 4 6 8 0 2 4 6 8 t t
Figure 2.7: Example 7
Example 8 An open box is to be made by cutting a square from each corner of an 11 inch by 11 inch piece of cardboard.
Figure 2.8: Example 8a a. Write an algebraic formula for the volume of the box. b. What values of x make sense in the context of the problem? c. For what value of x will the volume be maximized? d. When will the volume of the box be at most 50 cubic inches?
In part a, an equation that represents the problem must be created. This requires an analysis of the scenario and a translation of it into mathematical terms,
17 which is a skill fundamental to mathematics. Now in part b, the new equation must be evaluated to determine a suitable domain. Thus, the concept of domain is integrated into this problem. Next, in part c, the extrema value is asked for. Finally, part d is similar to part b of Example 7 because it facilitates an analysis of the graph. This problem is worthwhile because it requires the analysis of a cubic polynomial in a real life application.
100 x=1.8333 98.593 100
80 80
60 60 x=3.649 y=50 y y x=0.5 y=50 40 40
20 20
0 0 0 1 2 3 4 5 0 1 2 3 4 5 x x
Figure 2.9: Example 8b
2.4 Understanding relative growth of functions:
Relative growth is the comparison of multiple functions’ global behavior. In Example 9 one important aspect of this concept is illustrated.
Example 9 Below is an incomplete graph of y = 2x and y = x2. How many solutions are there to 2x = x2?
18 10.0
7.5
y 5.0
2.5
0.0 −2 −1 0 1 2 3 x
Figure 2.10: Example 9
This problem requires the recognition that the solutions to the equation will be the intersection points. At this point it is clear there are two visible solutions. However, an understanding of how these functions grow is necessary to get the correct answer. Since y = 2x will grow faster than y = x2, there will be another intersection, hidden outside the view of this graph. Thus, the correct answer to the problem is 3 solutions. This problem is challenging to students because of the graphical analysis required and the fact that students must deduce the behavior of the functions.
The majority of the instruction on relative growth that occurs before calcu- lus is found in the sections on rational functions. With rational functions students compare two polynomial functions through division. “Here students begin to build their own intuitive understanding of the behavior of di erent classes of important functions. The rich intuition developed in precalculus aides in the analytic study of functions in calculus” [13]. Example 10 was created to directly address the relative growth of these di erent families (classes) of functions.
Example 10 Sort the following functions from “slowest” to “fastest” (for large val- ues of x).
f (x) = ex g (x) = √x h (x) = x5 j (x) = x k (x) = 2x
19 For this problem, a graphing approach may not be the most e ective ap- proach. Certainly the correct ordering can be obtained graphically, however, with the graphical approach it may prove di cult to identify and keep track of which graph represents which function. For this reason a numerical approach is preferred.
Table 2.1: Example 10 x f(x) = ex g(x) = √x h(x) = x5 j(x) = x k(x) = 2x 1 2.71 1 1 1 2 5 148.41 2.24 3125 5 32 10 2.20 104 3.16 1.00 105 10 1.02 103 50 5.18 1021 7.07 3.13 108 50 1.13 1015 100 2.69 1043 10 1.00 1010 100 1.27 1030 150 1.39 1065 12.25 7.59 1010 150 1.43 1045 200 7.23 1086 14.14 3.20 1011 200 1.61 1060 250 3.75 10108 15.81 9.77 1011 250 1.81 1075
Thus, it can be seen from Table 2.1 that g(x), j(x), h(x), k(x), f(x) are in increasing order. This problem enables students to be more aware of how functions behave globally in relation to other functions.
2.5 Recognizing transformations:
“With the exception of trigonometric functions, very little attention was paid to the role of transformations on the basic families of continuous functions studied” [16]. Studying the transformation of functions allows for an elementary grouping of functions based on a parent function. Most continuous function are obtained from the transformation of a parent function. However, the section on transformations of functions is often brief and only covers the basic families of functions (i.e. y = x2, y = x ). Once the transformation section is over, with the exception of trigonometric | |
20 functions, the topic is rarely mentioned again. The idea of transformations is not connected to more complex functions (i.e. transcendental functions).
Recognizing the transformation of a function will allow students to apply what knowledge they have of the parent function to the function in question. Histor- ically, lessons on transformations of functions were intended to assist students with sketching the graphs of function. With today’s graphing calculators this is no longer necessary. However, better developing this concept may aid students in visualizing functions and properties of that function. This next problem shows a transformation applied to a cubic polynomial.
Example 11 The graph below shows the curve y = f(x) for some function f. Draw the graph of g(x) = 3 f(x 1).
10
8
6
4
2
0 −3 −2 −1 0 1 2 3 x −2
−4 y −6
−8
−10
Figure 2.11: Example 11
Transformation problems that involve functions with a degree greater than or equal to three are rarely found in textbooks. However, this problem is solved the same way as the common problems are. To obtain g(x), f must be re ected across the x axis, shifted one unit horizontally, and shifted three units vertically. This problem may help students see that all graphs are obtained from transformations.
2x 7 1 Example 12 Let f(x) = x 3 , and g(x) = x . What are the transformations of g(x) that yields the function f(x)? 21 The rational function f(x) was obtained from the transformations of the
1 parent function g(x) = x . To see the transformations of g(x), f(x) must be rewritten.
1 By using long division f(x) = 2 x 3 . Now, the three transformations are no longer hidden, and one could proceed with the solution. This example is bene cial because it connects the ideas of long division and transformations. In addition, this problem may help students to categorize families of functions.
5 5
4 4
3 3
2 2
1 1
0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x −1 x −1
−2 −2 y y −3 −3
−4 −4
−5 −5
Figure 2.12: Example 12
(x 2)2+1 x2 Example 13 Let f(x) = e and g(x) = e . What are the transformations of g(x) that yields the function f(x)?
This examples follows the same process as every transformation problem. However, transformations are rarely discussed with exponential functions. To solve
(x 2)2+1 (x 2)2 the problem, rewrite the function as f(x) = e = e e . Therefore, the transformations required to obtain f(x) from g(x) are a horizontal shift of 2 and a vertical stretch by a factor of e. Again, this example helps convey the presence of transformations throughout all families of functions.
22 2.6 Equation solving approach:
Solving algebraic and transcendental equations is a primary theme throughout high school mathematics [7]. With the integration of technology into the curriculum, a balanced approach for solving equations is needed [11]. Analytical, numerical and graphical solutions should be used in combination to build students understanding and to allow students to think about solving problems in multiple ways.
Solving problems analytically (by paper and pencil) is still a common way textbooks introduce speci c topics. However,
“There is growing evidence that paper and pencil manipulation skill does
not lead to better understanding of mathematical concepts. Indeed, the use of hand-held technology can provide more classroom time for the
development of better understanding of mathematical concepts by elimi- nating the time spent on ‘mindless paper and pencil manipulations’”[11].
With the graphing calculator, there are several approaches to solving equa- tions numerically. Each approach can be bene cial in developing students under- standing of trial and error, recursion and iteration, basic programming, and the In- termediate Value Theorem. Thus, each numerical approach has bene ts that exceed simply nding the solution [7]. Interpreting the graphs of functions is essential as well. The concepts of limits and integration rely heavily on a student’s ability to interpret graphs of functions. Not to mention, most secondary mathematics problems leading up to calculus can be solved graphically. In the next example graphical analysis is required to obtain the solution of the problem.
23 Example 14 The sum and di erence of two functions, f(x) and g(x), are provided below. Determine all values of x in the interval ( 5, 5) that satisfy the equation f(x) = g(x).
20 20
16 16
12 12
8 8
4 4
0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −4 x x
−8 −8 y y −12 −12
−16 −16
−20 −20 f(x) + g(x) f(x) g(x) Figure 2.13: Example 14
First, the observation must be made that if x satis es f(x) = g(x), than x satis es f(x) g(x) = 0. Therefore, the question can be answered by looking for the values of x where f(x) g(x) = 0. By examining the graph of f(x) g(x) it can be seen that the zeros are obtained when x = 2, 1, 3 . Thus, when x = 2, 1, 3 , { } { } f(x) = g(x). This problem is bene cial to students because it emphasizes graphical analysis as opposed to algebraic manipulation. In addition, this problem serves as an introduction to the f g = 0 approach.
Example 15 Solve the equation cos x = x using a numerical approach. Then ¡ 4 ¢ verify graphically.
There are multiple numerical approaches that can be used to solve this prob- lem. One method that latter appears in calculus is the Intermediate Value Theorem (IVT). For this reason, that will be the approach shown here. The (IVT) says, “Sup- pose that f is continuous on the closed interval [a, b] and let N be any number between
24 f(a) and f(b), where f(a) = f(b). Then there exists a number c in (a, b) such that 6 f(c) = N” [12]. Since the cosine function is continuous, we can use the IVT. To begin, consider the function cos x x on the interval [0, 2] with an interval step ¡ 4 ¢ of 0.1. This data is found on the left side of Table 2.2. Since there is a sign change between 0.9 and 1, this implies 0 falls in between those two values. Thus, use [0.9, 1] as the next interval to investigate, with an interval step of 0.01. This data is found in the right side of Table 2.2. This process can be continued to obtain the answer to the desired signi cance. The graph has been included to verify the answer. Two of the bene ts of this problem are the introduction of the IVT and the integration of numerical and graphical approaches to solve this problem.
Table 2.2: Example 15 x cos x x x cos x x ¡ 4 ¢ ¡ 4 ¢ 0.5 0.45955 0.96 0.0248 0.6 0.38286 0.97 0.0130 0.7 0.29636 0.98 0.0011 0.8 0.19989 0.99 -0.011 0.9 0.09344 1.00 -0.023 1.0 -0.0229 1.01 -0.035 1.1 -0.1491 1.02 -0.047 1.2 -0.2847 1.03 -0.060 1.3 -0.4295 1.04 -0.072
2
1
x=0.9809 y=0.9809 0 −2 −1 0 1 2 x
y −1
−2
Figure 2.14: Example 15
25 2.7 Solving inequalities:
Traditional mathematics instruction focuses on solving equations. Once students understand the basic mechanics behind solving an equation, they in theory, will un- derstand how to solve an inequality. Yet textbooks are still neglecting to incorporate inequalities with most families of functions. Solving applied problems that involve inequalities requires students to graph- ically examine the problem showing understanding of the problem scenario and the solution instead of simply applying a procedure to get an answer.
One example of an inequality in an applied problem is Example 7 part b. The students can examine the graph and determine what interval(s) satis es the
4t inequality 2 + 98.6 99.5. This additional analysis may be bene cial as students t +1 study more di cult topics that involve the concept of inequality, such as nding the area under a curve. This next example illustrates how inequalities can be utilized to provide more insight into trigonometric functions.
Example 16 Find all values of x such that sin (x) cos (x) on the interval [ , ].
Throughout the textbook analysis, trigonometric inequalities were rarely found. However, trigonometric inequalities may help students gain a better under- standing of the behavior of sinusoidal functions. For instance, this example may build intuition on the sine and cosine functions and how they relate to each other. Again, the f(x) = g(x) approach and the f(x) g(x) = 0 approach are contrasted. A bene t of the latter approach is only having one graph to examine.
26 2 2
cos(x) 1 sin(x) 1
0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x x
y −1 y −1
−2 −2 sin(x) cos(x) sin(x) cos(x) 0 Figure 2.15: Example 16
Example 17 What value(s) of x make the inequality e2x ex 2 a true statement?
To solve this problem analytically requires much mathematical skill. First by subtracting 2, the inequality now reads, e2x ex 2 0. Since e2x = (ex)2, the inequality can be rewritten as (ex)2 ex 2 0. Factoring (ex)2 ex 2 0 implies (ex 2) (ex + 1) 0. Thus ex 2 or ex 1. This implies, x ln2. This solution can be easily con rmed or obtained graphically. By establishing Example 17 as an inequality, instead of as an equation, students may be more likely to approach the problem with a global perspective, which may better their understanding of global behavior in general.
3
(0.693, 2) 2
y
1
0 −1.6 −0.8 0.0 0.8 1.6 x
−1
−2
Figure 2.16: Example 17
27 CHAPTER III RESULTS
3.1 Textbook analysis
The initial research for this thesis was to investigate how well precalculus textbooks incorporated the concepts discussed in Chapter 2 using a balanced technological ap- proach. There is no doubt that the content of textbooks has evolved over the past twenty years to become more technologically focused. However, the interest of this research was to identify general areas of weakness, to ensure that textbooks continue to progress as the pre-calculus curriculum progresses. To conduct the analysis a scoring rubric was established. Every concept area, except equation solving approach and local and global behavior, was scored in two parts. In the rst part, the main integration of the topic throughout the text was scored. To score this rst part a Likert scale of 1-5 was employed (to see the exact meaning of a particular score, see Appendix B). In general, if a textbook had great coverage of a concept through examples, instruction, and exercises, it would receive a score of 5. The second part of the score measured how technologically balanced the instructional approach was. A Likert scale of 0-2 was used. If the textbook covered a speci c area integrating analytical, numerical, and graphical approaches well, that concept area would receive a score of 2 for the second part. The scale described above was not used for equation solving approach and local and global behavior. For equation solving approach, the integration of graph- ical and numerical approaches was measured. Therefore, two Likert scales from 0-2
28 were used, with 2 representing the high score. One Likert scale measured the inte- gration of the graphical approach, and one measured the integration of the numerical approach. For local and global behavior, there was no apparent progression within each concept. For this reason, a Likert scale was not applicable. Thus, a check box system was utilized to determine whether or not local and global behavior concepts were integrated into the text. The analysis was conducted on twelve widely used1 precalculus books at the collegiate level. Although, the primary focus of this research was at the secondary level, there were several reasons that collegiate textbooks were analyzed. First, col- legiate textbooks were more accessible for the research. Secondly, most secondary precalculus textbooks are written by the same authors as collegiate textbooks, and thus, instructional approaches should remain very similar. Finally, secondary teachers take precalculus at the collegiate level, and how they learn precalculus will in uence how they teach it. The analysis for each textbook was conducted independently and then compiled into a database. The scores are reported in Table 3.1. This data shows that the lowest scores were in the concept areas of relative growth and domain and range. To clarify exactly what a low score implies, consider domain and range. Every precalculus textbook discusses these concepts. However, the de ciency of connecting the de nitions of domain and range with a graphical understanding of the concepts is re ected in a low score. The following conclusions for the textbook analysis were arrived at by comparing the average concept score
[Table 3.1] with the rubric used to assess the books [Appendix B]. For domain and range the average concept score was slightly less than 3. A score of 3 implies that the textbooks included examples on nding domain and range for most families of continuous functions. To improve this score textbooks can use
1Some of the primary authors include, Stewart, Larson, Sullivan, Bittinger, Cohen
29 Table 3.1: Book evaluations Books B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 Average Year 2005 2007 2001 2006 2005 2006 2001 2000 2005 2004 1997 2001 Scale Domain & Range (1-5) 3 4 4 3 3 2 3 1 2 4 3 3 2.92 Balanced Approach (0-2) 0 1 2 1 1 1 1 0 0 0 0 2 0.75 Extrema (1-5) 4 3 4 4 4 5 2 3 3 2 3 5 3.5 Balanced Approach (0-2) 1 1 2 1 1 2 0 1 1 1 1 2 1.17 Relative Growth (1-5) 3 3 3 3 3 2 3 1 1 1 1 4 2.33 Balanced Approach (0-2) 0 1 1 1 0 0 1 0 0 1 0 1 0.5 Transformations (1-5) 4 5 4 5 4 3 3 4 3 2 2 4 3.58 Balanced Approach (0-2) 1 1 2 1 1 1 0 0 1 0 0 2 0.83 Solving Inequalities (1-5) 3 3 4 4 3 3 2 4 3 3 3 4 3.25
30 Balanced Approach (0-2) 0 1 2 1 2 1 0 0 0 0 0 2 0.75 Eq. Solving Approach Graphically (0-2) 1 1 2 2 2 1 0 0 1 2 1 2 1.25 Numerically (0-2) 1 1 2 1 1 1 1 0 0 0 1 2 0.92 Local Behavior Graphically (0-1) 1 1 1 1 1 1 0 1 1 1 1 1 92% Numerically (0-1) 1 1 1 1 0 0 1 0 0 0 0 0 42% Rational Functions (0-1) 1 1 1 1 1 1 1 1 1 1 1 1 100% Piecewise (0-1) 0 0 0 0 0 0 0 0 0 1 1 0 17% Local Discontinuities (0-1) 1 0 1 0 0 0 1 0 0 0 0 0 25% Global Behavior Graphically (0-1) 1 1 1 1 1 1 1 1 1 1 1 1 100% Numerically (0-1) 0 1 1 1 1 1 0 1 0 0 0 0 50% Complete Graph (0-1) 0 0 1 1 0 0 0 0 0 0 1 1 33% Rational Functions (0-1) 1 1 1 1 1 1 1 1 1 1 1 1 100% Piecewise (0-1) 0 1 1 1 1 0 0 0 1 1 0 1 58% graphical and numerical approaches to address the concepts of domain and range. The average score for the concept of extrema was a 3.5. A score of 3 was given for textbooks incorporating use of technology to nd extrema values, and a score of 4 was given for textbooks that had optimization problems for some families of functions. To improve this score textbooks can incorporate extrema problems for most families of functions and include more optimization problems for most families of functions.
Of all the 1-5 Likert scores, relative growth scored the lowest. The average score from the textbook analysis was a 2.33. A Likert score of 2 was given for textbooks that discussed the relative growth of two functions graphically. To increase this score, textbooks can include exercises that require the knowledge of relative growth to obtain the solution of a problem.
The score for transformations was highest of all of the 1-5 Likert scores. The average score on transformations was a 3.58. This implies that on the average, the major area of improvement needed for transformations is applying the topic through- out the entire text. The solving inequalities score was a 3.25. This implies that textbooks, on the average, included instruction and exercises on inequalities for linear, quadratic, and perhaps rational functions. However, this score indicates that inequalities were rarely applied to transcendental functions. Most of the books addressed the concept of local behavior graphically, while fewer than half addressed the concept numerically. Also, there were few textbooks that included examples of local behavior for piecewise functions and functions with local discontinuities. All of the textbooks addressed the topic of global behavior graphically, while only half addressed the topic numerically. Approximately 60% of the books discussed global behavior with piecewise de ned functions. Suprisingly,
31 only one third of the textbooks discussed the idea of the complete graph. A complete graph is a graph that contains all of the important elements of the function. An awareness of the complete graph is essential for approaching problems graphically. To see only one third of the surveyed textbooks provide instruction on obtaining the complete graph of a function was unexpected.
To analyze each textbook, each concept area was compared against a scaling rubric using a Likert Scale. While this assessment was subjective, the goal of this portion of the research was to show general tendencies of current textbooks and to identify concepts that are missing or that could receive a more comprehensive treatment.
3.2 Teacher assessment analysis
An assessment was developed to examine 41 practicing secondary mathematics teach- ers’ uency in several concepts foundational to calculus, including the seven described in this thesis. The teachers were taking part in a continuing education workshop en- titled, “Mathematics Topics Foundational to Calculus.” The teachers represented 21 area schools, and of the 41 teachers, 35 had completed masters degrees in some eld. The assessment [Appendix A] contained 32 questions and had a time restric- tion of 90 minutes. Dr. A. Quesada and Dr. M.T. Edwards helped compile some of the test questions. The assessment was given at the beginning of the workshop to measure teachers’ understanding of topics foundational to calculus. The same test was administered at the end of the workshop to observe if there was any improvement between the pre-test and the post test. The scores were calculated for each test. Next, the teachers’ scores were determined for each of the seven concept areas. Finally, to display the data, the
32 concept area scores were averaged together to give the percentages found in Tables 3.2 and 3.3.
Table 3.2: Pre-test results Domain Local/Global Extrema Relative Transfor- Equation Inequal- Test Range Behavior Growth mations solving ities Average
Average 60.1% 64.4% 56.8% 39.6% 49.4% 56.1% 55.4% 53.9%
The scores from Table 3.2 provide an indication for how well teachers under- stand the seven concepts focused on in this paper. From this table it can be seen that the lowest scores were in the concept areas of relative growth, transformations of functions, and solving inequalities.
Following the initial test, teachers began their training through the workshop. The workshop was 11 days. Four of the days were Saturdays in April and May, 6 days were consecutive in June, and there was a follow up session scheduled in October.
The workshop lessons were primarily inquiry based followed by debrie ng sessions. The activities were team centered and involved the integration of technology. After completing this training the teachers took the test again. The results from this test are shown in Table 3.3.
Table 3.3: Post test results Domain Local/Global Extrema Relative Transfor- Equation Inequal- Test Range Behavior Growth mations solving ities Average
Average 66.7% 75.2% 67.7% 66.4% 71.4% 83.2% 65.9% 68.8%
From Table 3.3 it can be seen that the lowest scores on the post test were in the concept areas of solving inequalities, relative growth, and domain and range.
A paired t-test was conducted on the data from the pre-test and post test. Each teacher’s concept area score was calculated as a percentage. Then for each concept, the list of pre-test percentages was compared with the list of the post test 33 percentages. Thus, for the t-test the sample di erence is a percentage di erence between the pre-test and the post test. The results are found in Table 3.4.
Table 3.4: t-test results Domain Local/Global Extrema Relative Transfor- Equation Inequal- Test Range Behavior Growth mations solving ities Average Sample Di . 7.93 12.63 12.64 27.74 23.78 29.12 12.40 16.37 Std. Err. 5.12 3.01 2.68 5.77 5.29 4.21 5.32 2.95 T-Stat 1.55 4.19 4.72 4.81 4.50 6.91 2.33 7.81 P-Value 0.130 0.0001 <0.0001 <0.0001 <0.0001 <0.0001 0.025 <0.0001
To test signi cance an alpha level of 0.05 was used2. Thus, in all of the con- cept areas except domain and range, there was a statistically signi cant change from the pre-test to the post test. There was also a statistically signi cant improvement in the overall test scores. In general, there appears to be a correlation between teachers participating in the workshop and an improvement in their concept area and total test scores.
The initial teacher analysis revealed scores that were lower than expected. The group was diverse and large enough that this could be an indication of teacher preparation in general. This could be an indication for the need of more in-service training. Whatever the indication, there seemed to be a correlation to teachers at- tending the workshop and dramatically increased post test scores. Based on this apparent correlation, implementing workshops on the topics foundation to calculus might be bene cial for all secondary mathematics teachers.
Possible errors might have occurred by providing inadequate questions to su ciently measure teachers’ pro ciency in any speci c concept area. Also, the results might be biased because of teachers who were unfamiliar with a graphing calculator. However, knowledge of a graphing calculator is essential for a high school mathematics
2Note, that since there were 41 teachers, the degrees of freedom for each column was 40
34 teacher. While this could negatively bias the results, the lack of knowledge of a graphing calculator is also a problem.
In general, there is weakness in the pre-test post test design experiment. “The major weakness is that one can’t be sure that the treatment [instruction] caused the di erence between the pre and post measures” [17]. However, the main goal of this research was to obtain a general understanding of teachers uency of topics foundational to calculus. The post test provided feedback to the workshop organizers, and was valuable to assess the success of the workshop. It was included in this thesis for the sake of comparison.
35 CHAPTER IV CONCLUSION
This paper examined seven concepts that could be emphasized before calculus to better prepare students for calculus. The research of this thesis was conducted to see how well these concepts were implemented in textbooks and how well currently prac- ticing secondary math teachers understood these concepts. Further research could be conducted to determine the e ects of thoroughly implementing these concepts before calculus. Two possible outcomes are higher enrollment rates in calculus and better student success in calculus. Moreover, the seven concepts discussed in this thesis are not all-inclusive. There are several other concepts, such as matrix applications and the use of recursion, that could be integrated before calculus to enable students to better succeed in calculus. Further research could be conducted on these concepts.
Both of these additional research projects could improve the instruction of secondary mathematics. The instruction of mathematics is rapidly changing. “The estimate is that more than a quarter of the mathematics taught before the arrival of the scienti c calculator is not being taught today” [18]. Now, mathematics is at the next tech- nologically based instructional phenomenon, the Computer Algebra Systems (CAS). With the implementation of CAS, the mathematics curriculum will change drastically again. Integrating technology well throughout the entire mathematics curriculum will only become more essential. In addition, preparing students for calculus will continue to be a goal of the secondary curriculum. As the mathematics curriculum continues to evolve, may it be a continued focus to do both of these well.
36 BIBLIOGRAPHY
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39 APPENDICES
40 APPENDIX A TEACHER EVALUATION
Mathematics Topics Foundational to Calculus1 Summer 2006 Password (name )
1. Find the domain and range of the function h (x) = x √x. (4+i)2 2. Express 2+i in a + bi form. 3. Find the inverse of f (x) = x2 6x + 1, making the appropriate restrictions if necessary.
3x2+1 100 4. Approximate, to the nearest integer, the value of f (x) = x2 1 with x = 4 . 5. Suppose a > 0 and b > 0 and f (x) = log x. Which of the following is equivalent to f (a b)? a A. (f (b)) B. f (b) C. f (a) + f (b) D. f (a) E. f (ba) 6. Given the recursive function de ned by f (1) = 3 and f (n) = f (n 1) 6 for n 2, nd f (4). 7. Which of the following is a 1 1 function? A. y = x2 + 1 B. x2 + y2 = 4 C. y = sin (x) D. y = x3 E. y = x | | 8. Find the intersection points of the two circles with the equations (x 2)2 + y2 = 64 and (x + 2)2 + y2 = 49. 2 5x 1 9. Which of the following expressions is equivalent to 2 ? 5 x +x 2 2x2 x+1 2x2 x+1 x 3 x 3 2 A. 1 B. 5 C. 3 D. 1 E. 2x x + 1
1Minor formating changes have been made to some of the problems, but they remain the same in concept and di culty
41 10. The graph below shows the curve y = f(x) for some function f. Which of the four graphs below shows the curve y = 3 + f(x 1)?
10
8
6
4
2
0 −3 −2 −1 0 1 2 3 x −2
−4 y −6
−8
−10
10 10
8 8
6 6
4 4
2 2
0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x −2 x −2
−4 −4 y y −6 −6
−8 −8
−10 −10 A. B.
10 10
8 8
6 6
4 4
2 2
0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x −2 x −2
−4 −4 y y −6 −6
−8 −8
−10 −10 C. D.
11. What value(s) of x make the equation ln (x2 7x + 12) = ln (x 3)+ln (x 4) a true statement?
42 12. Consider the scatterplot depicted below. What type of function would appear to best model the data on the interval shown in the plot?
A. Absolute Value B. Quadratic C. Exponential D. Linear 13. Determine all values of x that make the following inequality true: 16 2x 3 + 9 | | 14. Solve for x, y, and z using matrices. (show some intermediate step) 1 2 3 x 11 4 2 3 y = 4 3 3 1 z 4 15. The height of a projectile after t seconds is given by the function f (t) = .012t2 + t + 2. Find the maximum height of the projectile.
When will the projectile be at least 19.5 feet high? 16. Find all values of x such that sin (x) cos (x) on the interval [ , ]. 17. Which of the following is equivalent to the function f (x) = log3 (x)? x A. g (x) = log( ) B. g (x) = log(3) C. g (x) = log (x) log (3) log(3) log(x) D. g (x) = 3 log (x)
2x 7 1 18. Let f (x) = x 3 , and g (x) = x . What are the transformations (horizontal shift, vertical shift, etc.) of g (x) that give the function f (x)? 19. The class of functions that best matches the behavior of decreasing followed by increasing is: A. Exponential B. Linear C. Quadratic D. Logarithmic
43 20. Consider the graph of the polynomial f(x)depicted below. Which of the following functions generates the graph of f(x)?
60
50
40
y 30
20
10
0 −3 −2 −1 0 1 2 3 4 −10 x
−20
−30
−40
A. f (x) = x (x 3) (x 1) (x + 2) B. f (x) = x2 (x 3)2 (x + 1) (x 2) C. f (x) = x2 (x 3) (x 1) (x + 2) D. f (x) = x2 (x 3)2 (x 1) (x + 2) 21. Describe all values of x that maximize the function f (x) = 5 sin (x) 6. 22. Which of the following is a decreasing function? x A. y = 2x 5 B. y = ln(x) C. y = 1 D. All of these ¡ 2 ¢ 23. What value(s) of x make the inequality e2x ex 2 a true statement? 24. Below is the incomplete graph of y = 2x and y = x2. How many solutions are there to 2x = x2?
10.0
7.5
y 5.0
2.5
0.0 −2 −1 0 1 2 3 x
44 25. Water in a tank will ow out of a small hole in the bottom faster when the tank is nearly full than when the tank is nearly empty. According to Torricelli’s Law, the height h(t) of water remaining at time t is a quadratic function of t. A certain tank is lled with water and allowed to drain. The height of the water is measured at di erent times as shown in the table.
Time(min) Height(feet) 0 5 5 2.8 8 1.768 10 1.2
Find the polynomial that best ts the data. Estimate, to the nearest second, how long it will take for tank to drain com- pletely. 26. Find the equation of a cubic polynomial with zeros x = 5 and x = 1 + 2i. 27. Find the sum of the following geometric series 1 + 1 + 1 + 1 + 2 4 8 28. Sort the following functions from ”slowest” to ”fastest” (for large values of x). (i) f (x) = ex (ii) g (x) = 2x (iii) h (x) = x5 (iv) j (x) = x! (v) k (x) = √x
A. v, iii, ii, i, iv B. v, iii, iv, ii, i C. v, iv, iii, ii, i D. v, i, iii, ii, iv E. None of these are correct x 2 29. Consider the function f (x) = | | . As x approaches 2, what value (if any) √x 1 1 does f(x) approach? | |
x+1 30. Consider the equation y = x . For very large positive values of x, values of y become: A. Very large positive values B. Very small positive values C. Very close to 1, but always greater than 1 D. Very close to 1, but always less than 1 E. None of these
45 31. The sum and di erence of two functions, f(x) and g(x), are provided below.
20 20
16 16
12 12
8 8
4 4
0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −4 x x
−8 −8 y y −12 −12
−16 −16
−20 −20 f(x) + g(x) f(x) g(x) Determine all values of x in the interval ( 5, 5) that satisfy the equation f(x) = g(x). 32. An open box is to be made by cutting a square from each corner of an 11 inch by 11 inch piece of cardboard.
Write an algebraic formula for the volume of the box. What values of x make sense in the context of the problem? For what value of x will the volume be maximized? When will the volume of the box be at most 50 cubic inches?
46 APPENDIX B BOOK EVALUATION RUBRIC
In this detailed rubric each possible score has comments (in italics) to elaborate what was required of textbooks to obtain a speci c score. The number in front of each statement represents the possible Likert score. In this section the term multiple im- plies a number greater than or equal to 2 and the term several implies 4 or more. The term example includes example problems that show the solution process as well as exercises that students are required to solve. A description of how the “Balanced [Technological] Approach” was evaluated is on the last page of this section. I. Solving inequalities with every type of functions
1. Multiple examples of solving inequalities with linear and quadratic functions The book mentions the idea of an inequality
2. Examples on solving inequalities with linear, quadratic, and rational functions The inequality problems are solved using only algebraic and/or numerical means for the listed families of functions
3. The graphical solution is introduced with linear, quadratic, and perhaps rational functions Inequality problems are solved in some manner that relies on the graph: either to con rm or to determine the solution
4. Examples of solving inequalities graphically appear in some other family of func- tions Several examples involving inequalities appear with other families of functions such as square root, polynomial, and absolute value
5. Example of solving inequalities appear in almost every family of functions There are several examples of logarithmic, trigonometric, or transcendental inequali- ties II. Domain and Range of functions
47 1. Contains theory of domain and range The book de nes domain and range using standard de nitions
2. Finding domain and range by looking at a graph The book illustrates multiple examples that show domain and range of a function graphically
3. Contain examples on nding domain/range for most families of functions The book contains several examples that require the identi cation of the domain and range (for the most part algebraically) for multiple families of functions (i.e. quadratic, square root, polynomial)
4. Examples of nding domain and range algebraically & graphically for some families of functions. The book contains several examples that require either the domain and range be found using algebra and veri ed graphically or examples that require graphical assistance to nd the domain and range for some families of functions (i.e. nding the range of a cubic polynomial)
5. Examples of nding domain and range using graphical means with most families of functions The book contains several examples, from most families of functions (i.e. logarithmic, polynomials, and rational function) of nding domain and range in some way that requires analysis of the graph. (i.e. problem #1 in Appendix A) III. Local and Global Behavior
Due to the lack of an obvious progression that textbooks might use to address the topics of local and global behavior, a checkbox system was employed to measure the overall tendency of textbooks.
Local behavior Technique
Numerically The book contains multiple examples of numerical tables that examined a certain in- terval of a function and sequentially (from the left and the right) got closer and closer to a xed point of that function
Graphically The book contains multiple examples of zooming in on a graph to examine the local behavior or of tracing a function in a small interval to examine a certain feature of
48 the graph
Examples of approaching a concrete domain value from either side with the following
Rational Functions The book contains multiple examples where students examine, numerically or graphi- cally, the function’s behavior at vertical asymptotes
Piecewise Functions The book contains multiple examples where students examine, numerically or graphi- cally, the local behavior of piecewise de ned functions
Local Discontinuities The book contains multiple examples where students examine, numerically or graphi- cally, the behavior of a function as it approaches a local discontinuity
Global behavior Technique
Numerically The book contains multiple examples that require students to examine the behavior of a function numerically, using a table of values, as the x values approach positive or negative in nity.
Graphically The book contains multiple examples that require students to examine the behavior of a function graphically, using the trace or zoom feature, as the x values approach positive or negative in nity.
Instructions on complete graph The book provides instruction on how to ensure all of the major attributes (tail be- havior, local max and mins, roots, discontinuities) of the graph of a function are contained in the view. This includes knowing how to change graphing windows as well as knowing the graph of the equation to ensure the major attributes are included.
Examples of students looking at the behavior of the functions as x goes to in nity with the following
Rational Functions The book contains multiple examples that require students to examine, numerically or graphically, the function’s behavior at horizontal asymptotes
49 Polynomials The book contains multiple examples that require students to examine a graphs tail (end) behavior based on the parity and degree of the leading term of the polynomial
With other functions The book contains multiple examples that require students to determine the global be- havior of other functions. (i.e. what happens to a logarithmic function as x goes to in nity) IV. Extrema (max/min)
1. Finding extrema values algebraically The book contains instruction on nding the vertex of a quadratic function alge- braically
2. Examples of applied optimization problems solved algebraically The book contains several examples that require the knowledge in #1 to solve applied problems
3. Examples of nding extrema values using graphical means The book contains examples on ways to calculate and approximate the min/max using a graphing calculator (i.e. 2nd calculate menu)
4. Examples of applied optimization problems graphically with some families The book contains several optimization problems that can be solved both algebraically and graphically (i.e. optimization of quadratic functions)
5. Examples of applied optimization problems graphically with most families of func- tions Several examples are found that require the optimization of functions, and that for the most part, can only be solved, at a level before calculus, with the graphing calculator (i.e. problem #32 in Appendix A) V. Relative Growth
1. Mention of the way a function grows The book discusses the idea of function growth as x goes to (+) in nity
2. Graphical comparison of the relative growth of functions The book has multiple examples that compare two graphs and the behavior of these graphs as x goes to in nity. (The comparison of y = x2 and y = x3)
3. Examples of relative growth to obtain a solution (hidden intersections)
50 The book contains multiple examples that require analysis of the way two functions grow to obtain the solution (i.e. problem 24 in Appendix A)
4. Several examples that require the knowledge of relative growth to obtain the solution The book contains several examples that require analysis of the way two functions grow to obtain the solution
5. Connection of relative growth to the quotient of the two functions The book discusses two functions’ relative growth in connection with the quotient of those two functions VI. Families with Transformations
1. Instruction on basic transformations of functions (quadratics) The book contains examples that examine how transformations (horizontal, vertical, re ection) a ect quadratic functions
2. Instruction on basic transformations of functions with some families of functions The book establishes #1 and has more examples for other families of functions (i.e. absolute value, square root, cubic polynomial)
3. All transformations of functions applied to some families of functions The book contains examples that require all transformations (vertical, horizontal, re- ection, stretch, shrink) be applied to some families of functions, and the book has a variety of examples to help students develop the knowledge of how di erent transfor- mations a ect functions
4. All transformations of functions applied to most families of functions The book establishes #3, but then applies transformations to other functions (i.e. piecewise de ned, polynomials)
5. Application of families of transformations mentioned multiple places outside of section The book connects the topic of transformations to exponential, logarithmic, and trigono- metric functions VII. Equation solving approach
The book provides a rationale and multiple examples that illustrate functions of the form f = g being solved f g = 0. Graphically
51 1. Examples of solving few to no equations graphically 2. Examples of solving some equations graphically 3. Examples of solving most equations graphically
Numerically
1. Examples of solving few to no equations numerically 2. Examples of solving some equations numerically 3. Examples of solving most equations numerically Implementation of Balanced [Technological] Approach
0. poor
The book did not or almost never alternated between analytical, graphical, or nu- merical methods for a speci c concept
1. good
The book contained multiple problem sections that alternated between analytical, graphical, or numerical methods for a speci c concept
2. better
The book contained several problem sections that alternated between analytical, graphical, or numerical methods for a speci c concept
52