FLORIDA STATE UNIVERSITY
COLLEGE OF EDUCATION
USING HISTORY IN THE TEACHING OF MATHEMATICS
BY .AYOKUNLE AWOSANYA
A Dissertation submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Spring Semester, 2001 The members of the Committee approve the dissertation of Ayokunle Awosanya defended on March 7,2001,
lh.afth Jahbowsku Profe sor Co-Directi Dissertation JUz2LAe-d Herbert Wills III Professor Co-Directing Dissertation sa.-. Q Emanuel Shargel Outside Committee Member
Akihito Kamata" Committee Member
Approved:
David Foulk, Chairperson, DepartAnt of Curriculum and Instruction This dissertation is dedicated to my parents and nll family members who encouraged me to realize this life's dream, but who are no longer around because they have passed to the patbeyond.
This dissertation is also dedicated to all those who have encouraged me throughout my career including my wife and chiIdren.
... 111 ACKNOWLEDGMENTS
I am deeply grateful to many individuals who made this stu-, possible. fin ' no words adequate to thank my major professors, Dr. Elizabeth Jakubowski and Dr. Herbert Wills El, for their valuable assistance, encouragement, and advice. They deserve the highest recognition for their contributions to my learning. So I acknowledge with pleasure my indebtedness to them.
I express my appreciation to Dr.Emanuel Shargel, who served as a committee member for his time, effort, and advice in preparation of this study.
Sincere appreciation is extended to Dr. Akihito Kamata, for his valuable assistance in statistics interpretations, and as a member of the committee. No acknowledgment would be completed without the appreciation of my family-my wife, Janet, who spent countless hours in supporting my effort, and my children, for their love and understanding that made my time much more enjoyable during this study. Also, I wish to thank the students who participated in this research study at Florida High School and their teachers for their cooperation throughout this research study.
iv TABLE OF CONTENTS
List of Tables ...... vi i Abstract ...... viii
Chanter
1. THE PROBLEM ...... E
Introduction ...... Uses of History ...... Benefits ofHistory ...... Purpose ofthe Study ...... Format of Dissertation ......
2 . REVIEW OF LITERATURE ...... 10
Curriculum Reform ...... 10 Biographies ofMathematicians and Their Contributions ...... 16 Teaching and Learning Mathematics ...... 19 Pedagogical Devices Used in Designing the Lessons ...... 24
3 . DE.SIGN AND PROCEDURES ...... 27
The Questions ofthe Study ...... 27 The Pilot Study ...... 28 Instructional Materials ...... 29 Lewis Carroll LessonsExercises ...... 29 Archimedes' LessonsExercises ...... 30 Triangular Numbers ...... 31 Oblong Numbers ...... 32 Sophie Germain LessonsExercises ...... 33 Instruments ...... 34 Content Validity of the Tests ...... 34 Internal Consistency of the Tests ...... 37
V Chauter P&€s
Follow-Up Interviews ...... 37 Experimental Procedures ...... 40 Data Analysis ...... 41 Hypotheses ...... 41 summary ...... 42
4 . RESULTS...... 43
Preliminary Analysis ...... 43 Hypotheses ...... 48 Item Analysis ...... 50 Interviews ...... 53
5. CONCLUSIONS AND RECOMMENDATIONS ...... 58
Conclusions and Discussions ...... 59 Educational Implications ...... 61 Suggestions for Future Research ...... 62 Mathematics Curriculum ...... 65 Mathematics Teaching ...... 66 Concluding Remarks ...... 67
APPENDICES ...... 69
A . Instructional Materials for the Experimental Group ...... 71
B . Instructional Materials for the Control GTOUP...... 132
C . The Instruments (Posttests and Solutions) ...... 192
D . Human Subject Committee (Approval Letter) ...... 201
REFERENCES ...... 202
BIOGRAPHICAL SKETCH ...... 206
vi LIST OF TABLES
Table Eazs
1 . Experiment Group: FCAT (Math) Test Scores Pretest ...... 44
2 . Control Group: FCAT (Math) Test Scores Pretest ...... 45
3 . Posttest: Experimental Group ...... 46
4 . Posttest: Control Group ...... 47
5 . Posttest Means and Standard Deviations ...... 48
6 . Confidence Interval ...... :...... 49
7. Tests of Within-Subjects Effects ...... 50
vii ABSTRACT
The results reported here are the product of the research titled: Using History the Teaching of Mathematics. The subjects are students in two classes of Algebra D course at Florida State University High School-36 students-males and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra Il is a course that is usually taken by high school seniors in 12' grade and a few 11' or 10' grade students which explains why the ages of the students are mostly 18 and a few 17 and 16 years old.
In this investigation, both quantitative study and qualitative study were employed.
The quantitative study was the main study-a teaching experiment using quasi- experimental methodology that involves two groupsgroup 1 and group 2. Group 1 is the control group, where various algebraichathematical concepts, or topics were taught or explained to students with the necessary formulas. Group 2 was the experimental group in which the accounts of the historical origin of algebraiclmathematicalconcepts and the mathematicians (Lewis Carroll, Archimedes, F'ythagoras, and Sophie Germain) who brought forward or created the concepts were used to augment pedagogical lessons and exercises used for this study as the main feature of pedagogy. The qualitative study augmented the main quantitative study; it was a follow-up interview for students to probe further an in-depth rationale for the research theme-using history in the teaching of mathematics.
viii The statistical analysis result indicated that there is a significant difference in the mean of score for the control group students and the mean of scores of the experimental group students. The mean of the scores of students' performance in the experimental group is greater than the mean of scores of students' performance in he control group; and the interview questions' responses indeed corroborate the fact that the use offistoy in teaching mathematics does improve learning and understanding of algebraid mathematical concepts.
ix CHAPER 1 THE PROBLEM
Jneoduction
There are various or many approaches devised to aid better understanding for students learning mathematics. One of such strategies is the use of history in the teaching and learning of mathematics.
Teachers of mathematics are continually looking for ways to improve instruction.
Several writers have suggested that using history in the teaching of mathematics will have a variety of educational benefits.
Although it is difficult to accept some so-called benefits due to history, this study shall focus on whether using history in teaching mathematics has an effect upon student achievement on those topics connected with history in their presentation.
In this study, both a quantitative study and qualitative study was employed. The quantitative study was the main study-a teaching experiment that involved two groups-group 1 and group 2. Group 1 was the control group, where the lessons without history on four characters-Sophie Germain, Lewis Carroll, Archimedes, and
Pythagoras-were used by the students to learn fundamental Algebra concepts. Group 2 was the experimental group in which the accounts of the historical Algebra concepts and the mathematicians associated with such concepts were used to augment pedagogical
1 2 lessons for the four characters used for the lessons as the main feature of pedagogy. The students in both groups were Algebra II students from a local high school. Indeed, many writers have eloquently suggested the use of history to teach mathematics because they claimed that it will engender the following uses and benefits:
uses of History
Among the possible uses of history cited by such writers are that the use of history may help to improve mathematics learning and make it more interesting and more meaningful to students.
0 Tying specific historical characters or events to a topic being taught.
0 Approach the teaching of mathematics consistent with its historical development.
That is, let history dictate the order in which students are exposed to the terms in their cuniculum.
As impressive as these claims appear to be, most writers fail to give actual evidence to support their profound claims. I wish to find out whether history can indeed enhance the learning of math for students.
Also, several writers have claimed that certain uses or advantages accrue from the use of history in learning mathematics. Among the claimed uses are: e Uses of history of mathematics in teaching and learning are useful for incorrect
conjectures, counterexamples, reference to famous mathematicians, application
to school mathematics, proofs, numerical patterns, geometrical representations
and visualization.
In a recent issue of Marhernnrics Teacher, Loretta Kelley argued eloquently on the uses and advantages of using history for teaching and leaning by stating that 3 mathematics has a place in history, and history has a place in mathematics.
Unfortunately, the rich history of mathematics is seldom included in either history or mathematics courses. It is up to mathematics teachers, to learn about the history of our subject both because it is interesting and because it can be useful in our teaching. The writings of these authors suggest three major reasons for bringing the history of mathematics into our classrooms are: (a) to put mathematics in its historical context; (b) to show the interaction of mathematics with the cultures of which it has been a part; and
(c) to improve the learning-and therefore teachinwf mathematics.
. History reveals that mathematics has been done by real people, some of whom were very interesting in addition to being successful in other fields. Integrating biographies of mathematicians into the relevant content is an effective way to
introduce students to these people. Students remember material better when thev
can relate it to a name. dace. or anecdote. For emule. when studvine the
Cartesian coordinate svstern, students may be interested to learn that Descartes
was a well-dressed, eccentric figure whose wealth allowed him to stay in bed
until 11 :00 A.M. every day. He came to the attention of Queen Christina of
Sweden, who invited him to tutor her. Descartes refused her request for a year,
until the queen sent a battleship for him. Chnstina required Descartes to instruct
her every morning from five o’clock to six o’clock, during the Scandinavian
winter, no less (Bergamini, 1963). . Biographies of female mathematicians can show that women have been involved in mathematics from its earliest recorded history to the present. Hypatia was a
fourth- and fifth-century Greek mathematician who contributed to the algebra of 4
her day. Despite several offers, she never married, claiming that she was
“wedded to the truth” (Osen, 1974, p. 29). One day, on her way to teach classes,
Hypatia was dragged from her chariot by a religious mob, slashed by sharp oyster
shells and killed.
Besides classroom use, writers have often claimed that history has more to teach
US. Parallels are observed between the historical development of mathematical concepts and the way that they develop in an individual mind. History, some claim, suggests ways of teaching. As George P6lya said, “Having understood how the human race has acquired the knowledge of certain facts or concepts, we are in a better position to judge how the human child should acquire such knowledge” (P61ya, 1962, p. 132).
In particular, studying the history of a given concept within the mathematical community may offer clues as to why that concept is not easy to understand. A connection exists between subject matter that is difficult for a student to grasp and that which was difficult for the mathematical community to accept.
Luetta Reimer and Wilbert Reimer (1999) in their research and writing titled,
“Connecting Mathematics with Its History: A Powerful, Practical Linkage,” argued the case for the uses and advantages of history as follows. Historv Motivates
A succinct example given by the Reimers in their research is as follows:
Amy rested her chin in her hands, but her eyes were wide open as she listened to Mrs. Cruz. Jose stopped drumming on the table with his pencil and Natasha quit twisting her hair. The roomful of children sat spell-bound, captivated by their teacher’s voice. It was mathematics period, and Mrs. Cruz was telling a story about Archimedes. When she got to the part where Archimedes was speared to death by a Roman soldier, the students winced and groaned. 5
Archimedes was so busy at work on a mathematics problem that he hardy noticed the soldier approaching,
Mrs. Cruz observed, “What kind of problem do you think Archimedes was working on?”
Mer hearing of Archimedes, Amy, Jose, and Natasha were more open to an introduction to geometry. They realized that mathematics is something that real people do-and sometimes even die for. Sophie Germain is an example of a person profoundly influenced by a historical tale.
A Human Endeavor
History integrated into the mathematics classroom reminds students that mathematics is essentially a human endeavor. In this increasingly technological world, students may begin to infer that mathematics is done only by calculators and computers; they may assume that they have no personal need to understand mathematics or perform computational tasks. Modem advances hove simplified many processes, but problem solving is still essentially a human task. Discoveries have been made because living people had need of them, and human problems are still the catalysts for mathematical experimentation today.
The Power of the Story
If mathematics is something people do, stories of mathematicians doing it may inspire others to do the same. Nothing communicates like a good story, even the most restless or seemingly uninterested student may listen to a story. Many of the figures in mathematics history lived fascinating lives. There was
Sophie Germain. who had to sneak candles into her room so she could study at night against her parents’ wishes. There was Galois, shot and killed in a duel at the age of 21, 6
but not before he left an important legacy in group theory, And there was Newton, who
started making good grades after a showdown with the school bully. Those were normal
people for the most part, although some had extraordinary powers of concentration. Like
everyone, they had obstacles to overcome. Their lives are outstanding testimonies to the
power of hard work and determination.
Where Did l7mI Come From?
For as long as mathematics has been taught, students have reacted to much of the
explanation with a puzzled “Why?” In many situations, mathematics history answers this question. History in the mathematics classroom counteracts students’ natural reluctance
to accept something simply because “that’s the way it is.” Understanding the origins of
certain ideas, such as algebraic notation or “imaginary” numbers, makes students more receptive to even difficult or abstract mathematical procedures.
Sometimes a brief anecdote becomes a perfect springboard to a new area of
study. The story of Galileo, for example, is a useful way to introduce the development of
the scientific method. The metric system makes more sense when one has heard the
story of Lagrange and the chaos in Europe when each city had its own system of weights
and measures. Why do we use (I, b, and c to label the sides of a triangle? Well, let me
tell you about Euler. Will students be more interested and more likely to remember what they study when they connect mathematical truth with real persons and practical
situations?
Mathematical Linkages
Exploring historical origins often opens a window to the interconnectedness of
mathematics itself Descartes’ revolutionary combination of algebra and geometry led to 7 a breakthrough in mathematical understanding. The story of how he applied one to the other to create analytical geometry shows students that one mathematical concept often enhances and strengthens another in a remarkable way.
Connectine with Tod ay
Finally, mathematics history creates a bridge from the past to the future. While still a boy, Blaise Pascal was determined to design a machine to help his overworked accountant father. It took him several years to perfect his invention, but finally he succeeded. Today, that machine is considered the first mechanical calculator. For this reason, a popular computer language is named Pascal.
In 1993, newspapers announced that “Fermat’s Last Theorem,” a problem that has intrigued mathematicians for hundreds of years, had at last been solved: Whether the proof was flawed or not, the headlines reminded the world that mathematics is not dead but still alive and developing.
Benefits of History
Many writers have claimed benefits that ensue from the use of history in teaching and learning mathematics. Some suggested benefits of history in this regard are: . Helps to increase motivation for learning..
0 Makes mathematics less frightening. . Pupils derive comfort !?om knowing that they are not the only ones who find mathematics problems difficult, that is, many other students find mathematics
difficult.
0 Gives mathematics a human face - making mathematics appealing to
study-associating mathematics with mathematicians of the past. 8
. Changes pupils' perceptions of mathematics as a dull subject to a more interesting one.
0 Less aversion to mathematical activities. . Knowledge of efficient approaches to current topics.
Realization that mathematicians are real peopleit is not uncommon to spend a
great deal of time on a single mathematical topic.
Purwse of the Study
As educators, we would like to see students develop their reasoning and thinking capabilities rather than their abilities to memorize meaningless facts. Emphasizing reasoning in the use and development of mathematical ideas means involving students in teaching and learning as they rarely have been before. I am using history of mathematics to evaluate students' understanding because in the field of mathematics education, the concept in lessons for the four mathematical characten-Archimedes, Sophie Germain,
Louis Carroll and Pythagom is fundamentally important in schools and widely used in real-world situations, Ernest (1998) and Porn (1998) recognized the role for the hlstory of mathematics in the teaching and learning of mathematics to reason as the conceptual watershed which plays at the borderline that separates elementary from more advanced concepts (Lesh, Post & Behr, 1988).
The purpose of this study is: (a) to investigate the use of history to understand mathematicaYalgebraic concepts as a teaching experiment using a controlled statistical experiment with two groups in an Algebra II class (quantitative study), and (b) to compare performance of the two groups in the given mathematical tasks. As Steffe
(1 994) wrote, research in mathematics education has turned to providing a useful 9 understanding of how a students mathematical concepts may be built up piece by piece by an educator whose aim is to foster the development of genuine mathematical understanding. This is applicable in using history for teaching and learning mathematicaValgebraic concepts.
Since this is an exploratory study, I will try to analyze the students' thinlcing strategies in solving a task in which possible learning models could be derived which would also be very useful for teachers. Studies accumulate evidence regarding the nature of students' reasoning and the critical factors to be taken into account in facilitating their development.
This study could be useful for curriculum developers and teachers, especially in evaluating teaching strategies and methods used in school. Furthermore, the importance of the use of history to teach, coupled with difficulties in acquiring the ability to teach, may give a rationale for studying the four mathematical characters in this research study.
Form at of DissertatiOQ Chapter 2 contains a review of literature, in which research is related to the use of history in teaching mathematicaValgebraic concepts associated with (mathematicians)-
Sophie Gemain, Archimedes, Lewis Carroll and Pythagoras. Chapter 3 describes the sample, the pilot study and the procedures. Chapter 4 presents the resulting data and the statistical analyses of those data. Chapter 5 discusses recommendations, suggestions for future research and conclusions. CHAPTER2
REvlEWOFTHELITERATuRE
A thorough review of International Dissertation Absmcts revealed that there are
353 dissertation abstracts out of 10,790 abstracts in Mathematics Education that used history as a key-word in their dissertation titles in all its ramifications that can be classified into the following categories: mathematics and cuniculum reform, biographies of mathematicians and their contributions, and teaching and learning mathematics at the secondary school level.
Curriculum Reform
Several studies concerning the use of history in teaching mathematics dealt with curriculum reform. Bell (1992) examined the contents of middle school mathematics teachers' curriculum from a historical perspective. The main aim of his dissertation was to develop a history of mathematics course appropriate for prospective or current middle school or junior high teachers. This course takes into consideration .the background and needs of these prospective or current teachers. Most history of mathematics textbooks assume that students have had calculus and are aimed at mathematics majors. A student's background for this course was assumed to be one year of high school geometry and two years of high school algebra. This course emphasized the history of the material
IO 11 that is included in the middle school mathematics curriculum. These topics included: number systems, computation, number theory, algebra, geometry, probability, and statistics. Attention was also paid to how mathematicians from different areas solved problems. Material not normally a part of the middle school mathematics curriculum such as trigonometry and calculus was covered to give students a taste of the overall flavor of mathematics. The course was taught in the fdof 1991 at Wingate College to current middle school teachers. The students’response to the course was favorable and indicated that the course was an appropriate course to teach to prospective or current middle school teachers. The course did prove to have too much material for a three hour course.
D~OW(1997) researched curriculum reform in terms of revitalizing and using original sources, history, and writing, in undergraduate mathematics. His research pointed out that curriculum reform has become a very prominent issue in school mathematics. One avenue of reform in which there is currently a great deal of interest is the use of original sources, mathematical history, and writing assignments in the teaching of school mathematics. His thesis was a study of this current movement. The early chapters of this thesis reported some of the major recent and current efforts from the mathematical community to move in these directions. In these chapters, the author surveyed some textbooks that are part of &is movement, some of which are in pre- publication form; the work of individual researchers, including course materials and
Courses that they have designed; the Institute on the History of Mathematics; and institutions such as St. John’s College that have implemented these ideas even beyond the mathematics curriculum. The later chapters contain a summary of the author‘s attempts 12 at using writing assignments and some mathematical history in his teaching at Idaho
State University. ln addition, these later chapters presented sample course outlines written by the author for three calculus courses using historical sources and for two courses on the history of mathematics. These later chapters also presented two units written by the author for use in undergraduate mathematics courses, one on Peano
Arithmetic and one concerning the Hardy-Weinberg Principle from genetics.
Lindsay (1998) examined in his dissertation the history of mathematical pedagogy by examining how American college and university mathematicians involved themselves with pedagogical questions and educational reform in the late 19" century and early 20* century. This was a period during which educational practices and institutions which had prevailed through tnuch of the nineteenth century were challenged both ideologically and demographically: new subjects, notably the natural sciences, were pressing for parity with the classical curriculum of Greek, Latin, and mathematics; the justification of education on the basis of mental discipline was being contested; the proper relationship between the secondary schools and the colleges was being debated; and the student population was increasing markedly, especially in the secondary schools. Coincident with such developments, there was emerging in the United States a national community devoted to the promotion of mathematical research, a community dominated by pure mathematicians. This dissertation investigated how the mathematicians responded to the ongoing educational ferment, The nineteenth-century period is analyzed through the careen andviews of educators associated with the Committee of Ten Report of 1893, the first major national study of secondary school curricula in the United States. Tlie pedagogical proposals of mathematical astronomer Simon Newcomb (1 835- 1909), 13
chairman of the mathematics subcommittee, were given special attention. The central
figure in the early twentieth-century period is E. H.. Moore (1 862-1932) of the University
of Chicago. Moore, an intellectual and organizational leader in the emerging community of pure mathematicians, advocated a pedagogical program for both secondary schools and colleges which would orient mathematics toward applications in science and engineering. In part this was a natural extension of the position of Newcomb.
Evans (1983) researched the brief history of mathematics with reference to a brief history of mathematics as responsible for demonstrating the psychological barriers these concepts pose. For most children, mathematical knowledge begins with counting. This dissertation addresses the issue of how early counting knowledge is extended to form new mathematical concepts. Two extensions of counting knowledge were considered:
(a) the child's knowledge of zero and its role as the identity element and @) the child's understanding of infinity in relation to closure. A brief history of mathematics is reported demonstrating the psychological barriers these concepts pose. However, historically, once these concepts became a part of the number system, mathematical thought was greatly enhanced, The close relationship of these two concepts to the counting numbers makes it likely that they will be among the first movements beyond counting that children will make. Their inclusion in the child's number system may similarly enhance mathematical thought. One group of children (kindergartners and first graders) were asked to solve zero and non-zero problems and then interviewed about their knowledge of zero. While the younger children often made errors, by first grade, children were virtually perfect at solving zero problems. Still, these children often said that zero was not a number. Error patterns as well as the children's comments suggested 14 that zero knowledge grows out of knowledge of “nothing,” exists somewhat separately from counting knowledge for some time and only later is coordinated with existing number knowledge. Another group of children (kindergartners, first, second and third graders) were interviewed to determine their knowledge of infinity. The results suggested that recognition that there is no greatest number is related to knowledge of numbers greater than one hundred, the ability to add systematically and the ability to induce that one may always add one to make a larger number. These conditions, however, were not sufficient for a child to recognize that there is no largest number.
Such a conclusion may require that the child is able to recognize and to resolve contradictions in his thought.
Eunsoon (1994) examined in his dissertation with reference to the use of history, instructional computers in high school mathematics reform-its theory and practice (curriculum reform). This study described and analyzed how computers were used and not used for high school math instruction in two high schools through a technology application project. The purpose of the study was: (a) to identify influences from the district’s community, history, and organizational structure that affect the development and implementation of the project, and (b) to describe the nature and effects of the implemented project as teachers experienced it. A case study approach was used to investigate the societal and organizational contexts that affected the implementation of instructional activities stimulated by the project. Observations, interviews, and document reviews were the primary means of data collection. This study found that the objective of the project to change the high school mat.h cdculum from students being eliminated from higher-level courses either by self selection or difficulty to one in which 15 computer experiences increased motivation and understanding rarely occurred. Instead,
existing organizational routines, faculty culture, and resources influenced not only the
degree to which instructional computer activities were implemented but also the form
and content of those activities. The faculty cultures in the two sites studied, especially
’ the teacher‘s perception of what to teach, how to teach, and whom to teach, were the
critical factors affecting the kinds of computer activities which were allowed or
encouraged. These faculty cultures grew aut of the history, community setting, and
organizational routines of each high school. Lacking power to change organizational
conditions through the technology application project, administrators and teachers
rationalized intended instructional computer use through bureaucratic curriculum
models.
Hayden (1981) in his dissertation titled: A Hisiov ofthe New Mufh Movernenf in
the United Sfures,studied the development and spread of the “new math” programs of
the 1960s and their roots in the revolution in mathematics in the nineteenth century. the
progressive education movement, and World War II. In the years 1850-1950, the changes in mathematics gradually filtered down into the colleges. The first high school
“new math” program was initiated in 1951 by the University of Illinois Committee on
School Mathematics. The most influential “new math” program, covering grades K-12,
was initiated by the School Mathematics Study Group, formed in 1958. By the mid-
1960s. “new math” programs had been widely adopted at all grade levels. In the
secondary schools, “new math” became the new stah quo. In the elementary school,
the “new math” programs were less successful, due to inadequate teacher trainin’g and
the opposition of many mathematics educators in the progressive tradition. At all levels, 16 the further development of “new math” programs ended with the 1960s. largely because the forces that brought the “new math” into being had ceased to exist.
muhiesI of Mathematicians and Their Contributions
This is another area most dissertations concentrated on in their research studies by examining the lives and educational works of mathematicians.
Ricardo (1983) focused his dissertation on Bertrand Russell and his origrn of the set-theoretic paradoxes. His research study’s main goal examined the role played by Bertrand Russell in the origin and development of the paradoxes of set theory. First, he described what he called “a standard interpretation of the origins of the set theoretic paradoxes.” Ricardo described the explanation of other historians of how the paradoxes were discovered. He also presented a discussion of the major reasons why this interpretation-with all its small variants-was important. Second, he analyzed Russell’s early philosophical and mathematical background to his writing of The Principles of
Mathematics. He tried to show how his study of Kant, Hegel, Cantor, among others, was related to how Russell became interested in philosophical thoughts, and how he originally studied the relations between mathematics and logic. Throughout this analysis, he described the stages of early drafts of the principles, and the influence of
Georg Cantor which most historians, influenced by Russell’s own later recollections, have overlooked. Next, he described how Russell discovered Cantor‘s paradox; his own paradox; and how he formulated the elements that would provoke the discovery of yet re .- another paradox. Russell‘s correspondence, especially with Louis Couturat, and the comparison between the manuscript of the principles with the printed version are used to 17 support his analysis. He maintained that Russell himself discovered at least two of the three most famous paradoxes of the theory of sets-and set forth the elements to provoke another one; that there are some inconsistencies contained in the principles related to the paradoxes; and, that Russell’s later recollections of these developments are inaccurate.
Fourth, he discussed the emergence of the nowadays called “semantic paradoxes.” The description of the polemics and debates concerning the Well-Ordering Theorem that helped clarify how these paradoxes originated and were widely accepted.
Marini (1998) examined the contributions of a woman mathematician-Mna
Spiegel Rees’ contributions with regard to her life and educational work, especially her contributions before and after World War Jl to the ofice of Naval Research and the City University of New York. The study examined her published papers, manuscripts, correspondence, and books, and included interviews with her co-workers. Resources for the study included the archives of the Graduate School and University Center of The City
University of New York and the Mina Rees Library. Dr. Rees’ importance in the fields of mathematics and mathematics education and how Dr. Rees thought about graduate education and its impohce in the future, and to women’s career opportunities in the field of mathematics were discussed. She was concerned about and predicted many problems and issues in higher education such as the open admission policy, setting national goals and standards, and maintaining the quality of education. The study also reviewed Dr. Rees’ war-time contributions at the Ofice Naval Research (Om)which were important in the development of mathematics and the development of computers.
The study concluded that Dr. Rees had a significant influence on mathematics in colleges and graduate schools, most notably through the motivation of women in the fields of 18 mathematics during and after the war. Dr. Rees' influence was important in the establishment of the Graduate School and University Center of The City University of New York. Her contributions and philosophy are models for the next generations of women in the mathematics professions.
Mohini (1990) studied in his dissertation the lives and contributions of selected non-Western mathematicians during the Islamic medieval civilization. His main god was to present materials in the teaching of history of mathematics as well as for teaching mathematics in a classroom using a historical approach. It is basically a iibrary study concerned with the literature on the history of Islamic mathematics during the medieval period. The time period was limited between the ninth and thirteenth century. Five outstanding mathematicians were selected. They were AI-Khwarizmi, Abu Jaafar
Muhammad Ibn Musa; Ibn al-Haytham, Abu Ali-AI-Hassan Ibn AI-Hassan; AIBi~uni,
Abu Ryhan Muhammad Ibn Ahmad; AI-Khayyarni (or Khayyam), Ghiyath al-Din Abud-
Fath Umar; Al-Tusi, Muhammad Ibn Muhammad Ibn Al-Hassan. The chapters devoted for each mathematician were basically divided into two parts: His life and his mathematical contributions. Each chapter first described a mathematician's life and his popularity in the world of science. Then his contributions in other fields other than mathematics were briefly described. His contributions in astronomy or geography were
focused only to those that are related to mathematics. Lastly, hls contributions in mathematics and where applicable, his influences were elaborated in considerable detail.
Nordmann (1997) in her dissertation titled: The Life undMulhemnrics ofJulia
Robmson, stated that the history of mathematics is long and rich; however, few women are part ofthat history. Until this century, social and political barriers prevented women 19 from obtaining a formal mathematics education. Access for women to mathematics education in the 20” century has led to the development of a number of prominetit and extraordinary female mathematicians, One of these mathematicians is Julia Bowman
Robinson. In recent years some material has been written about the lives of the few principal female mathematicians, but for the most part, little has been written about their actual ,work in mathematics. Thus, in this paper, the author explored both the life and the mathematics of Julia Bowman Robinson. This examination includes Julia’s extensive work on Hilbert’s Tenth Problem as well as her work in number theory and game theory. Particular focus was placed on her work in the area of decision problems which eventually aided in the discovery that there is no algorithm for determining whether a given arbitrary diophantine equation is solvable in the integers. The research reveals that
Julia Bowman Robinson was an amazing person and an incredible mathematician. Teachine and Leamine Mathematic5
Many other studies examined the use of history in teaching and learning mathematics.
Davis (1994) in his dissertation examined the analysis and the re-interpretation of several critical issues surrounding the practice of teaching mathematics-jncl uding the nature of mathematical knowledge, the place of education and the process of learning.
The dissertation focused on the notion of “mathematics teaching as listening,” a phrase which is used both figuratively and literally: to re-interpret various phenomena and as a practical basis of teaching action. The discussion of each issue begins with the broad considerations and moves to the more specific implications for teaching. It is intended to reflect the underlying theoretical and investigative frameworks which-drawn from recent 20 developments in such fields as Continental philosophy, ecology, biology, and cognitive
science-are rooted in post-Darwinian evolutionary metaphors rather than the analytic model of Descartes. As such, they help us to sidestep the sort of dualistic thinking that
give shape to much of current mathematics teaching. Wed(1986) in his dissertation focused on the changes in course content,
Cm-hIa and the teaching of secondary school first-year algebra from 1950 to 1985. Issues covered included trends in subject matter, changes in the school population, adoption of, and disappearance of certain teaching materials, conflict between theory- oriented mathematics and applications-oriented methods, and socioeconomic factors
influencing these changes. The research was qualitative. Data were obtained through
interviews with leading mathematics educators; a survey of cumculum guides issued by
supervisors to teachers in Maryland, Virginia, and the District of Columbia; and a topic- by-topic examination of texts produced between 1950 and 1985. Literature examined included publications of the National Council of Teachers of Mathematics (NCTM),
dissertation abstracts, yearbooks of the NCTh4, the daily press, book reviews, materials
listed in the Education Index, and data resulting from a search by Educational Resources
Information Center (ERIC) Clearinghouse on Higher Education. Throughout the period
of data accumulation, the researcher was a member of mathematics teaching staffs in the jurisdictions covered. The study found that major changes occurred in the content of basic algebra teaching between 1950 and 1985. Content has been in Constant motion,
initially because ofreports by the School Mathematics Study Group (1958) that gave rise to a movement popularly called the “new Math.” Some years after the introduction of these concepts a critical reaction occurred. Texts and study guides ofthe 1970s show 21 progressively less emphasis on the SMSG recommendations. An adjustment has taken
Place by 1985. There is a diversification in content and teaching of basic algebra, with a variety of texts available to suit differing Courses of study.
Foret (1998) studied in his dissertation an overview of historical development of mathematical induction. His dissertation surveyed literature in this area including papers and texts. Chapter 1 discusses the importance of induction in modem mathematics, including the NCTM recommendation to incorporate more of the topic in the curriculum.
Some material about great mathematicians beginning in the 1500s (Maurolycus) and later (Fernat, Peano, Pascal, Lucas, and DeMorgan) and their contribution to mathematical induction can be found in Chapter 2, which includes selected source material including some original correspondencebetween Fermat and Pascal. Lucas and the Tower of Hanoi, Pascal's Triangle, the Gamblers problem and applications are also discussed in Chapter 2. Chapter 3 reviews the literature on teaching induction and proof by mathematical induction. Chapter 4 presents an overview of different college and high school textbooks on teaching mathematical induction, looking at the different methods used to teach mathematical induction (geometric, drawings, verbal description, and formal proof). Appendix A shows an approach to teaching mathematical induction including using history to motivate students as well as Some geomemcal examples to reinforce the concept. A summary chapter is included which discusses the evolution of mathematical induction in textbooks and papers, and the influence of the new mathematics reform. Conclusions and suggestions for further research on mathematical induction were also included. 22 Canady (1983), in his dissertation titled: A Study of the Efects of rhe Essential
Elements of the Instructional Model on Mathematics Achievement researched the effects of a specific instructional model called the Essential Elements of Instruction Model, which is often referred to as EEI, and its effects on mathematics achievement in elementary school grades. Chapter 1 included an overview of the study with a brief discussion of the history of research on quality instruction, the purpose of the study, statement of the problem, limitations of the study, and definition of terms. Chapter 2, the review of the literature, was divided into three sections. Section I described the history of research on effective instruction. Section It included descriptions Of inShUCtiOM1 models that were designed to utilize research proven strategies. Included in this section were Bloom's Mastery Learning Models, the Beginning Teacher Evaluation Study's
Academic Learning Time Model and Rosenshine's Direct Instruction Model. Section Ill included the background and complete description of the EEI Model based on Huntefs
Clinical Theory of Instruction. Chapter 3 included a restatement of the problem, purpose of the study as well as a description of the procedures for the implementation of the study. As a result of the research conducted and reported in Chapter 4 of this study, a number of conclusions were drawn: (a) The EEIModel significantly affects scores on mathematics computation skills, mathematics application skills, and total mathematics achievement; (b) Students from grades four, five and six treated with the EEI Model achieve significantly higher scores in mathematics computation skills, mathematics application skills, and total mathematics achievement; (c) students from grades three, four, and six treated with the EEI Model achieve significantly higher scores in mathematics application skills; (d) male and female students treated with the EEI Model 23 achieve equally in mathematics commutation skills, mathematics application skills and total mathematics achievement. Recommendations were made in Chapter 5 of this
They included the encouragement of the inclusion ofthe Mode] in staff developmflt Programs and university education classes. Suggestions were also made for further mdY in Other grade levels and other subject area. Polehni (1995) in his dissertation titled: Teachers Perceptions of Chge:An hvninution of Mathematics Teaching Lij2 Histories examined teachers' perceptions of their own development. That is, teachers' perceptions of changes that occurred in their thought andor practice over time and their perceptions of what kind of experiences or challenges might have influenced those changes. It provided an opportunity for the participants to reflect on changes they believe occurred in their thought and practice throughout their career and the processes by which changes might have occurred. It took into account aspects of teachers' lives that might have influenced them to change their mathematics teaching, including personal, social, and professional influences (Teachers' thought) was interpreted as encompassing teachers' knowledge and beliefs about mathematics, mathematics teaching, and mathematics learnhg. The discussion of teachers' knowledge was concentrated on content knowledge, pedagogical content knowledge, and curriculum knowledge. The study considered a context of curriculum development in the state of Sao Paulo, Brazil. Topical life history was chosen as a research methodology, stressing the participants' mathematics teaching life histories.
F~~~ mathematics teaching life histories were conducted. TWO of the Participants Were secondary mathematics teachers and two were elementary teachers. The data included informal formal interviews, class observations, field notes, and written documents. 24
The mathematics teaching life histories showed that the participants felt the need for
change and started developing a commitment to change when they started reflecting more on their thought and practice through the curriculum change process. Close support was important at the beginning of their process of change. The secondary mathematics teachers changed the way they taught certain content, especially geomet
My research will focus on the effect of the use of history in the teaching and
learning of mathematics by comparing performances of two groupscontrol group and
experimental group. The control group will use lessons that have no history while the experimental group will use historical lessons to supplement their lessons for the study.
Pedagoeical Devices Used in Desioning the Lessons
In order for students to respond to lesson goals with ease, it is important that the pedagogical devices are relevant and easy to follow. The devices used range from:
Discovery, fill in the blanks, completing a task based on the given definition, and worked
examples; computing missing items, using given formulas, completing proofs of the
needed algebraic identities and the use of tables, providing for necessary prerequisite
concepts and skills, spiraling important facts, interrelating historical characters and the
relevant math, quizzes to assess and strengthen concepts, transfer items and application
of earlier topics in new lesson goals.
Stating goals is a way to promote appreciation of topics at hand and numerical
examples involving small familiar numbers which can facilitate applications involving
big numbers to illustrate the value of generalization at hand. Rules to be discovered are never provided lest a student be tempted to look ahead rather than engage in the necessary thought 25 Generally, each character is designed starting with a historical account that
includes the mathematical concepts associated with the historical character under
discussion. The facts are arranged in a logical order that are coherent with subsequent
goals and tasks required to be responded to by the students.
The goals are written and boxed, usually, beginning with goal 1,2,3 and so on...
Following the goals immediately are the numerous relevant tasks to be completed before
going back to respond to the relevant goal. This mode of pedagogy is repeated and
replete in all the lessons designed for Sophie Germain, Carl Gauss, Lewis Carroll,
Ramanujan, and Pythagom as well as Archimedes. Usually, emphases are laid on
' examples and the mathematical or algebraic concepts using various developmental
exercises for the tasks.
A good example can be found in the Sophie Germain lesson: An integer n is a Sophie Germain prime if the following conditions are met: (a) the integer n, itself, must
be a prime, (b) one more than twice the prime n must also be a prime. That is, 2n + 1 is
also a prime.
Consequently, the student should be able to determine if 11 is a Sophie Germin
prime.
In cases where completion of a proof is required by means of filling the blank spaces provided, the relevant developing exercises are certainly a usefd part of the
PedWxY. The use of cogent examples can also prove useful- as was the case in determining the prime numbers between 1 and 31 using the sieve ofEratosthenes. The student should 26 be able to cw out instructions provided to figure out the prime numbers between interval of given integers.
In the completion of developmental exercises, the student is encouraged to discover strildng relationships that come up that are usell in determining subsequent questions. This was the case in Archimedes’ Arbelos, and the Salinon exercises.
Using bold face, italics, underlining, or direct students’ attention in desired direction in lesson figures are also useful pedagogical devices.
Developmental exercises involve unambiguous data. Students need consistent data to identify a function. Mixing data for two or more functions not only lessens opportunities for discovering the desired function but makes such discovery highly unlikely. For example, in the Lewis Carrol example, we literally included the exercise:
2(12’+ 1’)=[ l’+( 1’ A student completed this as:
2 (12‘+ 1’) = [17’] + (1’) while the lesson wanted
2(122+ 12)=[135]+(112).
It turns out that 290 is he sum of squares in two different ways as shown above. Allowing this ambiguity detracts from the lesson. Thus, the item was changed to one having a unique representation as a sum of squares. Interestingly, there is a mathematical resolution as to which numbers are uniquely the sum of squares. Such an exploration might be unleashed among the more capable students in a normal classroom setting. A less troublesome exploration might be to find those numbers that can be expressed the sum ofa pair of squares. This is not trivial. The first few, in order, are:
2,4,5, 8, 9, 10, 13, 16, 17, 18,20,25,26,29,32,34,36,37,40,45, .... CHAPTER 3
DESIGN AND PROCEDURES
The various components of the design of the study, questions to be answered, the pilot study, the sample of the study, the instructional materials, and the instruments are described in this chapter. The discussion of the data collection and anaiysis is also provided.
The Ouestions of the Study
This study was designed to compare two distinct treatments for using historyin teaching mathematics in Algebra Il classes. History was used to augment lessons forth experimental group for the iessons on algebraic/mathernatical concepts for each of the four characters-Lewis Carroll, Archimedes, Pythagoras, and Sophie Germain. All traces of historical account or the use of history were excluded in the lessons for the control group for the four characters also. The pretest for both groups consisting of 36 students-18 students for the experimental group and 18 students forthe control group-was the (Florida Comprehensive Assessment Test) FCAT scores.
This study was a quasi-experimental design designated a “Pretest-Posttest Control
Group Design” by Campbell and Stanley (1963). The basic design of this study form:
27 28 Experimental group R 0, 0,
Control group R 0, x 0, where R = Random assignments of groups, 0, = Pretest, 0, = posttest, and x = experimental treatments.
The investigation was designed to find out whether:
1. Students in the experimental group that use history along with their lessons
perfom better than students in the control group whose same lessons were devoid of the use of history.
Pilot Study
A pilot study was conducted with three lZa grade students. The purpose was to learn of any necessary modification in the research plan. The main objectives were to:
1. Test whether the use of history in the teaching and leaming mathematics
(algebraic/mathematical concepts) facilitates understanding of the concepts of the
lessons/exercises more than when history was not used in the lessons for the research study.
2. Get accustomed to the techniques of data collection.
3. Detenine the time required for the written test/follow-up interview. The pilot study revealed that 50 minutes is the estimated time to be given. ,
4, Deternine the appropriateness of the task problems that were used for the assessment.
5. Clarify research questions.
The pilot sewed as a useful trial run for this researcher. It provided useful infomation insights which suggested a reconceptualizationof the research design. 29 After a description ofthe pilot study and its results, the reconceptualized methodo]ogy for the main study will be outlined.
Instructional Materials
The instructional units or lessons for both groups had the Same contents on algebraic/mathematical concepts on the works ofthe mathematical four characters- Lewis Carroll, Archimedes, qrthagoras, and Sophie Gennain that were used for the study except for the fact that the lessondexercises for the experimental group contain history of the mathematical characters that contained some origins of the algebraic or mathematical concepts while the control group lessons/exercises did not contain the historical account. -CarrollLessons/Exercises -CarrollLessons/Exercises Lewis Carroll lessons/exercises for both groups centered on using Carroll's famous theorem: Twice the sum of squares problems. Eight years before his death, Lewis
Carroll remarked in his diary that doubling the sum of squares of two integers is always the sum of the squares of some pair of integers. That is, if x and y are two integers, then for each positive integerx, and for each positive integery,
2 (2+J)= (x+y)2+ (x -y)?
Suppose x = 7 andy = 2
2 (72 + 22) = (7 + 2)2 + (7 - 2)2
2 (49 + 4) = 92 + 52
2 (53) = 81 + 25
106 = 106
theorem, 2 (2+ 9)= (x + y)2+ (X -yy was learned with exercises that involves finding squares ofnumben, and proofs that involve filling blanks to m*e *e resulting sentences to be me, The experimental group's lessondexercises were tided: 30 LEWIS CARROLL while the control group's Iessons/exercises with all traces ofhistory removed were titled: ADDING SQUARES (see Appendix A and B).
Archimedes' LessonsExercises Archimedes lessons contain two main sectionsthe Abelos and the Salinon. The Abelos (a curvilinear figure-no straight lines) lessodexercise involves finding the radii and area of semicircles that involved the use of the formula A = IT ? where A =area, x = 3.14, and r = radius. The Salinon is another curvilinear figure that involves finding radii and area of semicircles. Both lessondexercises in both groups retained the titled
Abelos and Salinon but with all traces of history removed from the control groups'
Pythagoras' lessons/exercises.
The Pythagoras lesson centered on the famous Pythagorean theorem that states that in any right triangle, d+b*=2
The exercises, therefore, involved Pythagorean triples in Various forms, manax numbers, and oblong numbers.
Pvthagorean mules in various forms.
Complete each Pythagorean mpk.
(47, 1104,-
Solution: (47, 1104, 1104+ 1)=(47, 1104, 1105)
(31,-, 481)
Solution: (31,481 - 1,481)=(31,480,481) L,9242 Solution: L,924,924 + 1) 31 924,925)
924 + 925 = 1849
1849 = 43
L924,925) = (43,924,925)
(9, L2 Solution: 9 = 81
81 =40+41
(9, 3= (9,40,41) Trianrmlar Numbers
The triangular numbers is a topic dear to the hearts of the phythagoreans and can be represented by Tetracyts. The Tetractys is a triangular array of dots in four rows. The first row consists of a single dot. The second row has two dots in it. The third row contains exactly three dots and so on.
0 0 0 00 00 000
TI TZ , T3 32
The first triangular number is 1.
The second triangular number is 3.
The third triangular number is 6.
Math books use a nice notation for the triangular numbers. Znstead of writing: The first triangular number, they simply write T, or T( 1). Using this notation,
T, = I
T,=1+2=3
T,=1+2+3=6 and so on and so forth. Oblofip Numbers
Just as each triangular number is the sum of consecutive positive integers, the oblong numbers are sums of consecutive even numbers starting with the integer two, thc first even number.
The first oblong number is denoted by B(1) thus, B(1)=2 OT ( 1) (2) = 2
B(2) = 2 + 4 =6 or (2) (3) = 6
B(3) =2+ 4 + 6= 12 or (3) (4) = 12
B(4) = 2 + 4 + 6 + 8 = 20 or (4) (5)= 20 and so on and so forth.
The use of oblong numbers is readily useful when finding the square of a number that ends in 5. For example, compute the following squares mentallv Without using the common multiplication algorithm: 33
4S2 = B(4) 52= 4(5) 57 = 2025
105’ = B( 10) 5’ = 10 (11) 52 = 11025
95’ = B(9) 52 = 9 (10) 52 = 9025
9995’ = B(999) = 999 (I 000) 52’ 99900025
Souhie Germain Lessons&xercises
The Sophie Ckrmain lesson deals with prime numbers. A prime number is a number that is only divisible by itself and 1. Examples are 2,3,5,7, ... on the other hand, a positive integer is said to be a Sophie Germah prime ifand only if (a) the integer n, itself, must be a prime, and (b) one more than twice the prime n must also be a prime. That is, 2n + 1 is also a prime.
For example: 3 is a Sophie Germain prime
since (a) 3 is a prime
(b)2p+ 1 =2(3)+ 1=7isaprime
In the control group lesson in which all various forms of history were removed,
Sophie Germain primes were referred to as “certain primes.’’ Another important topic in school mathematics that was in Sophie Germain lessons is the Sieve of Eratosthenes-a technique for finding prime numbers in any given range of numbers. Also, an ememely valuable and useful topic in the theory of numbers is factoring, both of integers and polynomials, these are included in the lesson for Sophie Germain. It involves using the use of factoring such expressions (special ProducWS:
(x + y)’ = 2 + w + y’ 34 Appendix A and B contain copies of the instructional materials or lessons which were used for the experimental group (lessons with the use of history) and control group (lessons without the use ofhistory).
Instruments The investigator used the following tests:
1. Florida Comprehensive Assessment Test-FCAT Math Test as Pretest.
2. Posttest on all the four character lessons as the achievement test.
truction of the Instrum ents
1. Pretest: Instead of designing a pretest, the Florida Comprehensive Assessment
Test Math scores were used as a measure of prerequisite knowledge of mathematical skills.
2. Posttest: Achievement tests on four lessons used fort the study were written to measure the mastery of the materials from each unit for both experimental and control groups. only one form of the posttest was given to both groups. The items in the posttest were split into two parts to determine its reliability. The posttests consist of 4 tests, one each of the four mathematicians (characters) used for the study.
Content Validitv of the Tests
In order for a test to be valid in an experiment it must be a valid test. A panel of experts was established to judge the validity of each item used in our tests. The panel consists of three people, one professor of mathematics (chairperson) from Florida State University, one professor from the Mathematics Education Department at Florida State, and one mathematics professor (chairperson) at Florida A&M University. The Panel members were asked to judge the adequacy with which items Ofthe test meaSuTe the 35 students’ understanding of the subject matter of the instructional unit of the study in a form provided on each lesson on Lewis Carroll, Archimedes, Pythagom, and Sophie
Germain. The four tests were each designed to measure various mathematical concepts and skills.
For example:
Skill Number 1
Evaluate: (a) B(11)
Solution: B( 11) = 11( 12)
(b) BU9)
Solution: B(19) = 19(20) = 380 Skill Number 2
Solution: T(199) = J99(200l= 199(100) = 19900 2
Skill Number 3
Calculate the following squares mentally without using the common multiplication algorithm.
(a) 1005*
Solution: 1005’ =B(100) 5’
= lOO(101) 25
= 1010025 36
(b) 95’ =B(9)5’
= 9( 10) 25
= 9025
(c) 252 =B(2)9
= 2(3) S2
= 625
Note: This technique only works for numbers that end in 5.
There were several items to measure the skill associated with each goal. The experts reviewed each item and established whether the item measured the target skill.
The test on Lewis Carroll consists of 21 questions, test on Archimedes consists of 8 questions, test on F’ythagoras consists of 40 questions, while the test on Sophie Germain consists of 10 questions. Experts were asked to rate their response as strongly agree, agree, and does not agree; and were asked to provide a suggestion for improving any item that was regarded as defective. The tests were administered to individual students in two different Algebra II classes at Florida Highne class was the experimental group and the other was the control group. There are 18 students in each class, making it a total of 36 students used for this study for approximately 5 weeks. This process was necessary because if a student could not read or understand the item, there might be no evidence that he understood the concept. Students’ reading and understanding of the test items were encouraging. All students who were involved in reading the test items had no difficulty or problem. 37 Jntemal Consistencv of the Tests
The internal consistency of the posttest w determined by using split-half reliability. Each test was split into two halves, the even-numbered items and the odd- numbered items. Then the score of the even-numbered half-test WBS correlated with the scores of the odd-numbered half-test using the Spearman-Brown formula:
Where, r, = correlation Coefficient between the two split-half-tests and n = number of test, and T,
the reliability coefficient
The reliability coefficient for skills items in the posttest were computed by using the Kuder-Richardson formula:
Where
K = the number of item in the test
M = the arithmetic mean of the test scores
s = the standard deviation of the test scores. Follow-UD Interviews
Interviews were conducted by the investigator with 4 students, 2 students from the experimental group whose scores in the posttest were very high and low; and also 2 38
'Om the control goup whose Scores were very high and low, The interview questions are as follows:
Wnie w oms(Fo IlOW-uD OU eshons)' for Exwn'mental Groue
1. Do YOU like Mathematics generally?
Mydo YOU like/dislike math?
Please explaidtell me more.
2. Do you like Algebra?
MY? Please tell me more,
3. Please rate your performance in Algebra as: (a) Excellent
(b) Good (c) Average (a) Below Average
WY? Please tell me more.
4. Do you think using a Historical account to teach Algebraic concept will help
students understand Algebra? Answer Yes or No, and then explain why.
5. Do the names in the historical account-Lewis Carroll, Archimedes. Pythagow
and Sophie Gennairkterest you?.
6, D~ you think a historical approach can help to improve perceptions of mathematics and attitudes to it by making it interesting, alive, and part of human
history adculture? (Paul Ernest, 1998) 39 7. Do YOU think the use of history of mathematics can help increase motivation for
learning? Give reasons for or against.
8. Do YOU think the use of history of mathematics can make mathematics less
frightening to students? Give reasons for or against.
9. DOYOU think the use of history of mathematics will make students derive comfort
fiom knowing that they are not the only ones with problems? Give reasons for or against.
10. Do you think the use of history of mathematics will give mathematics a human
face? Give reasons.
11. DO you think the use of history of mathematics can change students’ ~rceptiom
of mathematics? Give reasons for or agw bterview Ouest ions IFollow-UO 0uestions) for the Control Group
1. Do you like Mathematics generally?
Why do you likddislike math?
Please explaidtell me more.
3 1. Do you like Algebra?
Why? Please tell me more.
3. Please rate your performance in Algebra as: (a) Excellent (b) Good
(c) Average
(d) Below Average 40 WY'l
Please tell me more.
4. Do YOU think using a Historical account to teach Algebraic cOnc slp students understand Akebra? Answer Yes or No, and then expla- inteMews were tape. recorded.
Emrimental Procedures
The study involved 36 algebra II students (females and males) at Florida Hi@. School. One class consisting of 18 students represented the experimental group while another class consisting of 18 students represented the control group. FCAT (Florida
Comprehensive Assessment Test) math scores were obtained from the school's director.
Each group received their lesson as follows:
Week 1: Lewis Carroll Lessons: The first 3 days-usually from Monday-
Wednesday, students worked through the lessodexercises which were collected at the end of the class period which usually lasted 50 minutes. At the end ofthe lessodexercises, a quiz was administered, graded, and reviewed with students before administering the test Week 2: Archimedes Lessons: Exercises, qui& and the test Were completed in the same manner as described in Week 1. Week 3: mhqoras Lessons: Exercises, quiq and the test Were completed in the
Same manner as described in Week 1.
Week 4: Sophie Lessons. Exercises, qUk and test were completed as described in Week 1, finally. 41 There are different Algebra II cla~~esat the school that place daily-5 * week. The first Class begins from 7:30 a.m.-8:20 a.m. (50 minutes), and this is the
period for the experimental group students. The second class begins 8:30 a.m-9:20 a.m. (50 minutes), and this is the control group
Data Analvsis
Two aspects ofthe research study were considered in the data analysis. The first aspect w to plan the data analysis to test the null hypothesis of the study. The second aspect was to analyze the thought processes of the interviewers.
Prior to the instructional presentation, the pretests for both groups were examined. The Florida Comprehensive Assessment Math Scores for each of the 36 students for both experimental and control groups were used as the pretest to determine initial differences in the students' prerequisite skills which were found to be about the same for the average of both groups. J-hwtheses
The Statistical Package for the Social Sciences (SPSS) provided by the Florida
State University Computing Center was used in analyzing and testing the null hypotheses in this study.
A /-test program was used to test or compare the means of the experimental group and the control group.
There is no difference between the mean score in the control POUP and the mean
score in the experimental GOUP. There is a difference between the mean score in the control WUP and *e mean H,: Score in the expenmental group (claim). 42 H, = null hypothesis; H, = alternate hypothesis.
Summary In the fall of2000, two different Algebra II classes at a local high school (Florida High) were involved in this study, Using History in the Teaching of School Mathematics.
One class consisting of 18 students was the experimental group and the other class consisting of 18 students also was the control group. Four major lessons were used for this study. The experimental group lessons were augmented with history of mathematics indicating the original sources relating to the algebraic/mathematicalconcepts in the exercises while the lessons for the control group were devoid of any traces of the use of the history of mathematics. This study lasted 4 weeks-a week each for each lesson, and at the end of each week a posttest was administered to both groups. Each student's FCAT math score was obtained and was used as pretest at the beginning of this research study and the mean FCAT math score for each group was found to be about the same. Each posttest (4 in all) administered lasted 50 minutes. Preliminan, Analvsis This chapter reports the findings of this researchfirst, the quantitative study as well as discussion of the findings, and secondly, the qualitative study (interviews with students).
The Ouantitative Study The quantitative study addresses statistical analysis and descriptive analysis. The descriptive analysis was to give a description of students' understanding of algebraic/mathematical concepts of the lessons and exercises on the four characters used--lewis Carroll, Archimedes, Pythagoras, and Sophie Germain. The r-test is the statistical test that was used and appropriate for comparing the mean of students' scores in the control group and the mean of students' scores in the experimental group. Statistical Analvsis
The means of scores of students in the experimental group and control group as well as the information on he tables used were obtained using Statistical Package for
Social Sciences (SPSS) subprogram of t-test analyses. The results are presented in the appropriate corresponding tables.
43 44
Student FCAT Math Score 1. 315 2. 322 3. 287 4. 325 5. 332 6. 337 7. 303 8. 285 9. 338 10. 334 11. 396 12. 295 13. 321 14. 302 15. 289 16. 313 17. 294 18. 394
W.The mean of the experimental group's FCAT math test scores = 32 1.22. 45 Table 2.
Control Grow FCAT math) Test Scores (Pretest]
Student FCAT Score
~ 1. 3 74 2. 332 3. 342 4 309 5. 332 6. 313 7. 303 8. 308 9. 321 10. 298 11. 281 12. 332 13. 340 14. 332 15. 329 16. 297 17. 281 18. 298
m. The mean of the control group's FCAT math test scores = 3 18.55. Comparison of the Pretest Means
It is observed that the pretest mean in the control group and the experimental
group is about the same. 46 Table 3.
Student Lewis Arch Pytha Sophie Average % 1 90 10 90 40 57.50 2 81 100 73 100 88.50 3 90 90 90 100 92.50 4 81 100 88 50 79.75 5 76 100 90 90 89.00
NA.Lewis = Lewis Carroll Test; Arch =Archimedes Test; F’ytha = Pythagoras Test; Sophie = Sophie Germain Test. Average mean = 82.86. 41 Table 4.
Note: Lewis = Lewis Carroll Test; Arch = Archimedes Test; Pytha = pythagoras Test; Sophie = Sophie Germain Test. Average mean = 70.13. 48 HvDotheses
The following hypotheses were tested and analyzed in this study at LY; = 0.05. rhotheses One
There is no difference in the posttest mean of scores of students in the experimental group and the control group.
1-1, : p, - p2 = 0 Alternate Hvnothesis
There is a difference in the posttest mean of scores of students in the experimental group and the control group.
K:Pl # CLZ 1. The Mean
T-Test
Group N Mean Std. Deviation
Experimental 1.OO 18 82.86 10.43 Control 2.00 18 70.13 11.31
The mean and standard deviation for students' scores on the post-test were computed for each of the two goups. The mean of the scores in the experimental group
82.86 is greater than the mean ofthe scores in the control group70.13 (see Table 5). 49 The standard deviation measures the spread of a relative frequency dis~bution; therefore, since the standard deviation of the experimental group - 10.4330 is less than that of the control group - 113079, it means that students in the experimental group have more high scores than students in the control group (Table 1).
Table 6.
Confidence Interval I t-test for Equality of Means 95% Confidence Interval Std. Error of the Difference Difference
Average Equal variances assumed 3.63
& : PI- p2= 0 , the null hypothesis, I-& should be rejected because 0 is not within 95% confidence interval; that is, 0 is not in the interval 5.3663 and 20.0159.
Accordingly, H, : PI ?t p, ,should be accepted (do not reject), , $olit-Plot Factorial Desim
A consideration of the tests of within-subjects effects reveals that when topics are considered, there is a main effect oftopics due to the fact that the p-value was 0.039
(Table 7), which is less than a = 0.05 - indicating that there was a significant difference between the means of the four topics for this study. 50 Table 7.
Tests of Within-Subiects Effects
Source Typemsum df Mean F Sig. of Squares Square TOPIC Sphericity 1933.465 3 644.488 2.886 .039 Assumed
TOPIC * GROUP Sphericity , 797.854 3 265.951 1.191 .317 Assumed Error (TOPIC) Sphericity 22774.431 102 223.279 Assumed
However, there was no interaction effect between topic and group since the
Pvalue = 0.3 17 (Table 7) which is greater than a = 0.05. The reason lies in the fact that the difference in the means of the four topics as regard the performance of students were the same for both groups. The treatment effect was constant in the topic-group. The topic-group performed better than the control group on all of the four tests.
From the analysis of the mean, standard deviation, 95% confidence interval and the split plot factorial design, it is evident that students in the experimental group achieved significantly higher mean scores on the posttest than students in the control group on the post-test, regardless ofthe topic. Item Analvsis
Overall, students in the experimental group performed Slightly better than students in the con~olgroup on questions regarding tests on the k%sOnShercisesused for this study. This maybe due to the fact that they Were Well acquainted with the
52 Archimedes Test
Students in both groups performed very well in the first part-the Abelos in items
1-4, that require finding the radius of the big semicircle, if the radii of the small semicircles are 92 and 1400 (Appendix C). The solution is simply adding 92 + 1400 =
1492.
Most did not do Well in the Sdinon test that required finding the area of the Salinon, if the radii of the small semicircles are 11 and 43. The solution is (ll+43)’n:=54’1112916~.
PvthaPoras Test
About 70% of all the students did not get item 1 right; it involves drawing an may for the fourth oblong number, but majority of them performed better in items 11 and 12. Item 11 required listing in order, the first 11 triangular numbers: they are 1,3,6, IO, 15, 21,28, 36,45,55,66. Item 12: (1995, ,Lis a Pythagorean triple. The solution is (1995, 1990012, 1990013). Solution lies in (1995)’= 3980025
3980025 = 1990012 + 1990013 thus (1995, I3 = (1995, 1990012,1990013). Soohie Germain Test In the Sophie &main test (Appendix C) 90% of all students in both the Control
and experimental group got items 1 to 4 correct. It requires giving an example Of a number p such that p is a prime and 2p + 1 is also a Prime.
For example, item 1 : Is 23 such a ndm? (a) Yes (b) No a prime‘ The answer is yes, became 23 is a prime, and 2 (23) + 1 = 46 + 1 = 47 is,
54 PhilosoPhical SsuPtions underlying the case study draw from the qualitative rather than the quantitative research paradigm,
This Paofthe research was designed with the intention of focusing on students'
UdenmCh ofthe concepts in the lessons. It was designed so that the researcher could monitor a student's thinking as a case study, coupled with responses to follow-up questions to explore how students in Algebra II come respond to the questions on how effective the use of history in the teaching of school mathematics is.
There is a wide agreement (Hiebert & Behr, 1988) that if research is to inform instruction, it is important to analyze mathematical structures and students solution processes in light of developmental precursors (or sometimes, prerequisites) to the knowledge needed to function competently in a domain. These precursors or cognitive building blocks have been called by many names: key cognitive processes (Hiebm &
Wearne, 1991), key informal strategies (Heibert & Behr, 1988), theorems in actions (Vergnaud, 1983). They me mental constructions considered necessary for meaningful learning.
The goal of this analysis was to determine what sorts of experiences and understanding were critical for understanding algebraiclmathematical concepts in the lessons/exercises by using history in the teaching of the concept used for his study
(Appendix A and B). Indeed, Algebra II students' responses to follow-up quedons epitomized and reiterated almost all the responses of students in the control group as well as in the experimental group. 35 For example, students' response corroborated the fact that the names in the historical account were interesting and indeed interested to them and that it was from the stories about Lewis Carroll, Archimedes, magoras, and Sophie Germain that they were able to gain a much better appreciation towards mathematics. They stated that historical approach can help to improve the perceptions of mathematics and attitudes to it because it divesthem (the students) a historicid backpound knowledge of how each aspect of mathematics Or algebraic concepts in question came about,
They also contended that the use of a historical approach can help increase motivation for learning because if a student has both the background and the relevant mathematical concept, the student can be motivated to work more problems. The use of history can also make mathematics or algebra less frightening (Ernest, 1998) to students because it engenders the required background and confidence the students need in doing mathematics or algebra problems.
It is important to know that mathematics will help students derive comfort from the fact that they are not the only one with problems, explained the students. Another important response from the students was explained thus: the use of history in teaching algebraic or mathematical concepts will acquaint students with a lot,of historically important mathematicians that derived most equations or mathematical concepts that will in the ultimate engender positive response from students in a small Class and evoke interests among the students to like mathematics. conc~usion,students' response explained that the use of history in the teaching and learning mathematics or algebraic concepts will change dents'perceptions Of mathematics because Once the students know where mathematical ideas came from, then 56 they will start to appreciate mathematics a little more; on the other hand, there will
be students whose perception ofmathematics won't change, maybe because of fie
attitude them in the past of how the teacher explained mathematics or algebra to them Of from students not caring at all for math a subject; that is, the use of history in the teaching and learning certainly won't be an antidote for poor performance in school mathematics. Also,.in the follow-up questions (interview), all the students interviewed-both the high achievers and low achievers-readily agree in their responses that using history to teach school mathematics will help students understand algebra.
They contended that it will help students generally to know where the mathematical or algebraic concept came from so that students will have a better understanding of what they are doing. Another reason given is that if a person likes history, then he or she will like the use of history of mathematics to learn algebraic/mathematical concepts because it will engender a clear understanding of the origin of the algebraic concept he or she is learning and where the concepts come from. Another reason the students gave is that the use of history will bring in a much needed and related mathematical digression SO that concentration is not solely on working problems in mathematical/akebraic topics under discussion,
Students who expressed the aforementioned responses rate themselves mostly good or just average. The ones that rate themselves good defended their reason BS follows: That algebra or mathematics comes easily to them or naturally, and because they are doing so well in algebra courses they have had or are taking. The students who did not like mathematics or algebra gave the following reason: That they are repeating the 57 algebra course and that it depends on the teacher who is teaching the algebra course and that mathematics, or algebra, takes a lot of time. CHAPTER5 CONCLUSIONS AND RECOMMENDATIONS
study compared the results from teaching various dgebraic/mathematics concepts that centered on the works of four mathematiciwhwis boll, &chimedes, pythagomS, and Sophie eman, Using two groups of Algebra II students in two different classes. The experiment group used history of mathematics concerning the works of the mathematicians to augment their lessondexercises while the control group's lessonsfexercises were devoid of the using of history of mathematics. Comparison was then made of the posttests (achievement tests) given to both groups.
The current research was carried out in order to examine the following: Can the students in the experimental group perform better than students in the control group?
The quantitative study had two purposes. The first was to evaluate students' understanding of some algebraic/mathematical concepts on the works of notable rnathematiciansLewis Carroll, Archimedes, F'ythagoras, and Sophie Germain. In other words, to determine students' understanding of simple algebraic problems relating to school mathematics and to compare the means or arithmetic average bemeen the group and the expimental group on the test given (posttest) after studentshaveworked through the lessons and exercises for the SmdY (Appendix A and B).
58 59
A second purpose of this quantitative study was to determine student's evaluations with reference to the response to questions for the study on the four characters used for the study.
To achieve the above objectives, an instrument was constructed by the researcher to determine students' understanding of some algebraic/mathematical concepts. Students' performances on the tests were analyzed to investigate their understanding of the problems on the tests which were graded using a standard grade with scores ranging from
0% - 100%.
The study was conducted during the fall semester 2000 (firom September 18 to
October 20) at Florida State University High School (Florida High), A total of 36 students participated in the study. The data was obtained through the administered test on Iessons/exercises which were given during regular dass time. The tests were all graded by the researcher himself.
A measure analysis of standard deviation and the mean were used to test hypotheses. Further analysis of the significant difference was conducted using the t-test; the data analysis was tested at a = .05 level of significance. -s -s On the basis of the evidence in this study, the following conclusions were drawn from the quantitative study:
1. There is a significant difference between the mean scores of students' performance in the control group and the experimental group.
2. The mean scores of students' performance in the experimentalgroup were greater than the mean scores of students' performance in the control group.
61 relevant mathematical and algebra concepts, in the learning and teaching of topics connected to history or algebraic concepts in the classroom. It will help students to improve their performance based on the responses from the interview questions for this study.
F,ducstionalS This study indicates that solving various mathematical or algebraic topics is a little bit harder for some of the students: This is obvious by vime of the fact that most of the fundamental concepts in algebra should be taught before teaching topics on the mathematical characters that are in the school curriculum.
It is often the notion that mathematics as a subject is difficult for many pupils to the extent that mathematics suffers from this more than any other subject (Furinghetti,
1998). As a result, the mathematics image held by students is very poor; students think that mathematics is a very boring subject, without imagination, detached from real life
(Furinghetti, 1998).
Apart from the normal routine algorithms in solving mathematical or algebraic problems in the classroom, the other common strategy according to Hennigsen and Stein
(1997) is the problem-solving approach which can be summarized as follows. Henningsen and Stein (1997) summarize desirable features of the tasks using a problem-solving approach, which I believe the use of history can augment for a better teaching strategy as follows:
1. Genuine problems that reflect the goals of school mathematics.
2. Motivating situations that consider students' interests and experiences, local contexts, puzzles, and applications. 62 63
Students at a local high school-36 students, male and female-were used for the study. Algebra II is a course that is usually taken by high school seniors in 12* grade and few 1l* or lo* grade students which explains why the ages of the students are mostly 18 and 17 and a few 16 years old.
In this investigation, both quantitative study and qualitative study were employed.
The quantitative study was the main study-a teaching experiment that involves two groups-experimental group and control group, where the concepts or topics were taught or explained to students with the necessary formulas, and the experimental group in which the accounts of historical origin of concepts or topics connected to the works of mathematical characters-lewis Carroll, Archimedes, Pythagoras and Sophie Germain were used to design the lessodexercises, whereas the lessons/exercises for the control group were devoid of any traces of historical account (Appendix A and B)
As expected, students performed generally well in both groups. Students in the experimental group performed slightly better than students in the control group. The
only area most students in both groups did not perform very well in was in the problem
requiring proofs.
This is not surprising, since proofs as a topic usually required the use of identities and special products in Algebra IT course with the understanding that students would have been exposed to thorough fundamentals of algebra including special products of the
form (a + b)’ - a2 + 2ab + b2 and a? - bZ = (a+ b) (a- b). Teachina the Conceut
It can be seen that the algebraic/mathematical concepts used in this study might
have posed considerable difficulty for high school students. In general, how should 64 mathematics be taught in school? Thus, the questions that need to be answered are: (a) how can teachers help pupils learn mathematics or algebraic concepts meaningfully, and
(b) how can teachers be prepared to meet this challenge? Clearly, mathematics is a discipline students must experience and do in order to learn. That is, they must engage in mathematical thought and problem solving. There must be direct teachedstudent interaction as students construct their own mathematics. For meaningfd learning to occur, the subject or concept must be inherently meaningful. Through meaningful instruction and guidance, most students can understand and appreciate the underlying consistency, structure, and orderliness of mathematics and its beauty.
In helping students learn using history meaningfully, one must recognize the methods that they actually use, and help them to understand the relationship between what they are doing and what the teacher is presenting, and to appreciate the value of making ths connection, by helping them to recognize the limitation of their own approaches. Teachers need to learn not only “mathematical methods’ but also ways in which students go about making sense of mathematics. I would say that unless a student attempts to bring intuitive understanding to bear on mathematics tasks, a computational scheme would be expected to be based primarily on the memory availability of mathematical operations.
As a mathematics educator, I believe we can play a leading role in that development, but only on the condition that we learn to bring our teaching strategies into harmony with what students can do at any given point. If we want them to learn more than collections of “mathematical facts,” if we want them to understand the 65 concepts, we must provide a sequence of activities that explicitly lead their first intuitive attempts to the development of more sophisticated methods,
This study has made an attempt to explore a seemingly moflh&x mode of
~d%!OgY; that is, using history in the teaching and learning ofalgebmic/mathematicaJ concept. This is because research in the use of history has not dealt much with cenml ideas algebraic topics.
At the threshold of this new century, much attention should be focused on the future. In order to know where we are going, it helps to know our past, by means of focusing on the issues concerning history of mathematics and mathematics education
(Fauvel, 1986). I believe, based on my research study result that the use of history which has been around us and has always been useful as historical note in most textbooks should always be stressed to augment various teaching strategies in addition to the following suggestions for mathematics cuniculum and mathematics teaching guidelines by Henningsen and Stein (1997). Flathematics Curriculum
1, Topics to be given a more systematic indepth treatment, with far less Jumping from topic to topic. 2, clearlystated schemes ofwork for each year and differentiatedability ranges.
3. More emphasis on practical numeracy, particularly for students who will not be continuing their mathematical studies beyond high school level. 4, M~~~use of relevant applications, both for Course Work and for motivation with new topics concepts (Mathematics in School 1998). 66
Mathematics Teaching
I wsh to reiterate that these aforementioned guidelines are simply existing wholesome suggestions that the use of history can augment, for a better teaching strategy.
1. Emphasis on a clear, precise description ofthe basic idea or concept being taught, with worked examples and applications where appropriate.
2. Correct, precise, orderly, spoken, and written mathematics to be used at all times.
3. Limited, strictly controlled but effective, use of calculators.
4. Greater emphasis on whole class interactive teaching, with less individualized work during lessons and pupils regularly working on the board.
5. Homework to be used as a key component of learning.
6. Individual pupil mistakes to be used as teaching points for the whole class (Mathematics in School 1997, pp. 23-26).
In conclusion, I wish to reiterate that the use of history in the teaching of mathematical or algebraic concepts as an addition to the aforementioned guidelines of pedagogy will go a long way in helping students understand where most of the concepts came from as well as their origins to encourage interests among students. Using and laying emphasis on mathematical historical notes in the textbooks which are usually optional to stress fundamental concepts in algebra or mathematics in the curriculum should be of paramount importance; it will go a long way toward making mathematics or algebra less frightening to students (Ernest, 1998). My suggestion for future research is to lay emphasis on the use of history in addition to various methods suggested that are replete in the mathematics reform movement; and, indeed, a focus 67 research might be the question: Does the use of history to teach school mathematics augment a reform clmiculum?
Concluding Remarks
Possible future topics should include lessodexercises on the works of mathematical characters With interesting anecdotes such as 1, Ramanujan: A story often told about Ramanujan is that when the eminent British mathematician G.H Hardy visited Ramanujan as he lay ill in a hospital, he came in a taxi bearing the number 1729. He asked Ramanujan if there was anything interesting about this rather dull number, 1729. “NO,”Ramanujan replied without hesitation “It is a very interesting number.” It is the smallest positive integer that can be represented in 30different ways as the sum of two cubes: 1729 = 1’ +l2’ = 9’ + lo‘.
2. Karl Gauss: A story is often told about how the great mathematician Karl
Friedrich Gauss (1777-1855) at a very young age was told by this teacher to find the sum of the first 100 counting numbers. While his classmates toiled at the problem, Karl simply wrote down a single number and handed it in to his teacher. His answer was correct. When asked how he did it, the young Karl explained that he observed that there were 50 pairs of numbers that each added up to 101. (See below.) So the sum of all numbers must be 50 X 101 = 5,050. 68
Finally, if I were to repeat this study I would have chosen the same topic-but then, I will conduct this research as many times as possible in many different high schools on a larger scale with large number of students. 69 AF'PENDRA
Instructional Materials for the
Experimental Group
70 71 72
2 73
3 74
GdA , Finda~aLofpasitiveintegelatheaumofahosesquarra whom is twice the 8um of the squares of 647 md 311.
Part Exardnea. Without the use of n calculator m paper and pendl, we hm many of these spurnsYOU can give. You may tbinkabout any S@~IIM that puam Mlble to giva fit sway, but donot UM myphysical daviee
4 A. 1. 189 istheaquareof-, 2. 196 isthesquareof _. a. 961 iathasquaraof-. 4. 225 iathsequarsof-. 6. 144 iatheaquarsof-. 6. 256 iathsaquaraof-. 7. 121 iathesqunreof-, 8. 961 isthesquaraof-. 9. 626 &thequare&-. 10. 924 iathaspuarrnf-. 11. 441 isthssquarsof-. 12. 299 iathOBquaraof-. 19. 1444 iatheaquareof _. 14. 464 iathasquuraof-.
B. 1. a. 86 ie the ~umof &e squama 81 and -. b. 86 is the sum oftha aqum of - andtheanme of 2.
2.
3.
4.
6.
6.
5 16
In the hllowhgsrenisea, Iiatthekgsrnumberiht. 6. Find twa differtnt positive in- the sum of whose squnrea is twica the sumof the spuarrs of 6 and 7. - - 7. Find two diffmt poaitive integers the sum of whose aqumaie twica the aum of the squaren of 3 and 8. - -
(3.6) (6.4) p& 5 Exercises. Complete the table below. Put your mwm in damdingorder. That is, for each pair of integers listed,put the hignlrmberm
Part 6 Exmines. In this set of exercises, you are given n pair of integpa. YOUue ta give the atinteger of a pair of positive integm the sum of whose sqmsis twice the sum of squares of the given in-.
Part I Ezucisaa. In thin set of exerdaes, you mn given a pair of in- You rn to give tho laaat integer of a pair of poaitive integmthe BUIII of whose squares in tiesthe EU~ofaquarcsofthegivenintepa 78
9. 2(2131' + 5468') = (- Y+[-P 10. Nmgobaekanddothetaskof Goal 1. Youahoddnowbepreparedtodoitwith very little effolt. If you dl can= this, look owthe data in the fides of Part 6 and Part 7 above. Such resulta ledLenis Canon to his dismvq of the generalizetionthat renders auch taalrs into trividitiea
PROOF OF OUR DISCOVERY .. 79 80
10 81
ARCHIMEDES Archimedes was the most capable andent mathematician Indeed, he is one of the hetopmost mathematicians of all times. 'Ihese three, in &torid cuden, are Archimedes, , and Newton Archimedes was barn ht287 B. C. in Syracuse Sicily, a Greek mlmy off the coast OTItnly. He da~diedat Al- then the centar of tho 8d-a~world. There, he was a student d the followers of Edd During his atay at Alermdris, he invented the Archi& Sm.This device was wed to ruiw the water fmm &e Strams tu irrigatethe agricultural fields that wmahme the watar level. Archimedm SOsa ia baaed on two geomeHc forms, ths helix and the cyhder. After his return tu Syracuse he dwpted bel€to Sdena. During this time, Archimedes made aignificantmntrhutions to mathamatics. He extended the method of .rhouation tu determine the area of figura bounded by curved linaa and surfaces. zhe method of exhaustion rorashadowed a branch of mathematio MW know aa mlcuku. Archimeden used thismethod to maLe ray accurate appmdmationa to pi, the ratio of 10 tho drnrmfersnCa of a drde tu ita diameter. Heahmedthat pi is greater than 371 but I leasthan3-. Arhimsdea~dtheformulaafortheslnfacema andvolumeofthe 7 sphare and their dation to those of th~cylinder nnd mne aa hi3 greatest dismvdes. Ha Eound that the surface d a sphant had an ares qual tu the total area of four great drclea of the sphere He foundthat the vdume of a hemisphere wad axactly twice that of the its insmi mne. He than found that the -der dreumsmbed about the hemisphere snd mne had a vdnmo arady thrsa times that of the mne. ?his may be summarized by tbe statement
The vnlumen of E inacnhd in a hemiaphere, the hemis here and the~dmunsuibedaboutbothofthemarem eraho 1.2:3.
Archimedes was M enamored of this nlation that he asked that it be featured on.his tombstone. Wban his grave wm found after many years of neglee there carved upon it was a cylinder in which a hemisphere was imuibeb The hemisphere, in turn, was circumsmied about a mne. Arrbimedes request had been 82 83
THE ARBELOS
I
Part’ 1 &erdses. In the figureabove : 1. IfAE=G,thanAB=_ 2. IfBE=4, thenAB= - a. IfCD=s, then CBE- 4. IfDB=6,thanBC=- 3 83
THE ARBELOS
Our first god is to find at once, with no delay, the radiw of the big semidrde when we are tnld each rad& of the small aemidrde?~.hk at the ta& below. Ifjmu cop mer it at once wjth M pause whntsower, then go m to part 2. If you bnve to pause to determine the mmct answer, go rn to the ererises in part1 and then rebmn to do the fust god 'Ihe uerdsos of p& 1 'd-de you Kith a &e nay of doing & tzzerrises right away 4th.~delay.
Findtheradiwofthe bigsemidrdeiftberadii ofthe small rcmidrdea 1357 and 2000.
part 1 Ererdses. In the above : 1, UAE=6,tbanAB=- 2. JYBE.4, then AB= - 8, Um=3,theJla=- 4. IfDB=6, then BC= - 3 84
6. IfAE=2, ~11dBD=3,thenACi, andAO=- 6. IfAE=6, sndBD=7,tht~1AC=-,mdOC=- 7. Ifthesmallradiiare 1 andg, thenthediameterof thebigsemidrdeis _, anditamdimis _. 8. Ifthe amall rpdiiare 3 and IO, then the diameter of thebigsemidrdeis _, anditsradiusis-. . , 9. Ifthe mall radiiare I and 20, then the *of the bigsamidtde is 10. Vthesmallmdiiare 7 and IS, thsnthe&ofthabigsemidtdeis _. 11. Ifthemallradiiare6and14, thenthemalldiamet8rxare - and-. &,the bigdiameteris _. Hanee,thebigradiunia _.
AE 1 3 I I 7 7, 6 19 8 26 1492 1776 BD 9 IO 20 13 11 3 14 20 97 76 2000 5000 OF- -- _____-- - -
-Goal 2 Find the= ofthebigsemidrde ifths radii ofthe amall semiriden are 9 md 29.
Pat 2 berdaeS. 1. The wen of a 8emici.de is halfthe area of ita whale circle. Thus, ita circle has an area of 12, then each of ita semidrcles has an area of -. i 2. ThesreeofaEirdemaybefoundbyusingtheformula:, AI=? or: A=rrn
Iftheradiusofecircleis4, thenitsareais 4(-)x =-at.
4 85
8. Ifthe radius af a drde is 12 , then lts area is -. 4. If the radius of a &de is 22 , then ita anis -, and each of ita semicircles ban an nrea of -, 6. If the smdl semidrdes in en Arbelos have radiiof 2 and 6, then the radius of the bigaemidrdeia -,anditsareain-. 6. Ifthe am& Bemidfiles men arbelos hnve radiiof 3 end 'I, then the radius of the bigaemidrdeia -.andibareain -. 7. Iftheradiiofthslnndsemieirdeam 1 and 6, thenthemaofthebig semicide io -. 8. Iftheradiioftheamdsemicidesnre 8 and 30, thentheareaofthehk semidrde is -. 9. Return to Goalabaf0r-a CLm~Uillg.
You are XIOW ready to make a striking diaeoveq that will make the task of Goals amazingly easy. Once you have made your diamvary, you can do that task at once with no effort So, Sitwouldtake yau mom than a couple of aemnds to do it, aldp it anddotheexudmsofPart.3. ?h4~fuciceswiUguideyoutohdeieniiicfact expld eaensivelyby Archimedes YouwiUbedoaedtoreturnto~9~ththa exercises. At that time you willhave an appreciation far the elegance. ef6dmW, and usefulness of your dismary. r I Find the -.of the Arhelosifthe radii of the small aemidrdes nr~10 and 66. I part 3 Esercises . 1. Let's dderan Arbeloa wit& small semicircles having radii of 4 and 6. 'Ihe radiusof the big semicircle is -, end its mais -, Since the area of the mallest semidrde is -, ifwe remove it from the big semidrde we ere I& withan area of -, lfwe subtract the area of the other smd semidrde fmm that remainkarea, we dbeleft with the maof the Arbeloa. Sic+the area of theothersmdsddrdeis -, theareaoftheAtblosis -.
5 86
2. Let's consider an ppbelos with small semidrdsa havingrndiiof 2 and 10. ?be radiuaofthe big aamidrde is -, and its ares is -. Since the area of the smallest semicircle is -, if we remove it fmm the big semicircle we are left with an maof -. Ifwe subtract the ares of the other small semidrde from that remainingarea, we dheleft with the area of the Arbelas. Since the area of the other mnall ssmidrde is -, thsmaoftheArbelosis -,
3. Let our Arbeloa have mnaU &of 6 and 10. The area of the bignemidrde is - The ares of the smalleat nmnidrde is - The RmaindoaRn subtracting the ares ofthe nmallest semidrds - The ares of tho 0thmall semicirde is - The final remainder which is the maof the Arbelon - ' Sa, tbeareaofanArbeloswhoseBnallradiim 6 and 10 is -.
4. ht our Arbaloshave ddof4 and 20. The area of the big Mmicirde ia - "be maof the Bndaat aemicirde ia - 'Ihe remainder aRn subtracting the ares of the smallest semidrde - Ths ares of the other amall semidrde is - The finalremainder whi& is ths maof the WOE - Sa, the maof an Arbaloa whose emall radii= 4 and 20 is -.
6. ht'smganLeourdatato aasiat lwinmaldaga mqjmfmfindingthe areaof an whenwe have the rndiiof the mall semicircles.
Smallmu 4 2 6 4 I 100 OtherSmallRadiu 6 10 10 20 11 258 &eaofiheArbeloa ------
6. NOW,rem to the task of Gad 3. ?hen go on to the exenises ofpart 4.
6 87
Part 4Exerdses.
1. The 6gure below is a pair of "Crab Claws". It conaista of 3 drclea haviug allinearcentem. Each of the &ea in tangent to the othertro tides. Pokta D , 0,and E arethecenteraofthe3drdea.
Ifthe radiiof the small drdea are 92 and 98. What ia the area of the pair of crab elam? [Hint: DoyouaaeurArbaloainthe~71 2. If the radiun of a iddrdeis 13. them ib dinmeter is -. 3. Ifthediametmofa aemidrdsia48,thenitsradiusk_. 4. Uthe dimetam ofthe small aemidrdee ofan Arbaloe are 20 and 86, then the area of the Msin -. 6. Ifthe radiiof the small semidrelea of an &DS are 8 and 11, then the nreu of the Arbaloais -. 6. Ifthe diametma ofthe small aemidrdes of an Arbeloa are 23 and 32 , than !be diameter of the hig dcirdei3 -. I. IftheradiioftbeamallMlnicirderofanArhelosaR 16 and 51,thenthe diameter of the big semi&e ia -. 8. ?he maof a drdsbvinga diameter of 10 M -. 9. 'Iheareaofawmidrdebnvhgaradiusof 38 is -. 10. Themaofadrdehavingaradiusof 33 is -. 11. Themaofasemidrclehavingadiameterof64 is -.
7 88
THE SALlNON
figm 1 ,
8 89
Inordcr~mmpllfthcan.o~aSali~wcnccdwfindlhcarca~liubigsemidrdc~gionl3is.i~ nun. quimIU to know Ihc rddilu or the big scddrclc We can find lhc diu or tbc big Jcmidrcle ilwcbmwL+C dlradii. This is lhcgistolm. Lmk a1 e.If you cdn do that wk a~ mcg with no dclay. do so. If it would lakc you some time ID gal an anma. then go on ID do Ihe cxerd~s01 part 3 , When you have mmplcrcd thc cxdsa or parr 3 , you should mmm ID the lask 01 w3.
9 90
Rgurc 2. 91 92
.I 93
I a If Ihc radii Of the flanldng xmidrdcr and the middle wmicir+lc arcrrrparivcIy3md,4. lhcnthe~o~'thcbig~cmicirdeis_. b. When heamall radii arc 3 and 4 rrspccrively. tbe w,ofIhc Salinon is -.
6. 'Whm IIKmall radii arc 3 and 6 rcspdvcly, fhc ana or tbc Sdiuon is __ ,
13 94
8. Cmplek lhis QML Radii of Ranking Radii of Middle Amof SmallSemicirdcJ SrnallScmicida Wan
a 4 6 h 2 B C 5 6 d 4 8 L 8 4 f. 3 4
a 3 6 b. .I 4 c 5 8 d 6 8 L 14 .6 r. 8 30
11. Now. mmplcle this InMc : MmkingRadii 7 1 3 8 12 4 Middle Radiu 11 9 5 22 26 12 ~rca01 Salinon
14 95
15 96
PYTHAGORAS 97
&mnpancnlof ('L3,7) isapsitivcinlrgcr. wcmusfmoJidcrpivperty (4) abovr Dou 2'+3'=?7- 98
pythagoren triplc Sa. be idy included this triple in his sand d : 1 3 5 0 4 12 99
He condnucd rhis for svcral rnm cnmcs. FdI thc bl& klow VI cmdIhe mblc or Pyhagorar.
4 100
a13 57 91113 b 0 411226 40 60184 c 1 5 13 25141 61 115 101
way to do lhis would h to andour chat until we got 10 65 as Ihc last wrnpnmt This would not be M:bur it would be tedious. time mosurniag,aod boring. II wculd be nice if Ihm wma rrladon among thc fim and thc othcr rwo mmponcnu BS rhcm is knvm thc scwnd and thin! mmpnm&. You may wish m take time owto cxplm bson your own. If you find a Wdul parvrs & & e -vomclf. Announcing your diwovcry 10 your dsrrmacs would dcpve rhm or Ibc Satidzlion of rmding onc on their own. Cur ocw goal is u) gain heability 10 do larks like tha~in ‘24 2 with link clTon Youshouldlmowlhattbcnisonlyonc~answaforochtaslc Yatueprobablymat4e m do there lark right now. If lhis is hBY. EO ID thc following ucrdsa and ntum m thcm whco askuled&do.
- (-.924.- ) is a q.lbagorran niple b ( . ,174I)isaPylhagorcanUiple c (47. . ) isaPyrhaBacanniplc
a 3 5 7 9 11 U l5 17 19 21 23 25 b 4 12 24 40 60 84 112 I44 180 220 264 312 c 5u2541___-__-- b+c 2 2 __ - - - - - _- -
WcsacthatcachofourMns, b+c, iaapcrlcct , But the sqmof whaR
pan3 la thsc uc+. wc ahall rstria our mention 10 he F’ythagom eipls in our chkis abo~t-Sin= ~ytmgorardnixd way to gena hew specific tripla. we shall derID tbcm as thc Pythagom Tripla Of qrthag-. or PMP lor short We have yen other F’yIhagorrao rripls For uamplc, ( 6.8.10 ) is a PyhgOrran mplc. but it is not a Pymagorrvl Tripla Of Whagorar. lTOP. Try thac mthcqrlmking track ar yow cbanr or making ncy cbane
A 1. The wm of the last 2 mmpaneon of thc PTOP ( 3~.4,5 ) is -. 2 Thc sum of tbc last 2 mmpnncnts or tbe PTOP whmlint mmpnmr is 5 is -. 3. The sum of the lm 2 mmpancnts of the PTOP whcsc frst mmpncnt is 7 is -. 4. The swn 01 me lm 2 components of thc PTOP whrsc fim wmpncnt is l3 is -. 5. The sum of thc last 2 cmnponenu of the PTOP whmc fimmmpmmt is 17 is -. 6 102
1 103
0 00 000 OD00
8 104
1. T(l)+ T(2) = 1 + - = - 2 T(2)+T(3) = -+ 6 = - 3. TO)+ T(4) = 6+ - = - 4. T(4) + T(5) = -+ 15 = - 5. T(5J+ T(6J = IS+ - = - 6 T(@+T0=-+28=- 7. T(U) + T(l.3) = - 13 Tp0) + TPI) = - 9. Ouas thc sum of T@7)+ TO. - 105
A 1. 2 3.
..4. I 6. 7. a 9. 106
c. I. 2 3. 4. 5. 6. 7. a 9. 107 108 109
YOU efTofi rha! diaW found a couple or pga back The cmly dilrerem should be your diagam should not include thc shading round on htarray. 10. Move3 units lo right and mmplck your drawing lor herdtb oblong number. 11. Chsk your dnwings with a neighbor or your trachcr ID nukc sun Uw you were able IO fdlw didom comnly. This is impnant he rollowing umiss dcpcnd upon WICQ&IC drawings.
14 110
Nowthatwchavebsomcacquaintcdariththcoblongnumkrs. wemayuse thcminaveryeff~ciau ahoncutforsq~ngecrrainintcgen. Mow suggcsuhpwuofthisahatcur Ifyouarc unable to do the tasks of Gd 4, withcur any dday whatraver, lhcn go on to the exdathat follow. You ail1 be allowed to rem to UICse tarki Inbx when you M prcparcd to dancamak you newly gained p-.
I
Rn 9 Excrciaa.
A 1. a 49~- b. IO9 + - c 9s=- 2 Thcsquareofzwhintcgcrmdinginthcdigit 5 mdsinthe2digi1numaal: - 3. Thesquareof 75 e~~L~inthcZdigitnumSnl:- 4. The square of 75 m with uu 2digil numd - 5. ThefiIs2 digib. in &.Of lbs~ol anintcgucndingin 5 dsmrtingwitb 6b _, 6. The fint 2 digits. in &.of the square of an inwgcr cnding in 5 md dng: L with 4is. b. with 5 is -. c with 3 is -. d with Bia, c with Tis-. 1. with 9 is-. g withlob-. b. withWis, i with 6u,
B. 1. Arrangethcduagnincdinsxacika A.5md A.6 aboveinthetabletelow. Notchl 35 isamulnpleof 5 andthc firstdigitof 35 is _. The Squucof 35 is -md the nmbcrnamedbyihermz digirsormcsqmaf 35 ia _.
Fmldi~lofmultipleof5: 3 4 9 5 8 2 7 6 Fm2 digits of itn qm:------
z Earn number in tbc 2nd row of 0urlaMc above is an numbu.
3. How is e&h entry in Ihc scmnd mw nlatcd to the cwruponding entry in the lop row?
15 111
Ooal 5. Wifhoutdclay, mmpute Il- a T(1999) b. T(2W) c. T(39)
16 112
17 113
18 114 115
SOPHI E GERMAIN 116
SOPHI E GERMAlN PRIMES
wescethal7 a Sophie Oermain pimr is I is m1
2 117 118
a Which d&e rollowing 3 inegm arc prime? 88443 37 a1 428 787
4 119
5 120
1. whpl ia lhc remainder when p. 45 is dived by 97 - b. Any multiple d 9 is dividai by 9 - 121
caw and dispatch. Rupond IO his 8oal insuanlamausly wilhout delay. Remrd your quick rrsponsu before continuing. Youmay guw if you wish but mpnd LO %h withcut cwpltatim When you have finished thc cxcrci~rhat Idlow. yw may check your immcdir.tc rrsponta wih those mncdbut quick mpruer derived lmm using the dcnu rcnalcd in ihc rollowing exmira.
Withoutdelay. circle& pimstdow. 51 n 87 91 122
Is this a?- Lcl's se Consider the number' 75. Thedigitalalof75.7+5 is-. Is 12 amuldpleor6'7 - Is 75 amultiplcoi 61 - Now mnsidcr 41. WhatiriudigimIsum7 - 15 41 amultiplcof 57 __ We havcsrm Ihatncilhcr 6 nor 5 Worksas 6 and 3 do. Indced, nainkgcrolhcrlhan 6 and 3 wdulhcy do. Gausshasprovidedurwilhaveryurefulimllhatw~olllyforibcnumbw6 and _.
c 1. a 111 Exmriac 8.3. yaufomdthat 87 isamulti@cof _. b. 87=3x- c IS 81 aprirne7 __ d. Is 43 a Sophie Gennain prime? -
2 L In exercise E. 4. you fmdthat - is a multiple of 3. b. 3 x - = 51 c la 51 rpirne? __ d. Is 51 a Sophie Gcrmain prime? -
3. a In exercise E. 5. you found !ha1 - is a multiple or _. b. 3 x - = - c. IS n aprirne? __ d. Is SI aSophicGermainprimc? - 123
4. We see lhaldeciding which or Ihe 3 voublsomc inwgm is prime is caily dvedwing Gauss' device Can hrcmaining integer, 91. &judged by his dnicc? - U's ace. Is thedigidsum of 91 arnultiplcof 37 - Is 91 a multipeof 31 - We havedsidedthat 91 is notamultiplcof 3. Buiitmayhaveolberwntrivialfaclon.
We engage the hdp of another fmou historid charmtr to decide whethu 91 ia primr He isEralmlhenes ( 276- lb4 B. C.), a Gdmahematidan who vm rhc dimof &aatljbwy in the world during his lifetime I1 was !he library a1 Alaandria His Inrnds ddto him BO "Bern' [Befais thesecond lcnaof UK Greek alphaber ] lwausc he WBO @d in all branches of learning and stood a~ lesl Second in cach mdl field One or his mmt neewortby law using Euclidean GmmeW to get M ICCII~C m-msnt d hedmfacnu rA our plmEa& Our prrsmlinlprstin Eiatmlhcnuis hisdcviccfordeteminingprimcs. Ilia thcSimofhlosthmcs. We on explain it wilb lbe tarkof finding all pimu lcsll UW 31. We am by listing Ihc numbers hnvm 1 4.31 inmarray 23456 7 8 9 10 11 12 13. 14 15 16 17 18 t9m21222324 2s 26 n 28 29 30 Sincctkleastprimc is- wcmuout&multiplcof 2 thatisgramthan Z We havesiftcd ouiochcvminteguothw~2inelf. Wc uclcfiwi~I~thisivray. 23 5 7 9 11 13 IS 17 19 21' n 25 27 29
Next we rift out each multiple of Ibc nut pimc. 3, that is gmm thM 3 irulr. Tkc IIC ck
23 5 7 11 13 17 19 23 25 29
Since Ihc nut largcrprimc is -, wc delete cach multiplc of - lh%iis gMLa tban 5. This mulw in diminnling only the single number - . We nerd na( continue with thc nut prime 7
9 124
since we havealready deleted multiples of 5 and 5 tims 7 ex& 30 we do no1 met my llcw numbla rrrnove. We need only consider whpimc lhat is less than M equal 10 the sqw mi of the patat number in our ~r our ritPMcap- as 23 5 7 I1 13 17 19 23 29
Pan 6 Excrda
1. Cide each multiple ol 3. a 51 53 57 59 b. 81 5 87 89
2 Cide crch multiple 017. n 84 91 98
3. circle crch multiple d 13. 52 65 78 91
4. a 51=3x_ h. 57= 19x- , E. 81=-x3 d 17x- = 51 e. -x3=n f. 3x - = 81 g. 91 = 13 x - h 7x - = 91 i. - x7 = 91
5. You should now bz mdy lo attad: wilh dispatch and my. limphlW.
Is51 prime7 - Isnprimc? - Is 87 prime? - Is 91 prime1 -
10 . 125 126
B. 37 51 - b - i. 71 - j. 91 -
SOPHI E GERMAIN AND FACTORING
SWA, Circle i!&S those exprusimu below that are factatable. and Bivc their fanon. 8' + b' a' - b' a' + 4b' 2%' + 4b' 127
1. x'+2xy+? .2 x'+2xy-y' 3. 7.1-9 4. SS'- b' 5. %'-25b' 6. 4'-36b1 7. 36*+12xy+Y' 8 ?+14%y+19 9. a' - 4b' 10. 812-64b' 11. ?+4ny+4Y' 12 x'-bxy+W 13. x' + 4?y' + 4f 14. x4 - 4x4' + 4y'
Rn 10 Excrcius. Canplete cach mknceso that it is me or it holds for each nluc of thc KUinriabldS).
13 128
A 1. 4+-=4 2 -+0=9 3. at7b = a+%+-
4 x‘*W=X4+9y‘+- 5. 1%~4+ = +O
E. 1. 0=7-- 2 o=, -1444 3. 0=4*y‘-- 4. 1%x4+169f- =D S O=- - 1444xy
E 1. x4+4e+4y’=( )l 2 PK‘+ a*+ y‘ = ( 5 3. (x‘+4*Y‘+4!4-4xy = (*+Zr’)’- (-1’ 4. (x‘t4x+1‘+4y7-4x’y’= ( 1’- (W’
14 129
H. Ym should now be rrady to ioUw a developnmt of Sophie Clcmain's famintioa CmplaC the dmva!ion below by filling in all of thc blanks. We wish IO show that for Csm real nmber X end faah real numb y,
COnalUenuy, x'+4f = ' 130 APPENDIX B hstructional Materials for the
Control Group
131 132
ADDING SQUARES
I -7
A 1. Tbc.aquarCof7~-, 2. ThCsquarcOf4k-. 3. Tbcsqmof 9 is -. 4. '7besqumof 6 is _. 5. Thcsqmof 8 b -. 6. The aqm of 2 is -'. 7. Thesquareof 5 is -, 8. Thcsqumof3k-. cbnk your nnswm With ppn calculations md iddfymy rqvva htwsn Mfamlisr 10 you.
B. 1. ThCSqwOf12 is -. 2. Thcrquarrof 21 is-. 3. ThcaquucofZOis-, 4. 'Ihcspmof 70is-.
5. The sqwof 14 ir -, 6. Thesquareof 13 h _.. 7. Thesquareof 31 ir -, 8. Thcsquucof I5 ir -.
, Chmk your uuwm with paw ealculntionr md identify my squamthnt wmdiarto YOU.
9. Howisthc square of 14 rclatcd tothc quare of 137 10. How is the sqm of31 rdalcdtoibe sqImC of 131 11. Howisthemof 21 rclatcdtothewof 127
c. 1. Theaqmof I1 is -. 2. Thequarcot 19 is -.
3. Theaqwof 16 is -, 4. Thesquareof 38 is __. 5. The square of 18 is -, 6. Thesquareof 17 is _. 7. Thc sqm of 13 is __, 8. Thesquanof 31 is _. 9. ThCXqWOf 14 is _.. lo. The squarc of 25 is -.
Check your anrwWith pper calculations md idcdfy SAY aquarcs that wuc dmilinr 10 YOU 133
PIR 2 Exercises
A 1. I69 i.5thesqlbWCOf-. 2. I%istbe~Of,. 3. 961 isthesqirareof-. 4. 225isthcrqVanof-.
5. 144 isthesquareof _. 6; ' 256 isthcquarcof _. 7. 121 isthequareof-, 8. 361 istbcaquarcof _,
9. 625 u the squan of -, . IO. 324 isthesquareof _. 11. 441 isthequareof-. 12. 289 isthcq~ot-. 13. 1444 istbcsq~of_. 14. 484 utklquanof _.
9. 1. a 85isthesumofthesquara 81 d-, b. 85isthcaumofthcquarcof - lodthcsquanof 2.
5. 97isthcsumofthcsquof- mdthesqmaf _.
, 6. 1453 is the nun ofthe squan of -andthcsquarsof-.
Knowing the squarcs of familiar integm and the ability to cxpras certain integm u the sum of a pir of sq- will nssisf you 10 find the short cut to prove the gmdhtionconcerning doubling n sum of squarcs. 134
Par( 3 Exercises. Make sure that you do the exercise in dphbctjc order. Go from left to right rather than down the left column
1. a Thequareof 2 is _. b. Tbcsqmreof 3 is.-. c. 'Ihcsumofthcirquarais-. d Twicethissumis- e. Two qquam who& sum is 26 is - and _. f. These quara input e ucthc quarrrof - and of _.
1. a Thcmofaquaraof 1 and 5 is- , b. Wnthirsumis _. e. Tbedoublcsumofput b isthc~umofthcrquara- and _. d. Thcqunrcsofpan cucthc4uarcsof-andof-.
3.
4. L Twinthesmofthcquamof 2 and 5 is -. b. Thisisthesumofthequarcof -andhqqwof-.
5. Twiccthesmofthequaruof 6 and 3 isthesumofthcrquarsof- and thcqquanof-. 135
Pmt 7 Ererciu. In tbj, ut orcxmises. you ale a pair of intcpcrs. You are to giwthe rn mtcgm of a pair of podtivc mtcgcn tbc sum ofwbarc squara is twice the q of squares of the dm imCgcrJ.
Example: Givcnimegm (3.10). ?hcMintcgcrofthcpnk (%Y) mchW x’+y’- 2(3’+10’) is - Jincc 13’+7‘ - 2(3’+10‘). ~hcb~mpancntofthcpirofposirivcintegcrs (7y) suchthatx’+y’ -: a 2(5‘+ 4’) is - b. XI’+ 11’) is - c. Z(12’ t 3’) is - 136
PROOF OF OUR DISCOVERY 137
6. (6+y)+ - 2(6 + Y) 7. -+ (a+711) - 2(s+711) 8. (m+W + (m+W - 2(-) 9. 2(-) - 1492+14~ IO. 2(1776) - 1776+ - 11. 2( 1444) -- + 1444 12. (-XIOS~)- low+ 1089 13. 2(- ) - (a+b)+(a+b) 14. 2(c+ 34 -- + (c + 3x) 15. 2(ab+cd) - (ab+cd)+ - 16. (-XX’+$) - (XI+$) + (XI+$) 17. 2(n1+b’)- t
I. 7+0-- 2. 3+--3 3. -+0-9 4. 6-6+- 5. 1--+0 6. 8--+8 7. --0+4 8, 2--+2 9. 7+11 - 7+ O+ - IO. l4+ 92 - M+ -+ 92 11. 17 + 0 + 76 - 17 + - 12. (7+1) + (4+b)-(7+8)+-+(4+b) 13. (a’ + b’) + (c’+ d) - (a’ + b’) + - + (c’ + 8) 14. 2(s1+ 1’) - + IS. (XI + $1 + (2 + y’) - (x’ + $) + - + (2+ 9) 138
7. 2(c’+ P) - +- 8. (e’ + P) + (e’ + t) - (c’ + P) + - + (c’ + P) 9. 0-7Xy-- IO. (S!+?)+[26-2nI+ (SI+?) - [(S’+?)+ZStl + I-U+ -1
Pm 5 Excrciws. ?be bold Orpwions m uch memk MRspond to uchnhn. I. [(1+3)+21+ [-2+(1+3)] - (1+2+3] + [1-2+ -1 2. [(7+11)+21 + I-Z+cl+ll)l -[7+-+11] + [7--+11] 3. [(a+b)+21 + [-2+(a+b)l- [a+-+b]+ [a--+b] 4. [(c’+d‘) + 2m] + [-2m + (c’+d’)] - [cy+- +d’] + [c’--+d’] 5. 2(c’+ P) - + 6. (c’ + P) + (c’ + P) - (c’ + e) + - + (c’ + P) 7. 0-7Xy-- 8. (s‘+?)+[U-ZSt]+ (SI+?) - [(SI+?)+ -1 + [--+ (S’+?)] 9. [(x’+fi+ Wl + [-2xY+ (*‘+$)I - II1+2ry+j1+ [?-2xY+ -1
Pad 6 Excrciws. 1. [a’+ tb+b’] + [a’-2ab+ b’] - (a+b)’+ (-)’ 2. [XI+ 2xy+$1+ [r’-21y+ 91 - (-1’ + (1-y)’ , 3. [C’+Zrn+W’] + [c’-2m+w’]-(c+*)’+- 4. (e‘ + P) + (e’ + P) - (e’ + P) + - + (e’ + P) 5.0-7ay-- 6. (s’+?)+[U-kt]+ (s’t?) - [(SI+?)+-1 + [--+ (SI+?)] 7. [(X’+$,+ 2xyl + [-21y+ (x’+j)] - [XI+ 2xY+Y’l t 11’- -+ Y’1 1 8. [r’+ 2ry+y’] + [x -2w+ $1 - (X+Y)’ + (-1’ 139
We M now ready ta give a deductive derivation of Lhe gmdilarim at hand Fill in each blnnk klow lo unnplete the proof. 140
TEE ARBELOS
Find tbe dwofthe big dcirclciftbendii ofthe d annicircla m 1357 md 2W.
Pm 1 Excrcks. Inthcfip.uc~e: 1. If AE-6. tbm AB- - 2. IfBE-'4,thenAB-- 3. IfcD-3,thma-- 4. If DB - 5, then BC- -
r- 1 5. 6. 7.
a.
9. IO. 11.
12.
13. NowW'.un Lo the task iam,anddo itusing the infatmationpulcamcd 6om ihe uxacLa. If you still can? do ildth we,Iwk back ow your a& 10 find asimple way to do latk in W.
2 142
3 143
4 144
1. The figme below is a plir of "Crab Claws". lt ~nsisuof 3 ci101la having collinear centm. Each of the chla is laam lo the other two circles. Points D , 0. and E anthcc~ofthc3circls.
If the radii oftbc small klum 32 and 311. Whet is the area of the pair of crab claws7 [Him: hyouaer.m~osinthefigmc?] 2. Ifthe mdiu of n dcirclc is 13, lban iU diamcla is -. 3. If the dinmcvr of Isemicircle is 46. thm its ndiu i5 -. 4. If the dinmncrs of the mall acmiclnlcr of an Arbclw ye 20 md 86, thm the .M of the Arbelor is -. 5. Ifthe ndii of the dl remicircla of an hrbelos are 8 lad 11. lheo the ma of the Arbclo~is -. 6. If the diamapr of the dscmicirclu of m &los m 23 and 32 , then the diunclcr of& big semicble ir -. 7. If the fi of the small acmicirclcs of UI Arbclos uc I5 md 51 , thm the diun~of thc big wmicirclc is -. 8. Thcarcaofncimlciuvingadiunstaof 10 is -. 9. ?he area of a lcmicirclc having a diu of 38 is -. 10 Thcareaofncimloharingaradiusof 33 is -. 11. "be area of a semicircle having a diameter of 64 is -. 145
TEE SALINON
The Sdimwnsinr wtirely of collinurr sanicinlcs; n big serni~lc.2 mall sani~ird~ of the mesLe that !le.& another rmdl micircle in the middle. We shall dertnthew remichln U blg , Pdbg , and rnidde “pccfivcly. Homva, while the Arbclor wnsistr of - aunicinics, the Salinq &own bclowm Figm 1. consists of - remicircler.
Figure 1
Our main god m‘ll be IO find the uc14 of n ddyof Salinons and with lifflc effnri. The aclcilcs below will rev& the nest lo~utionOur firrt goal revim finding the amof a semicinle.
m,Find the ma of B semicircle nginn whore radius is 38.
PM I Exmiws.
Find the urn of n semicircle region whose radius is :
(LO 32 4. 12 1. 6 __ 2. - 3. __ - 146
Gpall &low concans finding the aras of the 2 small semicircle regions that flank the remaining small semicircle in the middle of the Salinon. This is important to us as m 6nd the area of the Salinon
1. Find the amofjun one of tbc mall fl&g semicircle regions ifits radius is 6.- 2. In each Sdhon. the flanking dlwmicircla have the meradius. ?bey m congrumt So. if one of the flankmg rrmicinla has a radius of 6. then the dusof the other flanking semicircle is also -, lad tbe w of its region is 3. ~r~O4if~OfthCfllnldng~circlcrrgi~huM~Of~p,thCnthCm of the otha flanking remicircle region is -. 4. If the area of OM of the flanldng micircle rcgionr has an w of 38 p , thm Ihe lob! Irraofbothofthrmis 5. If Ihc ndiu of one of the flanldng semicircle rcgiom is 8, then the total area of both of ihc flanking semicircle regions is , and the ma of a cide with that ndius is
6. Ifthc &us of om of the hlnking.. micircle m~onsis IO, then lhc total rn of bath of the flaaldng Mnicirclc rcgionc is , and the ma of Icircle with a radius of IO is 7. Make sure thai you do the mk of &&LZ bcfm You 50 on W Answer :
In ordn to wmputc the mea of a Salinon. wc need to find the arcs of its big micirele region. ?lis, in turq rrquim ys to how the radius of the big wmicirclc. We can find lbe mdiu of the big semicircle if wc how the mall radii. This is the gin of QQaL2 , Lwk at ShL?. If you w do thrt tuk at once. with no delay. do so. If it would lake you some time lo get an BNwcr. then go on to do the cxncixs of pafi 3 , When you have completed the exc~ciscsof pan3. youshould?ctumtolhe~kofW.
2 141
!&3L3. Find the radius ofthe big micircle in a Salinm whmesmall flanking semicircles have radii of 38 and whose middle snall Wcinlehu amdiluof~4.
In Fip2, AB is tbc diameter of the big micircle. The dim,- ofthe mall semicircles uc AC. W.md -. The cmtm ofthc semicircles M E, 0,and -,
I. The dusofthc big sanicinic is AE t EC + -. 2. If=-73, thenEC--, Recall that in each Salinon. the flanking wmicircler are exactly the WIC rk. They art wngluent. 3. If AC-47, hDE- _. 4. If CE-29. then BF - _. 5. If AE - 3. and CO - IO, then the &ius of the big wmicircle A0 - -. 6. If AE - 6, and CO - 8, thm the mdius of the big micircle is -. 7. If the radius of one of thc small flanking rhicixla is 12, and the radius of the other -11 middle semicircle is 20 , then the radius of the big micimie is -. [Look a!Figm2ifyouwish.]
3 148
S. Cumplckthischrt: AE IS 30' 6 21 9 m3 a co 8 39 6.4 16 2 7 k b A0 ------
9. It the radius of one of the smnU flanking semicircla is 21 , and the diu of thc d middle Wmidnlc middle is 30, then the ndius ofthc big rcmicirclc is -.
10. Return to tbc !a& of- before ping on aph2 .Alder:
Now that we ktknv how to iind the mdiw of the big scmichle. we arc pnparrd to had the w afihe big srmicircle region that is left Ifta taking amythe 2 m of the 2 flnnljsg ~Cirelcregians.ThisrrmainingrcgimisrhowninFipc3 Mow. 149
3. Suppose that the radii of the flankiag scmicinlu; and the'middle small &circle M 3 and 8 respectively. a The radius ofthe big semicircle is -. b. Its IM is -, c. So. the of the big wmicirclc thnt rcmainr &a both Ilnnldng d&lc rck5ons'have bcm taka away is -. 4. Thc radii of the flunking xmicirclcs md thc middle dlxmicirclc ue 11 and 16
rcspcctivcly. ' a The amof the big wmicircle ir , b. So, the area of the big sdcinlc that m'nratLr both 0&8 annichle regions have ban taken away is.-, 5. The radii of the flanking sanicirclcr and the middle small WmiCirelc m 4 and 6 mpcdVcIY. TIC LTC~of the big semicircle region (hal rcmh after both fl- micirclc regions have bem iaka amy is -. 6. RcNm to the tarlr ofuand thm mntmuc. WAnnVa:
Youan about to find n way to computc the LIC. of8 Winon given only the radii ofthe small micircla. Morswa. you will be &le to bdthis area with vay litllc &mi - much las effort than that spent in doing the task of-. Indeed, if you can no1 do the task of $id 2 below in 811 instant, then polltothc CxcrckS ofplrt 5. You will bc swto nnrm to completing the cxmi.us. A! that timc by to dlthe difficulty facing you now. Comparc this initial difficulty with the case to da the task lftcr learning thc approach suggmed by the following amim. This cantran will allow you IO myappreciate your ocwly gained skill.
m, If the radii of tbc flankingscmicinles md Ihc middle rcmicinlc IR: 17 md 21 rapenively, thcnthcamoftheSalinonis -.
Pall 5 Ehxmim. Figure 4 below shows the find Fcp in finding the urea of a Salinon. That is, once we have found the arc11 that is lefi aficr laking away the flanldng micirclcs. m necd only to add on the area of the middle scmicirolc region. This is show
5 150
beneath the dashed line in Figure 4 below.. We shall go through a few dies involving all the “gory“ ddh. Then you will bc allowed to do some OD your own Tbe early am& may bo Ued ll~a W’dc should you need them.
Fip4.
1. e. Ifthe xadii af the 5anking micircla and the middle rcmicirelc mx rupcctivciy 4 md 6, then the dmof the big rcmicinle is _. b. Thcamaofthebigdcireleis -, c. The maof the big dckcleregion that rrrminr aRn moving the 2 rmdl flmkin$ wmicircle ngim is -, d The maof the nudl middle dcirclc region is -. e. AddingthcueaofpM d toiblofprl c givaurthe8xaof the whole Salinon. So. the uea of the Winon is -. f Recapping when the dlradii M 4 and 6 rqcctivsly. the ma of the salinon is-. is-.
2. a lfthc radii ofthe flnnking rcmicircla rad the middle 3olicircle M rcspatiwly 2 and 8, tbm the ndiw of the big wmicirclc is -. b. Ibc UB of the big scm!kinlc is -. c. The M. of the big mnicirclc mgion that mnaia, &cr moving the 2 small Wigsemicircle rcgiom is __ . d. So, the ma of the Salinon is -. e. Recawing when the small radii M 2 and 8 respenively. the area of the Winon is -.
6 151
3. a If the radii of the flanking semicircle Mdthc middle dcixcle M rcspcCtivcly 5 and 6, then the ma of thc big scmicirclc is -. b. The BXB ofthe big semicircle region ha! nmainS aRcr moving be 2 snall DanLing Icmioirolc I+om is . c. So, the MB of the SJinon is -. d Rccappiug, ivhcn the dlradii ue 5 lad 6 rapcnivcly. the IRB ofthe Sllinon is -.
6. Whm the dlradii M 3 and 6 RspatiVely,&eues aftbc Srlinon is -.
i. 7 152
..
8. Complete this tublc. Radii ofhdg Mi of Middle Arcs of Small Semicircles Small Semicircles Mimu
a. 4 6 b. 2 8' C. 5 6 d 4' 8 e. 8 4 f. 3 4
9. a Lmk It the raulrr of the rauluofitcms d and e in ex& 8 above. i. Inacm'w 8d. tbedndii. in-m ( 4 ,-)while ieexcrcim 8e. knnrllndii. inordpuc ( , ). ii. How.doa the area ofthc Salmon in pn d wmpan With that in pn e? -
b. Find the area of1 Wonin which tbe ndii of tbe fl&g micircles uc : i 1 while the radius oftfie middle &circle h 6. ii. 6 ~lethed~ofthemiddlc~i&leis2.
c. WhtconjechmissuBgesrrdbypnrts a. md b above7
d Is'thewnjcchmofpm c me7 -
8 153
IO. Try a few man. Radii of Fkmking Radii ofMiddle P.m. of SdlSanicinlu Small Semicimlcs Salinaa
3 6 I 4 5 8 6 8 14 6 8 30
11. N~w.complNthisBblc: Flanking Radii 7 1 3 8 I2 4 Middlc R.diu 11 9 5 22 26 12 Area OfSalinon 154
PYTBAGOREAN TRIPLES 155 156 157 158
c. I. (3,4._)ham'OC. 2. (5,-,13) iaaPTOC. 3. (- .24.25) um.4oC. 4. (-.112.113) ha PTOC. 5. (-.144.145)haFTOC. 6. (-.180,181)isnPTOC. 7. (-.264,265) ha PTOC. 8. (-,4W0.481)kaproC. 159
E. Complete the table of q.lhagorcan biplu below wilhoull~akhgback ai yam ocher tabla.
. .
TRIANGULAR' NUMBERS 160
0 00 000 0000
7 161
1. T(l)+T@) - I+- - - Z T(Z)+T(3)- -+ 6-- 3. T(3)+T(4) - 6+-- - 4. T(4) + T((51 - -+ 15 - -- 5. T(S)+T(@ - IS+-- - 6. T(6)+T(7) - -+28 - - 7. T(lZ)+T(13) - __ 8. T(30)+T(31) - - . 9. GuartbcnrmofTl37)+T(38). __ 162
Some Writm uy dot diagmns to show the same relation You may find such an may of mtsmt It is
0000 000. oom. om..
A 1. E(l)+42-(1+1)+(2+2)-2+_-_ 2. E(l)+l%)+Y3)-(1+1)+(2+2)+(3+3)- 2+-+6-- 3. Yl) + Eo) + E(3) + E(4)- [( i + 1 ) + (2+ 2) + (3 + 3 )I +( 4 + 4 )- 12 +-- - [ r WCBOtthilfrmU2. t 1 4. E(1) + W2) + W3)+ E(4) + E(s)- [E(])+ W)+E(3)+E(4) 3 + E(5) --+ IO - - 5. E(1) + W)+ Y3) + E(4) + E(5) + E(6)--+ 12 -- 6. E(l)+W)+E(3)+ W4)+ E(5)+E(6)+EcT) --+ 14 -- 7. E(])+ €42)+ E(3)+ E(4)+ F@)+ E(b)+ E(7) +WE)--+ 16 -_ 8. E(1) + W2)+ y3) + E(4) + q5) +.&a) + ?2(7) + Ep)+ E49) - -+ 18 -_ 9. E(1) + E(Z) + E(3) + E(4) + WS)+ F.(6)+ ecr)+E(S)+ E(9)+ E(1D)- -
0 163
L
&VU. h Whbout delay. give the sum of the 6nt 999 even iutcgm. - b. Without delry, give the value of the IW-th Oblong numk. -
c. 1. 2 3. 4. 5. 6. 7. 8. 9. 164
fim is 8 geometric model for Oblong numbcn. Since each oblong numk is m of thc fyst y1 many cvm numh, simply bumblcd u an myby Mnbining rmral smys for c~~kutivc -htcgm. Below, we sec the result of mmb- mays for the fust 4 mn mhdting in an amy for chc fwnb oblong number. B(4).
El E2 E3 E4
B4
Figurc 2 Figurc 1 Figure 3
11 165 166
13 167
3. a Your nm bwi~gdirectly lo the right of the %and L an myfor B( - ). b. It hu - venial wlmnns. 0. It hrc - horLontal rows. d Ithumarmof- unitrquara e. B(3)--
8. a An myfor B( 24 ) hrr - coluum and - mm. b. hnyfmB(24)hanarraof - unitsquua. C. B(24) - 24x - 168
15 16 170 171 172
The in (he taMc below will p"c very uselot and izvdiag uith regard IOthc task3 al -6. Canplcw Ihc table while kccping in mind chc tasks of Goal6.Nok Ihy och shorl Ice in Ihc lablc Mowis n muidpic of - ( an iakpbewcen 1 and ID). a' 5 15 25 35 45 55 65 205 '245 195 b _------c------
19 173
CERTAIN PRIMES I I 8kd-L Is it the case ht9337 harthe 2 properties thae (a) 9337 is a prime, and (b) one more than twice 9337 is also iprime?
The task in' m,is rcmictcd to thac primes. n. having 2 properties. Thas M:
a. The integer n. iuclf, mmt be a prime.
b. One more thaD fice the prime n inW lllo bc 8 me. That is. 2n + I is also or piw.
Lcrs look at a few integers to decide whether uley uc such prima.
P.n 1 Exmiws, k 7 ~chiprimc7 Since 2( 7)+ I - _. and -- S (-1, mutthat 7 -such or prim. is / is not
Is S such a prime? Since 2( S)+ 1 - _. and -is a prime, weseethat s -such a prime. is / is not
Is IS such a prime7 Since 2( IS ) + I - -, and 2 is a prime. we re+ that IS satisfies propmy b of the definition of such primes above. Howcver. since 15 - 5(-) mwelhat IS iuclf -a prime. is /is not Thus. IS -such a prime is I is not 1 74
4. Exercise 3 mhdr us that in order for an integer n to be such a prime, n iuclf muJI be a prime. So. even though ln + 1 is a prime, n is not such B prime unless n iwlf is a prime. For cach integer lined below, uumrthuc 3 questions:
e. Irit aprimc? Example: 5 8. * b. Is 1 mmlimn its double B pime? b. w c. Is it such 8 prime? c. -lK4
5. With regard to cxzrcisc 4 Wlyabove, e. Will thcrc bee in pn c. If thm is a w in pi b. for that integer? - b. Will there be II in pan c. if thm is n ae in part a. for that integcr? __
e. Shouldthmka =inpan c. if thereis8 (~t in cithcrpn e. or pml b. for thaf integer?
d. Wha! rcsponre mlrn be in both pan a. md part b. in order for then w bc a in psn c. 7
e. Can there be& 2 mponscs for some politive integer? - 175
I @&2. Which of the following 3 integen arc ptimcl 88 443 37521 428 7a7 176
Parl 2 Exerciser. For each integer listed below. dvc irr digital nna
A 34- 2.a 57- 3.a. 62- 4.a 06- b. 349 - b. 597 - b. 962- b. 869 -
B. 1. p. 595 - 2. a 389 - 3. a. 799- 4. a 949 - b. 55 - b. 38 - b. I - b. 4 -
5. a. 991 697 - 6. a 5854 - b. 167 - b. 599 989 996 499 -
C. I. 2Mx) - 2. 1599 - 3. 78 - 4. 888 - 5. 1812 - 6. 31416 - 7. 678 - 8. 24 680 - 9. 695929s - 10. 999998979599 -
You an now frccd with a new god. A? before. if you UII not do it at ace with euc. then go on and came Luck to it whm you re dirrn+d 10 do so. At thar time you will rce bin0 1ubonOus division h Iiecuwy. Funhermore, you will then rso+ nnd IgPrsChic VlOthU IXe of urlicr di~sedtool.
I I
NOW hi you uc proficient at finding you may wonder, 'whrt gwd kit?" li Nms out tbat the digital sum of II positive integer is the rddalhat dttupotl that integer by 9. FM example, the digital rum of 23 is -, and the remainder when 23 is divided what the remaindm when it by 9 is _. What is the digital sum of 727 - is 72 divided by 9'7 - Since uch &dm that mula trom dividing an iatcgn by 9 is less lb8n 9. it is not possible to have a dnderof 9. Rcmemba "Caning out %I' SO, whenever 4 177
1. a WhntKthcrrmPindnwhm21 irdividedby 91 b. WbnththedigitslaumofZIT -
2. a whrfisthcremaindcrwhm352 isdividcdby 97 - b. Wbatisthcdigitnlstnuof 3521 -
3. a Wbatisthemlindcrwhm457295 isdividedby 91 - b. Wharistbedigitalrumof4572951 -
5. L whntistherrrmiadcrwhcn 8163Pn hdividedby 97 - b. Whatuthcdigitllnmtof 81639721 -
6. For each plirivc integer linal below, give tbe derwhen that integer 6 divided by 9. a 79- b. 136 - c. 4567 - . d 123456 -
7. You M now prrparrd to do the task of It is hafor your WWS~CWC. .@&lJ.What nmaindcr mdtr from dividing 711 l52 333 3€4 by 91 -
S 178
Pan 4 Excrciscs
I, What is the remainder when a 45 b dividcd by 97 - b. Any multipleof 9 isdividcdby 97 -
2. Can an integer that ir gl~tnthan 10 be a prime a If it is a multiple of 97 - b. Ifiadidlal nrm is 07 -
3. h i. Whsfk3thCdi&dlUlllof 367 - ii. Is 36aprimc7 - b. i. Whalk3thedigilalNmof 2977 - ii. Is297apime7 - c. i. Whatislhedidralsumof 1717 - ii. Isl’llnprimc? - d Forach hfcgnbelow, tell wbcthcr if is &e, Ifit is, wite ’ yer.‘ If it im‘t, dtc“no.’ i. 1998 - ii. 275463 - iii. 793452537 - iv. 42 183 -
4. 7hcuumrtothec~&~inparr3. d .bowshouldsuggcstanenrywaytotscWeW Thc goal is gatedhex: Which of the following 3 inregerr arc prime? 88443 37 521 428 787 - Put your mum to the mk of- in the blank rbove.
5. A positive integer that ir a multiple of ninc ir a pim+ Thus, eachpositivc sometimes I new integer whore digital sum ir 0 or a multiple of 9 a primc. isIBnot
We hsvc mn Ihnl L crucial skill in deciding whether an intcger is such a prime is Ihc abiliry to determine whcthcr an intcgn is I prime or not. lhis skill is needed in talk paru af lhc definition for ceMin primes. Expnicncc har sbowa that such intcgerr, each kss thm 100. nrc difficult for sNdenu to judge whahcr or not lhcy M primes. Formnafcly. 6 fdkular method comes to our aid in mosl of these. Goal 4 below ddrcS5umthc skill ncccuPr/ ill rcXrl~gthwe pitfalls wilh cbte and dispatch. Respond to this goal innanlancously without dclay. Rnord yaw quick rcsponrt bcforc continuing. You may guess if you wish but rcrpond (0 cach without computation. when you have finished (hc cxercirw that follow, you mychcck your immediate 179
nspnres With tho= rraMncd but quick raponru dcrived hm using the ddcc mmlcd m the following ucrcisa.
QwU, Without delay, circle ahprime klow. SI s7 87 91
Fomnutely. a dnmethod gives u1 mehelp with this to& lsmy oftbc imcp io. a multiple of 97 Even though no inegu abave h L multiple of 9. Amkidmmtion hu givm YI a device chac @lek our tool dahg with multiples of 9. Eulin wt lamed tM
an htcp is a multiple of 9 if its digid sum is -, It t dso the cyt thu the dim sum of an bger h 0 if (he integer is a multiple of -, The inlorrmrion of lhew 2 mews is commonly cxprrucd in thc single smtmcc:
An iatcger D bs mdtlple ot 3 U and only If the dlgiul inm of n t Immltipleof 3.
i. An iotcgcr n is a muhiplc of 3 ifthe digital sum of n is a multipleof 3. uui
ii Thc digital sum ofrn integer n is L multiple of 3 if n is a multipleof 3. 180
~n imponant warning is in order a Ibis time. We hvc sun that muliiples of 3 rod of 9 d@ upon UIeir mpeaive digital sums. You might swpa thut dl inlegm wok the meMY. Fw example, 24 is a multipleof 6 md the digital mol 24 ic _. WenMethst 6 isa multiple of 6, We &y k temptalto conclude : mrn isa~of6ifmddvif~ofn ,, is
Is Ibis me7 - ws sa. comidcr the numba 75. . Ibcdigicalrumof 75, 7+5 h - k 12 amultiplcof67 - II 75 amultiplcof 67 __
A 1. L Whsri~tbedigitalawnof187- b. It &e sum amultiple bf 31 - E. Is18 a multiple of 3? -
b. It the nnn a multiple of 37 _.
3. a Whatirthsdigild~mof427- b. h &e nnq a multiple of 37 _. c. Is 42 a multiple of 37 -
4. L Whal is Ihediaiml sum of 837 - b. Is the Nm 8 mdtiplc Of 31 __ c. Is 83 a multiple of 31 - 181
B. I. L Is Ihedigimlrumof84 amdtipleof 3? - b. Is84 amultipleof 3? - 2. L Is the digiiul nua of 95 a multiple of j? - b. Is 95 a multiple of 3? - 3. L hthedigitaImof87nmultipieof 3? - b. Is87imultiplcof 31 - 4. L h the digital sum of51 a multiple of 3? - b. Is 51 amultipleof 3? - 5. L Is the digital m of 57 a multiple of 3? - b. Is 57 imdtiplc of 31 -
C. 1; L laExenirB.3.youfoucdthat87isimultipleof_. b. 87-3x- c. h 87 iprime? - d Is 43 ruehapime? -
2. L Inex& 8. 4. youfoundthat - Lnmultipleof3. b. 3x _-SI e. Is SI aprime? - d h SI ruchapime? -
3. L IaacrcLc B. 5. youfoucdthrt- iramultiplcof-. b. 3x - - - c. h 57 aprime? - d h57nrhapirme? -
4. We w th.1 deciding which oftbe 3 aovblaame mtmn is Prime is cuily mold wing mother device. thc t~~i~hgiokgr, 91. k judeed by hir dsvi~=7 - Lets w. Is the &igiiul sum of 91 a multiple of 31 - Ir 91 amultipleof 37 - We hvc decided tha 91 h not I multiple of 3. Bd UUY ha= nher nonaivirl li!3on.
follom: 23456 7 8 9 io 11 IZ 13 14 I5 16 17 18 19 . 20 21 21 23 24 25 26 27 2a 29 30
Since the I& prime is - \K mu out cseh multiple of 2 btis mtn than 2. We have rifted out each mn imcgn other lhan 2 iuclf. We IIC Idt mlh Ibis my: 182
23 5 7 9 11 13 I5 17 19 21 23 25 27 29
So, each of (he rrmaining nnmbcn u iprime. Sinee we pc intacrtcd in deciding whabcr 91 is prime mnced 10 tat its @mt bycrchpbc Ius thanits aquuc root Thasnrc the primu 2, 3, 5, 4 _. since 11 cxcccdsthesquarcmtof 91. h 91 amultiplcof 27 No.since91ir anoddimcgerwhilcachmultiplcof 2 L . Is91 amultipleof 37 Ldsrcc. Urine rnathadmia,mfmdthuthcdigial~of91 ir -which imultipeof 3. So, WKDOt .-
10 183
I. Circleachmultiplcof 3. a. 51 53 s7 59
b. 81 83 '117 89
2. Circle each multiple Of7. n 84 91 98
3. Circle ach multiple of 13. 52 65 78 91
Put 7 Exncirn. 184
17. C.nsuahapimetbati3 grausrtbmtmmdh : L 27 - b. 41 - e. 67 - d 07 e. Anndkitl - f. 17 - s 37.- h 57 - i 77 - j. 91 -
17. Can nprimc thar is pentcrthm ten md m : a 87 - b. 07 - c. heValdigit7 - t 17 - g 37 - h 57 - i. 77 __ j. 97 -
18. Youshould now be prcpul to do thcMof m. Hcre it 3%
QQluk it the urCtb.19337h the 2pmpa-liaihat (n)9337uaprimc,md @) DOC more m'cc 9337 u tlso 1 +e7 Is 9331 da prime7
a. Write your anm in the blnnk following the quation above.
b. Did you regard the task ovawhlmiug Vmm you 6m saw it bcfore doing any of the cxcrciser? '
c. Wen you now able to do it with osc Wilh no &lay7 185
FACTORING somE EXPRESSIONS 186
1. 2+2xY+J 2. x'+ay-u' 3. 2-9 4. 9a'-b1 5. 92-2m' 6. 4a'-36b' I. 36x'+lW+$ a. ?+14xy+~ 9. a'-4b' IO. 81~'-64b' 11. 3+4*y+4J 12. ?-6q+9?
13. x' + 4# + 4Y4 14. x4 - 4xV + 4Y'
variablc(9). +a 9 3. .+%-a+%+- A 1. 4+--4 2. - - 4. x'+9y4 - x'+9Y' +- 5. 196x'+ 169f - -+O
14 187
B. 1. 0-7-- 2. 0-- -1444 3. 0-4Xy--
4. 196x' + l6V' - -0 5. 0---1444?$
c. 1. (7+ 11)+(13- 13) - (7t 13+ 11)- - 1. (7+ m)+(13-13)-(7+13+m)-- 3. (a+a)+(b-b) - (a+b+c)- - 4. (x'+4y')+(4#-4xy) - (x'+4@+4y3- - 5. (x4+4Y')+(4xy- -) - (x'+4xy+4y4)-4xy
E. 1. x4+4#+4y' - ( )' 2. 9X'+6x'y'+y4- ( )' 3. (x4+4?J+4y')-4# - (x'+Zyy- (-1' 4. (x4+4#+4y')-4x41 - ( 1'- (W)'
F. 1. m'-n' - (m+nXm--) 2. 9a'- 4x'y' - (3a1+2xyX ) 3. (x'+4xYt4y')- 9 - [(x'tzy3+-l[(z+~J,- -1 4. (x4+4x'y'+4y')- 4x'J - [(x'+zy')+-l[(x'+2J)- -1 5. (x' + 4x'y' + 4y4) - 4xY - [( -)+2xYlK-)- 2xyl
1s 188
K You should MW k mdy to follow a dmlopnmt of Ihe fmorimtim hplcte the daintion below by filling in 111 of the blmh. We wish to show IIMIfor clcb id number x, ud for acb red number y.
1. Fmctia writing the derintiar lbove until you uc able to do it with om tbc givm cues, You an allowed to omit Ihe let? mmkh cach line cxccpt the W Md last lher of the derivation. This raves you time uul you should k able to rcpmduec hue easily if deb
For wnple. the left member of the 4th line is simply 8 copy of the rial member of the - line. in a long proof, the let? member of the 38th line is L cogy of the rieht member of the -line. I6 189
Pan 11 Exercises. Use the theorem of Ccrtaio prima disewd lbovs lo fmar md npand the wrprryionr below.
ALrpnd 1. (n' + Zab + b'Xa' - Zab + b') 2. (a' - 2nb + blXal+ kb + b') 3. (9a'+6.b+b'X9a2-6ab+b') 4. (a'+4ab+4b2)(a'-4&+4b1) 5. 1(3a)' + Z(3a)b + b11[(3af - 2(3a)b + b' ] 6. I(7.1' + 2(7+ + b'l f(7ay - Y7a)b + b' 1 7. [(38a)'+ 2(38.)b+ b'] [(38.)'-2(3Ba)b+ b'] 8. [(Sa)' + 2(5aX7b)+ Clb)'] I(5n)' - 2(5aX7b) + clb)' 1 9. [(13a)' + 2( 13aX31b) + (73 I)'] I( 1%)' - 2( 13aX3lb)+ (31b)'I
B. F-I. Hint: surt by @nine the -ion in tht form ( l4+ 41 I' 1. a'+4b4 2. 4m4+ k' 3. pay + 4(%)4 4. (71)' + 4(1 lb)' 5. (3.k)' + 4(14b)' 6. (3lc)' + 4(17d)' 7. m'+4(16g') f 4(16g4) - 4(-)') 8. 81 +4(62Sw') ( l4 + 41 r' 9. 16x' + 324y' ( 1' + 41 I' 10. You should now be well prepad to do the t&3 of It h npcaad herc for your convcnimcc.
17 190 APPENDIXC
The Instruments
(Posttests and Solutions)
191 192
CARROLL TEST
CS Name DS.k
CP N1 the blnolci 90 that a proof ofLe& Cumll'r shm NL u displayed 193
AS 194 ' 195
'1.
2
3.
4. k 51 such nnumhdl a) ycs b) no
SGB 196
CS Nme .. -197
SOLUTIONS ARwE3 TEST AA 'Ihe figure Below is an Arbelas. It umsists of 3 mllinear semirides each of whichis tangant to the other 2 BJ ahm
1. Ifthe radiiofthe dsemicirclea 92 and 1400, thentheradiluafthehigaemidrdsis &.
.lQ 0 0 0 198 199
SOLUTIONS SOPHIE GERMAIN TEST
SGA
'1.
2
.3.
4. ..
SCiB
1.
2 3.
4.
5.
' 6. APPENDIXD
Human Subject Committee
(Approval Letter)
200 201
@@IFlorida state UNIVERSITY
REAPPROVAL MEMORANDUM from the Human Subjects Committee
Date: October 19,2000 From: David Quadagno, Chalrpenoad+ To: Ayokunle Awosanya P. Box 5365 Tallahassee, FL 32314 Dept: Currlculum & Instructlon Re: Reapproval of Use of Human subjear In Research Project entltlsd: Using Hlstory in the Techlng of Mathematics
Your request to continue the research project listed above involving human subjects has been approved by the Human Subjects Comrnlttee. If your project has not been xmpleled by October 22. 2001 please request renewed approval.
You are reminded that a change In protocol in this project musl be approved by resubmission of the projecl to the Committee for approval. Also, the principal investigator must report to the Chair promptly, and in writing, any unanticipated problems involving risks to subjects or others.
By copy of this memorandum, the Chairman of your department andlor your major professor are reminded of their responsibility for being informed concerning research projects involving human subjects in their department. They are advised to review the orotocok of such investigations as often as necessary to insure that the project IS being conducted In compliance with our institution and with DHHS regulations.
.hh cc. N Presmeg n,,,~!m,rsn.wal ns AWLCATION NO OO.178.R REFERENCES Beh, M., Harel, G., Post, T., & Lesh, R. (1990). On the operator concept of rational number: Toward a semantic analysis. Paper presented at the annual meeting of the Amxkan Educational Research Association, Boston.
Bell, M. (1992). The ontents of middle school mathematics “teachers” curriculumfrom a historical perspective. Unpublished doctoral dissertation, University of Houston, Houston.
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I BIOGRAPHICAL SKETCH
AYohnle 0.Awosanya was born in Lagos, Nigeria. Prior to coming to the United States for further studies, he worked as Senior Telecommunication Officer with the Federal Ministry ofcommunications, Lagos. He received the B.S.in Mathematics in 1991 from Florida A&M University. He later received the M.S. in Pure Mathematics from Clark Atlanta University in May 1995. Ayo has gained considerable experience
since 1995 as an Instructor of Mathematics (Adjunct) at Florida A&M University and
Tallahassee Community College, in the Academic Support Division.
He hopes to return to Nigeria to teach mathematics and mathematics education
courses in Polytechnic colleges or universities.
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