Book of Curves

E. H. Lockwood

Cambridge University Press | 2007 | 212 páginas | pdf (11.9 mb) // DjVu (2.1 mb)

link (pdf)
link (DjVu)

This book opens up an important field of mathematics at an elementary level, one in which the element of aesthetic pleasure, both in the shapes of the curves and in their mathematical relationships, is dominant. This book describes methods of drawing plane curves, beginning with conic sections (parabola, ellipse and hyperbola), and going on to cycloidal curves, spirals, glissettes, pedal curves, strophoids and so on. In general, 'envelope methods' are used. There are twenty-five full-page plates and over ninety smaller diagrams in the text. The book can be used in schools, but will also be a reference for draughtsmen and mechanical engineers. As a text on advanced plane geometry it should appeal to pure mathematicians with an interest in geometry, and to students for whom Euclidean geometry is not a principal study.

CONTENTS
Frontispiece. Section of the shell of a Pearly Nautilus fossil. (Reproduced by permission of
Dr A. M. Winchester of Stetson University, Florida.)
preface page vii
HISTORICAL INTRODUCTION ix
NOTATION Xi
PART I. SPECIAL CURVES
1 The Parabola . . . page 2
2 The Ellipse 12
3 The Hyperbola 24
4 The Cardioid 34
5 The Limagon 44
6 TheAstroid 52
7 The Nephroid 62
8 The Deltoid 72
9 The Cycloid 80
10 The Right Strophoid ... 90
11 The Equiangular Spiral . . 98
12 The Lemniscate of
Bernoulli 110
13 The Tractrix and
Catenary 118
PART II. WAYS OF FINDING NEW CURVES
14 Conchoids 126
15 Cissoids 130
16 Strophoids 134
17 Roulettes 138
18 Pedal Curves 152
19 Negative Pedals 156
20 Glissettes 160
21 Evolutes and Involutes . . 166
22 Spirals 172
23 Inversion 176
24 Caustic Curves 182
25 Bipolar Coordinates . ... 186
FURTHER READING 191
GLOSSARY 192
INDEX OF NAMES 196
INDEX OF SUBJECTS 198

Circles: A Mathematical View

(Spectrum)

Dan Pedoe

The Mathematical Association of America | 1997 | 138 páginas | pdf (5.7 mb)

link

This revised edition of a mathematical classic originally published in 1957 will bring to a new generation of students the enjoyment of investigating that simplest of mathematical figures, the circle. The author has supplemented this new edition with a special chapter designed to introduce readers to the vocabulary of circle concepts with which the readers of two generations ago were familiar. Readers of Circles need only be armed with paper, pencil, compass, and straight edge to find great pleasure in following the constructions and theorems. Those who think that geometry using Euclidean tools died out with the ancient Greeks will be pleasantly surprised to learn many interesting results which were only discovered in modern times. Novices and experts alike will find much to enlighten them in chapters dealing with the representation of a circle by a point in three-space, a model for non-Euclidean geometry, and the isoperimetric property of the circle.

 CONTENTS
Preface ............................................. vii
CHAPTER O ......................................... xi
CHAPTER I
1. The nine-point circle ................................. 1
2. Inversion .......................................... 4
3. Feuerbach's theorem ................................. 9
4. Extension of Ptolemy's theorem ................................. 10
5. Fermat's problem ................................... 11
6. The centres of similitude of two circles .......................... 12
7. Coaxal systems of circles .............................. 14
8. Canonical form for coaxal system ....................... 16
9. Further properties .................................. 19
10. Problem of Apollonius ............................... 21
11. Compass geometry .................................. 23
CHAPTER II
1. Representation of a circle ................................. 26
2. Euclidean three-space, E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. First properties of the representation .................... 28
4. Coaxal systems .................................... 29
5. Deductions from the representation ............................. 30
6. Conjugacy relations ................................. 32
7. Circles cutting at a given angle ........................ 35
8. Representation of inversion ............................ 36
9. The envelope of a system ................................ 37
10. Some further applications ................................ 39
11. Some anallagmatic curves ................................ 43

 CHAPTER III
1. Complex numbers .................................. 44
2. The Argand diagram ................................. 44
3. Modulus and argument ............................... 45
4. Circles as level curves ....................................... 46
5. The cross-ratio of four complex numbers ........................ 47
6. Mobius transformations of the a-plane .................... 50
7. A Mobius transformation dissected ...................... 51
8. The group property ....................................... 53
9. Special transformations ............................... 55
10. The fundamental theorem ...................................... 55
11. The Poincar6 model .................................. 58
12. The parallel axiom .................................. 61
13. Non-Euclidean distance ............................... 61
CHAPTER TV
1. Steiner's enlarging process .............................. 64
2. Existence of a solution . .............................. 65
3. Method of solution ....................................... 66
4. Area of a polygon ....................................... 67
5. Regular polygons .................................... 69
6. Rectifiable curves ....................................... 71
7. Approximation by polygons ............................. 73
8. Area enclosed by a curve ............................. 76
Exercises ............................................ 79
Solutions ....................................... 84
Appendix: Karl Wilhelm Feuerbach, Mathematician
by Laura Guggenbuhl ....................................... 89
Index .............................................. 101

The Algebra Teacher's Activity-a-day : Over 180 Quick Challenges for Developing Math and Problem-solving Skills

Frances McBroom Thompson 

Jossey-Bass | 2010 | 249 páginas | pdf | 10 Mb

link

Fun-filled math problems that put the emphasis on problem-solving strategies and reasoning

The Algebra Teacher's Activity-a-Day offers activities for test prep, warm-ups, down time, homework, or just for fun. These unique activities are correlated with national math education standards and emphasize problem-solving strategies and logical reasoning skills. In many of the activities, students are encouraged to communicate their different approaches to other students in the class.

The activities address a wide range of topics, skills, and ability levels, so teachers can choose whichever best suit the students' needs

• Filled with dozens of quick and fun algebra activities that can be used inside and outside the classroom

• Designed to help students practice problem-solving and algebra skills

Mathematics teachers in transition

(Studies in Mathematical Thinking and Learning Series)

 Elizabeth Fennema e Barbara Scott Nelson

Routledge | 1997 | 447 páginas | rar - pdf | Mb

link (password: matav)

This book addresses the need of professional development leaders and policymakers for scholarly knowledge about influencing teachers to modify mathematical instruction to bring it more in alignment with the recommendations of the current reform movement initiated by the National Council of Teachers of Mathematics. The book presents: * theoretical perspectives for studying, analyzing, and understanding teacher change; * descriptions of contextual variables to be considered as one studies and attempts to understand teacher change; and * descriptions of professional development programs that resulted in teacher change. 
One chapter builds a rationale for looking to developmental psychology for guidance in constructing models of reconstructing new forms of mathematical instruction. Another highlights the relevance to mathematics teacher development of research-based knowledge about how children construct mathematical ideas. Other chapters explore the relationships between the various contexts of schooling and instructional change. Included also are chapters that describe and analyze major reform efforts designed to assist teachers in modifying their instructional practices (Cognitively Guided Instruction, Math-Cubed, Project Impact, Mathematics in Context, and the Case-Based Project). Finally, the current state of knowledge about encouraging teachers to modify their instruction is discussed, the implications of major research and implementation findings are suggested, and some of the major questions that need to be addressed are identified, such as what we have learned about teacher change.

Contents: 
Preface. 
Part I: Introduction. 
B.S. Nelson, Learning About Teacher Change in the Context of Mathematics Reform: Where Have We Come From? 
Part II: Theoretical Perspectives on Studying Teacher Change. 
L. Goldsmith, D. Shifter, Understanding Teachers in Transition: Characteristics of a Model for Developing Teachers. 
M. Simon, Developing New Models of Mathematics Teaching: An Imperative for Research on Mathematics Teacher Development. 
T.J. Cooney, B. Shealy, On Understanding the Structure of Teachers' Beliefs and Their Relationship to Change. 
P.B. Campbell, The More Things Change…Gender, Change, and Mathematics Education. 
Part III: Context and Teacher Change. 
D. Jones, A Conceptual Framework for Studying the Relevance of Context to Mathematics Teachers' Change. 
M.K. Stein, C. Brown, Teacher Learning in Social Context: Integrating Collaborative and Institutional Processes With the Study of Teacher Change. 
W.G. Secada, L.B. Adajian,Mathematics Teachers' Change in the Context of Their Professional Communities. 
Part IV:Studies of Professional Development Programs in Action. 
C.A. Lubinski, P.A. Jaberg, Teacher Change and Mathematics K-4: Developing a Theoretical Perspective. 
M. Franke, E. Fennema, T.P. Carpenter, Changing Teachers: Interactions Between Beliefs and Classroom Practice. 
J. Stocks, J. Schofield, Educational Reform and Professional Development. 
P.F. Campbell, D.Y. White, Project IMPACT: Influencing and Supporting Teacher Change in Predominately Minority Schools. 
T.A. Romberg, Mathematics in Context: Impact on Teachers. 
C. Barnett, S. Friedman, Mathematics Case Discussions: Nothing Is Sacred. 
Part V: Summary and Synthesis. 
B.S. Nelson, Learning About Teacher Change in the Context of Mathematics Education Reform: Where Are We Going?

Magic Squares: Learning the Inverse Relationships Between Addition and Subtraction

(Math for the Real World)

Colleen Adams

Rosen Publishing Group | 2003 | 19 páginas |  pdf |

link (password: matav)

Contents
What Is a Magic Square? 4
An Ancient Magic Square 6
An Artist’s Magic Square 8
Changing the Numbers Around 11
Subtracting Numbers in a Magic Square 12
Benjamin Franklin’s Magic Square 14
Glossary 15
Index 16

In Eves' Circles

(Mathematical Association of America Notes and Reports, No. 34)

Joby Milo Anthony (Editor)

Mathematical Association of America (MAA) | 1994 | 209 páginas | DjVu (3.6 mb)

link

Howard Eves celebrated his eightieth birthday in 1991. To honor that occasion, the University of Central Florida sponsored a conference that focused on the life-long interests of this prominent American mathematician namely, the history of mathematics, the teaching of mathematics, and geometry. Howard is well-known for his contributions to all three areas. He is author of one of the best known books that has ever been writen on the history of mathematics, has a long association with the Mathematical Association of America, and has published extensive research in geometry.

The conference was unique. Conference participants included pre-college mathematics teachers, community college and university teachers, and research mathematicians. Papers were delivered in sessions devoted to the classroom teacher, to the history of mathematics,and to pedagogical and research interests in geometry. Many lectures combined these subjects. This book presents some of those lectures. anyone involved with teaching or producing mathematics can find something in this volume that will be interesting to them.

Some of these papers are speciically for the classroom teacher. They discuss a use of technology, or the organization of a class for some specific purpose. Other articles will provide teachers with examples of mathematical problems or historical episodes that can be used as part of a mathematics class. Still other papers deal with the philosophy of mathematics education. There are articles in this volume that present new insights into some of the history of mathematics, and there are other articles that deal with some new results in geometry. This is really the legacy of Howard Eves. He has been both a mentor for teachers at every level, and a colleague of research mathematicians. And so some part of this volume wil be appropriate for anyone interested in mathematics.

Contents
Howard Eves by Joby Anthony vii
Howard Eves—The High Elf by Donald J. Albers xv
Geometry, History, and Pedagogy
1. Geometry—an Innate Part of Mathematics Page 1
by Bruce Meserve
Professor Meserve begins this article with a short survey of the thirteen books of Euclid's Elements. He then uses Euclid to discuss the geometric representation of numbers and of algebraic concepts leading to the use of geometry to model the physical universe. He concludes with the roles of geometries in mathematics and science, and some objectives for the inclusion of geometry in the undergraduate mathematics curriculum.
2. Connecting Geometry and Number Theory Page 17
by Peter Hilton and Jean Pedersen
In this paper Professors Hilton and Pedersen show how an approximation with any desired degree of accuracy to any regular convex Af-gon (for example a regular pentagon) may be constructed from folding a straight strip of paper in a specific, repetitive pattern. In addition they show how the folding process for the regular convex Af-gons may be modified to construct approximations to all of the regular star polygons. The paper-folding process leads to some surprising results in number theory.
History and Pedagogy
3. The Meanings of Mathematical Proof: On Relations Between History and Mathematical Education Page 41
by Evelyne Barbin
Professor Barbin has been deeply involved with the development of mathematics education programs in France for some time. In this paper she examines carefully the historical notion of "proof" in mathematics, and draws some conclusions about the role of proof for mathematics teachers. She addresses the question "Is proving a means to convince or to enlighten?" She concludes that when teaching "proof" the question "How do you prove this result?" should be replaced with "How have you obtained this result?"
4. Some Mathematical Applications from the History of Cryptology for the Mathematics Classroom Page 53
by Helen Skala
This paper contains a brief history of some simple enciphering and deciphering techniques along with a collection of intriguing little "coding" problems. These problems could be used in a typical high school classroom as "enrichment" exercises that will absorb the attention of bright and inquisitive students.
Problem Solving
5. Reflections of a Problems Editor Page 63
by Clayton Dodge
For many years Professor Dodge was a colleague of Howard Eves at the University of Maine where he assisted Howard with the Elementary Problem Department of the American Mathematical Monthly. Presently he is the editor of the problem section of the Pi Mu Epsilon Journal. This paper contains a collection of problems, mostly in geometry, that have challenged Professor Dodge over the years.
6. Mathematical Creativity in Problem Solving and Problem Proposing II Page 73
by Murray Klamkin
Professor Klamkin is the former coach of the American Mathematical Olympiad team. He presents a very well organized list of problem-solving strategies, and then applies them to a collection of fascinating problems that have appeared in various contests, books, and papers.
Geometry
7. Bi-centric Quadrilaterals and the Pedal n-gon Page 97
by Gordon Bennett
A "pedal triangle" is formed in the following way. From a point P in the interior of a triangle, drop perpendicular line segments to the three sides of the triangle. The triangle whose vertices are the points of intersection of these line segments with the sides of the original triangle is called a "pedal" triangle. (A generalization of this concept produces the definition of a "pedal" n-gon.) A "bi-centric quadrilateral" is a quadrilateral with both a circumscribed and inscribed circle. Both bi-centric quadrilaterals and pedal n-gons have an extensive history. This paper uses advanced euclidean geometry to give some surprising results in both subject areas along with an interesting pedal n-gon conjecture and a new euclidean construction for the bi-centric quadrilateral.
The Fundamental Group as a Beautiful Functor Page 107
by Peter Hilton
Professor Peter Hilton is one of the best known of contemporary American mathematicians. This paper shows how techniques in algebraic topology can establish deep and beautiful links between geometry and algebra. Here are some comments by one of the referees. "It is not very difficult to learn a certain theory so that we are familiar with basic definitions and results and possibly aware of some applications. It is much more difficult to acquire a sufficient understanding of a theory so that we can effectively use its tools in new situations. The third stage is understanding how the studied theory fits within mathematics as a whole. This usually takes years of serious and devoted research and possibly some special qualities of mind. Most of us never really get there. Fortunately, it is possible to get glimpses of that world from mathematicians who have achieved that stage. This paper is in that category."
Calendars, Pacemakers, and Quasicrystals Page 117
by Marjorie Senechal
What is a "blue moon?" The answer may surprise you, and the mathematical connections between "blue" moons, pacemakers, and the study of crystals that don't quite have the symmetry that crystals are supposed to have (quasicrystals) will certainly surprise you. Professor Senechal shows how "circle" maps can be used to unify some aspects of the study of these apparantly disparate phenomena.
Two Families of Cubics Associated with a Triangle Page 127
by Peter Yff
A cubic curve is the locus of points in the plane that satisfy a third-degree polynomial equation. This paper in advanced euclidean geometry establishes some remarkable connections between ordinary triangles and families of cubic curves that are associated with that triangle.  

History
11. Symmetry Through the Ages - Highlights from the Page 139
History of Regular Polyhedra
by Benno Artmann
There are only five "regular" solids; the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Three of these, the tetrahedron, the octahedron, and the icosahedron, have faces that are equilateral triangles. One, the cube, has faces that are squares; and one, the dodecahedron, has faces that are regular pentagons. In this paper Professor Artmann discusses the history of these regular polyhedra from the time when they first appear in the writings of Greek mathematicians in antiquity to their modern connections with abstract algebra.
12. G. A. Miller: Mathematician, Historian of Mathematics, and Mathematics Educator Extraordinaire Page 149
by Richard Davitt
Certainly one of the most prolific and influential mathematicians of the first part of the twentieth century was Professor G. A. Miller. This paper concentrates on Professor Miller's 44-year career at the University of Illinois. Along with some anecdotes about the Professor, is a compendium of his contributions to the founding of the American Mathematical Society and the Mathematical Association of America. This is an excellent summary of the life and work of a brilliant researcher, a dedicated teacher, and an active, responsible member of the community of scholars.
13. The History of the Mathematical Representation of Electromagnetic Theory Page 159
by Domina Spencer and Uma Shama
Much of the mathematical representation of electromagnetic theory began in the last half of the nineteenth century. This paper traces the development of that theory from the early work of Oersted, Ampere, Gauss, and others through the publication of Maxwell's Treatise on Electricity and Magnetism in 1873. It is shown that current formulations of electromagnetic theory depend on the interplay of two distinct traditions. Gauss suggested the idea of developing the entire subject from a single retarded equation for the force between moving charges which depends on the relative velocity of source and receiver. Maxwell developed the mathematical formulation of the field concepts of Faraday. These mathematical representations are now a fundamental tool of contemporary physics.
14. From Howard Eves to Africa Counts to Multicultural Mathematics Education Page 171
by Claudia Zaslavsky
Before there was much attention being paid to multicultural education there was Africa Counts. This paper gives a brief description of how Professor Zaslavsky came to write one of the earliest and best known of the books now used as references for multicultural mathematics education. Pedagogy
15. An Alternative to the Conventional Wisdom of a Term Paper in History of Mathematics Survey Courses Page 175
by Joe Albree
We certainly want our mathematics students to be literate, to be able to use language to communicate accurately and correctly. And they must have something to say. How do they attain such knowledge and skill? Professor Albree lists a number of problems with conventional term papers that he has encountered. Then he states some of his goals for students in a history of mathematics course: (1) confidence with basic library research skills, (2) organizational and writing skills, and (3) succesful completion of a major piece of
writing. This paper then explains how these goals are met with a "progression of literature reports."
16. DERIVEing a New Approach to Calculus Page 183
by Steven Amgott and Jane Friedman
There is much discussion today about the use of computers in mathematics classes. This paper explains how computer laboratories and the DERIVE software can be used to teach the beginning calculus courses.
17. Alternatives to the Lecture in Teaching Page 191
by William Bonnice
Teachers usually teach the way that they were taught. Lectures have been the mainstay of mathematics teaching for many years. In this paper Professor Bonnice discusses a variety of teaching techniques that can supplement or replace lectures. Included are cooperative groups, journals, and "writing to learn."
Calculus in Context: Integrating Calculus Concepts and Finite Mathematics in a First College Course Page 201
by Stephen Turner
This paper discusses ways to introduce the concepts of finite mathematics and calculus in a mathematics course designed for business majors. The use of computer systems utilizing computer algebra software, graphics programs, and spreadsheets is a key element in the course design. A very nice example of building a mathematical model of a simple business decision situation illustrates most of the methodology. Professor Turner was a student of Howard Eves at the University of Maine. He concludes this paper with some recollections of Howard as a teacher.

Vita Mathematica: Historical Research and Integration with Teaching

(Mathematical Association of America Notes and Reports, No. 40)

Ronald Calinger (Editor)


Mathematical Association of America (MAA) | 1996 | 364 páginas | DjVu (10 mb)

link

Vita Mathematica will enable teachers to learn the relevant history of various topics in the undergraduate curriculum and help them incorporate this history in their teaching. It contains articles dealing not only with calculus, but also with algebra, combinatorics, graph theory, and geometry, as well as more general articles on teaching courses for prospective teachers, and describes courses taught entirely using original sources. Judith Grabiner shows us how two important eighteenth century mathematicians, Colin Maclaurin and Joseph-Louis Lagrange, understood the calculus from these different standpoints and how their legacy is still important in teaching calculus today. We learn from Hans Nils Jahnke why Lagrange's algebraic approach dominated teaching in Germany in the nineteenth century. Wilbur Knorr traces the ancient history of one of the possible foundations, the concept of indivisibles. This volume demonstrates that the history of mathematics is no longer tangential to the mathematics curriculum, but in fact deserves a central role.

 Contents
Foreword VB
I. Historiography and Sources
New Trends and Old Images in The History of Mathematics, David E. Rowe 3
The Role of Problems in the History and Teaching of Mathematics, Evelyne Barbin 17
Dramatizing the Birth and Adventures of Mathematical Concepts:
Two Dialogues, Gavin Hitchcock 27
II. Historical Studies:
From Antiquity to the Scientific Revolution
The Four Sides and The Area: Oblique Light on the Prehistory of Algebra, Jens HrjJyrup 45
The Method of Indivisibles in Ancient Geometry, Wilbur Knorr 67
Enigmas of Chinese Mathematics, Frank Swetz 87
Combinatorics and Induction in Medieval Hebrew and Islamic Mathematics, Victor Katz 99
The Earliest Correct Algebraic Solutions of Cubic Equations, Barnabus Hughes 107
From the Scientific Revolution to the Present
The Early Geometrical Works of Marin Getaldic, Zarko Dadic 115
Empowerment through Modelling: the Abolition of the Slave Trade, John Fauvel 125
The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and their Legacy, Judith Grabiner 131
The Development of Algebraic Analysis from Euler to Klein and Its Impact on School Mathematics in the Nineteenth Century, Hans Niels Jahnke 145
The Mathematics Seminar at the University of Berlin: Origins, Founding, and the Kummer-Weierstrass Years, Ronald Calinger 153
S. V. Kovalevskaya's Mathematical Legacy: The Rotation of a Rigid Body, Roger Cooke 177
Aspects and Problems of the Development of Mathematical Education at Technical Colleges in Germany during the Nineteenth Century, Susann Hensel 191
American Mathematics Viewed Objectively: The Case of Geometric Models, Peggy Kidwell 197
The Social and Intellectual Shaping of a New Mathematical Discipline:
The Role of the National Science Foundation in the Rise of Theoretical Computer
Science and Engineering, William Aspray, Andrew Goldstein, and Bernard Williams 209
III. Integration of History with Mathematics Teaching:
Fundamentals and Selected Cases
History of Mathematics and the Teacher, Torkil Heiede 231
Ethnomathematics: An Explanation, Ubiratan D'Ambrosio 245
The Necessity of History in Teaching Mathematics, \.t:' Frederick Rickey 251
Mathematical Masterpieces: Teaching with Original Sources, Reinhard C. Laubenbacher and David J. Pengelley 257
A History-of-Mathematics Course For Teachers, Based on Great Quotations, Israel Kleiner 261
Measuring an Arc of Meridian, Marie Franroise Jozeau and Michele Gregoire 269
From Egypt to Benjamin Banneker: African Origins of False Position Solutions, Beatrice Lumpkin 279
Mary Everest Boole (1832-1916):
An Erstwhile Pedagogist for Contemporary Times, Karen Dee Michalowicz 291
Origins and Teaching of Calculus
Pupils' Perception of the Continuum, Peter Bero 303
Historical Motivation for a Calculus Course: Barrow's Theorem, Martin Flashman 309
The History of the Concept of Function and Some Implications for Classroom Teaching,
Manfred Kronfellner 317
Integration in Finite Terms: From Liouville's Work to the Calculus Classroom of Today, SIU Man-Keung 321
How Many People Ever Lived? James Tattersall 331
Notes on Contributors 339
Index 345