DISCOVERT IN GEOMETRY THROUGH THE PROCESS OF VARIATION; Generation of New Theorems and Exercises In Geometry By Performing Certain Operations Upon Either the Data or the Conclusion, or Both, of a Known Theorem or Exercise DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State Uhiversity By CLARENCE h / h I&INKE. B.S.. M.A. The Ohio State Uhiversity 1953 Approved by* Adviser ACKNOWLEDGMENT The writer acknowledges with sincere gratitude the invaluable guidance and encouragement generously given him by Professor Nathan Lazar over a period of several years. He Is also indebted to Professor Paul Reichelderfer for helpful criticism and to Professor Harold Fawcett, Professor Earl Anderson, and Professor Everett Kircher, all of The Ohio State Uhiversity, for suggestions and in­ spiration. Without the cooperation of the group of students who were In the writer's class In plane geometry at Capital Uhiversity, an Important phase of this study would have been impossible. The writer expresses his deep appreciation to his wife, Jean Frances Helnke, for her understanding and consideration while the study was in progress, and to his children, Sally, Linda, and Karl, who do not yet fully understand why certain sacrifices were required of them. II A 0 9 7 9 5 TABLE OP CONTENTS* ffegft CHAPTER 1. INTRODUCTION............................. 1 CHAPTER 2. KINDS OF VARIATION IN WHICH THE CHANGES ARE MADE INITIALLY IN THE DATA........ 29 CHAPTER 3 . KINDS OP VARIATION IN WHICH THE CHANGES ARE MADE INITIALLY IN THE CONCLUSION... 88 CHAPTER *+. KINDS OF VARIATION IN WHICH THE CHANGES ARE MADE SIMULTANEOUSLY IN THE DATA AND THE CONCLUSION.................... 113 CHAPTER 5. MORE ILLUSTRATIONS OF THE USE OF VARI­ ATION IN A COURSE IN PLANE GEOMETRY TAUGHT BY THE WRITER.................. 127 CHAPTER 6 . SUMMARY? IMPLICATIONS OF VARIATION FOR THE TEACHING OF GEOMETRY.............. 208 BIBLIOGRAPHY............................. 2Mt AUTOBIOGRAPHY............................ 255 A detailed outline precedes each chapter. Ill DISCOVERY IN GEOMETRY THROUGH THE PROCESS OP VARIATION CHAPTER 1. INTRODUCTION 1.1 Origin of the Problem. P. 2 1.2 The Problem. P. 11 1.3 Definition of Variation and Other Term Used In This Study. P. 12 1.31 Definition of the term variation for purposes of this study. P. 12 1.32 Some examples of the use of the word variation In mathematical writings similar to Its use In this study. P. 12 1.33 Definitions of other terms used in this study. P. 15 The Problem Defined. P. 18 1.5 Limitations of the Study. P. 19 1.6 Procedure* Used In the Study. P. 21 1.61 Review of the literature. P. 21 1.62 Discussion with experienced geometry teachers. P. 23 1.63 Analysis of variation. P. 23 I.6I4. Exploration of variation with a class In elementary geometry. P. 2i|. 1.7 Methods Used In Reporting the Study. P. 25 1 2 1.1 Orl«ln of the Problem. Children, on a whole, are generally credited with being more curious than adults. The following Is typical of ques­ tions frequently asked by probing youngsterss "Mother gets new African Violet plants by breaking off little shoots and putting them In water until they grow roots. What If we try the same thing with roses? Can we get new rose plants In the same way?” A great deal of adult activity which Is described as "original" and "creative" also seems to result from asking specific questions of the sort."What happens in a given situation with which I am familiar certain changes are made?" Each system of non-Euclidean geometry, for example, may be viewed as the result of deliberately making certain changes in the set of assumptions on which the geometry of Euclid is based and then seeking an answer to the question, "What theorems follow from these new assumptions?" It seems altogether reasonable, moreover, to assume that the early geometers, who discovered most of the theorems studied by high-school pupils of the present day, frequently asked the question, "What if ?" The problem of this study originates with the questions! 1) Can students of elementary plane geometry be motivated to ask the question "What If..■?" In respect to propositions of plane geometry which they know? 2) Can they learn to discover new propositions in this manner? 3) Is such actlv- 3 ity worthwhile? Preliminary search for illustrations discloses a number of examples of two or more geometric statements which sure the same except for one or a small number of salient words. Several such examples follow. Example 1. Exercise 1.1. The midpoints of the sides of a parallel­ ogram. taken in order, are the vertices of a parallelo­ gram. Exercise 1.2. The midpoints of the sides of a auadrl- lateral, taken in order, are the vertices of a par­ allelogram. -1- Statement 1.2 is identical with statement 1.1 except that the word quadrilateral replaces the word parallelogram. A parallelogram is a quadrilateral with special properties. If these exercises are expressed in terms of figure, data, and conclusion, the relation between them is seen in another way; exercise 1.2 is the same as exercise 1*1 except for the deletion of two data. *A few textbooks which present these exercises in the order shown are the followings Ernst R. Breslich, Purposeful Mathematics. Plane Geometry, pp. 155 and 190. Walter W. Hart, Plane Geometry, pp. 98 and 107. Leroy H. Schnell and Mildred Crawford, Clear Thinking— AB APBTgftgfa XtiTQMh F l i M flf flutter* PP- 1^2 and 167. A.M. Welchons and W.R. Krickenberger, flemoetry. pp. 166 and 169. Samuel Welkowltz, Harry Sltomer, and Daniel W. Snader, Geometry— Meaning and Mastery, pp. 215 and 230. Exercise 1.1 (a) ABCD is a quadrilateral (b) AB Is parallel to DC (c) AD Is parillel to BC. (d) E, F, G, and H cure the mid­ points, respectively, of AB, BC, CD, and DA. Conclusion* (x) EFGH Is a parallelogram. It 2 Data: (a) ABCD Is a quadrilateral. (b) (Deleted). (c) (Deleted). (d) E, F, G, and H are the mid­ points, respectively, of AB, BC, CD, and DA. (x) EFGH Is a parallelogram. Exercise 1.2. except for the deletion of two data, Is the same as exercise and the figures Involved are closely related. The question may be asked: "Are there other pairs or groups of statements In geometry Involving figures that are only slightly different, such that If small 5 changes are made in one statement, the other Is obtained?" Preliminary search discloses relatively few examples of exercises having the same conclusion, in which one of the exercises can be obtained from the other by deleting one or more data. However, preliminary search indicates that there are quite a few groups of exercises involving figures not very different, such that, considered collectively, one exercise in the collection is the same as another exercise in the collection except for small variations in both the data and the conclusion. An illustration is provided by example 2. Example 2. Exercise 2.1. The midpoints of the sides of a ,££Lyfl££, taken in order, are the vertices of a aonare. Exerclse 2.2. The midpoints of the sides of a rhombus- taken in order, w e the vertices of a rectangle.* The manner in which the data for exercise 2.2 may be obtained from the data of exercise 2.1 is shown better if the data are itemized. Differences in the two statements are indicated by underlining the words Involved. This convention is observed throughout the paper. Sitrclitt Data: A £ B (*) ABCD is a quadrilateral y \ (b) AB is psurallel to DC. H < ; (O AD is parallel to BC. ^ ----- (d) AB is equal to BC. D G (e) Angle DAB Is a right angle. (f) E, F, G, and H sure the midpoints, respectively, of AB, BC, CD, and DA. Conclusion: (x) EFGH Is a square. fiawrSlg? 2.2 Data: B (a) ABCD Is a quadrilateral. (b) AB Is parallel to DC. (c) AD Is psurallel to BC. (d) AB Is equal to BC. (e) (Deleted). (f) E, F, G, and H sure the midpoints, respectively, of AB, BC, CD, and DA. Conclusion: (x) EFGH is a rectangle. Exercise 2.2 was obtained from exercise 2.1 by deleting one datum and making a chsmge in the conclusion. Moreover, exercise 1.1 can be obtained by deleting datum (d) in 7 exercise 2.2 and changing the conclusion from "EFGH Is a rectangle1* to ,(EFGH Is a parallelogram.11 Thus far the focus of attention has been upon exercises involving figures all of which are quadrilaterals and In which the one Is obtainable from the other by deletion of accompanied sometimes by changes 1» *** ftnneinMnn. Three questions arise: 1) Are there groups of exercises in­ volving figures other than quadrilaterals, such that one exercise may be derived from another by deleting one or more data and perhaps making changes In the condnslon? 2) Are there other variations beside deletion, which can be per­ formed upon the data in a given exercise such that a new ex­ ercise is obtained, provided appropriate changes are made in the conclusion If these are needed for the resulting statement to be true? 3) Might initial attention be focused on the conclusion, changes being made in it, followed by appropriate changes in the data if these are needed for the resulting statement to be true? Illustrations will be pro­ vided in the next three paragraphs to answer each of these questions in the affirmative. Two statements concerning triangles, rather than quad­ rilaterals, In which the second is the same as the first except for the deletion of a datum and a change in the con­ clusion are the followingt SMffllg ^ Exercise 1.» If two triangles have two angles and a side of the one equal, respectively, to two angles and a side of the other, the triangles are congruent.