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. Exercise 1.2. If two triangles have two angles of the one equal to two angles of the other, the triangles are similar- Bxemnles h and 5. which follow, illustrate substitution of data rather than deletion of data. ■BXftfflPl? k Exercise >4.1. The angle formed by two chords inter­ secting on a circle is measured by one-half the inter­ cepted arc. Exercise *f.2. The angle formed by a tangent and a chord intersecting on a circle is measured by one-half the intercepted arc. The exercises in axatbpIa h have the same conclusion. Example £ illustrates substitution of data which is accompanied by a change in the conclusion. EXMffliS S Exercise 1- If from a point outside an acute angle (but not between the sides of the angle extended) per­ pendiculars are drawn to the sides of the angle, the

*It is not felt necessary to distinguish between exer­ cise and theorem. angle between these perpendiculars Is ean»l to the given angle. ff.-re-roi 5.2. if from a point inside an acute angle perpendiculars are drawn to the sides of the angle, the angle between these perpendiculars Is supplementary to the given angle. To Illustrate a case In which attention Is Initially- focused on the conclusion, exercise 1.2. page 3, Is recalledi "The midpoints of the sides of a quadrilateral, taken In order, are the vertices of a parallelogram." The question is now askedl "Suppose It Is desired that the midpoint fig­ ure be a rhombus rather than a parallelogram; what changes in the data will bring this about?" Exercise 1.2 with the conclusion itemized, is repeated as exercise 6.1.

Example 6 Exercise 6.1 Datai (a) ABCD Is a quadrilateral (b) E, F, 0, and H are the mid­ points, respectively, of AB, BC, CD, and DA.

(x) HE Is parallel to GF. (y) HG Is parallel to EF. Exercise 6.2 is obtained In the following manner. Data (a) 10 and (b) and conclusions (x) and (y) are copied from exer­ cise 6.1s conclusion (z*) Is added: "HE Is equal to EF." Conclusions (x), (y), and (2*), taken together, are equiva­ lent to the single conclusion, "EFGH Is a rhombus." It Is apparent that one or more additional data are needed so that EFGH will be a rhombus. What datum will suffice can be discovered by studying the proof for exercise 6.1. Diag­ onals AC and HD are drawn. EH and FG are both equal to one-half of HD; likewise, EF and HG are both equal to one- half of AC. In order for HE to be equal to EF, It is evi­ dent that AC and BD must be equal. Hence, datum (c*) is added,

Sarerglg? 6.2 Data: (a) ABCD Is a quadrilateral. (c*) AC ■ BD. (b) E, F, G, and H are the mid­ points, respectively, of AB, BC, CD, and DA. Conclusion: (x) HE is parallel to GF. (y) HG Is parallel to EF. (:*) HE s EF. The word statement for exercise 6.2 is: "The midpoints of the sides of a quadrilateral having equal diagonals, taken In order, are the vertices of a rhombus." 11 1.2 The Problem A large number of geometry books contain exercise 6.1. but the writer found none In which exercise 6.2 Is given. ttrereige 6.2 was developed from exercise 6.1 by adding one Item to the conclusion and then determining the additional item needed in the data. In a similar manner, If one Is familiar with exercise 1.1. **The midpoints of the sides of a parallelogram, taken in order, are the vertices of a par­ allelogram,” but not familiar with exercise 1.2. "The mid­ points of the sides of a quadrilateral, taken in order, are the vertices of a parallelogram," he might discover the lat­ ter statement by deliberately deleting data (b) and (c) from exercise 1.1 and then determining the conclusion which follows, provided, of course, that he Is cognizant of this method for discovering new statements. Similar observations apply to the other groups of exer­ cises which have been given as illustrations. These obser­ vations suggest a teaching method to be used in geometry, whereby new statements are obtained by deliberately making changes In the data or conclusion df a familiar statement, followed by such other changes as are needed to make the new statement true. In this paper such a process Is called 12 1.3 Definition of Y«rlaMnther Terms Used In This Study. 1.31 Definition of the tarn variation for purposes of this atudv. Variation Is defined to mean a process of changing elements of the data or conclusion, or both, of a geometri­ cal statement, which has been proved to be true or Is accepted as true, with a view to obtaining a set of data, or a new conclusion, or both, resulting In a new statement. The new statement Is to be affirmed by deduction, or shown to be false, or adopted as a new assumption. In mathematics, the technical use of the word variation to denote a type of relation, such as Inverse, direct, or joint, between variables In a formula or equation Is well known. This meaning of the word is so different from Its meaning In this study that no confusion is anticipated. In this study the word variation denotes, as It does In ordin­ ary non-technical language, the notion of modification or change. It Is used In mathematical literature In this non­ technical sense, as well as in the technical sense mentioned above.

1.32 Some examples of the use of the word variation in math­ ematical writings similar to Its use In this study. Using as an Illustration the relations between the angle formed by two lines and the arcs which these lines intercept 13 on a circle, The National Committee on Mathematical Require­ ments suggests, "A great assistance to the Imagination Is gained In certain figures by Imagining variations of the figure through all Intermediate stages from one case to an­ other. However, the purpose of this remark was not to propose variation as a method of discovery but to show that the existence of functional relationships In geometry can be exhibited, for the committee adds, "Such observations are essentially functional In character, for they consist In careful observations of the relationships between the angle to be measured and the arcs that measure It."3 Butler and Wren, In their book The Teaching of Secondary Mathematics, use the word variation In connection with the many forms of the linear algebraic equation In the following manner: Linear equations In ninth-grade algebra assume a variety of forms, illustrative of which are such forms as: ax - b x 4 a 9 b ax Abac a - x ■ b x-aab Zsb ix A b i cx 4 d ax 4 bxs c a While these forms are all variations of a common form, the similarity is usually not Immediately apparent to

RtqrfiflisftU gB aC -Jtotatm U ci l a flgftfinfliry fifljaca- tion. Report of the National Committee on Mathematical Re­ quirements, The Mathematical Association of America, Inc., P- 69. 3IbH., PP. 69-70. lb children encountering them for the first time. Moreover, textbooks and teachers are frequently deficient In giving emphasis to this point.1* One chapter of G.M. Mertlman's book To Discover Ms the- nuatiis entitled "Theme with Variations."5 The author presents an Illustration from the calculus. The following four quotations convey his Interpretation of variations 1) "The theme Is the rate problem."6 Rate of change of £ with respect to £ Is meant. 2) "The first variation of the theme of Instantaneous change ... Is the extreme-value problem."7 Reference Is to problems Involving maxima and minima. 3) "For the second variation on the subject of derivative we turn to the area problem."8 Thus the inverse process of differentiation, namely Integration, Is obtained. b) "The finale will be a restatement of the theme. We re­ turn to the original rate problem."9 The problem then In­ dicated Is the one concerned with rate of change of related variables, each with respect to time. In Polya's book How To Solve It there Is a section w ^ Butler 2nd Wren Thg Iqacftiag Of, ggfiqftflMy Mathematics, second edition, p. 315.

10 ?Ga3rlord M* Merriman, To Discover Mathematics, Chapter

6Ibld■- p. 255.

7Ibid., p. 26M-.

8nna., P. 273. 9ibld.r p. 283. 15 titled "Variation of the Problem."10 In it the writer sug­ gests that when one is seeking a solution to a problem he might, as an intermediary step, try "variation of the prob­ lem" which has not been solved into a form which haa been solved or easily can be solved. Here variation is proposed as an aid in the solution of problems. In contrast, Bakst proposes variation as a means to the formulation of new problems. He says* The greatest advantage which may be de­ rived from the study of geometry is associ­ ated with the opportunity of variation of certain given conditions stated in a prob­ lem, whether this is an original or a con­ struction. If the pupils are encouraged to vary the conditions, they will eventually develop the tendency to "imagine" and to formulate problems of their own.H It is in the general, spirit of this comment that the word variation Is used in this paper.

1.33 Definitions of other terms used in this study. The word statement was used in the above definition of variation, where proposition might be expected by some readers. It is true that Russell uses the word proposition to mean "primairily a form of words which expresses what is true or false."12 in this study, statement means "a group of words,

10G. Polya, How To Solve It. pp. 182-186.

^Aaron Bakst, "Mathematical Recreations," The Mathe­ matics Teacher. XLIV (April, 1951), p. 270.

^Bertrand Russell, Introduction to Mathematical Philos­ ophy. p. 155. 16 or a group of words together with other symbols, which ex­ presses something that Is either true or false." Hence a statement may be expressed entirely In words, or in terms of a figure, data, and conclusion, or in both ways. Propo­ sition is avoided because In common usage, among both teachers of geometry and writers of textbooks, the word proposition means a "theorem or a construction," with theorem meaning "a statement to be proved."13 When the process of variation is applied to one statement, although it is true, the resulting statement might not be true; its truth or falsity must be determined. Not only is the purport of state­ ment. as just defined, familiar; it is also used with this connotation in mathematical logic. Quine, for instance, equates statement with declarative sentence.llf He also re­ fers to the "if - then" type of sentence as a statement which is "known as a conditional (or hypothetical)• Some of the sentences in geometry which express "something that is either true or false" are stated in the declarative form and some are stated in the conditional form. An example of the former is* "The altitudes of any triangle are concurrent." An example of the latter is* "If two sides of a triangle are equal, the angles opposite are equal." It is convenient to

^Joseph P. McCormack, Plane Geometry, p. 2*f.

llfWillard Van Orman Quine, Mathematical Logic, p. 11.

15Ibid.. p. 1^. 17 have a generic word which Includes both types. The word statement Is chosen for this purpose. The statement known to the Investigator, upon which one or more variations are performed, will be called the original the original WJTSlgfi, or simply the original. It will be expressed In words, or In terms of a figure, data, and conclusion, or In both ways. A statement obtained by varying the original exercise will be called a variant. For example, on page 3, exercise 1.1 Is the original exercise and exercise 1.2 Is a variant. An original exercise and one or more variants obtained from it will hereafter be referred to, as a whole, by the name illustration. Throughout the study, as in the examples already given, data refers to what Is given, both In an original exercise and In a variant, and conclusion refers to what is to be proved. It Is understood, of course, that until it is proved that the conclusion In a variant follows from the data, either the data or the conclusion Is formulated only tenta­ tively. That Is, If the variation Is performed Initially on the data, then the conclusion Is tentative until deduced; and If the conclusion Is varied Initially, then the data are tentative until It Is established by proof that the con­ clusion follows from the data. l.lf The Problem Difinad. The basic problem of this study is fourfold> 1) to explain the process of variation, illustrated in section 1.1 and defined in subsection 1 *3 1 ; 2) to explore vays of using the process as a teaching method for the presentation of new material in an elementary plane geometry course; 3) to ex­ plore vays of teaching students in a geometry class to use variation as a method for discovering new geometrical state­ ments; *+) to provide a large number of illustrations of the use of variation in elementary plane geometry. 19 1.5 Limitations of the Study. In order to define bounds for the study, the following limitations are imposed! 1. The application of variation, for purposes of this study, is limited to geometrical statements by the defin­ itions given for variation and for statement. Hence, for example, particular consideration of ways in which concepts and definitions may be modified is not emphasized in this study. 2. Although statements which are not geometrical in nature cam be changed in much the same way that geometrical statements are varied and although such a process is recog­ nized as being of importamce, implications of variation for areas other thaui geometry are only Indicated. 3. The subject material considered in the study is limited almost exclusively to that of elementary plame geometry. Thus applications of variation leading to state­ ments in solid geometry and in advanced Euclidean geometry and applications of variation to systems of assumptions for geometries other than two-dimensional Euclidean geometry are only mentioned. b. Construction problems can be formulated as state- merits In accordance with the definition given for statement.^ However, detailed consideration of variation in connection with construction problems in elementary plane geometry, use of variation to obtain statements in solid geometry and advanced Euclidean geometry, extension of variation to ob­ tain axiomatic systems for geometries other than Euclidean two-dimensional geometry, and extension of variation to statements in areas other than geometry, are Indicated In section 6.V, pp. 225 - 227.

Construction problems arefetated in geometry textbooks in two ways. These two ways are illustrated by the follow­ ing examples 1) To bisect a given angle; 2) It is possible to bisect a given angle. Most textbooks state construction problems in the manner illustrated by 1). This form is also used in T.L. Heath. The Thirteen Books of Euclid»s Elements: see, for example. Vol. I, p. 2$*f.Two textbooks which em­ ploy the form illustrated by 2) ares Hart, op. clt.T (see p. 6*f), and McCormack, on. clt.. (see p. 79)* 21 1,6 Procedure* Used In the Study

1.6l Review of the literature Literature dealing with the teaching of mathematics and the nature of mathematics was examined In the course of the study being reported. Some Items relevant to the study have been noted In section 1.32* pages 12 - l£. Other such Items are noted at appropriate places In connection with the Illus­ trations of variation presented and discussed In the chapters that follow. On the basis of conversations with creative mathemati­ cians, the writer concludes that some of the problems on which they did original research were deliberately obtained by the method of variation, applied to previously established theorems• Fawcett stressed the relation between assumptions In a postulatlonal system and the theorems which are implied by them. With special reference to the theorem "The sum of the Interior angles of a triangle is l8o°n, he reports, The way In which the pupils had been intro­ duced to the nature of assumptions was help­ ful In leading them to see that a change in any one of the three assumptions on which this conclusion depends was likely to change the conclusion.17 Fawcett’s students became acutely aware of the role of as­ sumptions and the results of adopting alternative assumptions.

— Harold P. Fawcett, The Nature of Proof. Thirteenth Yearbook of the National Council of Teachers of Mathemat­ ics, 193®* P* 73* 22 This is particularly evident from the papers they wrote on the subject "The Evolution of Proof.They realized that other assumptions may be substituted for Euclid's fifth pos­ tulate "Through a given point, one and only one straight line may be drawn parallel to a given line," and they viewed such substitutions as resulting in various non-Euolidean geometries, such as the geometries of Bolyai, Lobachevsky, and Riemann.19 However, the writer did not find any study or other presentation of a detailed and comprehensive nature con­ cerning a process like the one which in this study is called variation; nor did he find evidence in the literature of de­ liberate and extensive use of such a process in the teaching and learning of geometry or of any other branch of elementa­ ry mathematics. Hence no special section is devoted to a review of the literature related to the study, as is done customarily. The literature examined may be divided into the follow­ ing categories: 1) Reports of committees and associations. 2) Books on the teaching of geometry and of secondary mathematics. 3) Other research projects which appear to be related to this study.

1 ft Ibid. , pp. 126 — l4i+.

19Ibid., pp. 133 - 134, 136, 139 - ll+O, 142, lJ+3. U) Periodical literature. 5) General writings by qualified and thoughtful peop-le In such fields as nature of mathematics, nature of science, logic, philosophy of mathematics and of science, and discovery in mathematics. 6) Geometry textbooks In different recent periods, especially 19U5-1952. 7) Syllabi for courses in geometry. 1.62 Discussion with experienced geometry teachers. The writer spoke of this study with other geometry teachers, some of whom had had a large amount of experience. Five other teachers became particularly interested in the study and discussed It with the writer often enough to aid him materially In clarifying his ideas. They also provided a number of illustrations, which had resulted from the de­ liberate use of variation In their own thinking and with their classes.* 1*63 Analysis of variation. It became apparent early in the study that a given sit­ uation can be varied In different ways. For Instance, as noted In section 1.1, an Item of data or conclusion may be deleted, a new Item may be added, and a new item may be sub­ stituted for an Item that Is present In the original situa­ tion. A detailed analysis of the process of variation from

*They are: Mr. Harold Brockman, Capital University; Mr» Sheldon Myers, The Ohio State University; Mr. Oscar Schaaf, University School, The Ohio State University; Mr. John Schacht, Bexley, (Ohio) High School; Mr. Eugene Smith, University School, The Ohio State University. 2k this point of view la part of this study. The analysis is presented in Chapters 2, 3, and Ij..

1.61+ Exploration of -variation with a class in elementary geometry. It seemed advisable that the writer should try out the method of variation with a class in elementary plane geometry which he would teach. This was done for the following reasons: 1) It waa hoped that the notion of variation would thereby become clearer to the writer and that previously unnoticed facets thereof would be recognized, 2) Many illus­ trations were desired. Some, it was hoped, would be pro­ vided by students as they became familiar with the process of variation. Others would be discovered by the teacher in the course of the work. 3) Finally, it seemed desirable to find out whether or not students could learn to use the proc­ ess of variation and to determine some ways of teaching It. The class which the writer taught In connection with this study consisted of twenty—seven students at Capital University during the second semester, 195>0-£l. Most of the students were college freshmen. All of them were in the class to compensate for a oollege-entrence deficiency In plane geometry. With a few exceptions, none had previously had any experience with formal geometry. The material cov­ ered during the semester was approximately equivalent to that of a one—year high school course in plane geometry. 25 1.7 Methods Used In Reporting the Study

For purposes of analysis, variation may be divided into three main categories. In the first category are kinds of variation in which attention is focused initially on the data. In the second are those in which attention is focused initially on the conclusion. The third category consists of certain well-known operations in which data and conclusions are rearranged or changed simultaneously in accordance with prescribed procedures. These three principal categories of variation are considered in chapters 2, 3, and *+, re­ spectively. Chapter 5 consists of illustrations of variation organized according to the major subject headings of elemen­ tary plane geometry. A summary regarding variation is given in chapter 6, followed by implications for the teach­ ing of mathematics. CHAPTER 2 KINDS OP VARIATION IN WHICH THE CHANGES ARP. MADE INITIALLY IN THE DATA P. 29

2.1 One or More Data Art Deleted P. 30 2.11 deletion of data does not necessitate a change in the conclusion. P. 30 2.12 Deletion of data necessitates a change In the conclusion. P.38 2.2 One or More Data Are Added P.V5 2.21 Addition of data is not accompanied by a change In the conclusion. P.^5 2.22 Addition of data permits a change in the conclusion. P.51 2.3 Substitutions Are Made in the Data. P.55 2.31 The substitution involves numerical values assigned to elements of the figure. P.55 2.311 Substitution of data does not necessi- a change in the conclusion. P. 55 2.312 Substitution of data involving numeri­ cal values necessitates a change in the conclusion. P.58 2.32 Numerical data are replaced by data which are not expressed in numerical terms. P .60

26 27 2*321 There Is no change In the con­ clusion. P.60 2*322 The replacement of numerical data by data expressed in general terms necessitates a change in the con­ clusion* p. 61 2*33 Numerical data are substituted for data not expressed in numerical terms. P.62 2*331 There is no change in the conclusion. P .62 2.332 Substitution of numerical data for data expressed in general terms neces­ sitates a change in the conclusion. P. 63 2.3V The substitution involves elements of the figure. P. 73 2*3^1 There is no change in the conclusion. P. 73 2.3^2 Substitution of data involving ele­ ments of the figure is accompanied by a change in the conclusion. P.75 2.35 The substitution involves relationships. P. 77 2*351 There is no change in the conclusion P. 77 2*352 There Is a change in the conclusion when a substitution involving rela­ tionships is made in the data* P. 78 2.36 The substitution of data involves the basic figure. P. 79 2.361 There is no change in the conclusion. P. 79 28 2*362 There Is a change In the conclusion when the substitution In the data Involves the basic figure. p. 79 2.37 The substitution Involves combinations of some or all of the types of substitutions mentioned. p.80 2.371 A combination of kinds of substi­ tution in the data is not accompan­ ied by a change In the conclusion. P. 31 2.372 A combination of kinds of substi­ tution in the data is accompanied by a change in the conclusion. P.32 2A Combinations of Deletion. Addition, and Substltu- ttqn Are Fftrf9rawfl Data- P. 37 29 A variation performed initially on the data is some- times accompanied by changes in the conclusion and some­ times it is not. Variation may be performed by deletion, addition, substitution, and by combinations of these oper­ ations. Moreover, several different sorts of substitution are distinguishable. Prom a consideration of all of these types and combinations involving some of them, a large number of kinds of variation appear to be possible. A de­ tailed analysis of variation is made from this point of view. Deletion of data, addition of data, and substitution of data, respectively, are treated in sections 2 .1 , 2 .2, and 2.3. Combinations of the three basic types are consid­ ered in section 2.*f. In each of these sections two further cases are considereds 1 ) variation of the data Is not accom­ panied by a change in the conclusion; 2 ) variatbn of the data accompanied by a change in the conclusion. Consid­ eration is limited to those conclusions in a variant which bear a relation, first of all, to the conclusions in the original exercise from which the variant was formed through a change in the data and, secondly, to the change made in the data. 30 2.1 Qatt gr Mart Pat* Ac? Pslgtafl

It Is desired that a mathematical statement, In Its final formula tlon, should not contain more data than are required for establishing the conclusion. However, In a process of discovering them, geometric relationships are frequently recognized In situations which provide more than the necessary data. Then, as the result of further Insight and the discovery of new methods of proof, It Is found that some of the data may be deleted. Hence, In some exercises it Is possible to delete certain data without ef­ fect on the conclusion. In other cases, depending on the exercise and on the data deleted, deletion of data must be accompanied by a change in the conclusion In order for the result to be a true statement.

2.11 Deletion of data does not necessitate a change in the ggafilUgjgn.

ILLUSTRATION 1. Original Exercise 1. Statement! The midpctots of the sides of a parallelogram are the vertices of another parallelogram.*

YarlaflE 1*1 Statements The midpoints of the sides of a tranezold are

*As a convenience to the reader the key words involved In a variation are underlined in the statements of the original exercise and the variant. 31 the vertices of a parallelogram. Although the word trapeaoid Is substituted for the word naraiuingr«m the detailed analysis which follows discloses that the variation performed on the data may also be viewed as one of deletion. Original Exercise 1. Datai (a) ABCD Is a quadrilateral. (b) AB is parallel to DC. (c) AD Is parallel to BC. (d) E, F, G, and H are the mid­

points, respectively, of AB * BC, CD, and DA. £.gnalqajgQi (x) EFGH Is a parallelogram.

Variant 1.1

(a) ABCD Is a quadrilateral. (b) AB Is parallel to DC. (c) (Deleted). (d) E, F, G, and H are the mid­ points, respectively,of AB, BC, CD, and DA.

(x) EFGH Is a parallelogram. 32 Discovering the Conclusion Variation Is a method of discovering new statements. The data for variant 1.1 were obtained by deliberately de­ leting one of the data of the original exercise and retain­ ing the other. After this, the conclusion still remains to be discovered. A first question might be, **Does the orig­ inal conclusion remain true even after one of the data has been deleted; that Is, is a parallelogram obtained when the midpoints of the sides of a trapezold are joined, as was the case when the midpoints of the sides of a parallelogram were connected?” The investigator draws several trapezoids of different shapes, connects the midpoints of their sides, and inspects the resulting figures. All of them appear to be parallelograms. Hence he attempts to prove, deductively, that EFGH is a parallelogram.

Variant 1.2 Statementt The midpoints of the sides of a quadrilateral1 are the vertices of a parallelogram.

Quadrilateral, used without a qualifying adjective, means, in this itudy, as in general practice, convex quad­ rilateral. See, for instance, Frank M. Morgan, John A. Foley, and W.E. Breckenrldge, Plane Geometry. Revised Fdi- tlon, p. 102. Moreover, unless otherwise stated, a nlanur quadrilateral is meant. However, variants involving con­ cave, crossed, and non-planar quadrilaterals may"also be formed. These are presented in Chapter 5, pp. 173-185. 33 Datai (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. Conclusion* (x) EFGH is a parallelogram. D

Discovering the Conclusion A plausible conclusion for this variant might be dis­ covered, as for the one preceding, by drawing several con­ vex quadrilaterals of different sizes and shapes, and in­ vestigating the figures obtained by connecting the midpoints of their sides. However, if this variant is not formulated until after a deductive proof has been discovered for variant 1 .1 . it might be observed that datum (b) was not used in that proof and hence is not necessary. That is, it is perceived that a true statement can be obtained by delet­ ing the datum known to be superfluous, without making any change in the conclusion. Appearance in Textbooks of the Exercises in Illus­ tration 171 Variant 1.1 was not found in any of the textbooks in­ spected. Original exercise 1 and variant 1.2 are found in most textbooks but not always in the same order. In the footnote on page 3 , reference Is made to five textbooks In which the exercise "The midpoints of the sides of a par­ allelogram, taken In order, are the vertices of a parallelo­ gram," precedes the exercise, "The midpoints of the sides of a quadrilateral, taken In order, are the vertices of a parallelogram." Moreover, It Is to be noted that In all of the cases cited, the exercises are presented several pages apart. In some other textbooks these two exercises p are presented In the order opposite from that given above; In these cases the two exercises are found near together. Treatment in textbooks of the exercises presented in this Illustration, and others related to them from the standpoint of variation, is further discussed In Chapter 5, pp. 178-188.

Methods of Proving Original Exercise l.and Variants 1.1 and 172 Original exercise 1 (p. 30 ) may be proved by showing that triangles AEH and CGF are congruent and that triangles EBF and GDH are congruent and then applying the theorem, "A quadrilateral is a parallelogram if both pairs of opposite sides are equal.” However, this method of proof fails in variant 1.1 and also In variant 1.2. A proof may be obtained by drawing a diagonal of the original figure, as AC. In view of the theorem, "A line joining the midpoints of two sides

2J. Herbert Blackhurst, HutuGeometry, p. 79. D.T. Sigley and Wm. T. Stratton, Plane Geometry. pp. 112-113. Paul L. Trump, Geometry - A First Course, pp. 173-17*+. 35 of a triangle Is parallel to the third side and equal to one-half of It," EF and HG are each parallel to AC and equal to one-half of AC. Hence EF and HG are equal and parallel to each other, from which It follows that EFGH is a parallelogram.

gov Illugtaatlpftg gf Xhlf.glBfl gf YarlaUpfl Art Found in Itartlreatoa In this Illustration, after variant 1.2 has been proved, It Is apparent that neither datum (b) nor datum (c) in the original exercise is necessary for establishing the conclu­ sion. Hence, from the standpoint of mathematical efficiency, it is desirable to prove variant 1 .2 . which involves a gen­ eral quadrilateral, and then to perceive that variant 1.1 and the original exercise Involve special kinds of quadri­ laterals, for which the same conclusion is then also true. It is probably on tills account that original exercise 1 was

found to precede variant 1 -? in only approximately half of the textbooks that were inspected. Importance of Searching for Data That Can Be Deleted A high regard for mathematical elegance is perhaps, in general, the reason why only a few statements are found in textbooks in which data can be deleted without a change be­ ing made in the conclusion. Gut in the process of original investigation, a conclusion is frequently apprehended in a situation containing one or more data which are later de- 36 leted, as further study discloses that this may be done. It seems to the writer that sufficient experiences should be deliberately provided to keep the student alert to search for data which may be deleted without changing the conclu­ sions that his appreciation for this Important aspect of the evolution of mathematics may be developed and so that he will learn to Inquire about his own creative products, "Are there any data which I can delete?" It Is Important to notice that, although all of the data In a given exercise are used In the proof which Is known at the time, It might nevertheless be possible to delete some of the data and have a true statement result. A New Theorem Might Be Discovered Through the Heed For

When exercises and theorems are presented in textbooks, they are customarily presented only after the theorems needed in the proofs have already been established. This explains why, when original MWffilgg 1 precedes Yftflant It2 in a textbook, these two exercises are separated by several pages. Only a knowledge of parallelograms and of congruent triangles is needed for proving that "The midpoints of the sides of a parallelogram are the vertices of another par­ allelogram." But the theorem "A line joining the midpoints of two sides of a triangle is parallel to the third side and equal to one-half of It" Is needed In proving that "The midpoints of the sides of a quadrilateral are the ver- 37 tlces of a parallelogram."

The exercises In Illustration 1 are not presented In textbooks In a manner that suggests-variation by deletion of data. The exercises of Illustration 2. on the other hand, are presented In the order of their presentation In the ex­ ample which follows and in juxtaposition in nearly every geometry book that was inspected.3 However, they do not occur in this manner because an illustration of variation is intended but because original exercise 2 is used in the proof for variant 2.1.

ILLJSTRATION £. Original Exercise 2. Statementi An inscribed angle, one side of which Is a di­ ameter of thq circle, is measured by one-half of its intercepted arc. 3 Data: (a) Circle 0. (b) Inscribed angle ABC. (c) AB is a diameter of circle 0.

^The two exceptions noted are: David Relchgott and R. Splller, Today*s Geometry. 3rd ed., p. 239. David Skolnlk and Miles C. Hartley, Dynamic Geometry, pp. 150-151. 38 C9jwlualgB» (x) Angle ABC ® H c . h

Deletion of the datum that one side of the inscribed angle is a diameter of the circle leads to the following variant. Only the statement and figures are given.

Variant 2.1 Statement! An angle inscribed in a circle is measured by one-half of its intercepted arc.

In the proof for vari- q 0 ant 2.1 , two cases implied by the figure are considered. •O In textbooks these are pre­ C sented as cases 2 and 3, and the original exercise of this illustration is case 1 ; proof of all three cases estab- lishes the one statement, "An angle inscribed in a circle is measured by one-half of its intercepted arc."

2.12 Deletion of data necessitates a change in the con­ clusion.

ILLUSTRATION 3 . Original Exercise 3 . *— 11 ^ 1 ~ The symbol 41 is read, "is measured by." This termin­ ology, although not the symDol, is used by McCormack, oq. cit.. on p. 182 and in other places. Both the terminology and the symbol are used by Schnell and Crawford; see op. cit.. p. 205, and elsewhere. 39 The midpoints of the sides of an isosceles trapezold are the vertices of a rhombus.

Variant 1.1 Statement! The midpoints of the sides of any trapezold are the vertices of a PftElllglqgEMI- That deletion of a datum, rather than substitution, is involved in this illustration is shown by a detailed analy­ sis ; that the change in the conclusion is also one of de­ letion, rather than substitution, is also shown. Original Exercise Data: (a) ABCD is a quadrilateral. (b) AB is parallel to DC. (c) AD is equal to BC. (d) E, P, G, and H are the mid­ points, respectively, of AB, BC, CD, and DA. Conclusion! (x) EFGH is a parallelogram. (y) EF is equal to FG. bO

Data: (a) (b) AB Is parallel to DC. (c) (Deleted). (d) mid points, respectively, of AB, BC, CD, and DA. Conslualan* (x) EFGH is a parallelogram. (y) (Deleted).

ILLUSTRATION it, ftrlglna; Bareraigttit Statementi The bisector of the angle between the equal sides of an Isosceles triangle bisects the base. Data: G (a) Triangle ABC. (b) AC Is equal to BC. (c) CD bisects angle ACB. g&nalttalQB* A (x) AD Is equal to DB. Deletion of datum (b) leads to variant *f.l. Only the statement and figure are given. hi .Zurliat H-.l Statement! The bisector of m interior angle of jbx triangle divides the opposite aide into segments which are proportional

tq ths aides fldlaseat , . t a tto A O angle bisected, that is, s ^

The hopelessness of trying to prove that AD is equal to DB in variant lf.1 is apparent from the figure. The cor­ rect conclusion that is Indicated might be discovered by making measurements of AD, DB, AC, BC, and perhaps other parts in triangle ABC and then noticing from the data that AD/bB appears equal to AC/BC. However, it might also be discovered by way of observations somewhat like the follow­ ing: AD does not seem equal to DB if AC is not equal to BC; but AD is less than DB if AC is less than BC; hence, two relations appear possible, both of which are true in the original exercise. 1) BC - AC • DB - AD, and 2) AD/bB ■ AC/BC; only the latter relation can be substantiated by measurement and by proof. While most of the students who have been taught the process of variation might be expected to think of deleting datum (b) as a first step in obtaining variant h . 1r very few, if any, would be likely to discover the conclusion and the proof without a teacher's help. b2 Remarks Concerning Variation by Deletion of Data

1) The effect on the conclusion of deleting some of the data. There are situations In which deletion of data is not accompanied by any change In the conclusion, as has been shown in Illustrations 1 and 2. If, however, deletion of data Is accompanied by a change In the conclusion, that change consists either 1 ) in a deletion, as in 3 , or 2 ) in a substitution of a more general statement in the conclusion, as in illustration *f. In either case the entire conclusion of the variant includes the conclusion ◦f the original exercise as a special case. Formulation of the new conclusion before a proof is attempted is often a matter of experience. Frequently, it can be aided by study of several figures drawn in accordance with the data. 2) Reference to deletion of data in the literature concerning the methods of mathematics. Shaw says, "Much mathematical work of the present day consists in determining whether a conclusion can persist if the premises are made a little less restricted."5 in making this statement Shaw has reference to research in mathematics. When variation by deletion of data is performed in elementary plane geometry, the pupil has an opportunity to learn and to ---- g -■■■ ■■■ ■ ■■ James Byrnle Shaw, Lectures on the Philosophy of Math­ ematics. p. 172. if3 use In his own creative work a method which Is used by re­ search mathematicians. When deletions are made from the data In a true statement, a new statement Is frequently discovered. Sometimes the original conclusion remains un­ changed and sometimes It needs to be modified In order for the new statement to be true.

Presenting Variation to a Class It should not be Inferred that variation was presented to the class Involved In this study In the sequence In which it Is presented in this chapter. An awareness of the conception of variation and of the different kinds of vari­ ation emerged as the result of deliberately making changes in exercises that were assigned. At first no more was at­ tempted than to develop a favorable attitude toward making changes and to create an atmosphere of curiosity about what would happen If changes were made. Then, as different kinds of variation were recognized, the students were able to pro­ duce increasingly more numerous and more interesting vari­ ants themselves. The analysis made in this chapter Is presented in order to convey the Important ramifications of the process of variation in a brief and systematic manner. Illustrations are therefore selected according to their appropriateness, not according to any systematic geometrical sequence. More­ over, Illustrations are drawn from various phases of the Mf work In plane geometry, Including the Informal phase. Since emphasis Is on the development of new statements rather than on the proofs for them, the proofs which accompany these Illustrations are generally 6mltted except when some particular purpose Is served by reference to them. b5 2.2 One or More Data Are Added. 2.21 Addition of data la not accompanied hv * change In thf SQaclUglQB- It was noted earlier (p. 3^-) that In some textbooks the statement "The midpoints of the sides of any quadri­ lateral are the vertices of a parallelogram" is followed by the statement "The midpoints of the sides of any par­ allel Qgram are the vertices of another parallelogram." The second statement is the same as the first except for the addition of the data that both pairs of opposite sides are parallel. Of course, whatever Is true of a quadrilateral is also true of a parallelogram. Inclusion of the second statement after the first has been proved provides an exer­ cise for the student to prove; it emphasizes a particular implication of the first statement, but it does not present anything that is significantly new. Suppose, however, that a student suspects the truth of a statement but cannot prove it. Polya presents the follow­ ing suggestion as a possible aid in the search for a proof of a given statements We keep the conclusion and change the hypoth­ esis.We first!*irst try totc recollect such a the­ orems Look at the conclusion! Aflfl try tQ tfrlnfc Qf » fSflllap theorem having the same or a sim­ ilar eanelualon. If we do not succeed in recol .acting such a theorem we try to invent ones 1+6

fr oaTwhl ch "you c o ^ d e ^ l r derive the conclusion?*? Polya’s only specific suggestion, lay way of elabora­ tion, Is to delete part of the hypothesis and then to try to prove the conclusion, a procedure which, paradoxical &3 It seems, frequently terminates In success. This writer proposes a second explicit suggestion, recognizing that Polya might have intended to imply It alsox another way to Invent a statement related to the one under Investigation, which might be easier to prove, is to add to the data with­ out adding to the conclusion. Accordingly, special cases of a theorem under Investi­ gation can sometimes be produced by the addition of one or more properly chosen data. In the process of investigating a general statement thought to be true, it Is common strat­ egy to study special cases. If a special case Is discovered which is manifestly untrue, then the general statement also Is untrue. On the other hand, verification In special cases motivates search for a proof of the general statement. The writer suggests further that In some situations, study of the proof or study of several proofs of the special case might suggest how one of them may be modified so as to apply to the general statement.

Polya, o p . cit., p. 80. (Italics shown are in the original; henceforth, whenever italics are Indicated in a quotation they appear in the Original unless noted). It should be noted that In Polya's book emphasis Is on dis­ covering a proof for a given statement rather than on the discovery of a new statement. *f7 Consideration of tha Pythagorean Theorem Is particu­ larly Interesting In this connection and serves well to illustrate the coalments just made. Concerning the discovery and proof of this theorem Heath speculates as followss How did Pythagoras discover the general theorem? Observing that 3, k, 5 was a right-angled triangle, while 32 a lf2 ■ ^2y he was probably led to consider whether a similar relation was true of the sides of zlght-angled triangles other than the particular one. The simplest case (geo­ metrically) to investigate was that of the Isosceles right-angled triangle; and the truth of the theorem in this particu­ lar case would easily appear from the mere construction of a figure.7 Transition from the conjectured statement concerning any right triangle to a similar statement concerning an isos- celes right triangle Involves the addition of the datum that the legs are equal but does not Involve any change In the conclusion.

ILLUSTRATION Original Exercise 5 Statements The sum of the squares on the legs of anv right triangle Is equal to the square on the hypotenuse. Data: F (a) Triangle ABC with square AHDE on side AB, square BCFG In side BC, square ACJK on side CA. (b) Angle ACB Is a right angle. D ------'Heath, o p . cit.. Vol. I, p. 352 If8 Conclusion* (x) ACJK + BCFQ ■ ABDE.

Virlmt .5,1 Statement! The sum of the squares on the legs of an lso»- celes right triangle is ajual to the square on the hypotenuse. Data* (a) Triangle ABC with

square ABDE on side AB, K square BCFG on side BC, square ACJK on side CA. (b) Angle ACB is a right angle. (c*)^AC Is equal to BC. cqafilugjpai Cx) ACJK ♦ BCFG ■ ABDE

The truth of variant 5.1 is indeed, as Heath says, apparent from the mere construction of the figure, and the addition of lines KC, CG, AD, and BE. And it has been shown how variant 5.1 may be generated from original exer­ cise 5 by the addition of a datum but without any change in the conclusion. The earlier suggestion, that study of a figure and proof for a special case of a more general statement might ^ ------An asterisk accompanying the letter designating a datum or conclusion of a variant indicates that the datum or conclusion has been added in the process of formulating the variant. if9 suggest a method for proving the general statement, will now be elaborated. The writer Is aware, however, that his entire speculation might be Influenced by his knowing the proof for the general statement, In this case original exer­ cise 5. No more is known about how the Pythagoreans proved the theorem which bears their name than how they discovered the proof. The present conjecture is directed to the dis­ covery of the proof given by Euclid and attributed to him personally.® This proof makes use of the figure given in connection with original exercise 5 and repeated below with the addition of one line. The trend of speculation is the following. In figure 5.1 it is, of course, evident that since square ABDE is equal to the sum of squares BCFG and

F

N D E D Figure 5.1 Figure 5

g Heath, o p . cit.. p. 350. 50 ACJK, each of the smaller equal squares Is equal to half of square ABDE. Line CM, drawn perpendicular to AB and ex­ tended to ED, bisects square ABDE. The squares ACJK and BCFG are equal, respectively, to rectangles AMKE and BDNM. Drawing the line CM and extending It to ED In figure 5 suggests that perhaps also In this figure square ACJK Is equal to rectangle AMNE and that square BCFG Is equal to rectangle BDNM. This Is one of the Important observations necessary for the discovery of Euclid*s proof for original £2£I£l£S_i- In summary, when a given statement, conjectured to be true but not yet proved, is being investigated, a variation in which addition of data is not accompanied by a change in the conclusion may be studied with respect to truth or fal­ sity. Formulation of such special cases implies variation of the type being discussed. Moreover, due to the presence of more data, it is sometimes easier to discover a proof for a special case than for the original statement and pos­ sibly to discover several proofs. Careful scrutiny of such proofs, it is tentatively proposed, might reveal a suggestion that is helpful in devising a proof for the original state­ ment. On the other hand, when a given statement is known to be true, and additions are made to the data but no changes are made in the conclusion, most of the resulting variants repeat special cases of what has already been shown to be 51 true In a more general situation. Such a procedure may be used for the sake of emphasis or to provide exercises for pupils to prove, but it does not produce results of a new and interesting nature. Attention is now directed to variation in which addi­ tion to the data of a statement known to be true is accom­ panied by the question "What change can now be made in the conclusion?" Variation of this type is the subject of sub­ section 2.22, which follows. 2.22 Addition of data permits a change in the conclusion.

ILLUSTRATION £ Original Exercise 6 Statementi The bisectors of the interior angles of a par- ftTlttlnprram intersect in four points, which are the vertices of a rectangle.* Data: (a) Quadrilateral ABCD. (b) AB is parallel to DC (c) BC is parallel to AD. (d) AAf bisects angle DAB. (e) BB* bisects angle ABC. (f) CC* bisects angle BCD. (g) DD* bisects angle CDA. “''The reader may wonder why the illustration Is not begun with a general quadrilateral Instead of a parallelogram. There w e two reasonst 1) The corresponding theorem involv- (continued on next page.) 52 Conclusion* (x) AA* and BB*, BB* and CC*, CC * and DD*, and DD* and AA* intersect, respectively, In four points; (they are labeled M, N, 0, and P). (y) MNOP is a rectangle, that 1st (y.l) MN Is parallel to P0. (y.2) PM Is parallel to ON.

(y»3) Angle MNO Is a right angle.

Variant 6.1 Statement: The bisectors of the interior angles of a rectangle intersect in four points, which are the ver­ tices of a saiiAfft. Data: (a) Quadrilateral ABCD. (b) AB is parallel to DC. (c) BC is parallel to AD. (e*) Angle ABC is a right angle (d) AA' bisects angle DAB. (e) BB' bisects angle ABC. (f) CC' bisects angle BCD. (g) DD* bisects angle CDA. Conclusion: (x) AA* and BB', BB* and CC», CC' and DD', and DD' and AA* intersect, respectively, in four points;

(footnote continued from previous page), ing a gen­ eral quadrilateral is not found in elementary books on geometry. 2) The nature of the figure formed by the bi- sectlors of the interior angles of a quadrilateral is not readily apprehended; see pp. 192-19*K 53

(they are labeled M, N, 0, and P). (y) MNOP la a square, that 1st (y.l) MN is parallel to P0. (y.2) PM Is parallel to ON. (y.3) Angle MNO is a right angle. (y.l**) MN is equal to NO.

Variant. £ * 2 Statement! The bisectors of the angles of a square are concurrent. Datat (a) Quadrilateral ABCD. (b) AB is parallel to DC. (c) BC is parallel to AD. (e*) Angle ABC is a right angle. (f*) AB is equal to BC. (d) AA* bisects angle DAB. (e) BB* bisects angle ABC. (f) CC* bisects angle BCD. (g) DD* bisects angle CDA. Conclusion! (x) AA* and BB', BB* and CC», CC»and DD *, and DD' and AA' intersect, respectively, in four points; (they are labeled M, N, 0, and P). (y) Points M, N, 0, and P are coincident. 5^ The Effect Produced on the Conclusion by the Addi­ tion 9f Pl5

There are situations, of vhlch Illustration 5 is an example, In which addition of data In a statement Is not accompanied by a change in the conclusion. If, however, addition of data is accompanied by a change In the conclu­ sion, that change Is made In one of the following wayss 1) one or more Items are added to the conclusion, as in variant 6 . 1 : 2) a limiting case is substituted In the con­ clusion as in variant 6.2. It will be noted in the transi­ tion from variant 6.1 to variant 6.2 that as ABCD approaches being a square points M, N, 0, and P approach coincidence. 55 2.3 Substitutions Are Made in the Data* Substitution In a statement of new data for given data may be viewed as deletion of the given data followed by ad­ dition of the new data. If the conclusion is scrutinised after the deletion of data as well as after the addition of other data, it is, of course, necessary to distinguish the two processes. But if the conclusion is not studied until after both processes have been completed, it is more con­ venient to refer to the combined procedure as substitution. This convention is followed throughout the present section. Six kinds of substitution, differing with respect to the nature of the data involved, and the further possibility of combining these six basic types, are considered. 2.31 The substitution involves numerical values assigned to elements of the figure. 2.311 Substitution of data does not necessitate a ohange in the conclusion.

ILLUSTRATION 7. Original Exercise 7. Instructions: From any point P draw two rays forming an angle of 47°. On each ray locate a point 2 Inches from P. Letter these points M and N. Draw M N . Measure angles PMN and PNM.9 U " ...... ' 1,1 - ' .. ■ ■ This exercise is given in Welchons and Krlokenberger, op. clt., p. 15 56

The data provided by the Ins true tlone and the ra suit obtained by ooraplylng with then are Itemized as data and conclusion, accompanied by the appropriate figure* Data? (a) Angle APB s 47° I? (b) Points M and N are located on PA and PB respectively* (c) PM = 2" A (d) PN ■ 2" P M Conclusion: (x) Angle PMN is 66.5° (y) Angle FNM is 66*5° A student, upon completing this exercise, may wonder, "What would be the sizes of angles PMN and PNM if the lengths PM and PN were given some numerical value other than 2 in­ ches?" One such situation and the result of investigating it are presented as the following variant* 57 Variant 7*1

a) Angle APB = 47° b) Points M and N are located on PA and PB respectively (c* PM - 3" (d) PN * 3" Conclusiont (x) Angle PMN s 66*5° (y) Angle PNM = 66.5° An alert student may surmise that for a given angle APB the angle PMN Is 66.5° and the angle PNM Is 66.5° pro­ vided only that the lengths PM and PN are equal* Such a surmise may motivate him to Investigate a situation In which PM and PN are not equal* The result of formulating and Investigating one situation of this sort is presented in the variant which follows. Change of datum (d) in this manner necessitates a change in the conclusion* ^A prime (*} accompanying the letter Identifying a datum or conclusion means that the Item has been substituted for an Item ooeurrlng in the original exercise* Multiple primes lndloate that a total of two or more substitutions have been made In obtaining the Item so Indicated* 58 2.312 Substitution of data Involving numerical values necessitate a ehmg> In the conclusion.

Illustration 7. (confd) Variant 7.2 Da tat (a) Angle APB = 47° ” (b) Points M and N are located on PA and PB respectively. (c) PM s 2" (d*) PN = 3" P M

Conclusion! (x») Angle PMN = 91°# (y1) Angle PNM * 42°# Several other variants Involving changes In PM and PN, In which their equality Is not maintained, suggest no part­ icular relationship between the Items of the conclusion In the variants or between the conclusion In the variants and the conclusion In the original exercise, unless It is no­ ticed that the sum of angles PMN and PNM Is always constant. Such and observation might, together with later steps In which the angle APB is varied, lead to the very Important ^Iheae are approximate values, which a student would likely find by careful measurement with a protractor. 59 discovery that the sum of angles PMN, PNM and APB is always constant regardless of the size of angle APB and lengths PM and PN. But such steps are properly examples of vari­ ation performed initially on the conclusion, for although they may be suggested as the result of changes in the data, they may be made without benefit of such changes. This type of variation is illustrated later in Chapter 3. Attention may also be directed to datum (a), so that an inquiring student might ask, "What will be the size of angles PMN and PNM if the angle APB is given a numerical value other than k-7°?n The result of formulating and in­ vestigating one such situation is presented in the variant which follows. Change of this datum also necessitates a change in the conclusion.

Variant Data: (a*) Angle APB ■ 110° (b) Points M and N are B located on PA and PB respectively. (c) PM ■ 2" (d) PN * 2" P M A Conclusion: (x") Angle PMN - 35° (y") Angle PNM *35° 60 2.32 Numerical data tr> replaoed by data which are not expressed in nvunerical terms. 2*321 There la no change in the conclusion* Variation of this type la illustrated In the transition from original exerdtse 7 to the surmise expressed following variant 7.1. (p. 57) • This surmise is stated as variant 7*4.

ILLUSTRATION 7. (confd) Original exercise 7 may be stated verbally as follows: If on the sides of an angle APB of 47° two points M and N are located two inches from the vertex P, the angle PMN is 66.5° and the angle PNM is 66.5°. Variant 7A Statement: If on the sides of an angle APB of 47° two points, M and N, are located equidistant from the vertex P, the angle PMN is 66.5° and the angle PNM is 66.5°. In the transition from original exercise 7 to variant 7.4. a datum expressed in numerical terms was replaced by a datum not involving numerical terms, but there was no change In the conclusion* The reader has, of course, already recognised that the conclusion "the angle PMN is 66*5° and the angle PNM is 66.5°" may be replaced by the conclusion "angle PMN is equal to angle PNM" In original exercise 7 and in several of the variants that have been presented* Such a change, however, can be made without making a change In the data; hence it is 61 an illustration of variation performed initially on the conclusion, which is the subject of Chapter 3. 2.322 The replacement of numerloal data by data expressed in general terms necessitates a change in the conclusion. On the other hand, if in original exercise 7 the datum "angle APB is 47°" is replaced by the datum, "angle APB la any angle," angles PMN and PNM can no longer be erpressed in terms of numerical values. The conclusion that they are equal is nevertheless true. These changes are indicated by underlining the affected words in the statements of original

exercise 7 and variant 7.5 which is obtained by making the changes.

ILLUSTRATION 7. (confd) Original Exerolae 7. Statement: If on the sides of an angle APB of 47° two points, M and N, are located two inches from the vertex P, the angle PMN is 66.5° and the angle PNM is 66.5°. Variant 7.5 Statement: If on the sides of an angle APB of any size two points, M and N, are located two inches from the vertex P, angles PMN and PNM are equal. 62 2*33 Kumar* leal data are substituted for data not expressed in numerloal t < m . 2.331 There is no change In the conclusion. Suppose that after performing original exercise 7 and perhaps variants 7*1 and 7.2 and others like it, a student observes, "Apparently PM and PN can be any length, so long as they are equal. Then if angle APB always is 47°, angle PMN is 66.5° and angle PNM is 66.5°.n If this work is pur­ sued during the informal phase of geometry, the student should assign several numerical values in turn, each sim­ ultaneously to PM and PN, and in each case measure the angles PMN and PNM. Always each angle is found to be 66.6°. In such a procedure numerical data are substituted for data which are not expressed in numerical terms, but the conclu­ sion is unchanged. The statements in Illustration 8 are an example of this sort of activity. Hie words involved in the substitution are underlined.

ILLUSTRATION 8 Original Exercise 8 . (This statement is variant 7.4 in the preceding illustrationT • Statement* If on the sides of an angle APB of 47° two points, M and N, are located equidistant from the vertex P . the angle PMN is 66.5° and the angle PNM is 66.5°. 63 Variant 8 .1 . (This statement la similar to original exercise 7 and to variant 7*1 In the preceding Illustration)* Statement! If on the aides of an angle APB of 47° two points M and N, are looated so that PM Is 2 1/2 Inchea and PN la 2 1/2 Inches, the angle PMN Is 66*5° and the angle PNM la 66*5<>* 2*332 Substitution of nujnerloal data for data expressed In general terms necessitates a change In the conclusion* When, without necessitating a change in the conclusion, numerical values can be assigned to elements of a figure In accordance with specified relationships not expressed numer­ ically, the truth of the general statement is considered verified by each of the numerical cases* When this is not the case the general statement is thereby shown to be false* Hence in order to Illustrate variation of the type named In this subsection, the original exercise must necessarily be false* The statement used for this purpose In illustration 9 might be assumed by a careless pupil immediately after in­ vestigating original exercise 7 (p*55).

ILLUSTRATION 9 Original Exercise 9 6b

Statements If on the sides of an angle APB of any else two points, M and N, are located so that PM is 2" and PN is 2", the angle PMN is 66>5° and the angle PNM is 66>5°, Variant 9>1 In order to test the above statement, a particular figure is drawn, as shown, in accordance with the partic­ ular numerical datum that angle APB is 80? When the angles PMN and PNM are measured it is found that they are not 66,5° but 50°- Statements If on the sides of an angle APB of 80° two points, M and N, are located so that PM is 2" and PN is 2", the angle PMN is 50° and the angle PNM is 50°. A Statement. Not Involving Numerical Values. Results From Varying Original Exercise 7. Original Exercise 7 contained three data expressed in numerical termss (a) angle APB = 47°, (c) PM - 2", (d) PN = 2". There were two items expressed in numerical terms in the conclusions (x) angle PMN = 66.5°, (y) angle PNM - 66,5°. Investigation revealed that if datum (a) is replaced by datum (a1)* angle APB is any angle, and if conclusions (x) and (y) are replaced by the single conclusion, (s) angle PMN - angle 65 PNM, the resulting statement Is true. It was observed, more­ over, that if data (o) and (d) are replaced by the single datum, (e) PM - PN, the resulting statement, containing data (a*) and (e) and the conclusion (z), is a true statement* None of the data or conclusions in this statement are ex­ pressed in terms of numerical values* It may be formulated as follows: If on the sides of any angle APB two points, M and N, are located equidistant from the vertex P. angle PMN is equal to angle PNM.

The Role of Variation In the Processes of Generalization and Induction. The reader has probably thought, "This is an example of induction,11 or "This is an example of generalization," in connection with the kinds of substitution discussed in the three preceding subsections, 2.31, 2.32, and 2.33 (be­ ginning on p. 55) • In section 2.31 given numerical values were replaced by other numerical values* In sections 2.32 and 2.33 data expressed in numerical terms wire replaced by data of a more general nature, and vice versa* A review of pertinent observations will be of value* 66

All of the Illustrations given in sections 2*31, 2*32 and 2*33 are related to original exercise 7t If on the sides of an angle APB of 47° two points, M and N , are located two inches from the vertex P, P the angle PMN is 66.5° m and the angle PNM is 66.5°. One trend of development follows* In variant 7*1 the length of PM was changed to 3” and the length of PN was changed to 3 % but the conclusion remained true* In variant 7*2 the length of PM was 2" and the length of PN was 3M; in this case the conclusion was changed, angles PMN and PNM be­ ing found to be about 91° and 42°, respectively* Variant 7*4 was obtained, in view of a surmise which might be thought of at any stage of the investigation, by letting PM and PN te any equal lengths; no change in the conclusion was ne­ cessitated by this change* How the statement so obtained may be further tested to see if it is true for numerical values of PM and PM other than those considered previously is illustrated by variant 8 *1 * The above activity will now be described, step by step, in more general terms* 1) The beginning was a particular geometric figure with enough of its parts given numerically to determine ftt uniquely* A conclusion was expressed con- 67

corning two other parts of the figure. 2) Other speeific figures were obtained by varying, one or more at a time, numerical values in the data of the original exercise. It was found that If data were varied in certain ways the con­ clusion was not affooted, while if they were varied in other ways the oonolusion was affeeted. 3) A"generalisation" was recognised and stated. 4) How the validity of this gener­ alization may be tested again by means of speciflo figures constructed In accordance with the data was Illustrated with an example. In still more general wording, these four steps may be identified as follows: 1) Knowledge about a particular situation, 2) Observation of other particular situations in some ways similar to and in other ways different from the first, 3) Inductive generalisation,^0 4) Testing of the generalisation in further specific cases, that is, an ex­ tension of step 2 ) after the generalisation has been stated tentatively or as it is modified in the light of further knowledge.

^toductivc generalisation Is used here in the sense of the definition given tor generalisation by Morris R. Cohen and Ernest Nagel, Logic and Scientific Method, p. 277, as "the passage from a statement true of some observed Instances to a statement true of all possible Instances of a certain class•" For other distinct uses of the word generalIsatlon see pp. 151-57. Generalisation refers to the formulated statement, the product of the prooess, as well as to the process of genera­ lization, See, for instance Max Black, Critical Thinking. p. 256. 68 A total process consisting of steps 1), 2), and 3) Is denoted In this study by the vord Induetlon.* it Is under­ stood that steps 2) and 3) are not clearly defined chrono­ logically. It may be said that there Is some vague hunch, perhaps not verbalised, concerning a possible generaliza­ tion before any purposeful activity takes place in step: 2>. The process then alternates between step 2) and step 5) un­ til the generalization Is accepted for the time being; that is to say, further study and further modification (or re­ jection) of the generalization may occur at any later time. Induction, as used In geometry textbooks, denotes only steps 1), 2), and 3) in the above formulation, with steps 1) and 2 ) combined into a single step. The following three references Indicate this practice: In the last unit you drew conclusions in the various experiments by a process of Induc­ tive reasoning. That Is, you studied many specific cases In a single experiment and on the basis of observations and data collected were able to make a generalization (general statement) .i* In everyday reasoning we often argue from observation of a number of cases that a certain conclusion is valid,,,,. This kind of reasoning from observation is called Inductive reasoning.12 The process of reaching conclusions by ob­ serving examples or the results of experiments Is called Induction.15 *No confusion with the technical process mathematical Induction is anticipated. l)-Schnell and Crawford, op. clt., p. 64. 12Skolnlk and Hartley, op. clt., p. 77. 15Trump, op. olt.. p. 126. 69

According to these definitions, all of the evidence precedes the generalization. Of course, in demonstrative geometry, emphasis is on deduction rather than on Induction as the method of establishing truth. Inductive procedures, it is frequently pointed out, will be used as a means of discovery, but less often (or never), as a method of proof However, inductive procedures are used in the informal study of geometry, which may precede the demonstrative phase. At other stages of geometry and in other areas of mathe­ matics, when a conjecture is made for which a deductive proof has hot been found, inductive procedures are used for the purpose of Investigating the probable truth of the con­ jecture. If in the study of individual examples one is found which contradicts the conjecture, the conjecture has been proved false; if, on the other hand, the conjecture is supported by many examples, more diligent search for a deductive proof is thus motivated. Hence, testing of a generalization is an important con­ sideration. It seems well, therefore, to treat the complete process of obtaining and testing the generalization as a unit. Induction is an appropriate name for this complete process. John Stuart Mill defined induction as "the operation of dis­ covering and proving general propositionsBlack, like for instance, Bresllch, op. clt.. p. 11, Skolnlk and Hartley, ©£. clt.. p. 77, and Trump, o£. clt.. p. 127. 15John Stuart Mill, A System of Logic, p. 208. the authors of geometry textbooks, places emphasis on dis­ covery In his definition, "The process by which we pass from evidence concerning some members of a certain class of objects to an assertion concerning all members of that clas Is known as Induetion."*-® Cohen and Nagel, on the other hand, conceive the principal problem of Induction to be the discovery of "the basis of a generalisation when the In­ stances examined are not all the possible instances 7 Dubs also emphasizes determining the truth of a statement in his conception of Inductions "Induction may be defined as the method of establishing the truth of propositions, not from those more general, but from those less general or from mere experiences which have not been stated In the form of propositions."^® The principal value of variation in connection with inductive generalization Is In suggesting generalization. In Illustration 7 the original datum "PM = 2 in., PN - 2In• (p• 56 ), was changed to "PM = 3 In., PN = 3 In." in variant

7.1, (p. 57), and then to "PM * 2 In., PN * 3 In." In var­ iant 7.2, (p. 58 ) • If a student had not recognized Immed- ieBlack, oj^« | P+ 276 a Tills doflnltlon of lnduc^ tion, seemingly, Is no different from hie definition of generalization, p. 266, "By a generallzatlon we shall mean a principle which makes an assertion about some or all of the members of a class of objects...... the term 'general­ ize tion' can also be applied to the process by which we obtain statements true of the members of some class." l^Cohen and Nagel, op» clt.. p. 276. ^®Homer H. Dubs, Rational Induction, pp. 125-26. lately from original exercise 7 that the size of angle PMN Is 66.5° and the size of angle PNM Is 66.5° regardless of the lengths of PM and PN provided they are equal, angle APB remaining always 47°, It Is suggested that he would be more likely to recognize this generalization after formulating variants 7.1 and 7.2 and perhaps others In which numerical lenghths are assigned to PM and PN• So far as the statement of the generalization (variant 7*4, p.60 ) and the genera­ tion of further numerical examples (Illustration 8 , pp. 62 and 63) are concerned, variation Is no more than a way of describing these activities. In the generalization discussed In the preceding para­ graph, only the data are different from those In the partic­ ular examples; the conclusion Is exactly the same In the general statement as In each of the numerical examples. On the other hand, a statement may be different from statements referring to particular examples only with respect to the conclusion. For Instance, as was remarked on p.60, the con­ clusion, "angle PMN Is 66.5° and angle PNM la 6 6 .5°," In original exerolse 7 . may be replaced by the conclusion, "angle PMN Is equal to angle PNM." Hie resulting statement, however, may have the same data as original exercise 7 . There also are situations In which both the data and the con­ clusion In a statement are different from the data and con­ clusion In a particular numerical example. Such a situation may be viewed In two way#. 1) Consideration or orlnglal exercise 7 (pp. 55 - 56 ) * original exercise 9 ( p • 6*f ) * and Variant 9.1 (pp. 6*f — 65 )» illustrates the case where a change In the data necessitates a change in the conclusion* When the size of angle APB was varied, PM and PN being equal, the size of angles PMN and PNM also varied, although the angles remained equal to each other. Hius, substitution of the datum "APB la any angle" required substituting the con­ clusion "angle PMN Is equal to angle PNM." 2) On the other hand, if the change is made first In the conclusion, a change may be rendered possible in the data. For instance, if the conclusion "angle PMN Is 66.5° and angle PNM is 66.5° is re­ placed by the conclusion "angle PMN is equal to angle PNM, then the datum "angle APB is 47°" may be changed to iLFB Is any angle" (provided, of course, that PM and PN are equal). In both cases, the resulting statement is s "If on the sides of any angle APB two points, M and N, are located so that PM is 2" and PN is 2", (or, so that PM * PN), angle PMN is e- qual to angle PNM." In summary, variation may be said to promote inductive generalization. Any testing of a generalization after it has been formulated may be described in terms of variation. 73

2.34 The substitution Involves tlmwiti of th» figure.*

2.341 There 1» no change In the conclusion.

ILLUSTRATION 10. An exercise coasnonly proposed for deductive proof in geometry textbooks is the original exercise in this illus­ tration. Variant 10.1 also is frequently given. Original Exercise 10. Statement! The medians to the equal sides of an Isosceles triangle are equal. Variant 10.1 Statement: The altitudes to the equal sides of an Isosceles triangle are equal. Variant 10.2 Statement! The bisectors of the equal angles of an isosceles triangle are equal. Variant 10.3 Statement! The segments of the perpendloular-blaectora of the equal sides of sn isosceles triangle which lie wlthing the triangle K are equal. *Tbe 1?artsn of a triangle, for examplf, are the three sides and the three angles of the triangle. The word elements is used to denote the parts of a triangle and in addition to them any other lines and angles that may be associated with a triangle. 7*+ Further variations are possible on original exercise 10 and on aaoh of the variants that have been presented. One example, variant 10.4. Is given to show the nature of these variations. It Is preceded by original exercise 10 expressed In terms of data, conclusion, and the corresponding figure. Original Exercise 10. Datat A (a) Triangle ABC with AB = AC. (b) C* is the midpoint of AB. (c) B* Is the midpoint of AC. Conclusions (x) CC» - BB• Variants may be generated by substituting for data (b) and (d) other specific looatlons of C* and B», such as that they are the trisection points, respectively, of AB and AC which are nearer to vertex A. Two ways in which original exercise 10 or any of these variants may be proved are: 1 ) by establishing that triangles ABB* and ACC' are congruent, and 2) by establishing that triangles BB'C and CC'B are con­ gruent. If method 1) Is used, one step In the proof Is the deduction that AB1 = AC'. It Is this relation that is sub­ stituted as datum (b') In place of data (b) and (c) In order to obtain variant 10.4 from the original exercise. Proof of the original exercise by use of method 2) Involves the deduction that BC' - CB'. This relation stay be substi­ tuted for data (b) and (c) In order to obtain still another variant. 75

Variant 10.4 Statement: Llnea drawn from the vertices of the equal

angles of an Isosceles triangle to points on the

opposite sides which are equidistant from the

vertex common to these two 3ldes are equal.

Data: /\

(a) Triangle ABC with AB - AC.

(b») AC' ■ AB* S’

Conclusion: (x) C C ' = BB* B

2.342 Substitution of data Involving elements of the

figure Is accompanied by a change in the con-

clusion •

ILLUSTRATION 11.

Original Exercise 11.

Statement: In any triangle the line segment connecting the

midpoints of two of the sides is psLrallel to the third

side and equal to one-half of it.

Data:

(a) Triangle ABC. &

(b) C' is the midpoint of

AB, i.e., AC'/AB = 1/2

(c) B' is the midpoint of AC, ie., AH'/AC = 1/2. c 76 c : onclusion:

(x) C f3f is parallel to BC

(y) C»3* ■ 1/2 BC, i.e., C»3f/BC = 1/2.

Variants may be formed in which the ratios A C f/AB and

A3'/AC are given equal values other than 1/2, such as 1/3?

2/3, l/1-*-, and 3/*+* An observation indicating more maturity, and the type of observation toward which the student is to be guided as early as he is ready for it, is that the ratio

AC'/A3 is equal to the ratio A3'/AC in original exercise 11

(and in the variants just suggested.) This relation is sub­ stituted for data (b) and (c) in obtaining variant 11.1. Tt is found that conclusion (x) remains true but that conclusion

(y) must be changed. A suggestion for the correct change may be obtained from observing in original exercise 11 that the ratio C'3f/BC (in conclusion (y)) was equal to the ratios

ACr/AB and A B f/AC (in data (b) and (c)).

Variant 11.1

Statement: Tf points are located on two sides of a triangle

so that the segments between the two points and the

vertex common to the two sides are proportional to

the two sides. the segment joining these two points is

parallel to the third side and has the same ratio to

the third side as each of the segments of the other

taro sides has to the side of the triangle of which it is a part. (a) Triangle ABC.

(b *) AC */AB = AB'/AC.

Conclusion:

(x) C '31 is parallel to BC.

(y») c b '/b c = a c /a b .

Attention is now given to another type of substitution.

Ibis type involves relationships,

2.3-5 The substitution involves relationships,

2 , 3 5 1 There is no change in the conclusion,

I L L ’J STRATI 01: 12

Two familiar theorems concerning angles with their sides perpendicular and their sides parallel illustrate a substitution involving relationships in which the conclusion is not changed. The relation parallel is substituted for the relation perpendicular.

Original Exercise 12

Statement: If two angles have

their sides perpendicular,

right side to right side

and left side to left side,

the angles are equal B R A

*If an observer is located near the vertex of an angle outside the angle facing the interior of the angle, the side of the angle at the observer’s right is called the right side of the angle and the side at the left is called the left side of the angle. 73

Variant 12.1 Statement: If two angles have c their sides parallel, right side to right side and left side to left side, the angles are equal 0 A

2.3.52 There Is a change In the conclusion when a sub-

stltutlon involving relationships is made in the data. Substitution of the relation greater than for the re­ lation equal to In reference to two sides of a triangle is an example of substitution in the data which is accompanied by a change in the conclusion, as Is shown in the following illustration.

ILLUSTRATION 13

Original Exercise 13 Statement; If two sides of a triangle are equal, the angles

opposite these sides are equal. Variant 13.1 Statement: If one side of a triangle Is greater than a second side, the angle opposite the first side is

greater than the angle opposite the second side*

approach by way of variation to these statements and others related to them, together with certain observa­ tions, is presented later, in Chapter 6. See illustration 11, PP. 169-176. 79

The statement for \arlant 13.1 is expressed as it is here in order to emphasize the substitution of the word greater for the word equal. The statement is sometimes expressed as follows: "If two sides of a triangle are unequal, the angles opposite these sides are unequal, the greater angle lying opposite the greater side," 2.36 The substitution of data involves the basic figure. 2.361 There is no change in the conclusion.

ILLUSTRATION 14* In this illustration the word quadrilateral is sub­

tit1.! ted for the word triangle.

Original Exercise 14 Statement: The sum of the exterior angles of any triangle is 360°. Variant 14.1 Statement: The sum of the exterior angles of any quadril­

ateral is 360°.

The well-known general theorem, "The sum of the exter­ ior angles of any polygon is 360°", may be considered as an­ other variant in this illustration. 2.362 There is a change in the conclusion when the substitution in the data involves the basic figure.

1 — ■■ ■ In a class situation, the order of illustrations 14 and 15 probably would be reversed. 80

If the sum of the Interior angles, rather than of the

exterior angles, Is considered, substitution of the word

quadrilateral for the word triangle in the data results In a change in the conclusion. This Is shown in illustration 15.

ILLUSTRATION 15

Original Exercise 15 Statement; The stun of the interior angles of a triangle

is 180°.

Variant 15.1 Statement; The sum of the Interior angles of a quadrilateral

is 3 6 0 °.

The general theorem, "The sum of the Interior angles of any polygon of n sides Is (n - 2) straight angles”, may be

considered as smother variant In this Illustration. A de­ tailed presentation, culminating In this general statement,

is given on pages 133“13^-« 2.37 The substitution involves combinations of some or all

of the types of substitutions mentioned. Since one Illustration of combinations of kinds of sub­ stitution is indicative of other possible combinations, only one such Illustration will be presented in each of the sub­ sections 2.371 and 2.372. In each of these the changes in

the data are of the following two types; 1) substitution Involving elements of the figure, 2) substitution Involving relationships between elements of the figure. 81

2.371 A combination of kinds of substitution in the data Is not accompanied by a change In the conclusion .

ILLUSTRATION 16 Original Exercise 16

Statement: The altitude to the base of an Isosceles triangle bisects the base. Data:

(a) Triangle ABC (b) AB = BC

(c) AD Is perpendicular to BC Conclusions 3 r> C (x) BD = DC

Variant 16.1

Statement: The bisector of the vertex angle of an isosceles triangle bisects the base. Data:

(a) Triangle ABC {The figure is the same as (b) AB =■ BC for original exercise 16) (cf) Angle BAD Is equal to angle CAD Conclusion:

(x) BD =■ DC 82

That the operation of replacing datum (c) with datum

(o') involves two types of substitution is more apparent from a study of the itemized data than from the verbal

statements. The elements, "lines AD and BC", are replaced by the elements, "angles BAD and CAD"; the relationship,

"is perpendicular to", is replaced by the relationship,

"is equal to."

2 . 3 7 2 A combination of kinds of substitution in the

data 1s accompanied by a change in the

conclusion.

ILLUS^’RATTOTJ 17

Original Exercise 17

Statement; The altltude to the hypotenuse of a right tri­

angle is the mean proportional between the seg­

ments of the hypotenuse.

I a ta; (a) Triangle ABC A (b) Angle BAC = 90°

(c) Line AD, with D on BC (d) AD is perpendicular e t> c to BC •

Conclusion:

(x) BD/AD - AD/DC, or AD = \/BD * rt 83 Variant 17.1 Statement: The median to the hypotenuse of a right triangle is equal to one-half the hypotenuse* Data: (a) Triangle ABC A (b) Angle BAC = 90°

(c) Line AD, with D on BC (d«) BD is equal to DC B D C Conclusion: (x) AD = 1/2 BC, or AD = 1/2 {“D 4- DC)

Variant 17.1 may be obtained, with reference to origin­ al exercise 17, by inquiring, "If the word altltude Is re­ placed by the word median, what relation, If any, exists be­ tween the median of a right triangle and the segments into which ijfc divide* the hypotenuse?" It Is found that, whereas the altltude is equal to the square root of the product of the segments into which It divides the hypotenuse, the median Is equal to one-half of the sum of the segments into which It divides the hypotenuse. The changes in the data may be viewed as follows: 1) The elements BD and DC were substi­ tuted for the elements AD and BC, 2) The relation equal was substituted for the relation perpendicular. Thus, two kinds of substitution are Involved when obtaining the data for variant 17.1 is viewed In this manner. Sb

One or More Data May Be Replaced by One or More Data Which Are Implied by the Data That Are Replaced Sometimes a new and Interesting situation is obtained when one or more data are replaced by other data which are implied by the data which are replaced. Usually a substi­ tution of this type may be viewed as one of the types of substitution that have been presented, but frequently such a substitution may also be viewed as a combination of the types that have been enumerated. After such a substitution, as after substitutions of other kinds, a change in the con­ clusion is sometimes necessary if the resulting statement Is to be true, and sometimes no change in tho conclusion is necessary. An illxistration of both cases is provided by starting with the statement: "The midpoints of the sides of a square. taken in order, are the vertices of a square." It is desir­ ed to replace the word square by a phrase concerning a quad­ rilateral having diagonals with certain special properties; these properties must also be possessed by the diagonals of a square. One such phrase which might be selected is: quadril­ ateral with diagonals which are equal and which are the per­ pendicular bisectors of each other. But this phrase is equivalent to the word square, for any quadrilateral with diagonals which are equal and which are the perpendicular 35 bisectors of each other is a square, and the diagonals of any square are equal and are the perpendicular bisectors of each other•

Another phrase with which the word square may be replac- is: quadrilateral with diagonals which are equal and perpen­ dicular » This phrase is not equivalent to the word square. for not every quadrilateral with diagonals equal and perpen­ dicular is a square* However, the figure obtained by con­ necting the midpoints of the sides of such a quadrilateral is the same as the figure obtained by connecting the mid­ points of the sides of a square; namely, a square. The new statement is: "The midpoints of the sides of a quadrilateral with diagonals which are equal and perpendicular, taken in order, are the vertices of a square."

On the other hand, if the word square in the data of the original statement is replaced by the phrase quadrilateral with perpendicular diagonals, the figure obtained by connect­ ing, in order, the midpoints of the sides is not always a square, but it is always a rectangle. The new statement is:

"The midpoints of the sides of a quadrilateral with perpen­ dicular diagonals, taken in order are the vertices of a rectangle." It is noticed, of course, that the data for this statement may also be obtained by deleting from the

statement In the preceding paragraph the datum that the

diagonals are equal* Moreover, by substitution In the

original statement or by deletion from the data In the

statement in the preceding paragraph, the following state- rent may be obtained: "The midpoints of the sides of a

Quadrilateral with equal diagonals, taken In order, are

the vertices of a rhombus." 87

2.1+ Combinations of Deletion. Addition, and Substitution Are Performed on the Data.

Any substitution may, of course, be viewed as a combin­ ation of deletion and addition. Another example, one in which variation of the data involves both dentation and sub­ stitution, will be given. The theorem "The bisector of an angle of any triangle divides the opposite side into seg­ ments that are proportional to the adjacent sides" may be viewed as a variant of the theorem "The altitude to the hypotenuse of a right triangle divides the hypotenuse in such manner that either lee of the triangle is the mean proportional between the whole hypotenuse and the segment of it which is adjacent to that leg." So viewed, the datum that the triangle contains a right angle is deleted, and bisection of an angle of the triangle is substiLuted in the data for perpendicularity to a side. Ordinarily, nowever, variations consisting of two operations are performed in two steps. Deliberate variation proceeds more successfully if performed slowly. Thus, this example is presented only in order to illustrate a type of variation which it is possible to conceive, but one which is not of great impor­ tance otherwise. CHAPTER 3

KINDS OF VARIATION IN WHICH THE CHANGES ARE MADE INITIALLY IN THE CONCLUSION

3.1 One Or More Conclusions are Deleted P. 93 3.11 Deletion of a conclusion does not make

possible a change In the data. P. 93 3.12 A change in the data Is made possible by the deletion of a conclusion. P. 9b

3.2 One Or More Conclusions Are Added P. 99 3.21 There Is no change In the data. P. 99 3.22 The addition of a conclusion requires a change In the data. P. 100

3.3 Substitutions Are Made In the Conclusion P. 103 3.31 The substitution Involves numerical values assigned to elements of a figure. P. 103 3.311 Hiere is no change In the data. P. 103 3.312 Substitution in the conclusion of numerical values assigned to ele­

ments of a figure necessitates a

change In the data. P. 103 3.32 A conclusion expressed in numerical terms is replaced by a conclusion which is ex­

pressed In general terms. P. 105

83 89

CHAPTER 3 (Continued) 3.321 A conclusion expressed in numerical terms is replaced by a conclusion which is not expressed in numerical terms, but no change is made in the data. P.lo6 3.322 A conclusion expressed in numerical terms is replaced by a conclusion which is not expressed in numerical

terms, and a change is made possible

in the data. P.106 3.33 A conclusion expressed in general terms is replaced by a conclusion which is ex­ pressed in numerical terms. P.106 3.331 There is no change in the data. P. 107

3.332 lhere is a change in the data. P. 107 3.34 The substitution involves elements of the figure. P.107 3.341 There is no change in the data. P. 107

3.342 There is a change in the data* P. 107 3.35 The substitution involves relationships. P. 103 3.351 There is no change in the data. P.loS

3.352 There is a change in the data. P. 108 3.36 Hie substitution involves the basic figure.P*1^9 90

CHAPTER 3 (Continued)

3*37 The substitution In the conclusion in­ P. 109 volves combinations or the types of sub­ stitutions mentioned. 3.4 Combinations of Deletion. Addition, and Sub­ stitution Are Performed on t2ie Conclusion. P. Ill

Summary. Chapters 2 and 3 . P.112 91

In the preceding chapter variations were considered which initially involved the data. Chapter 3 deals with types of variation performed initially on the conclusion* The possible types that need to be considered are analogous to those that were delineated in the preceding section* Some types again are productive of more interesting results than others. In connection with each kind of variation Involving the conclusion two further cases will be considered: 1 ) variation of the conclusion .is not accompanied by a change in the data; 2) variation of the conclusion ^ accompanied by a change in the data. Consideration of data for a var­ iant is limited to those data which bear a relation, first of all, to the data in the original exercise from which the variant was fonned through a change in the conclusion and, secondly, to the change made in the conclusion. The word conclusion* it is understood, may refer to the entire conclusion or to one or more items of the conclusion if it contains several items. Of course, as Lazar has pointed out, "Every theorem that contains many conclusions can always be subdivided into as many theorems as there are conclusions, all having the same set of hypotheses." ^

■‘'Nathan Lazar, The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry, p. 27• This work is also published in The Mathematics Teacher. XXXI (March, April, and May, 1938) pp.99-113, 156-74, 216-40. 92

It Is, however, advantageous in this discussion not to limit attention to statements having one conclusion* For instance, if the only statements considered were those having but one conclusion, it would be impossible to have a situation in which a statement Is obtained from another by the deletion of a conclusion* 93

3.1 On* Or Mort Conclualona Are Deletod. It surely would not bo anticipated that deletion of a conclusion from a statement having more than one conclusion would necessitate addition of a datum. Hence, It Is approp­ riate to consider whether or not deletion of data, or some other change In the data, Is made possible by deletion of a conclusion. 3.11 Deletion of a conclusion does not make possible a change In the data. Variation In which deletion of a conclusion does not make possible a change in the data does not lead to any sig­ nificantly new statements. However, by deleting one or more conclusions from a statement having a multiple conclusion, it is possible to obtain several new statements each having the same data as the original statement but having fewer conclusions^ Illustration 18 is an example of this kind of variation•

ILLUSTRATION 18 Original Exercise 18 Statements Hie bisector of the vertex angle of an isosceles

bisects the base and la perpendicular to the b a s e . 2

■' T| — ■ i ■ ARie proof of this statement is proposed as an exercise in Breslich, op. clt., p. 68 . 9*+ Data: (a) Triangle ABC. (b) AB - AC. (c) Angle BAD z angle DAC• Conclualon:

(x) BD ■ DC. 3 D O (y) AD is perpendicular to BC• Variant 18.1 is obtained by deleting conclusion (y) in original exercise 18. Variant 18.2 is obtained by deleting conclusion (x) in original exercise 18. As the reader read­ ily recognize#, all of the data of the original exercise are necessary to establish the conclusion in each of the variants. Only the statements of the variants are given. Variant 18.1 Statement: The bisector of the vertex angle of an Isosceles triangle bisects the base. Variant 18.2 Statement: Hie bisector of the vertex angle of an Isosceles triangle la perpendicular to the base.

3.12 A change in the data is made possible by the deletion of a conclusion. Two illustrations of this type of variation will be given. In the first, deletion of a conclusion permits the deletion of a datum; in the second, deletion of a con­ clusion permits substitution of a datum which is more 95 general In nature than the corresponding datun In the orig­ inal exercise.

ILLUSTRATION 19. Original Exercise 19. Statement: Hie midpoints of the sides of a rectangle are the vertices of a rhombus. Variant 19.1 Statement: Hie midpoints of the sides of a parallelogram are the vertices of a parallelogram.

From the verbal statements It appears that variant 19.1 is obtained from original exercise 19 by making a sub­ stitution. In order to make clear that this change may also be viewed as being one of deletion, the data and conclusion of the original exercise are Itemized. Original Exercise 19 Data: (a) Quadrilateral ABCD. (b) AB Is parallel to D C •

(c) BC Is parallel to AD. (d) Angle ABC is a right angle. (e) E, F, G, and H are the mid­ points, respectively, of AB, BC,

CD, and DA. 96

Conclusion* EFGH is a rhombus, that is, (x) EP is parallsl to HG* (7 ) EH is parallel to FG*

(z) EP - PG. Conclusion (z) is deleted to form variant 19.1. which follows; after this has been done, by scrutinizing the proof of original exercise 19 it is discovered that the re­ sulting conclusion can be established without the use of datum (d)* Consequently, datum (d) is shown as having been deleted in the Itemised form of variant 19.1« Variant 19.1 Data: (a) Quadrilateral ABCD* (b) AB is parallel to DC• A (c) BC is parallel to AD* / (d) (Deleted)• T (e) E, F, G, and H are the / midpoints, respectively of D AB, BC, CD, and DA• Conclusion: EPGH is a parallelogram, that is, (x) EP is parallel to HG* (y) EH is parallel to FG* (z) (Deleted)• In illustration 19, the conclusions are itemized by delineating definitive properties of particular types of 97 quadrilaterals• In Illustration 20, which follows, the con- elusions of the original exercise are not related in the sense that they are both properties of a particular type of figure.

ILLUSTRATION 20. Original Exercise 20 Statement! The line connecting the midpoints of two sides of a triangle is parallel to the third side and is equal to one-half of it. Data: (a) Triangle ABC. . r \ (b) c* is the midpoint of AB. B 1 is the midpoint of AC. Conclusion! c (x) C*Bf is parallel to BC. (y) C'B» = 1/2 BC. Variant 20.1 is obtained by deleting conclusion (y). It is readily apparent that conclusion (x) can still be established if datum (b) is replaced by a properly chosen more general datum. A satisfactory datum is: C* and B' are points on AB and AC, respectively, such that AC*/C,B - AB*/BTC. Only the statement of the resulting variant will be given. Variant 20.1 Statement; The line connecting points which divide two sides 98

of a triangle proportionally la parallel to the third aide * nils la, of course, a useful statement* Moreover, as the reader has noticed, without requiring any change in the data a conclusion may be added, replacing conclusion (y), which was deleted, as follows: C*B,/BC = AC*/AB. But, as the reader also remembers, it is deletion of a conclusion that is being illustrated in the present example* Deletion of conclusion (x) from original exercise 20 may also be mentioned* If this deletion is made and no other change is made in the conclusion, the resulting statement is, of course, true* However, as the reader can readily verify, datum (b) is not necessary for conclusion (x); that is, there are many pairs of points, other than the midpoints, on each aide of the two sides of a triangle such that the distance between them is one-half of the third side of the triangle* In other words, B 1^ s 1/2 BC does not imply that B* and C* are the midpoints of AC and AB.®

^For a formal treatment of the concept "necessary condition" the reader is referred to Lazar, op* clt*, pp. 58-59, and to Carl A* Garabedian, "Some Simple Logical Notions Encountered in Elementary Mathematics," The Math­ ematics Teacher, XXIV (October, 1931), pp. 345-52. 99

3.2 One Or More Conclusions Are Added* 3.21 There la no change In the data. An opportunity for variation in which conclusions are added without a change in the data is implied when a teacher challenges his students, "Here is a figure with these and only these data. I will suggest one conclusion. Mow dis­ cover as many other conclusions as you can." Illustration 21 is an example.

ILLUSTRATION 21.

Original Exercise 21 Statement: The bisector of a central

angle of a circle is perpendicu­ lar to the chord which subtends the arc intercepted by the cen­ tral angle. Conclusions which may be added to this statement are indicated by underlining them in the statement of the var­ iant which follows. Variant 21.1 Statement; The bisector of a central angle of a circle is perpendicular to the chord which subtends the arc in­ tercepted by the angle, bisects the chord, and bisects

the arc. In this Illustration the student is challenged to dis­ cover conclusions that follow from specified data. The 100 situation la somewhat different when the teacher suggests, or the student wonders, "How will the data need to be chang­ ed If we wish to add this or that (specified) conclusion?" An example is furnished by illustration 22. 3.22 The addition of a conclusion requires a change in the

data,

ILLUSTRATION 22. Original Exercise 22 Statement: The midpoints of the sides of a quadrilateral are the vertices of a para­ llelogram . Suppose, now, that the add­ itional conclusion desired is that two adjacent sides of the parallelogram (EFGH in the above figure) shall be equal, that is, that EFGH shall be a rhom­ bus. Perhaps the student recalls that if ABCD is a rec­ tangle or an isosceles trapezoid, the figure obtained by connecting the midpoints of the sides will be a rhombus. (See original exercise 19, p. 95, and original exercise 5 . p. 39; . Either of the statements implied In the preceding sentence suffices to illustrate the type of variation under consideration. But further investigation is fruitful, for it reveals that ABCD need be nlether a rectangle nor an Isosceles trapezoid in order for EFGH to be a rhombus. If 101 the student has proved original exercise 2 2 . he will have made use of the diagonals AC and BD. Study of their function in the proof leads to the observation that. If they are equal, then EFGH Is a rhombus* One question remains: Is It poss­ ible to have a quadrilateral which has equal diagonals, but which Is neither a rectangle nor an Isosceles trapezoid? In order to provide a basis for an affirmative answer, It Is only necessary to draw two lines AC and BD, equal In length, and connect the points A, B, C, and D In order,* Variant 22.1 has now been formed, first by adding a conclusion, then by adding the needed datum. Variant 22.1

Statement: The midpoints of the sides of a quadrilateral with equal diagonals are the vertices of a rhombus. Somewhat similar variants are possible In which EPGH Is a rectangle and a square. These are stated explicitly on p. 18*+ . The reader might notice that variant 22.1 may also be considered a variant of original exercise 19. "The midpoints of the sides of a rectangle are the vertices of a rhoraUUs. "

"’‘The lines AC and BD will Intersect if the quadrila­ teral ABCD Is convex. Quadr11ateral without a qualifying adjective means a convex quadrilateral. (See p. 32). 102 This la Indeed tx*ue and Illustrates variation in which a substitution Involving elements of the figure Is made In the data but is not accompanied by a change In the conclu­ sion* (See section 2.341, pp. 73-75* ) 103

3*3 Substitutions Are Made In the Conclusion* In section 2*3 substitution made Initially In the data is discussed under six distinct categories, each of which is further divided Into two subcategories* Analogous types in which a substitution Is made Initially In the conclusion are not all of great significance; not all of them are even possible* Since neither the possibility nor the Im­ portance of a supposed type of variation can be perceived without some reflection, each type will be considered* 3.31 The substitution Involves numerical values assigned to elements of a figure. 3*311 There is no change in the data. If a geometrical exercise is given In which all or some of the data are expressed numerically and in which the con­ clusion Implied by the data is also expressed in numerical terms, then any numerical change In the conclusion will necessarily be accompanied by a change in at least one of the data. Hence, variation of the type Indicated in this subsection Is Impossible*

3*312 Substitution In the conclusion of numerical values assigned to elements of a figure necess­ itates a change In the data* This type of variation might be encountered during the Introductory phase of geometry or in an Investigation at some later stage In which particular figures are used. The example given Is a simple one* 10W ILLUSTRATION 23. Original Exercise 23. A student finds that, if the angle between the equal sides of an isosceles triangle is 100°. then each of the base angles is 40°. Variant 23.1 The student then inquires, "What must be the size of the angle between the equal sides of an isoaoeles triangle If eaoh of the base angles is to be 50°?" By construction of a figure or by computation if he knows that the sum of the angles of a triangle is 180°, he finds that the angle between the equal sides must be 80°. As the reader has no doubt observed, the problem of variant 23.1 may also be phrased as follows: "In triangle ABC, AB is equal to AC and angle CBA is 50°. Find the size of angle BAC." In similar examples the situation is the same • If numerical changes are made in the conclusion of an exercise, changes must also be made in the data. But de­ termining the changes to be made in the data may be viewed as solving a different problem, that is, finding the conclu­ sion when certain data are given. Hence, no great importance is attached to the type of variation which is performed on the conclusion of a statement by changing numerical values assigned to elements of a figure. 105 3.32 A conclusion expressed In nunTlcal terms Is replaced by a conclusion which is expressed in general terms. 1311a typa of variation occurs whan during an investi- gation involving apaoific figuraa an insight is experienced which leads to the formulation of a general statement. Hi is is a very Important process. It may be aided by encouraging the student to scrutinize every conclusion which involves numerical values and, In doing so, to be guided by two ques­ tions: 1) "What general relation, If any, is implied by this conclusion?" 2) "What changes, if any, may be made In

the data without disturbing the general relation appearing

in this conclusion?" Question 1) Implies a change In the

conclusion but none in the data; question 2) Implies a

change in the conclusion and the possibility of a change in

the data. A particular example will illustrate both cases.

ILLUSTRATION 24, Original Exercise 24. A Statement: If an angle in­ scribed in a circle £ intercepts an arc of 140°, the angle is 70°. The observation may now be made that angle BAC contains one-half as many degrees as arc BC, in the particular case where arc BC is 140°. This leads to variant 24.1, which illustrates variation of the type named in section 3.321, which follows. 106 3.321 A conclusion expressed In numerical terma la replaced by a conclusion which la not expressed In numerical terms, but no change la made In the data. Variant 24.1 Statement: If an angle inscribed in a circle Intercepts an arc of 140°, the angle is measured by one-half of the Intercepted arc. Recognition of the relationship expressed in variant 24.1 might motivate the question, "Is the number of degrees in the inscribed angle equal to one half the number of de­ grees in the intercepted arc regardless of the size of the arc?" Further numerical examples may be investigated, or search for a deductive proof may be begun immediately. In either case, variant 24.2 is implied. It illustrates var­ iation of the type named in section 3.322, which follows. 3.322 A conclusion expressed in numerical terms is replaced by a conclusion which is not expressed in numerical terms, and a change is made possible in the data. Variant 24.2 Statement: If an angle Inscribed in a circle intercepts an arc of any size the angle is measured by one-half of the intercepted arc. 3.33 A conclusion expressed in general terms is replaced by a conclusion which is expressed in numerical terma. 107 Variation of this type may be performed during the In­ vestigation of a statement conjectured to be true* However, a more effective procedure is to substitute numerical data which illustrate a given relationship and then to determine whether or not the relationship expressed In the conclusion Is also Illustrated. This type of variation Is discussed In section 2*33, pp. 62-72. 3*331 There Is no change In the data* This kind of variation is illustrated if variant 24.1 is the original exercise, and original exercise 24 Is the corresponding variant* 3*332 There Is a change In the data* This type of variation is illustrated If variant 24.2 is the original exercise} then original exercise 24 Is an example of a corresponding variant* 3.34 The substitution Involves elements of the figure* 3*341 There Is no change In the data* Variation In which no change In the data is required when one conclusion Is substituted for another may be viewed as addition of a conclusion, except that two statements re­ sult Instead of one* Addition of a conclusion Is discussed In section 3.2,. pp. 99-102. 3*342 There Is a change In the data* On the other hand, the substitution of one conclusion for another might make necessary or make possible a change 108 in the data. In illustration 25, which follows, a change in the data is made possible by the substitution of a new con­ clusion for the original conclusion* If original exercise 25 and variant 25*1 were interchanged, the necessity of a change in the data would be illustrated*

ILLUSTRATION 25. Original Exercise 25.

Statement: In a parallelogram having a right angle, the diagonals are equal•

If the conclusion Is to assert that opposite angles, rather than the diagonals* are equal, as Is done In variant 25.1, then the datum that the parallelogram contains a right angle may be deleted* Variant 25.1

Statement: In any parallelogram, opposite angles are equal.

3.55 The substitution Involves relationships* 3.351 There Is no change in the data*

Variation of this type, like the type of variation in section 3*341 may be viewed In terms of addition of a con­ clusion* 3.352 There is a change in the data. In illustration 26, which follows, a change Is made necessary in the data by a change in the conclusion. 109

ILLUSTRATION 26.

Original Exercise 26.

Statement: In a parallelogram having a right angle, the

diagonals are equal.

Variant 26.1 is obtained by asserting that the di­ agonals are perpendicular, rather than equal. It then be­ comes necessary to have the datum that two adjacent sides of the parallelogram are equal instead of the datum that one angle of the parallelogram is a right angle.

Variant 26.1

Statement: In a parallelogram having two adjacent sides

equal, the diagonals are perpendicular.

3.36 The substitution involves the baalo figure.

It is not possible to make a substitution in the con­ clusion which involves the basic figure, for the stipula­ tion of the basic figure is part of the data.

3.37 The substitution in the conclusion involves combin­

ations of the types of substitution mentioned.

Ordinarily a variant differs from the original exer­ cise or from the preceding variant with respect to only one change. This is true particularly when each of the variants is obtained by making a substitution. The writer was unable 110

to find a situation In which the simultaneous employment of two or more kinds of substitution Is recommended in pre­ ference to achieving the same result by making one substi­ tution at a time. Therefore, illustration of the type of variation in which substitutions of different kinds occur

simultaneously in the conclusion, is omitted. Ill

3.4 Combinations of Deletion. Addition, and Sub­ stitution Are Performed on the Conclusion.

The writer did not discover a situation In which var­ iation of the type indicated In the heading of this section would be as desirable as performing the variation In dis­ tinct steps, each involving only deletion or only addition or only substitution of a conclusion. Hence variation of this type is not illustrated. 112

Summary, Chapters 2 and 3 Two reciprocal processes tinder lie the concept of var­ iation presented In chapters 2 and 3: 1) A change Is made Initially In the data of a statement which Is known; then it is determined whether or not accompanying changes In the conclusion are implied, and what these changes are* 2) A change Is made Initially in the conclusion of a known state­ ment. This Is followed by an Investigation of the changes, if any, which may or must be made in the data In order for the new conclusion to be implied by the data. The basic emphasis In this study Is upon obtaining a new set of data or a new conclusion, or both, by making changes In a known statement. Different types of changes are discussed and Illustrated in chapters 2 and 3, and to some extent the relative Importance of the various types is Indicated. These changes are made In a deliberate man­ ner. In choosing which changes to make initially and in determining what other changes, if any, should accompany the initial changes, the student is guided by experience, gradually developed through continued utilization of the process of variation. CHAPTER 4

KINDS OF VARIATION IN WHICH THE CHANGES ARE MADE SIMULTANEOUSLY IN THE DATA AND THE CONCLUSION

4.1 Loglaal Transformations P.115 4.11 Converses P.115 4.12 Inverses P.116

4.13 Contrapositives; The Law of Contraposition. P. 119 4.2 The Principle of Duality; The Principle of Reciprocity. P.122 4.3 The Law of Duality. P.125 Summary. P.126

113 Ill*

A new statement may be obtained from a known statement by Initially varying the data, as is discussed In Chapter 2, and by initially varying the conclusion, as is discussed In Chapter 3. Although such variation is performed In a delib­ erate manner, it is not performed in accordance with any precise rules of procedure.

Several other processes which may be viewed in terms of variation will be mentioned in Chapter 4. In all of them new statements are generated from known statements, but all of them differ from variation as presented in Chapters 2 and 3 in the following respects: 1) Changes are made simultan­ eously in the data and the conclusion of the known statement.

2) The changes are made in accordance with definite rules of procedure. 3) Although some of these processes are better known than others, all of them are mentioned in the liter­ ature • 115 4 .1 Logical Transformations. Only the three logical transformations treated by Lazar In his work are considered In this study.1 They are the converse, the Inverse, and the contraposltlve. 4.11 Converses. Lazar's published definition for converse Is: "Hie converse of a theorem may be obtained by Interchanging any number of conclusions with an equal number of hypotheses."2 His more recent, revised definition is: "The converse of a theorem may be obtained by Interchanging any number of con­ clusions with any number of hypo theses . "3 t w o Important observations are to be made: 1} Lazar's revised definition may be viewed as a variant of his original definition of a converse. 2) For a given theorem, the revised definition allows more converses than does the original definition. For Instance, a theorem with two data and one conclusion has two converses under the original definition, and three under the revised definition. Formation of converses is a rich source for new state­ ments, some true and some false, hence requiring investiga­ tion. Lazar computed that a theorem with four data end three conclusions has 34 converses in accordance with his original definition. In accordance with the revised defin­ ition, even more converses are possible. Nevertheless,

■^Lazar. o p . clt. 2Ibid. . p. 16 3Conversation with Dr. Lazar. 116 under any circumstances the number of converses depends on the number of data and the number of conclusions in the original theorem. A student knows when ha has formed all of the converses; he has a definite mechanical method for forming them, but no opportunity to change any item except to change It from data to conclusion or vice versa. Var­ iation of the types presented In Chapters 2 and 3 has nei­ ther of these limitations as a method for forming new state­ ments • 4.12 Inverses. With the explanation, "Every theorem that contains many conclusions can always be subdivided Into as many theorems as there are conclusions, all having the same set of hy­ potheses,"4 Lazar's original definition for Inverse is: "An Inverse of a proposition having one conclusion may be formed by contradicting one of the hypotheses and the con­ clusion."5 He has also revised this definition as follows: "An inverse of a proposition having any number of conclusions and any number of hypotheses may be formed by denying at least one conclusion and at least one of the hypotheses."®

The observations made in the preceding paragraph with res­ pect to the formation of converses in contrast to variation as a process for generating new statements apply equally to the formation of inverses.

^Lazar, op. clt. , p .27. 5Ibid., p. 25. ^Conversation with Dr. Lazar. 117 There is, however an important relation between var­ iation and the formation of inverses. It is Implied, but not enunciated, in Chapters 2 and 3. An illustration will be given to make this relation understood and to emphasize

it. ILLUSTRATION 27.

Original Statement: The midpoints of the sides of a square are the vertices of another square. Inverse: The midpoints of the sides of a quadrilateral which is not a square are not the vertices

of a square. Two remarks are to be made about this inverse. For one thing, it is not a true statement, as may be seen from the figure at the right, in which diagonals AC and BD are equal and perpendicular. Secondly, even without knowledge as to ifeether the Inverse is true or not, two questions follow directly: 1) If ABCD is not a square, what other kind of quadrilateral with special properties might it be fruitful to consider? A suggestion for beginning the investigation of this ques­ tion is given in exercise 2.2, pp. 5 a*1** 6, where it is *18 shown that "The midpoints of the sides of a rhombus, taken in order, are the vertices of a rectangle2) Under the conditions of ABCD being some quadrilateral other than a square, if EFGH is not a square, what is it? One answer to this question is also given in the above exercise* The occurrence of two propositions related as above by way of an implied Inverse can be found in Euclid* His Proposition 4, Book I, is: "If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal;"17 Proposition 24, Book I, isj "If two tri­ angles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them. of the other, the base of that which has the greater angle jshall be greater than the base of the other*"8 However, the inverse which relates the two statements does not occur in Euclid nor is it mentioned by Heath*®

In summary, whereas the inverse of a statement is based on a negation of one or more conclusions and at least one datum and is itself a kind of variation, other statements 7I. Todhunter, The Elements of Euclid, p. 9. 8Ibid., p. 26. ®Heath 0£* clt., Vol. I 119 obtainable by the process of variation are expressed In positive terms by comparison with the original statement* A statement expressed In positive terms Is often more use­ ful than an Inverse even though the Inverse Is a true state­ ment* For Instance, the Inverse of "If two sides of a tri­ angle are equal, the angles opposite are equal," is "If two angles of a triangle are unequal * the sides opposite are unequal»" A variant of the original statement Is: "If one angle of a triangle Is greater than a second angle, the side opposite the first angle is greater than the side opposite the second angle*" The Inverse states only that the sides In question are unequal; the variant tells which one is greater• 4.13 Contraposltlves; The Law of Contraposition An Inverse of a true statement sometimes is a true statement, and sometimes it is not* The same is true of a converse of a true statement. There is a transformation, however, which, when applied to a true statement, always yields a true statement. It is contraposition: the re­ sulting statement is called a oontraposltlve of the orig­ inal statement* Lazar offers the following definition to convey the traditional meaning of contraposltlves ,,The con- trapositive of a theorem is another theorem obtained from it by contradicting both the hypothesis and the conclusion and by interchanging their respective places*" 19Lazar. op. cit** p* 37 120

The Law of Contraposition Is: WA proposition and the con- traposltlve derived from It are equivalent — If one Is true, the other is true, and if one Is false, the other is false An especially Important extension of the Law of Con­ traposition applies to theorems having more than one hypoth­ esis but only one conclusion, Lazar's statement of the ex­ tended law is the following: "If a theorem contains more than one hypotheses and only one conclusion, it is equiva­ lent to any theorem formed from it by the interchange of the contradictory of one of the hypotheses with the con­ tradictory of the conclua Ion.”12 An illustration of contraposition will be given,

ILLUSTRATION 28. Original Statement: If a line bisects the vertex angle of an isosceles triangle, it bisects the opposite side. Contraposltlve: If the b isector of an angle of a triangle does not bisect the opposite side, then the other two sides of the triangle are not equal. The original statement Is a familiar, true statement. In view of the Law of Contraposition, the contraposltlve is also a true statement. This the reader may easily verify by means of a geometric proof.

1^Ibld., p. 37. 1SdIbld. , p. 42. 121 As Lazar emphasizes, although a eontraposltlve and ths corresponding original statement are logically equivalent, the meanings conveyed are different*^9 It follova then that there are situations in which It Is nore convenient to apply one than the other of two oontrapositlve statements, al­ though logically the two are equivalent# Moreover, differ­ ent proofs are possible for a given statement and its con- trapositive, and often one of these proofs is more easily discovered and understood than the other.^ Hence, although contraposition does not lead to statements vhich are new from the standpoint of logical truth, it is nevertheless an important transformation.*6 ISlbld.V p. 47. x*5bid., pp. 43-46. l°Further references relating to converse, Inverse and con traposltlve may be found in the bibliography given by Lazar, op. clt. A recent article concerning converses of statements in geometry 1st Frank B. Allen, "The Multi-Con­ verse Concept in Oeometry," The Mathematics Teacher. XLV {Beeea*er*195Si), ppcwt588-84~ A recent textbook on math­ ematics in which converse and oontrapositlve are explained is: Kenneth 0. May, Elementary Analysis, pp. 26-26. 122

4.2 The Principle of Duality* The Principle of Reciprocity. In two-dimensional geometry the Principle of Duality refers to the Interchange of the words point and line In a statement containing both. Sometimes additional but slight modifications of the wording are necessary. The new state- ment thus formed is called the dual of the first. 1 ft ° An ex­ ample of a statement and Its dual is; Two Lines determine a point; two points determine a line. Enlargement of the number of pairs of words which may be Interchanged results in the principle of Reciprocity, which apparently received more attention on the high school level a half century ago than it does at the present time.17 |U' .. — Levi S. Shively, An Introduction to Modern Geometry, p. 40. E. H. Taylor and G. C. Bar too. An Introduction to College Geometry, p. 113. 17That this excellent example of symmetry in geometry is not emphasized on the high school level seems unfortun­ ate. The writer consulted the Indices of a number of Plane Geometry texts of the past twenty five years and searched a smaller number page by page without finding reference either to dual1ty or the related concept, reciprocity. A situation not greatly different exists with regara to books on the Teaching of Mathematics. Several mention neither duality nor reciprocity. Butler and Wren give the principle of dual ity passing note as follows: "The teacher of secondary math ematlc8.... should know something.... of Gergonnefs (1771-1859) ’principle of duality' which so definitely en­ riches the theory of the united position of point and line" (Butler and Wren, o£. clt., p. 235). 123 D. E. Smiths excellent section on Reciprocal Theorems Is reproduced In full. There Is an Interesting line of propositions, early met by the pupil, in which one theorem may be formed from another by simply replacing the words point by line, line by point, angles of a triangle by (opposite) sides of a triangle, sides of a triangle' by (opposite) angles of a triangle. This is seen in the following propositions: If two triangles have If two triangles have two sides and the included two angles and the Included angle of the one respectiv­ side or the one respectively ely equal to two sides and equal to two angles and the the included angle of the Included side of the other, other, the triangles are the triangles are congruent. congruent.

If two sides of a tri­ If two angles__ of a tri- angle are equal, the angles angle are equal, the sides opposite those sides are opposite those angles are equal• equal• Of course the teacher may pass over this relationship, as most textbooks do, without comment. But there is great advantage in recognizing the parallelism early in the course for two reasons: (1) It adds greatly to the pupil's inter­ est to see this symmetry of the subject, to note that cer­ tain propositions have a dual; and (2) It often suggests new possible theorems for investigation — the pupil has the interest of discovery This is seen in the following simple exercise: In a triangle ABC, where a - b, the bisector of angle C, produced to c, bisects side c. The pupil who is led to discover the reciprocal theorem, and to investigate its validity (for reciprocal statements are not always true), has opened before him a field of perpetural Interest, a field in which he is an independent worker.18

^David Eugene Smith, The Teaching of Elementary Math­ ematics, pp. 275-77. 12b

Wentworth mentions reciprocity in the preface to Plane and Solid Geometry with the added note that the application of it "is usually left mainly to the discretion of teachers!!^® It Is to be noted that Wentworth does not distinguish be­ tween duality and reciprocity* In the following quotation, §139 and §143 refer to the theorems on the congruence of triangles frequently abbreviated by s.a.a« and a.a.a. Reciprocity: "In §139 we have two angles and the included side, in §143 two sides and the included angles; hence, by interchanging the words sides and angles, either theorem is changed to theother* This is called the Prin­ ciple of Duality* or the Principle of Reclproc- i*2> " The reciprocal of a theorem is not always true, just as the converse of a theorem is not always true.®0 Use of the Principles of Duality and of Reciprocity, like the formation of converses, Inverses and contraposi- tives, is included in the concept of variation.

^yG • A. Wentworth, Plane and Solid Geometry, p. 4. 20Ibid.. p.35. 125

4.3 The Law of Duality. Under the conditions that it Is established as a law, the Principle of Duality is of Interest sufficient only to be mentioned in connection with variation* In Projective Geometry, for instance, it is possible either to postulate or to prove the dual of every postulate concerning point and line * It is then possible to assert the Law of Dual­ ity as a theorem: "Any proposition deducible from ((the original set of assumptions)) concerning the points and lines of a plane remains valid, ((with appropriate con­ ventions as to terminology)), when the words •point' and pi 'line' are interchanged".

210swald Veblen and John Wesley Young, Projective Geometry, Vol I, p. 29. 126

The transform&tionf, converse. Inverse and contra- positive, the Principles of Duality and Reciprocity, and the Law of Duality may be viewed in terms of variation. In all of these processes changes are made simultaneously In the data and conclusion, and these changes are made according to definite procedures. The contraposltlve of a true statement is a true state­ ment. In a postulational system In which the Law of Dual­ ity applies, the dual of a true statement is also a true statement. In both cases it Is unnecessary to prove the new statements. Use of the Principles of Duality and Re­ ciprocity and the formation of converses and inverses lead to new statements. The truth of these in each case is in­ dependent of the truth of the original statement. Hence each new statement obtained needs to be investigated for truth or falsity. An Inverse of a statement is obtained by negating one or more conclusions and one or more data in the given state­ ment. Hie more inclusive process, variation, Includes sub­ stituting new data and new conclusions for the negated Items^ thereby providing a way of obtaining a new statement ex­ pressed entirely in positive terms. CHAPTER 5

MORE ILLUSTRATIONS OP THE USE OF VARIATION IN A COURSE IN PLANE GEOMETRY TAUGHT BY THE WRITER.

5.1 Korls P. 13o 5.2 Triangles P. l*frO 5.3 Parallel and Perpendicular Lines P. 162 5 A Eg.lY.g9ng. P. 177 5.5 Circle? P. 195 5.6 Ratio. Proportion ^Similari ty P. 2ob

S.wamary p . 207

127 128 The Illustrations In chapters 2 and 3 are presented for the specific purpose of explaining the particular kinds of variation which were outlined there. The purpose of the present chapter is to indicate, by means of further illustrations, how variation may be utilized at various stages in a high-school course in plane geometry. In many cases several of the kinds of variation outlined In chapters 2 and 3 will be used in one illustration, for emphasis In this chapter Is on the discovery of new statements and upon the development of the material in geometry through the use of the process of variation. Each illustration begins with an original exercise, which Is assumed to be known, and is followed by one or more variants. Many of the original exercises are chosen from the textbook used with the class which the writer taught in con­ nection with this study.^ Many of the variants were discovered by students in the class. In the writer's method of procedure with his class, duplicated sheets entitled MThings to Do for Next Time11 were distributed daily.* Usually these were not prepared until after the previous class meeting. They called attention to material that would be studied by the class as a whole. With respect to some of the assigned exercises, specific sugges­ tions for possible variations were given, sometimes in

1 Welchons and Krlckenberger, ojd. cit. * An indication of the content of these sheets is given in connection with illustrations pp. 137-139. l6*+. 169-17U. 179-30. ’ ’ 129 considerable detail* Written work was collected at the begin­ ning of each class period and read by the teacher before the next meeting. Sometimes additional suggestions relating to an exercise previously assigned were incorporated in a later assignment sheet.

During class periods, variants discovered by individual pupils were shared with the other members of the class. Some­ times variants were worked out by the group as a whole. Also during these periods, the teacher actively attempted to teach the process of variation, illustrating and pointing out dif­ ferent types from among those presented in earlier chapters of this paper.

In this chapter comments about the process of variation are enunciated in connection with some of the illustrations.

Also included are comments concerning ways in which the exercises involved are treated in geometry texts and in the literature on geometry and the teaching of geometry. This chapter is divided into sections roughly following the organi­ zation of the material covered in the text that was used in the class taught in connection with this study. 130 5.1 Introductory Work

Variation as a deliberate process was not brought to the attention of the class in a formal manner until the course, which was to run for approximately seventeen weeks, had been in progress for about three weeks. But even in the intro­ ductory work, as he recognized opportunities for doing so, the teacher began to create and develop favorable attitudes toward change, which he hoped would later help to make the students receptive to variation. Suggestions made by the teacher often were no more than proposals that changes be made in given situations. Out of these informal beginnings, variation emerged as a specific procedure for changing given exercises to obtain new ones.

ILLUSTRATION 1* After perpendicular lines had been Illustrated with several examples of their occurrence In experience but before a formal definition of perpendicularity had been stated, the notion of adjacent angles was developed. In the process, the teacher requested someone to draw a figure showing two angles having a common vertex and a common side between them. The figure shown below labeled original figure was drawn by one of the students. The teacher then asked, "Can you draw a slightly different sort of figure that will also illustrate the situa­ tion?" Other figures, such as those shown, were then drawn ^It will be noticed that a new series of numbered illustra­ tions is begun in this chapter. No confusion with illustra­ tions bearing like numbers in preceding chapters is anticipated. 131

by other students and by the teacher. / Original Figure IZ

Illustration 1. Is not presented as an example of varia­

tion as It was defined for purposes of this study on page 12

of this paper. That definition is:

Variation is defined to mean a process changing elements of the data or conclusion, or both, of a familiar geometrical statement, which has been proved to be true or is accepted as true, with a view to obtaining a new set of data, or a new conclusion, or both, result­ ing in a new statement. The new statement is to be affirmed by deduction, or shown to be false, or adopted as a new assumption.

But illustration 1 is Included here to show how the making of changes was used as a preliminary to the variation of state- ments.

This Illustration, however, brings out a matter of great

importance In the teaching and learning of concepts: although the verbalized definition of a concept implies many kinds of specific situations, pupils frequently do not apprehend the extent of the situations Included in the definition without 132 actually seeing many different kinds of examples. For

Instance, some students seeing only Illustrations of adjacent angles like those given In the first line of examples above,

In which all of the rays are drawn either horizontally or upward, might not quickly recognize that, In the figure at the right, angles a and t) also are adjacent angles. It is, therefore, urgent for a teacher to seek to remove the possibility of the students1 unconsciously adding condi­ tions to a definition which are not at all Included therein.

An effective way of accomplishing this Is to start with one

Illustration of a concept and then to change those aspects of the Illustration which are not relevant to the concept.

ILLUSTRATION 2

An experience sometimes suggested to students in junior- high-school mathematics Is finding the sum of the angles of a triangle. This experience also may be suggested in a class in demonstrative geometry at some t5me previous to the proof that the sum of the interior angles of a triangle Is 180°.

The sides of each triangle that Is drawn will be certain definite lengths. Suppose that the first triangle drawn is an equilateral triangle with sides one inch In length. The results of drawing this triangle and finding the sum of the interior angles may be expressed as a statement. 133 Original Exorcise 2

The sum of the interior angles of an equilateral triangle with aides one inch long is 180°.

The student suspects, or the teacher suggests, that there might be good reason to find the sum of the angles of other triangles. The question arises, ”In what way can the triangle of original exercise 2 be changed?” One way of doing this is to change the length of the equal sides of the equilateral triangle. The statement of each result is a variant of original exercise 1. After investigation of a number of equi­ lateral triangles of different sizes, the inductive generali­ zation (see p. 67) stated as variant 2.1 may be verbalized.

Variant 2.1

The sum of the Interior angles of any equilateral triangle is 180°.

But the triangle may be varied in another way: by chang­ ing the relative lengths of its sides. With respect to the relative lengths of the sides, a total of three types of triangles are commonly named, equilateral, Isosceles, and scalene. Triangles are also named according to the size of the angles, as follows: acute, right, and obtuse. Hence a total of seven different types of triangles, according to common nomenclature, may be distinguished: equilateral, isosceles-acute, isosceles-right, isosceles-obtuse, scalene- right, scalene-acute, and scalene- obtuse. For each of these 134 typos, with aides specified numerically, variants of original exercise 2 may be stated; and for each of these types a generalization may be stated, which is a variant of variant

2.1. Interest is Increased if a guess is made as to the sum of the angles each time before they are measured. Finally, the generalization stated as variant 2.2 may be verbalized.

Variant 2.2

The sum of the interior angles of any triangle is 180°.

Scrutiny of variant 2.2 reveals that It may be varied also. One way in which it may be varied Is by substituting quadrilateral for triangle. It Is necessary then to determine whether or not the sums of the interior angles of all quadri­ laterals have the same numerical value, and, if so, what that value is. Students may be permitted to classify quadrilaterals in a manner similar to the way in which the triangles were classified. Of course there are many more types. However, before all such types are Investigated, at least some of the students will recognize that any quadrilateral may be divided into two triangles by drawing a diagonal. Then, on the basis of variant 2.2. they deduce the statement of variant 2.3.

Variant 2.5

The sum of the Interior angles of any quadrilateral Is 360°.

P'urther variants are obtainable by substituting pentagon. 135 hexagon, etc., for quadrilateral. But soon a pattern emerges.

The sum of the angles of any polygon Is a multiple of 180°, and the multiplier Is the number of non-overlapping triangles into which the polygon can be divided. This generalization can be verbalized In the way it is stated in variant 3.4.

Variant 2.4

The sum of the Interior angles of any polygon is equal to (n-2)180°. where n is the number of sides in the polygon.

Another series of variants may be obtained by substitut­ ing in all of the preceding statements the word exterior for the word interior. Suppose that variant 2.2 is the first statement chosen for this substitution. The sum of the exterior angles for several triangles of different sizes and types is determined. Always the sum is found to be 360°.

Variant 2.5 is then stated.

Variant 2.5

The sum of the exterior angles of any triangle is 360°.

By this time it will hardly be necessary to suggest substitution of quadrilateral for triangle in variant 2.5.

Guessing the resulting sum of the exterior angles in this case injects special zest if the answer is not known. There is confusion about what pattern to follow in arriving at a plausible guess. Moreover, the surprise is great when it is found that the sum of the exterior angles is the same for a 136 quadrilateral as for a triangle. Variant 2.6 results.

Variant 2.6

The sum of the exterior angles of any quadrilateral Is

Finally, perhaps after Investigating several other polygons, variant 2.7 may be stated.

Variant 2.7

The sum of the exterior angles of any polygon Is 360°.

ILLUSTRATION 3

One more illustration from the introductory work will be given. It will al3o be indicated how, by means of written instructions, the teacher suggested variations of an original exercise and led the students to verbalize general conclusions.

Original Exercise 3

An exercise given In the text is:

Draw a line segment AB. Choose any point £ in AB and a point D not in JCB. Draw CD. TTsing tilie protractor, bisect angle ACD and angle BCD. What is the size of tKe C angle formed by the two bisectors?2 (Figure Supplied)

^ftuadr11ateral, without a qualifying adjective, means convex quadrilateral. (See footnote, p. 32). However, variant 2.6 is true also for concave quadrilaterals, and these may well be included in the investigation. Similar remarks apply to polygons having more than four sides.

^Welchons and Krickenberger, op. clt.. p. 18, exercise 8. 137 In assigning this exercise the teacher added the suggestion,

'‘Imagine DC rotating about point C. Draw the bisectors of angles BCD and DCA for various positions of DC. What do you conclude about the size of the angle between the bisectors of any two adjacent supplementary angles? Write out a state­ ment of your conclusion." The following statement or its equivalent was submitted by most of the class.

Statement; The angle between the bisectors of any two adjacent

supplementary angles is 90°.

Also given were the suggestions which follow. In each case the variant stated by the students is given immediately after the written suggestions made by the teacher.

a) Repeat the above exercise, but with these changes.

Start with a right angle ACB. Choose any point D "inside" the angle ACB. Draw DC. (What is the relationship between angles

BCD and DCA?) Bisect the angles BCD and DCA. What is the size of the angle formed by the two bisectors? Repeat this exercise for several other positions of point D inside a right angle ACB. Make a general statement concerning the bisectors of two complementary adjacent angles.

Variant 5.1

The angle between the bisectors of two adjacent comple­ mentary angles Is 45°. 138

b) Repeat the above exercise, starting with two adjacent angles whose sum Is 40°.

Variant 3.2

The angle between the bisectors of two adjacent angles whose sum is 40° is 20°.

c) Repeat the above exercise, starting with two adjacent angles whose sum is 170°.

Variant 3.3

The angle between the bisectors of two adjacent angles whose sum is 170° is 85°.

d) Repeat the above exercise, starting with angles whose sum is 220°, 300°, and 360°, Explore any other cases you wish to investigate. Can you now complete the following state­ ment? "The angle between the bisectors of any two adjacent angles is equal to------."

Variant 3.4

The angle between the bisectors of two adjacent angles is equal to one-half the sum of the original angles.

No attempt was made with the class, and the attempt is not made in this paper, to exhaust the opportunities for variation in connection with any one exercise. For example,

^In an elementary course in geometry consideration is generally limited to angles not greater than 360°. Euclid seems to have considered only angles less tlian 180°. See Heath, ojd. clt.. Vol. I., p. 176. 139 in the illustration just given, bisector may be replaced by trisector, etc*; the reader may explore the possibilities for himself. As familiarity with the process of variation in­ creases, suggestions for variation should be stated less specifically than in illustration 3 . It was the writer’s experience, later on in the course, that a suggestion such as,

"Now explore the results of making changes in the given exercise,” was sufficient to elicit the variants of which the teacher was aware. Frequently the students produced significant variants which had not occurred to the teacher at all.

The reader is also reminded that only a few illustrations of variation are presented In this chapter. kany more were produced by the students in the writer's class in which variation was emphasized. 140 5.2 Triangles

In most textbooks the formal work in demonstrative geometry begin9 with the study of triangles, and one of the first con­ cepts Introduced Is that of congruence. By definition, two

triangles and also any two polygons are congruent if they can be made to coincide. From this It follows that two triangles are congruent If every part of one is equal to the correspond- ing part of the other. But theorems for proving the congru­ ence of triangles and of polygons as well may be stated which

involve fewer than all of the parts of the figures concerned.

An approach to the concept of congruence and to propositions having to do with congruence is suggested by the following statement made by a committee of the British Mathematical

Association in 1939:

The 1923 Report enunciates the Principle of Congruence thus: 'If instructions point to a single definite figure at one place, not only do they point to a single definite figure at another place, but there are no internal differences between the two figures.1

The possibility of such instructions rests on the assumption that there is meaning, as every one but a philosopher takes for granted, in the terras and statements characteristic of metric geometry: 'the line AB ■ the line X y * , 'AB - 3 inches', 'the angle ABC is a right angle', 'circle', etc.

Hence any proposition established by Euclid by the method of superposition can be established with precisely equal validity by noting whether, when it is re-stated as an order to construct a specific figure, the specification Is such as points to a definite unambiguous result.^

The sides and angles of a polygon are the parts of the polygon. 4A Second Report on the Teaching of Geometry in Schools, p • 23. 141

Congruence of Triangles

ILLUSTRATION 4

Statements concerning the congruence of triangles may be obtained In accordance with the above suggestions by asking

the students to copy a given triangle and to determine how many and which parts of the triangle they used in the process.

Of course, the exercise is to be repeated with several

triangles of different types and sizes. Then the students

are encouraged to find as many different criteria as possible

for the congruence of triangles. In this manner the following

three statements, labeled original exercise 4 t variant 4.1.

and variant 4.2. are discovered but not in the same order by

all of the students. However, it has been the writer's observa­

tion that most of the students either discover original

exercise 4 before they discover the other statements concern­

ing congruence or else know it before they begin the work in

geometry.

Original Exercise 4

Two triangles are congruent if the three sides of the

one are equal to the three sides of the other.

Variant 4.1

Two triangles are congruent if two sides and the included angle of the one are equal to two sides and the included angle

of the other. 142 Variant 4.2

Two triangles are congruent IT two angles and the included side of the one are equal to two angles and the Included side of the other.*

The writer prefers to adopt variants 4.1 and 4.2 as assumptions and to establish original exercise 4 by deductive proof. Of course, this proof is usually postponed until after the proof of the needed theorem "In an Isosceles triangle the angles opposite the equal sides are equal."

By adopting the term "consecutive parts" to denote sets of any number of parts of a triangle or of any polygon such as side-angle-side and angle-side-angle, variants 4.1 and 4.2 may be combined Into a single statement. It is presented as variant 4.3.

Variant 4.5

Two triangles are congruent if three consecutive parts of the one are equal to three consecutive parts of the other.

Variant 4.3 may be viewed as a variant of original exercise 4 or of variant 4.1 or 4.2. However, It is not likely to be discovered as a variant of any of these statements.

’**The statements of variants 4.1 and 4.2 have been men­ tioned previously in this paper as illustrating the Principle of Reciprocity (p.1 2 V ) . They may, however, be obtained, as is Indicated here, by means of variation without reference to reciprocity.

^ 4!1 The concept of "consecutive parts," and this statement, were suggested by Mr. Lazar. 143

It is more likoly to be formulated upon recognition of the opportunity to combine variants 4.1 and 4.2 after both of them have been stated separately.

In addition to being concise, the above statement has the advantage that it can readily be extended to the congru­ ence of polygons other than triangles. This is done in illustration 6 , under the title, Congruence of Polygons, rp. Like original exercise 4 and variants 4.1 and 4.2. variants 4.4 and 4.5, which follow, are found in most geometry texts.

Variant 4.4

Two triangles are congruent if a side and any two angles of the one are equal to the corresponding side and two angles of the other.^

Variant 4.5

If two right triangles have the hypotenuse and a leg of one equal to the hypotenuse and a leg of the other, the triangles are congruent.

The three variants which follow are found in some plane- geometry texts, usually as exercises or corollaries.

”r0f course this statement cannot be proved without having available, either as a theorem or as an assumption, the state- aent "The sum of the angles of a triangle is 180°." It Is not intended that all of the variants in this and other Illustra­ tions should be encountered in immediate succession. But it is recommended that as each one is taken up, If It is not dis­ covered by the students, the teacher should indicate Its rela­ tion to the others by way of variation. 144

Variant 4.6

If two right triangles have the two legs of one equal to the two legs of the other, the triangles are congruent.

Variant 4.7

If two right triangles have a leg and an acute angle of one equal to the corresponding leg and acute angle of the other, the triangles are congruent.

Variant 4.8

If two rlght triangles have the hypotenuse and an acute angle of one equal respectively to the hypotenuse and an acute angle of the other, the triangles are congruent.

The ’'Ambiguous Case”

All of the statements In Illustration 4 are commonly found In geometry textbooks with the exception of variant 4.3; and, as has been remarked (p.1^2), the statement of variant

4.3 is not dependent upon the process of variation*

But there are additional statements, related to those of illustration 4 . which are not found In most geometry texts.

These may be obtained very conveniently by the use of variation.

They are presented in Illustration 5 .

ILLUSTRATION

In order to show how the variants In this illustration are obtained deliberately by the process of variation, the data and conclusion of the original exercise and each of the 145 variants will be itemized. Since, in complicated situations, the variant is sometimes obtained in itemized form before it is stated in words, the verbal form of each variant follows the itemized form.

Original Exercise 5

(Variant 4.4, repeated)

Statement: If two right triangles have the hypotenuse and a

leg of one equal to the hypotenuse and a leg of the other,

the triangles are congruent.

Pa ta :

(a) Triangles ABC and A'B'C*.

(b) AC is equal to A' C 1.

(c) AB is equal to A' B ’.

(d) Angle B is a right angle.

Angle B 1 is a right angle. A

Conclus ion;

(x) Triangle ABC is congruent

to triangle A ’B’C ’. B' c

The variants which follow are obtained as the result of first noting, in original exercise 5 . that, since angles B and B ’ are both right angles, they are equal and then asking the following questions: 1) If angles B and 3’ were either acute or obtuse, but equal, would the triangles necessarily be congruent? 2) If not, what additional data would make them congruent? 3) If the triangles are not congruent, 146 what other significant conclusions can bo discovorod?

Variant 5.1 Data: A (a) Triangles ABC and A'B'C*.

(b) AC is equal to A'C*.

(c) AB is equal to A'B'. a

(d1) #Angle B is equal to angle B' •

(tft #Angle B is obtuse.

(Then angle 3' also is obtuse). £

Conclusion:

(x) Triangle ABC is congruent

to triangle A'B'C*.

Statement: If two triangles have two s ides of the one equal

to two side3 of the other respectively, and the angles

opposite to a pair of equal sides equal, and if one of

the given angles is obtuse, the triangles are congruent.

Variant 5.2

Data ►> A

(a) Triangles ABC and A ’B'C*

(b) AC is equal to A'C'.

(c) AB is equal to A'B'. r a* f (d*) Angle B is equal to angle B*.

(e-M-') AC is equal to or greater than AB.

T,The reader is reminded that a prime (*) indicates the substitution of a datum or conclusion and that an asterisk (■*»•) indicates an addition. 147

(Then A'Cf also Is equal to

or greater than A'B1).

Conclusion:

(x) Triangle ABC Is congruent

to triangle A'B'C*.

Statement: If two triangles have two sides of the one equal

to t wo sides of the other respectively, and the angles

opposite to a pair of equal sides equal, and if in one

of the triangles the side opposite the given angle is

greater than the other given side, the triangles are

congruent.

Variant 5.3

Data A

(a) Triangles ABC and A'B'C'.

(b) AC is equal to A'C'. a (c) AB is equal to A'B'. C (d1) Angle B is equal to angle B'. A (e-;:-") Angle B is acute.

(Then angle B' also is acute.)

Conclusion:

(x*) Triangle ABC is congruent to

triangle A'B'C', or angle ACB

is the supplement of angle A'B'C'.

Statement: If two triangles have two sides of the one equal

to two sides of the other respectively, and the angles

opposite to one pair of given sides equal, and if one of 148

the given angles Is acute, then either the triangles are

congruent or the angles opposite the other pair of given

sides are supplementary.

The reader has no doubt already noticed that variants 5.1,

6.2, and 5.3 are related to the so-called "ambiguous case" generally mentioned in books on trigonometry but not usually found in geometry textbooks. However, the ambiguous case has received some mention in the literature dealing with geometry.

A committee of the British Iwathematical Association recommends, "A theorem of which more use should be made is the ambiguous case in the congruence of triangles, which can be stated thus: If two triangles have a = a* , b = _bT , A = A* , and are not congruent, then B and B' are supplementary."^

Euclid does not include a proposition dealing with the ambiguous case, but Heath states it as follows: "If two tri­ angles have two Bides equal to two sides respectively, and if the angles opposite to one pair of equal sides be also equal. then will the angles opposite the other pair of equal sides be either equal or supplementary; and, in the former case, the g triangles will be equal in all respects.

Heath says further:

If It is desired to avoid ambiguity and secure that the triangles may be congruent, we can — The Teaching of Geometry in Schools, p. 65.

^Heath, og, cit.. p. 306. 149

introduce the necessary conditions into the enunciation, on the analogy of Eucl. VI. 7.

If two triangles have two sides of the one equal to two sides of the other respectively, and the angles opposite to a pair of equal sides equal, then if the angles opposite to the other pair of equal sides are both acute or both obtuse, or If one of them is a right angle, th

The proof of the three cases (by reductlo ad absurdum) was given by Todhunter

It will be noticed that variants 5.1 and 5.2 satisfy the con­ ditions of the last quoted theorem, for In both cases it can be shown that "the angles opposite to the other pair of equal rides are both acute."

Congruence of Polygons

It is possible to formulate statements about the con­ gruence of polygons with more than three sides. Such state­ ments may be considered as variants of variant 4.5 (P. l*f2), as is shown in illustration 6 . which follows.

ILLUSTRATION 6

Original Exercise 6

(A repetition of variant 4.5)

Two triangles are congruent If three consecutive parts

*Eucl. VI. 7. Is: If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal, the sides about which are proportional. ^All of the paragraph Is Italicized In the original 7Ibid.. p. 307 150 of the one are equal to three consecutive parts of the other.

.Yith original exercise 6 as a point of departure, similar criteria may be discovered for the congruence of two quadri­ laterals, two pentagons, and so on. The statements for quadri­ laterals and pentagons are given as variants 6.1 and 6.2.

Variant 6.1

Two quadrilaterals are congruent if five consecutive parts of the one are equal to five consecutive parts of the other.

Variant 6.2

Ty/o pentagons are congruent if s even consecutive parts of the one are equal to seven consecutive parts of the other.

Study of original exercise 6 and variants 6.1 and 6.2 reveals that in each case the number of consecutive parts which must be equal in two polygons in order for the polygons to be congruent is always three less than the total number of parts. Variant 6.5 may be stated as an assumption in view of this observation.

Variant 6.5

Two polygons, each having n sides (and thus having 2n parts) are congruent if 2n - 5 consecutive parts of the one are equal to 2n - 3 consecutive parts of the other. 151

Variation as a Formal Process for Discovering New

Statements Developed with a Class

The process of variation is presented to the reader in a formal manner In preceding chapters of this paper. Its relation to some other processes, such as induction and cer­ tain logical transformations, have been pointed out. Such a procedure would obviously not be satisfactory for use with a group of students beginning the study of geometry. The writer lias already indicated how, in the class that he taught in connection with this study, he emphasized the making of changes in given situations preliminary to Introducing the process of variation formally.

Variation was first Introduced formally as a process of deliberately making changes of distinguishable types In given exercises during the work with triangles. Illustration 7 , which follows, shows how this was done. The variants were all made in class under the guidance of the teacher.

ILLUSTRATION 7

Original Exercise 7

Statement: The line segments drawn from the midpoint of the

base of an Isosceles triangle to the midpoint of the

Q equal sides make equal angles with the base. Q Welchons and Krlckenberger, 0 £. c i t .. p. 83, ex. 12. (Italics are not in the original.) 152

The data and conclusion were itemized and the figure was drawn as is shown.

Data: A

(a) Triangle ABC.

(b) AB is equal to AC.

(c ) 3D is equal to DC.

(d) AF is equal to FB. AE is equal to EC.

Gonclusi on:

(x) Angle BDF is equal

Proof: (Proofs are omitted

One step in the proof for original exercise 7 consists of showing that BF = CE. Hence, the proof is possible if the datum "BF = CE" is substituted far the datum "AF s FB"and

AE = EC." Variant 7.1 results from this observation, which was pointed out by the teacher. The change is made first in the itemized form of the exercise; then, after a proof, the new verbal statement is formulated.

Variant 7.1

Data:

(a) Triangle ABC.

(b) AB is equal to AC.

(c) BD is equal to DC .

(d?) BF is equal to CE. B d C 153

Conclusion:

(x) Angle BDF Is equal to angle CDE.

Proof: (In class work, the proof Is placed at this point; In

this presentation, proofs will be omitted hereafter with­

out comment.)

Statement: The line segments drawn from the midpoint of the

base of an Isosceles triangle to points located on the

equal sides, equidistant from the ends of the base, make

equal angles with the base.

After variant 7.1 had been completed, the teacher asked,

"Is there another change we can make in the data without

fir;rearing to affect the conclusion?" A student suggested that

if the datum "E and F are equidistant from the ends of the

base" was replaced by the datum "E and F are equidistant from

the vertex," the conclusion would follow without change.

Variant 7.2 Is the result.

Variant 7.2

The line segments drawn from the midpoint of the base of

an Isosceles triangle to points located on the equal sides,

equidistant from the vertex, make equal angles with the base.

In response to the teacher*s question which elicited

variant 7 . 2 , another student suggested that the lines could

be drawn from two points on the base equidistant from the ends

of the base Instead of from the midpoint of the base. Although

only one of the possible variants is given as variant 7 . 3 , 154 others are possible and were developed with the class, P’or instance, with reference to the figure for variant 7 . 3 , one datum may be either that AF -> AE or that BF ■ CE. Another datum may be either that BQ * CR or that BR m C4i. Moreover, either the lines FG and ER or the lines EQ, and FR may be drawn.

Variant 7.5 A The lines drawn from points

on the base of an isosceles tri­ angle, equidistant from the ends

of the base, to points on the

equal sides, equidistant from the vertex, make equal angles B Q R C with the base.

All of the variants produced up to this point are of the

type in which changes are made initially in the data. All

of the changes were substitutions, and in no case was a change

in the data accompanied by a change In the conclusion. The

class became aware of these observations, and that apparently

other specific types of variation are possible. Other types

of variation, in which the initial change is made in the data, which are discussed in chapter 2, were identified as they were encountered at later times.

The opportunity of making changes in the conclusion was, however, brought to the attention of the class in connection 155 with this illustration. Returning to original exercise 7 , the teacher asked, "Now is there anything else that we can prove with the data we have? In other words, can we add any Items to the conclusion?" Someone said, "We can prove that DE Is equal to DP.” Addition of this conclusion results In variant

7.4.

Variant 7.4

Statement: The line segments

drawn from the midpoint of

the base of an Isosceles

triangle to the midpoints

of the equal sides are

equal and make equal B D C angles with the base.

It was noted that the conclusion added to original exer­ cise 7 to form variant 7 . 4 , may also be added to variants 7 . 1 , 7.2. and 7.3.

Referring to variant 7.4 again, one of the students said,

"DR appears to be equal to BP, and if that is true, then DE is also equal to CE," but was unable to devise a proof. The teacher informed him that a proposition that comes later can be used conveniently in the proof and encouraged everyone to keep this variant In mind for proof later. The proposition referred to is "A line connecting the midpoints of two sides of a triangle Is equal to one-half of the third side." The 156 suggested variant Is stated as variant 7,5.

Variant 7.5

The line segments drawn from the midpoint of the base of an isosceles triangle to the midpoints of the equal sides make oqual angles with the base, are equal to each other, and are each equal to one-half of one of the equal sides of the given triangle.

Variants 7.4 and 7.5 were obtained by adding to the con­ clusion. The added conclusion may also have been substituted for the original conclusion. Loreover, none of these changes

.■/ere accompanied by changes in the data. Other types of variation, initiated by making changes in the conclusion, discussed In chapter 3, were identified as opportunities for using them arose.

Numerous other variants may be added to this Illustra­ tion. For instance, the Isosceles triangle may be replaced by an Isosceles trapezoid. Perpendiculars may be drawn, either to the base at points on the base which are equidistant from the ends of the base, or to the equal sides at points on the equal sides which are equidistant from the vertex. If perpendiculars are drawn to one side of a triangle from points on the other two sides of the triangle which divide those sides In the same ratio, the perpendiculars are equal 157 even If the triangle is not isosceles. This case is shown in the figure on the preceding page.

Variation Is Not Encouraged in Textbooks

The process of variation is not presented or developed in textbooks on geometry. Moreover, there is good reason to believe that it is not recognized, for it is common to find separated in textbooks exercises which are related from the point of view of variation. For example, in the text contain­

ing original exercise 7 . the exercise “The line segments connecting the midpoints of the equal sides of an isosceles

triangle with the midpoint of the base are equal'1 is found

two pages after original exercise 7 , "Prove that the line segments drawn from the midpoint of the base of an isosceles

triangle to the midpoints of the equal sides make equal angles v/ith the base," and In a different set of exercises. The other variants are not found at all in the textbook from which the original exercise was taken, and they are not suggested in any way.

Another Illustration of Variation

Illustrations are presented in chapters 2 and 3 which concern the work on triangles. They are: illustration 10.

PP. 73*75 ; illustration 16. pp. Sl-82 ; and illustration 18.

PP* 93*9*+ • Qn® more illustration of variation will be given

In this section on triangles. It Is selected because the original exercise and some of the variants are commonly found

in geometry texts, though not always juxtaposed, while the 158 general statement achieved as the final variant is not found O in many texts,

ILLUSTRATION 8

Original Exercise 8

Corresponding angle bisectors of two congruent triangles are equal.

Variant 8.1

Corresponding medians of two congruent triangles are equal.

Variant 9.2

Corresponding altltudes of two congruent triangles are

equal.

(Proof of this variant depends, of course, upon a theorem

or corollary like, f,Two right triangles are congruent if the

hypotenuse and an acute angle of the one are equal to the hypotenuse and the corresponding acute angle of the other.” )

Variant 8.5

The lengths of vhe perpendicular bisectors of correspond­

ing sides of two congruent triangles are equal. (It is under­

stood tiiat by the phrase, length of a perpendicular bisector,

the segment of the perpendicular bisector which lies within

the triangle is meant.) (Figure on next page). 159 (Figuresfor variant 8.3) A

B Q C B' Q ' a

It is convenient to Introduce the notion of correspond­ ing points of congruent figures in connection with this illustration. Variant 8.5 may be considered as providing the clue, for both the concept and the definition. A definition nay be stated in the following manner: Corresponding points of congruent polygons are points which lie on corresponding sides of congruent polygons and are equidistant from corre- Q sponding vertices. Using the concept of corresponding ooints, corresponding line segments are defined as follows:

Corresponding line segments of congruent polygons are line segments which connect corresponding points in each of two congruent polygons. Varlait 8.4 is now stated.

The notion of corresponding points as presented here is included in Forder*s basic definition for congruence: "Two figures F, F* are congruent when the points of F can be put into one-to-one-correspondence with those of F* in such a way that, if A, B in P correspond to A 1, B 1 in F f , ... then the point couple (A, B) Is congruent to the point couple )» an<* we treat the congruence of point couples as a new undefined relation." Henry George Forder, The Foundations of Euclidean Geometry, pp. 87, 88.

This definition may be modified slightly so as to apply to similar figures. See p. 206. 160 Variant 8.4

Corresponding line segments of two congruent triangles are equal. A A'

I I

A general statement can also be expressed concerning angles formed by corresponding line segments of congruent riolygons. Such angles shall be called correlative angles of congruent polygons^ and are defined as follows: If polygon

P is congruent to polygon P1 , and if Qli and RS of polygon P correspond, respectively, to QfR' and R'S* of polygon P* , then the angles GiRS and ii’R ’S1 are called correlative angles of the congruent polygons P and P'.

Variant 8.5 may now be stated.

Variant 8.5

Correlative angles of two congruent triangles are equal.

Variants 8.4 and 8.5 may be extended to congruent polygons in general. The result of doing so is stated in variant 8 . 6 .

The word correlative is introduced because there does not seem to be a word in common usage which describes angles formed by corresponding line segments of congruent polygons. Applying the word corresponding to angles of this type is unsatisfactory because corresponding angles has the established meaning of angles formed by corresponding sides. 161 Variant 8.6

Corresponding line segments and correlative angles of

congruent polygons are equal.

Many ramifications of variation were explored in the

writer's class during the early work of the course which are

not recorded in this presentation. Gradually the students

came to use the process with facility, often without prompting.

Although new aspects of it remained to be discovered, the

basic procedure had been adopted by the time the next section

of work, on parallel and perpendicular lines, was taken up. 162

5,3 Parallel and Perpendicular Lines

One of the first statements which is either proved or adopted as an assumption in the work on parallel lines is a criterion for determining when two lines are parallel. In many textbooks this statement is original exercise 9. Often one or more of the variants presented in illustration 9 are proposed as corollaries.

ILLUSTRATION 9

Original Exercise 9

If two lines form equal alternate interior angles with a transversal, the lines are parallel.

Variant 9.1

If two lines form equal corresponding angles with a transversal, the lines are parallel.

Variant 9.2

If two lines form equal alternate exterior angles with a transversal, the lines are parallel.

Variant 9.5

If two lines form supplementary interior angles on the same side of a transversal, the lines are parallel.

Variant 9.4

If two lines form supplementary exterior angles on the same side of a transversal, the lines are parallel.

The set of statements which are converses of the statements 163 in Illustration 9 may also be considered as variants of each other. The one given In many textbooks as a theorem Is a converse of original exercise 9 . It Is "If two parallels are cut by a transversal, pairs of alternate Interior angles are equal." Variants may then be obtained by making substitutions like those in Illustration 9 .

Pupils Should Be Led to Discover Many Statements in

Geometry By Using the Process of Variation

Illustration 9 and the set of statements alluded to in t;re paragraph which follows it are groupings of exercises and theorems with which it is assumed every geometry teacher is familiar. Several illustrations In this chapter consist solely of statements found either as theorems or as corollaries

In practically all geometry textbooks and there are other illustrations which are not mentioned. In these cases the writer suggests that after the students have be on led to the discovery of the original exercise or have had it presented to them directly, they may be expected, with whatever guidance is found necessary, to discover some or all of the variants themselves•

On the other hand, in some of the illustrations Interest­ ing results are achieved, which, although no doubt well known, are not commonly found in textbooks. Such discoveries par­ ticularly increase the interest of both pupil and teacher. Variation Sometimes Produced Unexpected Results

Frequently, with one or more particular variants in mind

the teacher emphasized the possibility of variations when

assigning an exercise found in the textbook. The variety of

variants produced by the students was sometimes surprising.

Illustration 10 is an example.

ILLUSTRATION 10

Original Exercise 10

"A line intersecting one

of the equal sides of an

i303cales triangle and paral­ A

lel to the other forms a tri­ angle having two equal angles.'*1"0

In Assigning this exercise

the teacher added, "Formulate, JD ^ either verbally or In terms of a figure and symbols, any variants which you can make and which seem to you to be true. Give as many proofs as you can." All of the variants which follow were presented by one or more students. The teacher had thought of only variant 10.1 when he made the suggestion to the class.

^0Welchons and Krickenberger, ojo. clt. p. lol, exercise (The italics have been added.) 165

Variant 10.1

A line which Intersects both of the equal aides of an isosceles triangle and IB paral­ lel to the remaining side forms, with the two sides it intersects, a triangle having c two equal angles.

Variant 10.2

If two lines are paral­ lel to the equal sides of an isosceles triangle, they form, with a segment of the third side, a triangle having two B Q d c equal angles.

Variant 10.5 A line which intersects H one of the equal sides of an isosceles triangle and is parallel to the other, forms with the side it intersects and a line through the vertex parallel to the base a triangle B D C having two equal angles. 166

Of course in all cases the triangles formed have equal sides as well as equal angles. Some of the students mentioned substituting equal sides for equal angles in each of the con­ clusions. While such a substitution may be viewed In terms of variation, the result can also be obtained by deduction by use of the theorem "If a triangle has two equal angles It is isosceles•"

Substitution of Parallel for Perpendicular, and Vice Versa

Soon after the study of parallel and perpendicular lines was begun, it was noticed that some of the statements concern­ ing parallel lines have corresponding variants Involving per­ pendicular lines, and vice versa. Illustrations of such occurrences are mentioned in the next three paragraphs.

Attention was first drawn to the substitution of perpen­ dicular for parallel. This happened In connection with the following corollaries: 1) "Two lines parallel to a third line are parallel to each o t h e r . " ^ 2) "Two lines perpen­ dicular to a third line are parallel.

A corollary, "Lines perpendicular to Intersecting lines will Intersec t, was assigned for proof by the indirect method. On the following day, although no suggestion as to a

^ Ibid. , p. 95. (The entire statement is italicized In the original.)

^ Ibid. , p. 97. (The entire statement is italicized in the original.)

1 ‘ ’ . 99. (The entire statement is italicized in the 167 possible variant had been made by the teacher, a member of the class made the conjecture "Lines parallel to Intersecting lines will intersect," and a day later he submitted a proof.

The latter statement is not found in the text that was being used. However, it is needed for a rigorous proof in some exercises, variant 1 0 . 2 , for example.

The following two statements also illustrate substitution of perpendicular for parallel: 1) "If two angles have their sides parallel. right side to right side and left side to left side, they are equal." 2) "If two angles have their sides perpendicular, right side to right side and left side to loft side, they are equal." A detailed discussion of these stateinents from the standpoint of variation follows.

Two Angles with Their Sides Parallel or Perpendicular

Theorems about angles with their sides parallel or per­ pendicular are found in nearly all, if not all, plane geometry textbooks In use at the present time. This is, however, a rather recent development. In his section on parallel lines

Heath reports:

De Morgan adds that a proposition is much wanted about parallels (or perpendiculars) to two straight lines respectively making the same angles with one another as the latter do. The propo­ sition may be enunciated thus:

If the sides of one angle be respectively (1) sides either equal or supplementary

-*-4IIeath, o£. cit.. Vol. I, p. 306 168

Two remarks may be made about the statement proposed by neath: 1) The precise function of the word "respectively” is not evident, 2) A statement more desirable than the one stated would specify conditions under which the angles are equal and conditions under which they are supplementary.

Two ways, typical of how De forgan*s suggestion is put to practice in current textbooks, are the following:

1) Trump uses two theorems, nine pages apart. They are,

"If two angles have their sides respectively parallel, they are either equal or supplementary,"^ and, "If two angles lying in the same plane have their sides respectively perpen- 1 f iicular, the angles are either equal or supplementary•"

2) V/elchons and Krickenberger present four statements.

The proposition "If two angles have their sides parallel. right side to right side and left side to left side, the angles are equal" Is followed by an exercise, " If two angles have their sides parallel, right side to left side and left side to right side, the angles are supplementary. 7 hour pages later, the proposition "If two angles have their sides perpendicular, right side to right side and left side to left side, the angles are equal" Is followed by an exercise, "If two angles have their sides perpendicular, right side to left — Trump, cl t . , p. 162.

loIbid.. p. 171.

17 Welchons and Krickenberger, oja. clt . . p. 101. side and left aide to right side, the angles are supplementary .

In the writer's class, theorems concerning angles with sides parallel or perpendicular were developed as the result of variations of a single exercise. (Strangely enough, this exercise Is located In the text that was used by the class after the statements Involving angles with their sides parallel and preceding the statements concerning angles with cheir sides perpendicular.) The extent of the teacher's guidance, the discoveries made by the students, and the final results obtained by varying this one exercise are reported in detail In the illustration which follows.

ILLUSTRATION 11

Original Exercise 11

’• If from any point outslde an acute angle perpendlculars are drav/n to the sides of the angle, the angle between these perpendiculars is equal to the given angle.

In assigning this exercise the teacher reproduced it from the class text on one of the duplicated sheets distri­ buted each day, underlining certain words as shown. The following suggestion was added: f,This exercise appears to be

18Ibid., p. 105.

^9Ibld.. p. 104, Ex. 2, GroupC. (No italics In original.) 170 rich in opportunities for variation. You will be able to discover m a n y variants. Hints for making variations of the given exercise are suggested by underlining certain key words. Try to prove each variant that you form which seems to b e true." Nearly all of the variants which follow were produced by students In the class; exceptions are Indicated.

Variant 11.1

If from any point inside an acute angle perpendiculars are drawn to the sides of the angle, the angle between these perpendiculars is supplementary to the given angle.

As soon as variant 11.1 had been completed, one member N of the class asked, "What happens if point P is on a side of the given angle?" By means of a figure like the one R A at the right, this situation was investigated. There was confusion, not present in original exercise 11 or In variant

1 1 . 1 . as to whether angle RPS or angle KPS' should be the one considered. At this point another member of the class observed that the same sort of ambiguity arises If the point from which 171 the perpendiculars are drawn is the vertex of the given angle.

The teacher then called atten- tion to the situation por­ c trayed in the accompanying figure, in which the given point is ’’outside" the given A angle but the angle between the perpendiculars is the supple­ ment of the given angle. After seeing this figure, members of the class wondered whether there might be other locations outside the given angle which

3houid be investigated. Perhaps, they mused, it makes a difference if one of the perpendiculars intersects on a side of the given angle, while the other perpendicular intersects an extension of the remaining side of the given angle. Or oerhaps an unusual situation results if one of the perpendi­ culars passes through the vertex of the given angle and if the other perpendicular intersects either an original side of the angle or one of the sides extended through the vertex.

All of these cases, which may be considered variants of ori­ ginal exercise 11, were investigated. In each case it was proved that the angle between the perpendiculars is either equal to or supplementary to the given angle.

Locations of the given point P relative to the given angle ABC which had been investigated with the given angle an acute angle were also investigated with the given angle a 172 right suable and with the given angle an obtuse angle. In every case It was proved that the angle between the perpendi­ culars Is likewise either equal to or supplementary to the bivan angle.

Pointing out the desirability of having statements specifying precisely when the given angle and the angle between perpendiculars drawn to its sides from a given point are equal and when these angles are supplementary, the teacher reempha­ sized the notion of right and left sides of angles. The notion

is the following: "Viewed from the vertex, the 3ide of an angle at the right is called the right side of the angle and

she side at the left is called the left side of the angle."^^

It thus became possible to combine into two statements original exercise 11 and all of the variants which had been

investigated. These are named variants 11.2 and 11.3; for

the sake of convenience, changes in the wording have been made in addition to those resulting directly from the use of

variation., , , -it

Variant 11.2

If two angles have their sides perpendicular, right side to right side and left side to left side, the angles are equal.

Ibid.. p. 100.

**If changes in wording Indicated only variation of ori­ ginal exercise 11. variant 11.3. for example, would be stated as follows: If from any point in the plane of an angle less than 180° perpendiculars are drawn to the sides of the angle, the angle between these perpendiculars is equal to the given angle, provided the two angles have their sides perpendicular right side to right side and left side to left; side. 173 Variant 11.3

If two angles have their sides perpendicular, right side to left side and left side to right side, the angles are supplementary.

It must be definitely understood that only angles less than 180° are meant in variants 11.2 and 11.3. It Is possible to interpret these statements satisfactorily If eitlier or both of the angles concerned are 0° or 180°; but both of the statements are emphatically false if one of the angles is greater than 100°. This is shown In the figures below,

hght and left sides of the angles concerned are labeled (r) and (1), respectively.

C D A (r)

Precise and correct statements may of course be stated in case one or both of the given angles Is greater than 180°.

'They are, however, omitted from this report.

Thus far in this illustration the changes made have In­ volved only the position of the vertex of one of the angles relative to the vertex and the sides of the other angle. It is possible also to vary original exercise 11 by substituting 174 the word parallel for the word perpendicular. This may be done In all of the possible variants of original exercise 11 tnat have thus far been indicated. >*hen considering angles with their sides parallel, in order to include the cases in which the vertices coincide and the cases in which the vertex of one angle lies on one side of the other, it must be under­ stood that lines which coincide are considered as being 21 parallel. With this understanding, variants 11.4 and 11.5 are stated. They may conveniently be considered as variants of variants 11.2 and 11.3, respectively, obtained by substitut- in ; tho word parallei for the word perpendicular.

Variant 11.4

II' two angles have their sides parallel, right side to right side and left side to left side, the angles are equal.

Variant 11.5

If two angles have their sides parallel, right side to left side and left side to right side, the angles are supple­ mentary .

^Itfith his definition of parallel lines, "Two lines a.,t> in the same plane which do not meet are 'parallel1'1, border includes the statement "A line is also considered to b e paral­ lel to itself." 0£. clt., p. 138. The possibility of coincident lines being considered parallel is allowed in a definition for parallel lines such as the one given by Myron ^ . Rosskopf, Harold D. Aten, and William D. Reeve, Mathematics -- A Second Course, p. 104, "If two lines AB and CD In the same plane have the same amount of turning and the same direction from a given line EF .... the lines AB and CD are said to be parallel." 175

As in variants 11.2 and 11.3, It Is understood that only angles less than 180° are meant In variants 11.4 and 11.5.

Further interesting variants of orlr.Inal exercls e 11 are obtained if one of the lines from the given point Is drawn

- er-endicular to one side of the given angle while the other line is drawn parallel to the other side of the angle. One of the possible variants is stated as variant 11.6.

Variant 11.6

If from any point Inside an acute angle two lines are drawn, one perpendicular to one side of the given angle and the other parallel to the other side of the given angle, the angle between these lines is c oraplamentary to the given angle.

Application of Variant 11.6

Variant 11.6, in addition to being novel and interesting,

1 s not without application. An application in the field of

Physics is shown in the accompany ing figure. The angle between the vector denoting the weight of a load on an inclined plane and the vector denoting the com­ ponent of force parallel to the plane is the complement of the angle which the plane makes with the horizontal.

Use of the process of variation in connection with 176 parallel and perpendicular lines, especially the impressive results obtained in illustration 11. captured the interest of the students. Further results of utilizing the process deliberately are reported in the following section, which deals with the study of polygons. 177 5. *+ Polygons.

That some theorems concerning polygons may be discovered as variants of related theorems concerning triangles is in­ dicated in preceding sections of this chapter. For example, statements concerning the sum of the interior angles of a polygon and the sum of the exterior angles of a polygon are presented (pp. 13*f - 136 ) as variants of similar statements concerning triangles. A general statement concerning the congruence of polygons is achieved (p. I5u ) as a variant of a statement concerning the congruence of triangles.

In geometry courses generally, triangles and quadri­ laterals are studied more intensively than are other types of polygons. This was done also in the writer*s class. A separate section was not devoted to quadrilaterals, however, as was done in the case of triangles. Quadrilaterals were studied as one kind of polygon in the section of work dealing with polygons in general. However, quadrilaterals received more emphasis than polygons having more than four sides.

This is reflected in the illustrations of the use of vari­ ation given in this section.

Most geometry texts contain an exercise like the follow­ ing: "The midpoints of the sides of a quadrilateral, taken in order, are the vertices of a parallelogram." If the given quadrilateral is a particular type of quadrilateral, the figure obtained by connecting the midpoints of its sides is, 173 in most cases, a particular type of parallelogram. The process of variation is very effective in making a detailed study of the results obtained by connecting the midpoints of the sides of particular types of quadrilaterals.

Before illustrating how this may be done, it is neces­ sary to state definitions of particular types of quadri­ laterals. In the writerfs class the following definitions were used:

A Quadrilateral is a polygon having four sides.

A convex quadrilateral is one in which each of the interior angles is less than 180°.

A concave quadrilateral is one in which One interior angle is greater than 180°.

a grassed .Quadrilateral is one in which two of the sides intersect without being extended. An example of a crossed quadri­ lateral is shown in the accompany­ ing figure.

(As a convention, convex quadrilateral is meant when the word "quadrilateral" is used without a qualifying ad­ jective) .

A trapezold is a quadrilateral with one, and only one, pair of opposite sides parallel.

An Isosceles trapezoid is a trapezoid in which the non­ parallel sides are equal.

A parallelngram is a quadrilateral in which both pairs of opposite sides are parallel.

A rectangle is a parallelogram with one right angle.

A rhombus is a parallelogram with two adjacent sides equal. 179

A fffliiA-ro is a rectangle with two adjacent sides equal.*

Figures obtained by connecting in order the midpoints of the sides of quadrilaterals are presented in illustration

12, viiich follows. Some of the variants are used to illus­ trate particular types of variation in earlier chapters.

In illustration 12 they are presented in a systematic manner.

ILLUSTRATION 12

Original Exercise 12

The midpoints of the sides of a square, taken in order, are the ver­ tices of a square.»*

Although the statement used as a point of departure was selected by the teacher, the students were guided to discover it. After discovering the original statement, they were expected to vary it, thereby obtaining new state­ ments. The teacher's written suggestions distributed to the class are the followings

a) Draw any square. Locate the midpoints of the sides of the square. Connect these midpoints In order. What sort of figure is formed? Complete the statement: "The midpoints of the sides of a square, taken In order, are the vertices of a ." Prove the statement.

*A square may. of course, also be defined as a rhombus with one right angle.

Discussion of the reason for selecting this state­ ment as original exercise 1 2 . instead of one of the vari­ ants which follow, is postponed until some of the variants are presented. The discussion appears on pp. 182-1 8 3 . 180

b) Look for variants of the above statement. There are several stages. The key words are square and the word you supplied. Try replacing the word square with the names of other types of quadrilaterals. Find out what sort of figure is obtained by con­ necting the midpoints of the sides. Prove as many of the resulting statements as you can.

Practically all of the class completed the statement suggested in a), which has been designated original exer­ cise 12. Each of the variants which follows was produced by one or more members of the class as a result of sugges­

tion b). Of course all of them were shared with the entire class.

YfrElaiU The midpoints of the sides of a rhombus, taken in order, arc the vertices of a rectangle.

The midpoints of the sides of a rectangle, taken in order, are

the vertices of a rhombus.

Variant The midpoints of the sides of a parallelogram, taken in order, are

the vertices of a parallelogram. 181

Variant 12A

The midpoints of the sides of an isosceles tranezold. taken in order, are the vertices of a rhombus.

Variant 12^5

The midpoints of the sides of a trapezoid (which is not isosceles), taken in order, are the vertices of a parallelogram (which is not a rhombus.)

Variant 12.6

The midpoints of the sides of a convex quadrilateral, taken in order, are the vertices of a par­ allelogram. *

Variant ig,7

The midpoints of the sides of a concave quadrilateral, taken in order, are the vertices of a parallelogram.

* This statement is also true for a "skew" quadrilateral, that is, if A, B, C, and D are not coplanar. 182

Variant 12.8

The midpoints of the sides of a crossed quadrilateral, taken in order, are the vertices of a parallelogram.#

Which Statement in a Set Related by the

Process of Variation Shall 3e the Original Exercise.?.

The question may be asked, T*Why was not variant 12.6 or variant 12.3 or perhaps any of the other variants in

Illustration 12 chosen as the original exercise rather toan the statement in which the given quadrilateral is a square?” Several reasons may be given in reply. 1) Stud­ ents are likely to be more familiar with the square and

Its properties than with any of the other types of quadri­ laterals, with the possible exception of the rectangle.

2) It seems to the writer that original exercise 12 is more likely to be discovered, in the absence of a teacher*s guidance, than any of the subsequent variants. It might occur to an observer simply as the result of looking at a

Under certain conditions, for instance If the sides of a crossed quadrilateral are equal and parallel, the mid­ points of the sides are collinear. However, with the under­ standing that coincident line segments may be considered to be parallel, the figure obtained by connecting the midpoints of such a crossed quadrilateral may be viewed as a "degen­ erate” parallelogram. The consecutive vertices of the crossed quadrilateral are A, B, C, D, and the consecutive sides are AB, 3C, CD, DA. Consecutive midpoints of the sides, taken in ordey, are E, F, G, H). 133 design such as the one in the

accompanying figure, which cer­

tainly is more common than de­

signs that might suggest any of

the other statements. However,

in independent investigation, the situation, whatever it

might he, in which a particular relationship is first dis­

covered is the one from which further investigation should

proceed. 3) The two types of variation utilized in the

process of developing the statements of illustration 12.

if the given quadrilateral in the original exercise is a

square, are deletion of data and substitution of data,

hence, the investigation proceeds in the direction of

greater generality of the given figures. While from tne

standpoint of mathematical elegance it is often desirable

to present the more general situation first, it seems to

toe writer that from the standpoint of learning and dis­ covery it is desirable to begin at the point of greatest

familiarity.

Variants Obtainable bv Connecting the Midpoints ■all.the Sides of Quadrilaterals Other Than Those Previously Considered

Special types of quadrilaterals other than those al­ ready mentioned may be investigated with respect to the fig­

ures obtained by connecting the midpoints of their sides. 18*+

Some of them are quadrilaterals with one right angle, quad­ rilaterals with two right angles, quadrilaterals with two sides equal, quadrilaterals with three sides equal, quadri­ laterals with two pairs of adjacent sides equal, inscript- ible quadrilaterals, and quadrilaterals possessing combin­ ations of the aforementioned properties. However, once diagonals are drawn in a given quadrilateral it is noted that the figure obtained by joining the midpoints of the sides of the given quadrilateral is determined by relations between the diagonals of the given quadrilateral. Variants

12U, 12.C10). and 12. (11) then follow immediately.

Variant 12.9

The midpoints of the sides of a quadrilateral with equal diagonals, taken in order, are the vertices of a rhombus.

Variant 12.110)

The midpoints of the sides of a quadrilateral wi th perpendicular diagonals, taken in order, are the vertices of a rectangle.

The midpoints of the sides of a quadrilateral with diagonals equal and perpendicular, taken in order, are the vertices of a square.

All three of the above variants are true for convex 135 and concave quadrilaterals provided the meaning of "sides and diagonals of anv quadrilateral" is the following! If

A, 3, C, and D are consecutive vertices of a quadrilateral, then AB, BC, CD, and DA are the sides, and AC and BD are the diagonals of the quadrilateral. Only variant 12.9 has meaning in respect to a crossed quadrilateral, for It Is impossible to have a crossed quadrilateral with perpendicu­ lar diagonals.

Variants Obtainable _by Connecting Points on the Sides of

Quadrilaterals Other Than Midpoints

In figures 1 and 2 below the trisection points of t n e sides of a square are shown connected in two different ways. A

/ \ i /

/ F i g u r e .2 : 'j s / _____u —

I f trisection points of the sides of a quadrilateral are connected in a manner similar to that shown in figure 1. the resulting figure is, in general, not a parallelogram unless the given figure Is a parallelogram. However, if trisection points of the sides of anv quadrilateral are connected in the manner shown for a square In figure 2. the resulting figure Is always a parallelogram. Moreover, the remarks made concerning trisection points may be ex­ tended to points which Internally divide the sides of quad­ rilaterals in any given ratio. Variant 12.(12). which 186 follows, is the only one of the several possible variants that have been indicated which is stated.

The points E, F, G, and H ufalgft divide the sides AB, AD, CB, and CD, respectively, of any quadrilateral

A BCD (convex, concave, crossed or shew) in a given ratio are the ver­ tices of a parallelogram.*

Further variation involves assigning one ratio to one pair of opposite sides of a given quadrilateral and another ratio to the other pair. however, the figure ob­ tained by connecting the points of division in the particu­

lar manner stated in variant 12.(IB), which follows, is not

in general, a parallelogram unless the given quadrilateral

is a parallelogram. Variant, 12..CUJ, The points E,F,G, and H on the

sides of any parallelogram A3CD, such

that E and G divide AB and CD in a

given ratio r i . and such that F and H divide BC and DA in a given ratio are the vertices of a parallelogram.

*The ratios ares Ad, and EB HD FB GD 137 Variants Obtainable by Connecting the Midpoints of the Sides of Polygons Other Than Quadrilaterals

Results of connecting the midpoints of the sides of pentagons were investigated briefly. No interesting results were obtained except for regular pentagons. Attention was therefore focused upon regular polygons, beginning with considerations involving triangles and quadrilaterals.

It is of course well known that the midpoints of the sides of any triangle are the vertices of a triangle which is similar to the given triangle; hence, in particular, the midpoints of the sides of an equilateral triangle, which may also be called a "regular triangle", are the ver­ tices of another equilateral triangle. It Is not true, however, that the midpoints of the sides of any a uadri- lateral are the vertices of a quadrilateral which is simi­ lar to the given quadrilateral. But If the given quadri­ lateral is a square, that is, a regular quadrilateral, the midpoints of its sides are the vertices of another square.

In view of these observations it seemed worthwhile to the class to investigate whether or not the polygon obtained by connecting the midpoints of the sides of a given regular polygon is always another regular polygon. It was discovered that if the midpoints of the sides of a given regular poly­ gon are connected in order, a regular polygon is obtained naving the same number of sides as the given polygon. It 188 was discovered also that if the number of sides of the given polygon is greater than four and if the given poly­ gon has an even number of sides, a regular polygon having half as many sides as the given polygon is obtained if the midpoints of alternate sides of the given polygon are joined. This notion was then extended to polygons for which the numbers of sides are divisible by numbers greater than two. If midpoints of alternate sides of a regular polygon having an odd number of sides are connected incrder until the starting point is reached, the figures obtained are not regular polygons, but are "star-shaped" figures. This notion may also be extended with interesting results by connecting the midpoints of every third side of any given polygon, and so on. For example, if the midpoints of every third side of a regular ten-sided polygon are connected in order until the starting point is reached, a ten-pointed star is obtained. But if the midpoints of every fourth side are connected in this manner, a five-pointed star is obtained.

A large number of significant results, most of them easily understood and proved, have been indicated in illus­ tration 12. All of them were obtained by the process of variation with one specific statement as the starting point, however, use of the process of variation did not always 189 lead to a large number of variants of a particular state­ ment. An example is provided by Illustration 18. which follows.

ILLUSTRATION H .

Original exercise 18 and 1,^,1 were proposed by a member of the class without any suggestion on the part of the teacher.

Qrifiiflal . U The area of an equilateral triangle

Is four times the area of the equilateral triangle obtained by connecting the mid­ points of the sides of the given tri­ angle.

Variant 12.1

The area of a square is twice the area of the sai^a/re obtained by con­ necting the midpoints of the sides of the given square.

Original exercise 1^ and variant 1^.1 are proved easily by means of congruent triangles. The student who proposed them thought that the ratio of the area of a given regular polygon to the area of the polygon obtained by connecting in order the midpoints of the sides of the given polygon might be a "simple" number for all or some of the polygons 190 having more than four sides as well as for those having four or three sides. It can be shown by the methods of elementary geometry and algebra that the ratio of the area of a regular hexagon to the area of the hexagon obtained

by connecting in order the midpoints of the sides of the given hexagon is b/$. Elementary methods fail in attempts to investigate similar area ratios for regular polygons other than triangles, quadrilaterals, and hexagons. It was observed, however, that for regular polygons of three, four, and six sides the ratios obtained by comparing the areas of the original polygon witn the polygon whose ver­ tices are the midpoints of the sides of the given polygon are, respectively, b/1, k/2y and *+/3. The next area ratio

In this sequence likely to be thought of is h/b or 1/1;

this ratio is approached as the number of the sides of the given polygon becomes very large.

Neither the student who had so hopefully proposed the first two exercises of illustration 1^ nor the other mem­

bers of the class were discouraged by the disappointment

that was experienced in this case, for similar disappoint­ ments were encountered frequently. An important lesson to be learned concerning any sort of original investigation is

that the number of spectacular successes are few compared to the number of inquiries that are initiated.

Illustration l*t. in contrast to iUjlgtaafriPfl 13, is 191 again an example in which a great number of statements can be obtained by varying one given exercise.

ILLUSTRATION Hi

A great many geometry textbooks contain either or both of tne following exercises, which are labeled original exer­ cise l*f and variant !**.!.

Original Exercise l b

The bisectors of the interior angles of a rectangle intersect in four points which are the ver­ tices of a square.

Variant 1^.1

The bisectors of the interior angles of a parallelogram intersect in four points which are the ver­ tices of a rectangle.

It may be mentioned also that the bisectors of the interior angles of a square and a rhombus are concurrent.

The bisectors of the interior angles of a ’‘kite-shaped1* quadrilateral, which has two pairs of adjacent sides equal, are likewise concurrent.

Moreover, the bisectors of even a general quadrilateral, either convex or concave, have an interesting property. This 192 property can be discovered and proved by a bright and persistent student of elementary geometry. In discover­ ing this property an important lint may be obtained by first studying the bisectors of the in­ terior angles of a special quadri­ lateral such as an Isosceles trapezoid. A figure like the one at the right is drawn. It is noticed, first of all, that the bisectors of the interior angles intersect in six points. However, the bisectors of pairs of consecutive angles of the given figure intersect in four points. These four points are the vertices of a quadrilateral with two pairs of adjacent sides equal; the angles between non-adjacent sides are right angles. Vari­ ant 1U-. 2 is a statement of these observations.

The bisectors of pairs of consecutive interior angles of an isosceles trapezoid intersect in four points which are the vertices of a quadrilateral in which two pairs of adjacent sides are equal and in which the non-eaual sides are perpendicular. Other special types of quadrilaterals may also be in­ vestigated, such as quadrilaterals with one right angle, quadrilaterals with two sides equal, quadrilaterals with 193 three sides equal, quadrilaterals possessing combinations of these properties, inscriptible quadrilaterals, and of course trapezoids. For the purpose of this presentation, however, the general convex quadrilateral may be considered next.

Study of the figure at the right reveals that the figure formed by the bisec­ tors of the interior angles of a general convex quadri­ lateral has none of the spe­ cific properties noted in pre­ ceding cases. However, it is true and probably more easily noticed than in the general figure that in all of the preceding cases tne quadrilateral formed by the bisectors is inscriptible in a circle. This suggests the conjecture that the quadrilateral obtained in the general case is also inscriptible. That this is true can be proved quite easily. Moreover, an inscriptible quadrilateral is also formed by the bisectors of the in­ terior angles of a concave quadrilateral. The results dis­ covered are stated as variant l*f.^.

Variant 1*+. ^ The bisectors of successive interior angles of any con­ 19^ vex or concave quadrilateral, taken in pairs, intersect in four points which are the vertices of an lnscrintible quad­ rilateral.

The writer found only one textbook in elementary geometry which gives an exercise similar to variant 1^.1. Chauvenel presents the exercise: ’’The four bisectors of the angles of a quadrilateral form a second quadrilateral, the opposite angles of which are supplementary." He adds the comment, "If the first quadrilateral is a parallelogram, t-ie second is a rectangle. If the first is a rectangle, the second is a square."^ Consideration of the bisectors of the exterior angles of various types of quadrilaterals results in other variants associated with 1 Thistration lW. For instance, the bisectors of the exterior angles of a square form another square. The bisectors of successive exterior angles of any convex or concave quadrilateral, taken in pairs, intersect in four points which are the vertices of an inscriptible quadri­ lateral.

Other illustrations of tne use of variation in connec­ tion with the study of polygons can be found, which are not mentioned in this paper. Opportunities for the use of variation in connection witn the study of circles are considered next. 195

5.5 Circles Two illustrations will be given in connection with the study of circles. Both include familiar theorems which are usually grouped together in textbooks.

ILLUSTRATION 15.

The statements In this Illustration concern the measurement of angles in terms of arcs intercepted on a circle. The writer prefers to develop the first four state­ ments in the order in wtlch they are presented, leaving to the students the discovery of the remaining statements and ience also the order in which the statements are discovered.

Original Exercise 15. An angle formed by two diameters of a circle is equal in degrees to one-half the sum of the arcs Intercepited^by rLand its vertical angle. This statement follows Immediately from the statement, usually presented as a postulate, that "A central angle of a circle is equal In degrees to its intercepted arc.11 Original exercise 15 is enunciated in order to provide a convenient point of departure for the variants that follow.

gariant-15.1 An angle formed bv a diameter and any other chord in­ tersecting within a circle is equal in degrees to one-half the sum of the arcs intercepted by it and Its vertical angle. 196 Variant 15.1 can be proved without use of any of the

statements which follow.* However, proof 6f it and also

of variant 15.2 may be delayed until after the proof has

been given for variant 15. 3. which in most geometry texts

is the first statement that is proved concerning the meas­

urement of angles in terms of their intercepted arcs.

Variant 15.2

An angle formed by anv two chords intersecting within

a circle is equal in degrees to one-half the sum of the

arcs intercepted by it and its vertical angle.

Variant 15.2 can be proved in the following manners

(a) Circle 0. (b) Diameter AB. (c) Cnord CD. (d) A3 and CD intersect at P. Conclusions

(x) Angle x 3 £ (arc AC 4 arc B D )

Proof:

(1) Draw radii 0C and 0D. (2) Angle x « angle BOD - angle CDO (3) Angle x ■ angle AOC 4 angle OCD (*+) Angle OCD " angle CDO (5) 2(angle x) * angle BOD 4 angle AOC (6) Angle x ■ £(angle BOD 4 angle AOC) (7) Angle BOD g arc BDJ angle AOC d arc AC (8) Therefore, angle x 2 £(arc BD 4 arc AC).

Notes The symbol £ is read "is equal in degrees to." 197

An angle formed by a diameter and a chord intersecting

the circumference of a circle is equal in degrees to one-

half the intercepted arc.

Variant 15.*+

An angle formed by anv two chords intersecting on the

circumference of a circle is equal in degrees to one-half

the intercepted arc.

Variant 15.5

An angle formed by a tangent and a chord intersecting on tne circumference of a circle is equal in degrees to one-

half the intercepted arc.

Positions of the vertex of the given angle inside the circle and on its circumference having been investigated, it seems appropriate also to consider cases in which the vertex of the angle is outside the circle. The convention of using the word secant to denote a chord extended outside a circle is observed in the variants that follow.

Variant 15.6

Aii angle formed by two secants intersecting outside a circle is equal in degrees to one-half the difference of the intercepted arcs.

In all of the statements preceding variant 15.6. the 198 angle is equal in degrees either to one intercepted arc or to the sum of the two intercepted arcs. In variant 15.6 the size of the angle is expressed in terms of the difference of two intercepted arcs. Of course it is possible to associ­ ate algebraic signs with the angle and the arcs in such a manner that the size of the angle between two secants may­ be expressed in terms of the algebraic sum of the inter­ cepted arcs; this procedure will be discussed after the presentation of two more variants.

Assuming, however, that the convention regarding alge­ braic signs is not introduced and recognizing that the students are not likely- to do so without suggestion by the teacher, the question may fairly be asked, "How can the con­ clusion for variant 15.6 be discovered?" One way of dis­ covering a tentative conclusion is to draw several figures in accordance with the data, measure the angles between the secants and the arcs intercepted by the secants, tabulate the data, and then search for a relation which appears to exist in every case.

Another way of discovering the conclusion in variant

15.6 is by drawing a figure labeled like a corresponding figure for variant 15.^. which concerns the angle formed by two chords intersecting within a circle, and then attempt­ ing to carry through a proof which is analagous, step by step, to the proof used for variant 15.2. The method is indicated 199 by the figures below. In figure 1?.2. angle 1 ■ angle 2

+ angle 3; in figure 15.6. angle 1 - angle 2 - angle 3.

Angles 2 and 3* in both cases, are equal in degrees to one-half of arcs AC and BD, respectively.

D B /j

Fl&urfLJ-5*2 Figure 19.6 3y replacing one secant in variant 15.6 by a tangent, variant 15.7 is obtained. Variant 15.3 is obtained by sub­ stituting tangents for both secants.

Variant 19.7

An angle formed by a tangent and a secant intersecting outside a circle is equal in degrees to one-half the differ­ ence of the intercepted arcs.

Variant 15.8

An angle formed by two tangents which intersect is equal in degrees to one-half the difference of the inter­ cepted arcs. 2 0 0

References Concerning Illustration 15

In the Literature

The statements included in illustration 15. as the reader knows, are not found in most textbooks in even approx­ imately the order here presented. Wentworth, however, gives a series of figures suggesting the statements in illustra­ tion 15 and in very nearly the same order, but he does this after the theorems and their proofs have been presented in the usual sequence.^ In his book concerning teaching pro­ cedures Schultze presents a series of diagrams waiea with hut minor differences suggest the sequence of statements in illustration 15.

A report of The School Committee of the City of Boston presents the statements concerning the measurement of angles in terms of intercepted arcs in such an order that variation could be used effectively in developing them. In that re­ port but without mention of variation the following proposi­ tions and exercises are presented consecutively:

Proposition: "The angle between a tangent

and a chord is equal to the— gga.tr al angle inter- g.qaUftfi W-f. fti&•" (A novel proof is presented).

23 Wentworth, op. cit.. p. 107.

2U Arthur Schultze, The Teaching of Mathematics. In Secondary Schools, p. 185. 2 0 1

Proposition. "An Inscribed angle is

equal to the central angle standing on half the

same arc.

Exercise. "An angle formed bv two tan­

gents that meet Is the supplement of the central

angle on the same arc."

Exercise. "Find and state a relation be­

tween the number of degrees in an angle and in

the Intercepted arcs when two lines wnich cut

a circle intersect (a) within the circle; (b)

on the circumference; (c) without the circle.’'2^

By adopting proper conventions, all of the statements i-a illustration 15 may be summarized in one statement. The following statement is typical of those found in textbooks w u c h include such a summary statement: "lkie angle formed jy two intersecting lines cuttlng^or. .touching a circle Is measured by half the ALGEBRAIC sum of the Intercepted arcs; both arcs being considered POSITIVE when the lines inter­ sect WIT BIN the circle ; one as ZERO when th^v intersect

O N the circle: and the greater arc positive and the lesser arc NEGATIVE when they intersect WITHOUT the circle."26

^"Outline of Geometry," Report of the School Committee of the City of Boston. School document No. o - 193^> p. 31**

Bruce, Elements of Plane Geometry, p. 112. Another series of familiar statements concerning lines and circles, but involving segments instead of angles, is presented in Illustration 16. which follows.

ILLUSTRATION

Original exercise 16. not usually enunciated in textbook as given here, Is presented as a statement which might be recognized somewhat more easily than the statements which follow and from which those that follow may be obtained by variation.

Original Sxerclse 16

If a chord and a diameter of the same circle are per­ pendicular. either segment of the chord is the mean propor­ tional between the segments of the diameter.

Variant 16.1

If a chord and a diameter Intersect within a circle, tne product of the segments of the chord is equal to tne product of the segments of the diameter.

Variant. M>uZ If two chords intersect within a circle, the product of the segments of one chord is equal to the product of the segments of the other.

Variant 16.1

If two secants intersect ou'cside a circle, the product 203

of one secant and its external segment Is equal to the

product of the other secant and Its external segment.

Varlam If a tangent and a secant Intersect outside a circle,

tne tangent is the mean proportional between the secant and

its external segment.

Figures corresponding to all of the statements in illus­

tration 16 can be labeled in such a manner that, except for

minor differences in the reasons, the proofs for all of the

statements are the same. This is not surprising, for all

of tne statements are special cases of a single general

statement, which is given by Smith as follows: "If a pen­

cil of lines cuts a circumference, the rectangle (product)

of the two segments from the vertex is constant whichever

line is taken.M^ /

Other illustrations of the use of variation in con­ nection with the study of circles can be found but are not

Included in this report.

°7 D.E. Smith, The Teaching of Geometry, pp. 231-232. 20*4-

5.6 Ratio, Proportion. Similarity

Variants involving ratio and proportion are easily ob­ tained from statements involving equality. An example of tills procedure was given in connection with illustration 12

In the section entitled "Variants Obtainable by Connecting

Points on the Sides of Quadrilaterals Other Than the Mid­ points," pp. 183-185 * Also shown in that discussion is that when in a true statement involving bisection, bisection is replaced by division in a given ratio, the resulting statement might and might not be true.

The concept of similarity is closely related to the concept of congruence. Cpngruent polygons may be defined as follows: "Two polygons are congruent when their corres­ ponding angles are equal and their corresponding sides are equal." Similar polygons may be defined in the following manner: "Two polygons are similar if their corresponding angles are equal and their corresponding sides are propor­ tional. " Hence it is to be expected that certain theorems concerning congruence and the equality of sides may be viewed as being related by way of variation to other theorems concerning similarity and the proportionality of sides. A number of illustrations, with which the reader is familiar, are found in textbooks. Two examples are given in illus­ trations 17 and 18. ILLUSTRATION 1£

QiXrXw I F«rgl?g 17 If two sides of one triangle are equal to two sides

of another triangle and if the angles included by these

sides are equal, the triangles are congruent.

Variant 17.1

If two sides of one triangle are proportional to two

sides of another triangle and if the angles included by

these sides are equal, the triangles are similar.

ILLUSTRATION IS

Original R-xerclse 13

If two angles and a side of one triange are equal to

two angles and a side of another triangle, the triangles

are congruent.

Variant 13.1 is obtained by deleting from original

exercise 18 the portion of the data which is underlined and

substituting the word similar for the word congruent.

Variant 18.1

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Before the next illustration is presented, it is nec­ essary to state definitions of corresponding points and corresponding line segments of similar polygons. They are 2 0 6 obtained by modifying related definitions given for corres­ ponding points and corresponding line segments of congruent polygons, (pp. 159-160 ). The new definitions ares

1) Corresponding points of similar polygons are points which lie on corresponding sides of similar polygons and whose distances from corresponding vertices are propor­ tional to any two corresponding sides.* 2) Corresponding line segments of similar polygons are line segments which connect corresponding points in each of the two polygons.

Illustration 19 can now be presented.

ILLUSTRATION lg.

Original F.xercise 19

Corresponding line segments of congruent polygons are equal.**

Variant 19.1

Corresponding line segments of similar polygons are proportional to corresponding sides of the polygons.

Additional statements involving eaualitv or congruence. or both, can be found from which other statements involving ratio, proportion,, and similarity can be obtained by the process of variation.

♦ This definition can be extended to apply to points not on the sides of the polygons, but for purposes of ele­ mentary plane geometry, the definition given is considered adequate.

See variant 3.6, p. 161. 207

Summary

Numerous opportunities for using variation in a course

In elementary plane geometry have been shown. Illustra­ tions have been given under the following headings: in­ troductory work; triangles; parallel and perpendicular lines; polygons; circle; ratio, proportion, and similarity.

A large number of particular statements, many of them not found in geometry texts, were obtained by varying cer­ tain given statements. Several groups of important theorems in geometry were developed by varying one statement, se­ lected as a point of departure.

The extent of the students' participation and the nature of the teacher's guidance in connection with delib­ erately using variation in the study of geometry were indi­ cated in a small degree. CHAPTER 6

SUMMARY; IVPLICATIONS OF VARIATION FOR THE TEACHING OF MATHEMATICS Tie ^roceaa of Variation. P. 21'J Emphasis on Discovery In the Literature on the Teaching of Geometry: Variation

Is Propoaed As a Method of liscovery. P. 215 ,'arlatlon and Generalization, P. 218

■' ,01 Inductive Generalization P. 218

6 .52 Generalization by Turin- r 1 za t * on P. 219 6.55 Generalization by Removal of Lata P. 222

6.-34 Number of Special Cases that Should P. 223 be Considered. Possible Extensions of Variation as LePlnea and Illustrated In This Study. P. 225 6.41 Uses of variation In reference to

advanced Euclidean Geometry and to P. 225 Solid Geometry.

5.42 Variation of Construction In Geometry. P. 227 5.43 Variation of Concepts and Definitions

In Geometry. P. 228

208 209

CHAPTER 6 (Continued)

6.Mf satgflgigfl qf Variation tg Arsaa of Mathematics other than Geometry. P. 228 6.If5 Variation of Sets of Assumptions. P. 229 6.1+6 Extension of Variation to Areas Other than Mathematics. P. 229 6.5 Values Associated With Using the Process of Variation in a Plane Geometry CJLass. P. 233 6.6 Limitations of the Process of Variation. P. 23S 6.7 Specific Recommendations for Using Vari­ ation In the Teaching of Geometry. P. 2*+l 6.3 Problems for Further Study. P. 2*+2 210

6.1 The Process of Variation.

The following definition of variation, given on page 12 applies to use of the word In a strict sense for purposes of this study:

Variation is defined to mean a process of changing elements of the data or conclusion, or both* of a geometrical statement* which has been proved to be true or is accepted as true* with a view to obtaining a new set of data* or a new conclusion, or both* resulting In a new statement.

Statement is defined aswa group of words, or a group of words

together with other symbols, which expresses something that

is either true or false."

As is stated by the definition, variation is always applied to a statement which is known to the investigator.

Moreover, the original statement either has been proved to be true or is accepted as being true. But, although the original statement is true, it is necessary to investigate each statement obtained by making changes in it to deter­ mine whether the new statement Is true or false.

Usually, the new statement is either proved to be true or shown to be false. However, particularly in the intro­ ductory work* the new statement may also be adopted as an assumption, provided it appears to be consistent with ob­

servation and with other assumptions.

The types of changes which are made in varying a state­ ment are: deletion, addition* and substitution. These

changes may be made in the data only* in the conclusion only* 211 or in both. Changes made initially In the data sometimes are accompanied by changes in the conclusion and sometimes are not; an analysis of this type of variation into several

subtypes is presented In Chapter 2. Changes made initially in the conclusion sometimes are accompanied by changes in

the data and sometimes are not; this type of variation is analysed in Chapter 3. Certain formal processes, involv­ ing changes of specific kinds in the data and conclusion simultaneously, may also be viewed in terms of variation; a discussion of these processes is presented in Chapter 4.

The relation of variation to the formation of inverses was noted particularly. An inverse of a statement is obtained by denying one or more data and one or more conclusions in the given statement. The more inclusive process, variation includes substituting new data and new conclusions for the negated items, thereby providing a way of obtaining a new statement expressed entirely in positive terms.

When variation is performed initially on the data of a statement, there remains the problem of finding a conclu­ sion such that the new statement seems to be true and is thus worthy of search for a proof. There are several ways of searching for a conclusion, not all of which apply in every case. The following procedures frequently are profit able: 212

1. Determine whether or not the original

conclusion can be proved by means of the new data*

2. Draw one or more figures in accordance

with the data, and search for relations that seem

to be true, either by making direct observations

or by making measurements and studying them.

3. Uake guesses. As experience increases,

hunches tend to become increasingly reliable*

4. Try discovering the new conclusion by

the synthetic method of proof. Studying the

proof . for the original exercise from which the

variant was obtained sometimes discloses a way

to proceed with the proof of the variant and

thus also discloses the conclusion*

If the conclusion of a known statement is varied init­ ially, the following procedures for discovering appropriate data are suggested;

1* Try to prove the new conclusion by

means of the original data*

2* Using the knowledge of theorems which

might apply, make a guess concerning the data.

3. Use the method of analysis* This may

include drawing a figure in which relations ex­

pressed in the conclusion exist and then study­

ing the figure in an attempt to find other re­

lations which might serve as data* Some Principles Associated With the Use of Variation

It is not possible to present a set of specific pro­ cedures which apply in all instances where the process of variation may be used. The total number of variants which can be obtained from a given original statement and from variants of it is not fixed. Imagination and experience are important assets in the application of the concept of variation. It is possible, however, to indicate four gen­ eral principles that have been found useful. They are the following:

1. If an investigator is to make maximum use of

variation, he must deliberately scrutinize statements

which are known to him for possible opportunities to

vary them. He must be aware of the different kinds

of variation that can be used.

2. Usually it is advantageous to make Initial

changes in the data or the conclusion by varying one

item at a time. For example, If a given statement

concerns a square, often it is better to change square

to rhombus or rectangle than to change square to an£

quadrilateral•

3. In many cases It Is fruitful to attempt var­

iation of the verbal form of a statement, but some­

times more opportunities for variation are disclosed 21*+ by study of the itemised data and conclusions than are readily perceived by study of the verbal form of the s tatement*

4. Sometimes an opportunity for varying a given statement is disclosed by study of a proof for the statement. For instance, if a certain datum is not used in a particular proof, a variant, usually of a less restricted nature, may be obtained by deleting that datum from the original statement* 215 6.2 fiaphaala on Discovery in ths Litgrat.urg on the Teaching of Geometry: Variation la Proposed As a Method of Discovery. The process of variation is explained and Illustrated in this study, and Is proposed as a method whereby students may be guided to the discovery of many statements In elemen­ tary plane geometry. Both discovery and variation, as the latter term is used in this paper, have received attention in the literature concerning the teaching of geometry. Student discovery in geometry has received increasing consideration during the past century. At first the em­ phasis with respect to discovery was placed entirely upon the discovery of proofs. Shibli reports, "The first text book containing originals that the writer has seen Is an edition of Simson's Euclid published in 1 8 5 3 Wilson, studying the period 1918 to 19M3, notes emphasis on orig­ inality and discovery as a stated aim of textbook writers and interprets it, particularly between 1918 and 1928, as being largely directed toward the proving of suggested originals.2 Writing in 1932, Shibli states:

1J. Shibli, Recent Developments In the Teaching of Geometry, p. 156.

2Jack Douglas Wilson. "Trends in Elementary and Secondary School Mathematics," p. 208. unpublished D. Ed. dissertation, Stanford University, 19*+9. 216

M^ny of the best teachers today look upon the study of geometry as an opportunity for the pupil to do his own thinking, to cultivate the spirit of investigation and discovery...... Work in geometry should consist largely not of statements of theorems and exercises to be prov­ ed, but of specific problems to be Investigated with the purpose of discovering some property or theorem which can be demonstrated.3

A number of textbooks stressing exploration by the pupils have appeared in recent years.4

Variation, like discovery, is mentioned in the litera­ ture on the teaching and study of geometry, and of mathe­ matics in general, with much the same meaning that is given to the word in this study, but variation has not received as much attention as has been given to discovery. The writ­ er did not find any detailed study or other presentation of a process like the one which in this study is called varia­ tion ; nor did he find evidence in the literature of delib­ erate and extensive use of such a process in the teaching and learning of geometry or of any other branch of elemen­ tary mathematics. Important references which the writer found in the literature are mentioned in Chapter 1, pp. 12-15.

^Shibli, op. clt., pp. 223-24.

4Among those which may be mentioned are the followings

McCormack, o£. clt. , (1934). Schnell and Crawford, o£. clt., (1943). Trump, oj). clt., (1949TT Rosskop?7 Aten, and Reeve, oj>. clt., (1952). 217

Variation la presented here as a process for discover­

ing many of the Important theorems of elementary geometry,

and other statements In elementary geometry to which less

importance Is attached. Deliberate use of variation re­

sults also in the discovery of statements found in text­ books only infrequently or not at all. 213

6.3 Variation and Generalization

The word generalization Is used in a variety of ways.

Sometimes the meaning apparently associated with general­ ization is so broad that the word Is nearly useless. In this study a statement G Is considered a generalization of another statement S If the statement S can be considered a special case of the statement G_. In connection with the process of variation it is possible to identify three uses of the word generalization. They are presented and dis­ cussed briefly, with reference to variation. In the follow­ ing three sections, 6.31, 6.32, and 6.33.

5.31 Inductive Generalization.

This type of generalization Is discussed In detail under the heading, "The Role of Variation In the Processes of Generalization and Induction,” pp.6 5“ 72 » In connection with illustrations 7 , 8 , and 9, pp. 55 -65« The term In­ ductive generalization is used in the sense of the defini­ tion by Cohen and Nagel in which generalization means "the passage from a statement true of some observed instances to a statement true of all possible Instances of a certain class.f,5 in summary, It is noted that "variation may be said to promote Inductive generalization," and that "any testing of a generalization after it has been formulated may be described in terms of variation."*

^Cohen and Nagel, oj£. clt., p. 277. * P- 72. 219

6.32 Generalization by Summarlzatlon.

Illuatratlon 15. PP-195-397# concerns the measurement of angles In terms of arcs Intercepted on a circle. In

Illustration 15 and In most geometry textbooks, several sep- g arate statements are enunciated. Following this, as Is done in about half of the textbooks that were inspected, a single statement is given of which all the separate statements may be considered special cases- Three books on the teaching of mathematics may be cited in which this particular example, concerning angles and the tree intercepted on a circle, Is presented as an illustration of generalization. Smith says, in a section entitled "Generalization of figures";

This generalization of typical figures materially lessens the detail of geometry# For example, the propositions concerning the measure of an Inscribed angle, an angle form­ ed by a tangent and a chord, ...... are all special cases of a single theorem.”

Under the subject "Generalization of certain theorems,

"Schultze says, " ... we may summarize all theorems relat­ ing to angles as follows; An angle Is measured by one half the algebraic sum of the intercepted arcs."®

®An exception, in which only the one general theorem Is stated, is; Skolnik and Hartley, oj>. clt., p. 151.

^Smlth, loc. clt.

®Schultze, ojd. clt., p. 185. 220

In a paragraph entitled, "Generalization of theorems on

angle m e a s u r e Davis states, "The preceding theorems may be generalized into one if provision is made with respect

to positive and negative arcs, ...... " 9

Generalization of this sort Is also noted In connec­

tion with Illustration 16, pp• 2 0 2 -2 0 3 , which concerns the

product of segments of lines which intersect each other and

also intersect a circle* This type of generalization Is

also utilized in combining the two theorems, "Two triangles

are congruent If two sides and the included angle of the one

are equal to two sides and the Included angle of the other"

and "Two triangles are congruent if two angles and the In­

cluded side of the one are equal to two angles and the in­

cluded 3 lde of the other" into the single theorem"Two Tri­

angles are congruent if three consecutive parts of the one

are equal to three consecutive parts of the other," as was

done in Illustration 4, pp. H+l-lMf.

In each of the illustrations which have been cited,

in order to verbalize the generalization, it is necessary

to introduce terms or adopt conventions, or both, which

are not necessary In order to state any of the individual

theorems included in the generalization. For instance, in

order to generalize the theorems concerning the measurement

of angles In terms of intercepted arcs, It is necessary to

^David R, Davis, The Teaching of Mathematics, p, 61. 221

stipulate a convention concerning negative and aero arcs.

Such a procedure la not, however, always Involved when sev­

eral statements are collected Into one* In most geometry

texts, one theorem concerning Inscribed angles la stated, but three cases are considered In the proof. Statements for each of the three separate cases are Implied, although

they are not always verbalized explicitly* These state­ ments are: 1) "If the center of the circle lies on one

side of an angle inscribed In the circle, the angle is measured by one-half the intercepted arc;" 2) "If the

center of the circle lies Inside an angle inscribed In a

circle, the angle Is measured by one-half the intercepted

arc"; 3) "If the center of the circle lies outside an angle

inscribed In a circle, the angle Is measured by one-half the

intercepted arc." The general theorem proved by proving the

three Individual theorems is: "Any angle inscribed in a

circle Is measured by one-half the intercepted arc.”

7/hen a generalization is achieved by summarizing part­

icular statements, variation can be used to generate all of

the particular statements except the one which was the or­

iginal statement. The generalization Itself may be consid­ ered a variant of any one of the special cases. 222

6.33 Generalization bv and. If Necessary,

Making Appropriate Changes In the Conclusion.

Discussing generalization In mathematics, Shaw says,

Usually the process of generalization takes place by means of the various analog­ ies present. Hie observation of these Is necessary to generalization. But there is another mode of generalization, and that Is the removal of premises In arguments, or at least parts of premises. Much mathematical work of the present day consists In determin­ ing whether a conclusion can persist If the premises are made a little less restricted . 1 0

In many cases, when one or more data are removed from

a gemetrical statement, changes in the conclusion are also

necessary. If the resulting statement is to be a general­

ization of the original statement, the changes made in the

conclusion must be of such a nature that the original state­ ment Is a special case of the statement finally obtained.

A number of examples of generalization of this type can be found among the Illustrations In Chapter 5. Other examples are given In section 2.11, pages 29-3^, where de­ letion of data Is presented as a particular type of varia­ tion. It is In this sense that the statement "The bisector of an interior angle of any triangle divides the opposite side into segments which are proportional to the sides ad­ jacent to the angle bisected" Is considered a generaliza­ tion of the statement "The bisector of the angle between the equal sides of an isosceles triangle bisects the base.”

^uShaw, loc. clt. 223

In this sense also, the statement "The area of a parallel­ ogram is equal to the product of one side and the perpendic­ ular distance between that side and the side opposite" is a generalization of the statement "The area of a rectangle Is equal to the product of two adjacent sides," Similarly, the statement "The bisectors of the interior angles of any quad­ rilateral intersect to form an inscriptible quadrilateral" is a generalization of the statement, "The bisectors of the

Interior angles of a rectangle intersect to form a square."

Finally, in this sense of generalization, the theorem "The sum of the interior angle of any polygon is equal to n - 2 straight angles, where n Is the number of sides of the poly­ gon" may also be considered a generalization of the theorem

"The sum of the angles of a triangle is one straight angle".

6.34 Number of Special Cases that Should Be Considered.

If a generalization is asserted on the basis of induc­ tive argument, particular Instances must be exhibited which verify the generalization. Moreover, these instances must be representative of the totality of types of Instances to which the generalization applies.For Instance, in est­ ablishing the theorem "The sum of the angles of a triangle

Is 180°" by measuring the angles in particular triangles, large and small triangles of different sizes and shapes should be Investigated: in particular, triangles of each of

^Cohen and Nagel, 0£. clt., pp. 280-86. 22b the following types should be drawn and their angles meas- ured: equilateral, isosceles-right, Isosceles-obtuse,

Isosceles-acute, scalene-right, scalene-acute, and scalene- obtuse. These different types of triangles may be consid­ ered as being obtained by the process of variation.

If the generalization Is obtained by combining a num­ ber of particular statements, these particular statements may be obtained by the process of variation. If the gen­ eralization Itself Is to be established by means of a separate deductive argument, the role of the particular statements and of variation in generating them Is primarily that of suggesting the generalization. With respect to the number of special cases that are affirmed Smith recommends careful judgment.

... elementary geometry offers a field, interesting to teachers and pupils alike, for simple generalizations. The danger lies on the one side In always attempting to give the general before the particular (a fatal error), and on the other in cut­ ting out all of the Interest which comes from generalization, thus falling Into the old humdrum of multiplying theorems to fit all special cases. 1 2

^Sml th, » P* 282 225

6.4 Possible Extensions of Variation as

Defined and Illustrated In This Study

6.41 Uses of variation In reference to advanced Euclidean

Geometry and to Solid Geometry.

For purposes of this 3 tudy, the application of variation is limited to geometrical statements. The subject material considered in the study was confined almost exclusively to the field of elementary plane geometry. Without any extension of the definition of variation, statements which are usually thought of as belonging to the area of advanced Euclidean

Geometry and also statements in Solid Geometry can be obtained by varying well-known statements in elementary plane geometry.

For example, the theorem in advanced Euclidean Geometry often called "The Converse of Ceva's Theorem" may be viewed as a variant of the elementary theorem "The medians of any triangle are concurrent or as a variant of the other concur­ rency theorems." The Converse of Ceva's Theorem may be stated as follows: "If lines are drawn from the vertices of any triangle to points on the opposite sides, such that the six segments formed with the vertices of the triangle, taken in order, determine three ratios whose product is unity, then these lines are concurrent."*

"^Although this illustration is in accordance with the de­ finition of a variant given in this study, no claim Is madd that an average student can be expected to rediscover Ceva's Theorem In this manner without guidance, even If he knows the process of variation. 226

An example of obtaining a statement in Solid Geometry by varying a statement in Plane Geometry was noted on p. 181.

Variant 12.(12) is: "The points E, F, G, and H which divide the sides AB, AD, CB, and CD, respectively, of any plane quadrilateral ABCD (convex, concave, or crossed) in a given ratio are the vertices of a parallelogram." This statement

Is also true if ABCD is a "skew" quadrilateral, that is, if

A, 3, C, and D are not coplanar.

Another example of obtaining a statement in Solid Geometry by varying a statement in Plane Geometry is the following:

Original Exercise (Plane Geometry)

In any triangle the lines drawn from the vertices

to the midpoints of the opposite sides are concurrent

in a point which is two-thirds of the distance from each

vertex to the midpoint of the opposite side.

Variant (Solid Geometry)

In any tetrahedron the lines drawn from the

vertices to thecentroids of the opposite faces are

concurrent in a point which is three-fourths of the

distance from each vertex to the centroid of the

opposite face.

It Is not only possible to vary certain theorems in elementary plane geometry and obtain variants In advanced

Euclidean Geometry and In Solid Geometry. There are oppor­ tunities also for varying statements in either of the last mentioned fields, thereby obtaining other statements In the 227 same field. One example from each field follows:

Solid Geometry

Original Statement: Through a given point one and only

one line can be drawn perpendicular to a given plane.

Variant: Through a given point not on a given plane one and only one plane can be drawn parallel to the given plane.

Advanced Euclidean Geometry

Original Statement: The inverse of a straight line

t.irough the center of inversion is the line itself.

Variant: The inverse of a straight line not passing

trirough the center of inversion 13 a circle tlirough the center of inversion.

G.42 Variation of construction problems in geometry.

If the aspect of change is retained in the concept of variation but the entity being changed is not limited to con­ ditional statements, the process can be applied to construc­

tion problems in geometry. Illustrations of this sort are

the following:

1) Original Problem: Prom a given point outside

a given line construct a line perpendicular to the given

line. Variant 1: At a given point on a given line,

construct a line perpendicular to the given line.

Variant 2 : Through a given point outside a given line

construct a line parallel to the given line.

2) Original Problem: Bisect a given ary.le. 228

Variant 1: Bisect a given line segment. Variant 2;

Trisect a given line segment.

6.43 Variation of concepts and definitions In geometry.

As has been remarked (p. 20*t), the concept of similarity is closely related to the concept of congruence. Congruent figures, it is pointed out in many textbooks, have the same size and shape, while similar figures have the same shape but not necessarily the same size. Congruent polygons may be defined as polygons having corresponding angles equal and corresponding sides equal; s1mllar polygons may be defined us polygons having corresponding angles equal and correspond­ ing sides proportional.

The definition of a sphere may be viewed as a variant of the definition of a circle, as follows: A circle is a (plane) curve all noints of which are equidistant from a fixed point;

1 sphere is a 3 urface (in three-dimensional space) all points of which are equidistant from a fixed point.

>.44 Extension of variation to areas of mathematics other than

gaometry.

The observation by Butler and Wren that the different forms of the linear algebraic equation "are all variations of a common form" is quoted earlier. Iuerriman's illustration of variation in connection with the calculus is also noted.^

*See pp. 13-1 *f.

^See p. l*f. 229

6.45 Variation of aata of assumptions.

hew systems of assumptions may be obtained by varying a particular system of assumptions. The first "non-euclidean" geometry was obtained by denying Euclid's parallel postulate and substituting an alternate assumption. "Hilbert's Axioms" for Euclidean Plane Geometry, with slight changes, become sets of axioms from which several non-Euclidean geometries and four-dimensional Euclidean Geometry can be developed . 1 3

6.46 Extension of variation to areas other tlian inathema ti c s .

Variation was defined for purposes of this study in such a -lanner that consideration was limited to statements in geometry. Two kinds of extension of this definition have been indicated. One kind of extension is obtained by removing the limitation of applying variation only to statements. The second kind of extension Involves applying variation in areas other than geometry. With these two extensions, variation has the meaning of deliberately changing one or more elements in a given situation, the aspects of which are understood, with a view to obtaining new knowledge; the basic ways of making such changes are addition, deletion, and substitut ion.

When variation is not limited to the subject material of geometry, the process appears to have a wide range of useful­ ness. Examples from three academic fields follow.

1 3 C. R. Wylie, Lectures, hathejnatlcs 655. The Ohio State Iniverslty, Summer, 1940. Paul L. Weaver, ”A Geometry of Pour Dimensions," Raster's Thesis, The Ohio State University, 1940. 230

Chemistry. Organic chemistry Is presented by some teachers from the point of view that all organic compounds are obtained by substituting atoms and groups of atoms Into the one compound, methane, for which the chemical formula Is CH . To cite two simple examples, 4 the substitution of three atoms of chlorine for three atoms of hydrogen In CH^ results In CH CI3, chloroform; the substitution of an additional chlorine atom for the remaining atom of hydrogen in chloroform results in C Cl4 , the well-known cleaning agent, carbon-tetra- chloride. A great number of steps are involved in the production of some organic compounds, and sometimes a rearrangement of atoms accompanies the substitution, but

methane can be considered the starting point for all of them. Many organic compounds were produced for the first time In the laboratory after their chemical formulas were expressed symbolically by making particular substi­ tutions into the written formulas for known compounds. Physics. Many illustrations of the process of vari­

ation can be found in the field of physics. Two will be mentioned here. 1) After It is learned that the move­

ment of a conductor in a magnetic field will produce a current In the conductor, it seems reasonable tc ask whether moving the magnetic field relative to the con­ ductor or changing its intensity will produce the same result. Experiment reveals that a current is produced in either case. 2) Perhaps the "simple machine" closest 231 to the beginner*a experience is the inclined plane. The wedge and the screw can be and often are treated as variants of the inclined plane. Other simple machines can be considered variants of the inclined plane in the sense that the same formulas for theoretical mechanical advantage apply to all of them; namely, that the mechanical advantage of a frictionlesa machine can be expressed either as output force divided by input force or as input distance divided by output distance. Economics. In 1857 Ernst Engel, then head of the Statistical Bureau of Saxony, on the basis of surveys he had made, stated four principles. These have become known as Engel*s Laws of Consumption; they may be stated briefly as follows: As the income of a family Increases In amount, the percentage or proportion of the Income expended: 1) For food decreases. 2) For clothing remains approximately the same • 3) For rent, fuel, and light remains constant. 4) For education, health, religion, recrea­ tion, amusement, etc., Increases. ^ Engel*s studies were based on studies involving

Saxon families with incomes ranging from $300 to $1000 per year. Variation is illustrated when studies similar to Engel*s are performed at different times and In

x ? ------See: H. H. Maynard and T. N. Beckman, Principles of Marketing. p. 74. C. S. Wyand, The Economics of Consumption, p. 219. 232

different pieces, when different Income ranges are con­ sidered, when the apportionment of Income la studied with regard to Items of expenditure othsr than those Included by Sngel, and when factors other than total Income are studied for their effect on the apportlonment of Income, such as the following mentioned by Morgan: 1) effect of a change in income, and 2) effect of a change in the 15 Income of friends and neighbors* Geometry appears to be an excellent mediun for learning

to use the process of variation with facility and Imagination, ^ne of the principal values to be desired is that learning the process in connection with geometry will result in its trans­ fer to other fields if opportunities for using it in other fields are afforded in the geometry class. Investigation of this sort of transfer is of course outside the scope of this s tudy•

Theodore Morgan, Introduction to Economics, pp. 535-36 233

6*5 Values Associated with Using the Process of Variation In a !Plane Geometry ClasTT The teacher may utilize variation as an effective process for developing some of the subject material of the course In plane geometry. Using either written materials or the spoken word he may lead the class to the discovery of specific theorems found in nearly every textbook for plane geometry. In order to accomplish this he will need to make suggestions which are quite specific. The teacher may also encourage variation without making specific suggestions regarding the items of data or conclusion to be changed and without even suggesting specific exercises to be varied If he develops in his students a disposition to investigate every statement with a view to developing variants of it. Some of the statements discovered Independently of suggestions by the teacher will be new to the teacher as well as to the student and some of

them will not be found In many geometry textbooks. The study was not designed to establish merits of varia­ tion in an objective manner. However, the writer believes that the following tentative conclusions may be stated on the basis of the study.

6.51 Interest In a subject Is stimulated when the students

themselves discover and develop some of the material with

which the subject is concerned. Ever expanding horizons can be disclosed to the learner who has learned to use the process of variation and does so consciously and deliberately. 23*f

6.52 Variation fosters security at each stage of learning in which it is used. Emphasis is on beginning with statements which are known and understood and then changing them a little at a time.

6.53 Use of variation affords means of providing for individual differences of ability, time, insight, and interest. Use of variation can offer opportunities for nearly every member of a class to make significant contributions to the group experi­ ence; yet the more capable students always have a challenge before them. Once the process of variation has been intro­ duced, there need never be a feeling that an assignment has been completed; hence, the students who have extra time between class meetings can always apply it to geometry. Students in the class which the writer taught pursued their own insights in varying given statements; this is shown particularly by illustration 7 , pp*151 - 157* and illustration 13« pp.189-190, out was evident on many occasions. Members of the class con­ tributed the most variants in connection with those areas of geometry which appealed to them most. Association of geometry with interests not specifically geometrical, by means of an extended notion of variation, also was noted.

6.54 When students themselves discover possible relations

between geometrical elements, they can be expected to have a more vital interest in proving them deductively and to acquire a better understanding of the proofs than if they are simply given the relations to prove. «?35 t .55 i

#'nich are rolotod sc that largo bodies of l

6.56 Mathematics is sometimes criticized because for a very problem as it is encountered by the student there is * solu­ tion; thus mathematics provides training ifi j^nsonir.g but not

in the making of judgments concerning the aec®83^'/ or adequacy of certain data. Stated with specific reference to plnrae geometry, the criticism is sa follows: The student Is not asked to prove anything which cannot be proved» and impossible constructions are not proposed; hence, in elea»antary geometry, the student is not required to judge a proposed conclusion as probably true or probably not true, and thus he is not chal­ lenged to reject a conclusion as untenable. Since it is n°t the nature of problems as they are met in n f e situations to

have enough and Just enough data available, and to be always capable of solution, mathematics fails to provide complete 236

experiences in dealing with problems* 16 ■Phis condition is

remedied when the pupil actually discover and develop*

ma thematic a 1 notion* himaelf. In particular,. i extensive

experienoe is had in deciding whether to hrMr y t0 prove or* dis-

prove a statement formulated tentatively and in rejecting It if it i« found to be false or changing l*. ^ Iur^,,-thor so t h a t It

is Judged more likely to be true when the tne students* g e n e r a t e s new statements by the process of variation,

6.57 One of the values commonly accorde the study of tor the-

natics, particularly geometry, is the dev . mant oi’ H a b i t s of

precise expression. Reference to this o hij tive ig m a d e in the prefaces of a high percentage of georae . textbooks •

fhis objective is highly regarded also by, a i ar«« number* of

teachers. J. Shibli, in connection w i* t h M s study, r e c e i v e d

replies from lai teachers regarding the. and values o f

the teaching of demonstrative geometry* shibli reports that these 181 teachers rated the aim "to develop th

clear thinking and precise expression ti as Kaini.' of g r e a t e s t importance by an over-whelming margin* 17 Exactly bow the aim of precis# expression is to be achievedj *, , however, s e l d o m stated. One might infer that achievemen . f this ob j e c t i v e

TH’”' a 1s Dewey** conception Illustrative of this point “f th« oompl*te , c t of of the role of mathematics in relation --theniatica to the reflective thought. The contribution o thinking of a high development of the ability to do ref1*®“* . c0 j.n one phase order is limited largely to providing th# narrower of the total process; namely, "reasoning v sense)." John Dewey, How We Think, PP* A

17J, Shibli, oj>. cit.. pp. 214-16* is to result from repeating memorized statements found In the textbook being used; any value gained In this manner Is, at best, very limited, when compared w i t h the challenge to a student of putting Into words the results of his own discoveries

Frequently, when the process of variation Is used, a new state­ ment Is obtained by just making changes In a figure or In the data and conclusion stated In t erms of a figure. Formulating a corresponding word statement after the changes have been made In the latter manner provides very significant experience

in the art of precise expression. 238

6.6 Limitations of the Procesa of Variation

Certain values tentatively associated with, the process of variation have been presented. There are also certain

limitations to the process. The writer recognises the follow­

ing limitations as being apparent.

6.61 Variation is not a complete method of discovery in the

sens® that all statements of geometry can be obtained by its use. Each opportunity for using variation depends upon the possession of a statement which has been proved to be true or

Is assumed to be true. Hence some of the statements to which variation Is applied must be discovered or obtained by methods

other than variation.

6.62 Over-enthusiasm for variation on the part of the teacher might result In attempts by him to use variation in the pre­

sentation of new material or in guiding his students to dis­

cover particular statements when some other method of develop­

ing the material would be just as effective or perhaps even

more effective.

6.63 There is a possibility that if variation is over-empha­

sized, the students might overlook relationships which are

present in a given situation but which are not suggested In

terms of variation.

6.64 The principal motivation for the use of variation is

that by means of It students can discover statements previously 239 unknown to them. Hence a student who does not desire to make discoveries and in whom the teacher cannot instill such a desire is not likely to be motivated to use the process of variat ion*

6.65 In many cases, when a student applies the process of variation he does not know immediately whether a variant he has obtained is true or false. Even if the student succeeds in proving that a statement he has obtained is true, he will seldom know the degree of its importance. However, these comments apply not only to discovery through use of the process of variation; they apply to research generally. A student who is disappointed that spectacular success does not accompany a large percentage of the attempts to use variation is not likely to look for opportunities to use the process. On the other hand, the teacher can use the method of variation to emphasize the fact that original discovery requires penetra­ tion into unknown areas and that the results of original investigation are not usually predictable.

6.66 Early in the work, in spite of specific instructions to try to prove each variant obtained, some of the students pre­ sented deductive proofs for relatively few of the statements they had obtained. The writer therefore points out the possibility that some students might become so engrossed with the generation of new statements that they neglect proving them. 2*fO

6.67 Much of the emphasis upon discovery in geometry in the past has been upon the discovery of proofs. Variation is a method for discovering new statements; although hints for the proofs of some variants can be obtained by studying one or more proofs of the associated original exercise and although hints for the proof of a given statement can sometimes be obtained by varying the given statement and studying proofs of the resulting variants, variation is not primarily a method for discovering proofs. 2*+l

6.7 Specific Recommendations for Using Variation In the teaching of (geometry

Recommendations for using variation in the teaching of geometry have been made at various places in this study.

They are summarized in the following sentences.

1. The process of variation should be introduced in­ formally early in the course in geometry.

2. Later on, the students should be aided gradually to identify and discover particular kinds of variation and to organize the different types in a formal manner.

3. Variation should be used both by the teacher in pre­ senting new material to the class and by the students in their individual work.

4. The teacher should give many hints of opportunities to use variation, but the students should be encouraged to exhibit initiative by applying variation in situations not mentioned by the teacher.

5. Use of variation in connection with statements from fields outside mathematics should receive some attention in the geometry class. Students should be encouraged to use variation deliberately in fields other than geometry and to share their results with other members of the class. 2b2

6,8 Problems for Further Study

The following problems, remaining to be examined, are recognized as being suggested by the study reported In this disserta tlon:

1) Extension of the process of variation to construction problems and to concepts and definitions in elementary plane geometry should be studied in detail.

2) The process of variation as presented In this study, together with extensions indicated in item 1), should be employed and evaluated by other teachers.

3) Arrangement of topics and principal propositions of plans geometry should be studied from the point of view of increasing the opportunities for using variation most effec­ tively.

4) Opportunities to develop topics and theorems of solid geometry as variants of topics and theorems in plane geometry should be studied.

5) Opportunities for guiding able and interested students to the discovery of theorems In advanced Euclidean geometry as variants of theorems In elementary geometry should be studied.

6) The study by Fawcett and subsequent studies by Ulmer and G-adske show how a comprehension of the logical structure of geometry can be used to help students understand "The Nature of Proof" as It may be applied in many situations encountered I -ssi w y wp y

2b 3

In daily life* A similar study involving variation should

be made, in which deliberate use of variation is taught in

connection with plane geometry and exercises in the use of

variation in non-academic areas developed; the value of

using geometry as the medium for learning how to use the pro­

cess of variation and how to apply it to situations of daily

life should be appraised*

^®rlarold P. Fawcett, The Nature of Proof, Thirteenth Yearbook of the National Council ot Teachers of Mathematics, 1938.

Gilbert Ulmer, "Teaching Geometry to Cultivate Reflective Thinking; An Experimental Study with 1239 High School Pupils," Journal of Experimental Education, VIII (September, 1939), pp. 18-25.

Richard Edward Gadske, "Demonstrative Geometry as Means of Improving Critical Thinking.” Unpublished Doctor's thesis, Northwestern University, 1940. BIBLIOGRAPHY

Allen, Frank B. "Teaching for Generalization in Geometry," The Mathematics Teacher. XLIII (October, 1950), pp. 24-5-251.

. "The Multi-Converse Concept in Geometry," The Mathematics Teacher. XLV (December, 1952), pp. 582-847

Archer, Allene; Hartley, Miles C . ; and Schult, Veryl. Plane Geometry Experiments. New York: D. Van Nostrand Company, Inc., 1949*

Bakst, Aaron. "Mathematical Recreations," The Mathematics Teacher. XLIV (April, 1951), 269-270.

Barber, Harry C.; Hendrix, Gertrude; and Brown, Bancroft H. Plane Geometry and Its Reasoning. New Yorks Harcourt Brace and Company, 1937.

Beatley, Ralph. "Third Report of the Committee on Geometry." The Mathematics Teacher. XXVIII (November, 1935), pp. IfOl-frJO.

Bell, E.T. The Development of Mathematics. New York: MeGraw-Hill Book Company, Inc., 1940.

Betz, Wm.; Miller, A. Brown; and Miller, F. Brooks. Work Book in Intuitive Geometry. Cleveland: The Harter School Supply Company, 1928.

Birkoff, George David, and Beatley, Ralph. 3asic Geometry. New York* Scott, Foresman and Company, 1940.

Black, Max. Critical Thinking. New Yorks Prentice-Hall, Inc., 19 .

Blackhurst, J. Herbert. Humanized Geometry (An Introduction to Thinking). Des Moines, Iowa: University Press, 1935. Blumberg, Henry. "On the Technique of Generalization," The American Mathematical Monthly. XLVII (August - September, 19**0), 451-62.

244 2h5 Bibliography Continued.

Bluraberg, Henry. "On the Change of Form," The American Mathematical Monthly. LIII (April, I9U6 )~ lSl-92.

Breslich, Ernst R. Purposeful Mathematics. Plane Geometry. New York! Laidlaw Brothers, 19*+3. Xfrg JTgag.hl.ag Arithmetic. Lancaster, Pennsylvania: Lancaster Pennsylvania Publishing Com­ pany, 1880.

Bruce, W.H. Elements of Plane Geometry. Dallas, Texas: The Southern Publishing Company, 1929.

Butler, C.H., and Wren, F.L. The Teaching of Secondary Mathematics. 2nd ed. New York: McGraw-Hill Book Company, Inc., 1951. Carmichael, R.D. The Logic of Discovery. Chicago: The Open Court Publishing Co., 1930.

Chauvenet, Wm. A Treatise on Elementary Geometry. Phila­ delphia: j7b. Lippincott and Company,1o8mT

Christofferson, H.C. Geometry Professionalized for Teachers. Oxford,Ohio:(Copyright by author), 1933.

Clark, John R.; Smith, Roland R.; and Schorling, Raleigh. Modern-School Geometry. New Edition. Yonkers-on- Hudson, New York: World Book Company, 19*+3.

Cohen, Morris R., and Nagel, Ernest. An Introduction to Logic and Scientific Method. New York: Harcourt, Brace and Company, Inc", 193>+.

Coleman, Robert. The Development of Informal Geometry. Contributions to Education #865* New York: Bureau of Publications, Teachers College, Columbia university, 19^2. Columbia Associates In Philosophy. An Introduction to Re- .ectlve Thinking. Boston: Houghton Mifflin Company, f t>23. Comstock, C.C. "A Few Deficiencies In the Teaching of Bibliography Continued.

Mathematics," School Science and Mathematics, VII (January, 1907), 12-18,(February, 1907), 97-101.

Coolidge, Julian L. A History of Geometrical Methods. Oxford: The Clarendon Press, 19*+0.

Courant. Richard, and Robbins, Herbert. What Is Mathe­ matics? New York* Oxford University press, iy*f-L.

Course of Study in Mathematics for Grades Ten Through Twelve- Prepared by Hillsborough County Mathematics Workshop, Tampa, Florida, August, 19*+3. Course of Study in Mathematics for Secondary Schools. Bulle­ tin 3«0, Commonwealth of Pennsylvania, Department of Public Instruction. Harrisburg: 1952.

Dadourian, H. M. How To Study-How To Solve. Cambridge: Add! sondes ley Press, Inc., 1951. Daily, B.W. Ability to Select Data-in Solving Problems. Contributions to Education #19

Davis, David R. Modern College Geometry. Cambridge: Addisbn-Wesley Press, Inc., 19*+9.

. The Teaching of Mathematics. Cambridge: Addison-Wesley Press, Inc., 1951. Dewey, John. How We Think. New York: D.C. Heath and Company, 1933• Dubs, Homer H. Rational Induction. Chicago: The Univer­ sity of Chicago Press, 1930. Failor, Isaac Newton. Plane and Solid Geometry. New York: The Century Company, 1911. Fawcett, Harold P. The Nature of Proof. Thirteenth Year­ book of the National Council of Teachers of Mathe­ matics. New York: Bureau of Publications, Teachers College, Columbia Uhiversity, 1938. Bibliography Continued

Forder, Henry George. The Foundations of Euclidean Geometry. Cambridge!At the University Press, 1927. Gadske, Richard Edward. "Demonstrative Geometry as Means of Improving Critical Thinking." Unpublished Doctor's thesis, Northwestern university, 19^+0. Garabedian, Carl A. "Some Simple Lbgical Notions Encount­ ered in Elementary Mathematics," The Mathematics Teacher. XXIV (October, 1931), pp. 3*5-52. A General Survey of Progress in the Last Twenty-Five Years. First Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University, 1926. Hadamard, Jacques. The Psychology of Invention in the Mathematical Field. 2nd ed. Princeton, N.J.: Princeton Uhlversify Press, 19>t9. Haertter, Leonard D. Plane Geometry. New York: The Century Company, 1923. Hamley, Herbert Russell. Relational and Functional Think­ ing in Mathematics. Ninth Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University, 193*. Hart, Walter W. Plane Geometry. Boston: D.C. Heath and Company, 1950. Hassler, Tasper 0., and Smith, Rolland R. The Teaching of Secondary Mathematics. New York: The Macmillan Company, 1930. Hawkes, H.E.; Luby, W.A.; and Teuton, F. C. Plane Geometry. Boston: Ginn and Company, 1920. Heath, T.L. The Thirteen Books of Euclid*s Elements. Three Volumes. Cambridge: The Uhiversity Press, 1908. Henderson, Archibald. "The Teaching of Geometry." The Uhiversity of North Carolina Record. October, 1920, Number 181. (*9 pp). Bibliography Continued.

Johnson, W.E. Logic - Part II. Cambridge! At the uni­ versity Press, 1922. Kaplan, Harry Gordonson. "Reasons Advanced For and Against the Teaching of Geometry and Their Significance." Vol. I. Unpublished Ph. D. Thesis, New York Univer­ sity, 1926.

Kattsoff, Louis 0. A Philosophy of Mathematics. Ames, Iowax The Iowa State College Press, 19*+3. Kemmerling, H. "Device to Aid in Generalizing Geometrical Ideas,” School Science and Mathematics. XXVII (June, 1927), 606-9. Kilpatrick, William Heard. "The Next Step in Method," The Mathematics Teacher. XV (January, 1922), pp. 16-25. Kinney, Lucien Blair, and Purdy, C. Richard. Teaching Mathematics in the Secondary School. New York! Rinehart and Co., Inc., 1952. Keniston, Rachel P., and Tully, Jean. Plane Geometry. Columbus* Ginn and Company, 19*+6. Lazar, Nathan. The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry. New York: Copyright by the author, ±93^. Also published in The Mathematical Teacher. Vol. 31, pp. 99-113, 156- 7*f, 216-m-O, March, April, May, 1933.

The Learning of Mathematics* Its Theory and Practice. Twenty-first Yearbook. The National Council of Teachers of Mathematics. Washington, D.C.!, 1953. Leonhardy, Adele; Joseph, Margaret; and McLeary, Ralph D. New Trend Geometry. First Course. New York* Charles E. Merrill Co., Inc., 19*+0. Lloyd, D.B. "Developing the Principle of Continuity in the Teaching of Euclidean Geometry." The Mathematics Teacher. XXXVII (October, 19^), pp. 253-62. Lowenstein, Norman. "What Is Scientific Method? An Inter­ pretive Study of Opinions on The Nature of Scientific Thinking.” Unpublished Master*s Thesis, College of the City of New York, 1930. 2b9 Bibliography Continued

MathematicsT Grades Nine and Tenr Ohio Curriculum Guide. Prepared by The Mathematics Committee, H.C. Christoffer- son (Chairman), H.P. Fawcett, R.L. Norton. Bulletin No. 53, 193^-. Columbus, Ohio* The Ohio Scholarship Tests, State Department of Education. Mathematics in General Education. A Report of the Committee on the Function of Mathematics in General Education for the Commission on Secondary Education of the Progressive Education Association. New York? D. Apple- ton-Century Company, Inc., 19*+0. May, Kenneth 0. Elementary Analysis. New York: John Wiley and Sons, Inc., 1952. Maynard, H.H., and Beckman, T.N. Principles of Marketing. H-th ed. New York: The Ronald Press Company, 19l+b. McCormack, Joseph P. Plane Geometry, rev. ed., New York: D. Apple ton-Century Company, Inc., 193*+. Merriman, Gaylord M. To Discover Mathematics. New York: John Wiley and Sons, Inc., 19^2. Mill, John Stuart. A System of Logic. 8th Ed. New York* Harper & Brothers, (No copyright date given).

Minnlck, J.H. Teaching Mathematics in the Secondary Schools. New York* Prentice-Hall, Inc., 1939. Mitchell, F.W. The Nature of Mathematical Thinking. Mel­ bourne, Australia* Melbourne University Press, 1938. Moore, E.H. "The Foundations of Mathematics," Science. N.S., XVII (March 13, 1903), pp. ^10-16. Morgan, Frank M.; Foberg. John A.; and Breckenridge, W.E. Plane Geometry. Revised Edition. New York* Houghton Mifflin Company, 19^3•

Morgan, Theodore. Introduction to Economics. New York* Prentice-Hall, Inc.,1950. Nightingale, A.F. "Report of the Committee on College En­ trance Requirements," Proceedings and Addresses of National Education Association.38th Annual Meeting, i 599, 6«+8-51. 250 Bibliography Continued.

Nunn, T.P. (Sir Percy). "Notes on the Place of Similarity in School Geometry," The Mathematical Gazette. XXII (July, 1938), 235-49.

"Outline of Geometry," Report of the School Committee of the Cltv of Boston. Boston* City Printing Department, 1931*—3 5 . School Document No. 8, 193^.

Peak, Philip. "Continuity in Geometry," The Mathematics Teacher. XXXVII (December, 19M+), 360-62.

Perry, John. "The Teaching of Mathematics." Educational Review. XXIII (January - May, 1902), 158-81. Picket, HAle. "Functional Thinking in Geometry," The Mathe­ matics Teacher. XXXIII (February, 19^0), pp. 69-72.

The Place of Mathematics in Secondary Education. Fifteenth Yearbook of the National Council of Teachers of Mathe­ matics. New York: Bureau of Publications, Teachers College, Columbia University, 19l+0.

Poincarfc, Henri. The Foundations of Science. Authorized Translation by G.B. Halsted. New Yorks The Science Press, 1929. Polya, George. "Generalization, Specialization, Analogy," The American Mathematical Monthly. LV (April, 1 9 W , pp. 24-1-1*3.

How To Solve It. Princeton, New Jersey: Princeton university Press, 19^+5

"Let Us Teach Guessing," The American Mathe­ matical______Monthly. LVII (August - September, 1950)?

"Provisional Report of the National Committee of Fifteen on Geometry Syllabus." School Science and Mathematics. XI (April, 1911), 329-55. Quine, Willard Van Orman. Mathematical Logic. Rev. Ed. Cambridge: Harvard University Press, 1951.

Reichgott, David, and Spiller, Lee R. Today♦s Geometry. 3rd ed. New York: Prentlce-Hall, Inc., 1950. 251 Bi bli ography C ontin ued.

The Reorganization of Mathematics In Secondary Education. Report of the National Committee on Mathematical Re­ quirements under the auspices of The Mathematical Association of America, Inc. The Mathematical Associ­ ation of America, Inc., 1923- "Report of the American Commissioners of the International Commission on the Teaching of Mathematics." United States Bureau of Education, Bulletin No. 1*+. Washing­ ton, D.C.: Government Printing Office.

Report of the committee of Ten on Secondary School Studies. Chicago: American Book Company, lo9*+. (This is a reprint of U.S. Office of Education Publication, Bulletin No. 205). Rosskopf, Myron F. ; Aten, Harold D.; and Reevs, William D. Mathematics - A Second Course. New York: McGraw- Hill Book Company, Inc., 1952.

Rupert, C.E. Famous Geometrical Theorems and Problems WUft Itolr fllatory. Boston: D.C. Heath and Company, 1900. Russell, Bertrand. Introduction to Mathematical Philosophy. London: George Allen & Uhwin, Ltd., 1919. Salit, C.R. "Geometry As a Natural Way of Thinking," Education. LXVIII (September 19*+6), 23-33.

Schaaf, William L., (editor). Mathematics.- Our Great Heritage. New York: Harper & Brothers,19*+^.

Schacht, John F., and Kinsella, John J. "Dynamic Geometry," The Mathematics Teacher. XL (April, 19^-7), 151-57. Schnell, Leroy H., and Crawford, Mildred. Clear Thinking - An Approach Through Plane Geometry. (Rev. Ed). Nev/ York: Harper and Brothers, 19*0. Schultze, Arthur. The Teaching of Mathematics in Secondary Schools. New York:The Macmillan Company,1916. A Second Report on the Teaching of Geometry in Schools. Pre­ paredfor the Mathematical Association (3ritish). London: G. Bell & Sons, Ltd., 1939. 252 Bibliography Continued

Seymour, F. Eugene. Plane Geometry. New York* American Book Company, 1925. Shaw, James Byrnie. Lectures on the Philosophy of Mathe- Chicagoi The Open Court Publishing Company,

Shibli, J. Recent Developments in the Teaching of Geometry. State College, Pennsylvania! Published by the author, 1932. Shively, Levi S. An Introduction to Modern Geometry. New York* John Wiley & Sons,Inc., 1939. Sigley, D.T., and Stratton, Win. T. Plane Geometry. New York* The Dryden Press, 19^8,

Skolnik, David, and Hartley, Miles C. Dynamic; Plane Geometry. New York* D. Van Nostrand Company, Inc., 1950. ------, • Dynamic Solid Geometry. New York* D. Van Nostrand Company, Inc., 1952. Smith, David Eugene. The Teaching of Elementary Mathematics. New York* The MacmlDLan Company, 1900. ______• The Teaching of Geometry. Bostons Ginn and Company, 1911. Spencer, William George. IflYgftttonal ggpfflgtrr* New York: American Book Company, 1876. Stebbing, L. A Modern Introduction to Logic. New York: Thomas Y. Crowell Company. Publishers, 1930. Stone, John C., and Mallory, Virgil S. Stone- Mallory Modern Plane Geometry. Chicago* Benjamin H. Sanborn and Company, 1929* Strader. Wm. W., and Rhoads. L.D. Modern Trend Geometry. Chicago* Winston, 191*0. Swallow, Kenneth P. Investigations into Geometry. Columbus: Educational Enterprises,1952. Swenson, John A. Integrated Mathematics with Special Appli­ cation to Geometry. Ann Arbor, Michigan* Edwards Brothers, Inc., 1936. 253 Bibliography continued.

Syllabus for Secondary Schools 1910. Albany, New York* Dhiveraity of the State of New York Bulletin No. 607, January 15, 1916. Taylor, E.H., and Bartoo, G.C. An Introduction to College Geometry. New York: The Macmillan Company, 19*+9. The Teaching of Geometry. Fifth Yearbook of the National Councilcf Teachers of Mathematics. New Yorks Bureau of Publications, Teachers College, Columbia Univer­ sity, 193O. The Teaching of Geometry in Schools. A Report Prepared for the Mathematical Association (British). Londons G. Bell & Sons, Ltd., 1923. (Fourth ed., 19M+). "Tendencies of Modern Mathematics." (From the Presidential Address of Professor E.W. Hooson before the mathemati­ cal section of the British Association at Sheffield, 1910). Educational Review. XL (December, 1910), pp. 52^30. Tobey, Wm. S. "Presentation of Plane Geometry Through Pupil Discovery." Uhpubllshed Master's Thesis, New York Uhiversity, 1928. Todhunter, I. The Elements of Euclid. New Yorks Mac­ millan and Company, 1896. Trump, Paul L. Geometry - A First Course. New York: Henry Holt and Company, 19^9. Ulmer, Gilbert. "Teaching Geometry to Cultivate Reflective Thinkings An Experimental Study with 1239 High School Pupils," Journal of Experimental Education. VIII (September, 1939), PP. 18-25. Veblen, Oswald, and Young, John Wesley, Projective Geometry. Vol. I. Bostons Ginn and Company, 1910. Weaver. Paul L. "Geometry of Four Dimensions." Tlhpub- lis hed Master's Thesis, The Ohio State Uhiversity, 19^0. Welchons, A.M., and Krickenberger, W.R. Plane Geometry, Rev. Ed. New Yorks Ginn ana Company, 19^9. Welkowltz, Samuel; Sltomer, Harry; Snader, Daniel W. Geometry - Meaning and Mastery. Philadelphia: The John C. Winston Company, 1950. 25V Bibliography Continued. Wentworth, G.A. Plan* *nd Solid Geometry. Bostons Ginn and Company, 1899. Wertheimer, Max. Productive Thinking. New Yorks Harper and Brothers, 19V5. Whewell, William. The Philosophy of the Inductive Sciences. Vol. I. Londons John W. Parker, West Straud, 134-7. Whitehead, A.N. An introduction to Mathematics. New Yorks Henry Holt and Company, 1911. Williamson, A.W. Plane and Solid Geometry on the Sugges­ tive Method. Rock Island, Ill.sLutheran Augustana Book Concern, 1901. Wilson, Jack Douglas. "Trends In Elementary and Secondary School Mathematics." Unpublished D. Ed. Dissertation, Stanford university, 19^9. Winger, R.M. "Generalisation as a Method in Teaching Mathe­ matics." The Mathematics Teacher. XXIX (May, 1936), pp. 2V1-50. Wyand, C.S. The Economics of Consumption. New Yorks The Macmillan Company, 193o. Wylie, C.R. Lectures. Mathematics 65^. The Ohio State uni­ versity , Summer7l9Vor^^^^^^^^ Young, John Wesley. Lectures on Fundamental Concepts of Algebra and Geometry. New Yorks The Macmillan Com­ pany, 1911.

Young, j „w .a . The .iwhliuL pf ttetUefaaUca lfl tiK filcnwUary and the Secondary School. New Yorks Longmans, Green and Company, 1920. AUTOBIOGRAPHY

I, Clarence Henry Heinke, was born In Marathon County, Wisconsin, July 13, 1912. My secondary education was ob­ tained at Wausau High School, Wausau, Wisconsin. From Capital Dhlverslty I received the degree Bachelor of Science in 1933. During the ensuing three years I taught in the high schools of Kings Mills and Ashtabula, Ohio. While in residence at The Ohio State University during the year l?1*-!-^, I was an assistant in the Department of Mathematics; the Master of Arts Degree was granted me in June 19*+2. In the summer of 19^2 I was commissioned an officer In the Uhited States Naval Reserve and served on active duty as an Aviation Electronics Officer until October, 19^5. From December, 19*+5, until August, 19^6, I was an Instructor of Mathematics and Engineering Drawing at the State Teachers College, Eau Claire, Wisconsin. Since then I have been on the faculty of Capital Uhiversity, Columbus, Ohio, as an Instructor in Mathematics and Engineering Draw­ ing from 19*+6 to 19^8 and as Assistant Professor since 19^3.

255