SUGGESTIONS TO TEACHERS CHAPTER I is given for providing a concrete approach to the subject, for acquainting the student with the uses of the tools of geometry, for developing a body of experience and imagery as a basis of demonstrative geometry, and for introducing definitions, axioms, and other fundamental ideas. Impress upon the student the fact that all axioms and other principles printed in italics must be memorized for future use. The amount of time to be devoted to this chapter depends upon the previous training of the students. Each student should be provided at the beginning of the course with a straightedge, compasses, protractor, and triangle. These instruments may be obtained from supply houses at small cost. The use of these instruments by the student in the accurate constructions of the figures needed throughout the course should be insisted upon. An accurately constructed figure often will suggest the proof of a theorem, while a poorly constructed one often is misleading. In beginning the study of proofs in Chapter II, teach thoroughly the matter in § 30. Emphasize the dependence of the proof of a theorem upon a knowledge of the definitions, theorems, etc., which have preceded it in the text. Hence, urge the student to memorize, not only definitions, etc., but also the exact wordings of theorems (not the proofs) in the text as they are studied. In the proofs of theorems, the student should be required to give the reason for each step in full, and not make merely a reference to a previous section. In oral proofs, when previous theorems are referred to, they should be quoted exactly. When studying a theorem demonstrated in the text, encourage students to seek other proofs, if such exist, thus developing originality and independence. Progress slowly with the work, until the nature of a proof is well established. Three types of proof are employed in every geometry text: (1) simple direct proof; (2) indirect proof; and (3) proof involving the superposition of figures. It is in order to avoid the introduction of all three of these types of proof in the first few theorems that, at the beginning of Chapter II, the present text assumes from observation and measurement the equality of corresponding angles made by a transversal of parallel lines. This leads to an easy sequence of demonstrations by the simple direct method of proof alone. Thus the student encounters and overcomes only one difficulty at a time. The traditional assumption, in textbooks, of the so-called Axiom of Parallels—that through a given point only one line can be drawn parallel to a given line-is believed to be unpsychological, for it leads to a sequence of theorems that require all three types of proof at the beginning of the subject, and in very close succession. This makes geometry so difficult at the outset that a student is apt to become discouraged and get a dislike for the subject. In the present text, the simple direct proof is introduced in § 27; the indirect proof is not encountered until § 52; while the proof by superposition, which to most students seems the most subtle of all, is not introduced until Chapter III. In addition to being more psychological, the plan of this text is as mathematically sound as that which employs the Axiom of Parallels, for in each case a single mathematical assumption is made. If either assumption is made, the other may be proved. (See Corollary, § 46.) No student is expected to solve all of the exercises in the text. Select those which are best suited to the powers of the class. Those which are omitted at first may be used later, in connection with reviews, if desired. Have the student study carefully the viii SUGGESTIONS TO TEACHERS "Methods of Attack" given in Chapter VII (§ 149 and the following exercises). Much that is given in this chapter should be presented from time to time by the teacher, as need of it arises in the early exercises. Special study, however, is expected to be given to this matter before taking up the extensive list of miscellaneous exercises for review at the end of Chapter VII. Teachers are urged to use as many of the applied problems as time will permit. None of them is too technical for the average boy or girl. The construction of instruments and their use in out-of-door problems, suggested at different points of the book, will prove quite practical and of great interest and value to the student. It is through this practical work in geometry that the subject is to be made to function extensively in the life of the student. The plural of any symbol representing a noun is obtained by affixing the letters. Thus, & represents angles. ABBREVIATIONS Ax., axiom. Ext., exterior. Alt., alternate. Comp., complementary. Corres., corresponding. Def., definition. Hyp., hypothesis. Rect., rectangle. St., straight. Supp., supplementary. |