Abilene High School - Orange and Brown Yearbook (Abilene, KS)

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completing the square, Q33 The use of a formula developed from solution by square root. In addition to this, a study is made of the features of the equation which determine the nature of the roots. The graph is used, not only as an introduction to the study of equations, but also as a means of relating Algebra to Geometry. Considered in this light, the first degree equation corresponds to the straight line, and the quadratic becomes the Algebraic interpretation of two-dimensional space. The solution of equations by the use of graphs, that is, the determination of points which satisfy the required conditions, gives a new meaning to the locus of Geometry. The object of the course is not to present Algebra as a complete and finished subject, but rather to impress the pupil with the idea that while he has gained enough knowledge to be of valuable assistance, he really has mastered only the a, b, c of mathematics, and that there lies ahead of him a world of thought which will challenge every faculty of mind and imagination to conquer it. PLANE GEOMETRY Geometry is both a mental and a physical science. It is concerned with every- thing which occupies space, for it is the study of forms. In this respect it is the business of Geometry to investigate and classify figures according to their common properties, and to formulate the laws which determine their relationship to each other. From this standpoint Geometry serves to point out the common laws which are in force in all forms of nature. For instance, take the regular hexagon, which may be made up of six triangles, the sides and angles of which are all equal. This figure is the basis for all the various forms of snow-flakes, the cell of the bee is in- variably hexagonalg the blood vessels of the human body under unusual pressure are forced into such a shape that a cross-section takes the shape of a hexagon, and an orchard may be planted to the best advantage by laying out the ground in a series of equilateral triangles with a tree at each vertex. A close observation will discover the fact that wherever in nature conomy of space is needed, the hexagon is the form chosen. Furthermore, Geometrical forms are the basis of architecture, of painting, and in fact, almost every other art, and a knowledge of them is necessary for an aD- preciation of the elements of proportion and symmetry wherever these occur. Ac- cordingly, drawing is an important part of the work in Geometry, as there is no bet- ter method of becoming familiar with the properties of a figure than by constructing it accurately. But Geometry is also a mental science in that the truth of its theorems- is es- tablished not by observation or measurement, but by a rigid process of reasoning, commonly called the demonstration Certain assumptions are made, and on these as a foundation, principles are carefully and logically worked out. Nothing is ad- mitted to be true merely because it looks reasonable, and on the other hand, nothing is accepted contrary to the guidance of common sense merely because a proof con- vincing in appearance has been worked out. From this point of view the object of Geometry is, as it has been for several hundreds of years, to train the brain to an appreciation of clear cut, logical thinking, unprejudiced by feeling. This is not ac- complished, of course, by the mere memorizing of proofs, but rather by original work, so arranged that the argument consists! of but one simple step at first, but gradually becomes more difficult. The demonstration of theorems- given in the text-book is required, in order to see how well the pupils have followed and under- stood the reasoning, but exact reproduction is neither demanded nor encouraged. In fact, some of the more complicated theorems are discussed in class, and the student required to master only certain points of the proof. Numerical exercises, While they do not call for any careful or sustained reasoning on the part of the pupil as a rule, are of value in that they help to make clear and fix in mind the more important theorems, and also show the applicability of the algebraic formula to the work of Geometry. The note book is a feature of the work enjoyed by neither teacher nor pupil, but is useful in that it emphasizes concise, accurate statements and neatly drawn figures. SOLID GEOMETRY The Geometry of two dimensions is studied today before that of three dimen- V TWENTY-THREE

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girl enter High School. At High School age the most important factor in education is the development of the newly awakened self-consciousness and individuality of the pupil. Algebra and Geometry function largely in this development in that they set before the student simple, definite examples of clear, accuiate, logical reasoning, and train him to rely on his own judgment rather than the authority of some one else. ln a more particular sense, they form an excellent introduction to scientilic study in general, for the reason that of all the sciences, 11l2LLil8ll1HLlcS is the most nearly exact in method and by far the simplest in material. r'tli-thermore, mathe- matics furnishes the language in which the results of all scientific investigation are finally stated, namely, the algebraic formula. The Freshman course in Algebra is an introduction to mathematics considered in this larger sense as a science and not as a body of rules by which problems may be Worked. 'lhe numbers and concrete terms of Arithmetic are replaced by letters, and the laws of the four combinations familiar in Arithnletic, addition, subtraction, multiplication and division, are applied to them. The use of letters instead of nunl- bers brings out the tact that expressions tend to group LHQIHSGIVGS under a lew dis- tinct types-for example, a may represent any whole number whatever, a over b any fractional number. 'lo classify problems under their type forms, and to discover and put to use the principle by which each type is handled, then becomes a very large part of the work. Fractions are treated very Illlllliil as in Arithmetirs, and about the same share ot time is given to tllein as in the average year s work in Arithmetic. The statement OL problems in Algebraic terms is emphasized in the chapters on the equation, and is very inlportaiit ln that it emphasizes the necessity or a definite, precise understanding OL what one has 1'6k:Lll. rowers and ioots are also discussed, as these are employed in the S0111L10I1 of the quadlatic equation, which is the last new topic studied in the year. 'l he solving of 13100161115 is really the least important pa1't of the course, as this is largely 11l6C1li:l.1l1C2l.1, and may DS purely iinita- tive. ALGEBRA II Algebra in the High School should be taught from two standpoints, first, as a tool, and second as a type of thought. From the first point of view, it is the most powerful instrument yet devised for handling the problems of science. Accordingly, the formation and use of the formula and equation should become so familiar to the student as to be automatic. To this end, constant repetition and drill are neces- sary, and consequently are a large and important part of the third year of Algebra. One point in which the state text is weak and which must be guarded against is the constant use of the same letters. Very frequently a pupil can think and work only in terms of 'xv and y, and has to translate the and V of Physics into those letters before he can proceed any further. For this reason, the literal equation such as D-RT, in which each letter is to be expressed in terms of the others, is a topic which requires particular emphasis. Another topic which is valuable on account of its usefulness and applicability is that of logarithms. ln logarithms, addition and subtraction take the place of multiplication and division, while the latter take the place of powers and roots. Thus the cube root of a number may be found by dividing its logarithm by 3. Very fre- quently, the work in Algebra is so crowded that there is not sufficient time left for logarithms. In this event the topic may be included in the course in Solid Geometry. Viewing algebra as a type of thought, its chief characteristic is the substitution of symbols for concrete things. The parts of Algebra in which this idea is brought out most clearly are those dealing with the theory of exponents and the quadratic equation. In the former, powers and roots, radicals, fractional and negative expo- nents and imaginaries are shown to be merely a development in various forms of such simple expressions as A.A equals A squared, and K-AJ squared equals A squared. To be sure, these topics are discussed in Algebra I, but the student is much better able in his Junior year to get a clear understanding and appreciation of the process of reasoning by which their relationship is established. The quadratic equation, however, is the most striking instance found in elementary science of the Way in which a single expression may serve as a type for a multitude of forms. The solu- tion of the quadratic is developed in three ways Q19 Factoring, Q21 Square root-or, TWENTY-TWO

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TWENTY-FOUR g ,.,1 SNAPSHOTS IN THE BUILDING