PENNSYLVANIA, v. L. L. KUHN, PRINTER TO THE COMMONWEALTH WH Avie” YEARS VII-XII HABRISBURG, COMMONWEALTH OF PENNSYLVANIA DEPARTMENT OF PUBLIC INSTRUCTION ~ Y) fal O =) pow im FH v2 < Sea % 1) ae Y) Sam aS =) OQ = . we J 732 > =, eee rte r7 eee pessceeerecsrsctces! SSSeieSsrssrecsssetes ess seca tos sale sbasiebecs sestseieieds | ¢ COMMONWEALTH OF PENNSYLVANIA DEPARTMENT OF PUSLIC INSTRUCTION comics STATE COUNCIL OF EDUCATION President and Chief Executive Officer Thomas E. Finegan Term Expires (Marion: haweards | ParkiPn ay eto vote ria es att eleameters Bryn Mawr July 1928 Mrs. Edward W. Biddle, BALLARD SAS EARL a ah ci oh an .--Carlisle July 1928 Homer: De VV STS ore Se ee re an ine an eMart Oe neg Pittsburgh July 1927 John C., Saye EA aR Bea) ) La rau eae OS paw iotahs Sept hua Philadelphia July 1926 © Mire! cB) Si. Ee Me Cantley eens ous beech UtOR e mere Beaver July 1926 Mere) TS. Rrentisss Nichols aes OU ee Or Re ve ia Philadelphia July 1925 HM roderiks RASMUSSEN Pees. oni ENING bonds wie eae ee eiaere Harrisburg July 1924 Fons) 2 MEATCUS WARPOT ae oes vk ti SGiede al POON Woaia! atediel oe Pittsburgh July 1924 rnests Laplace, NEI Dale) yee le a as eit ace ek ve Philadelphia July 1923 Thomas E. Finegan, M.A., Pa:D., Litt.D., L.B.D., LU.D. Superintendent of Public Instruction J. George Becht, M.A., SeD.. LL.D. ‘ James N. Rule, B.S., M.S. Albert W. Johnson, B.A, Helen J. Ostrander Deputy Superintendent, Higher Education Deputy Superintendent, Secondary Education Assistant to Superintendent (School Law) Secretary to Superintendent — DIRECTORS OF BUREAUS PV HASETA ELON coi U vase Sica Made le wate Wintwia: aa A ; .-Francis B. Haas, B.S., M.A. PATTON ANCE, oi nak ones glee Qiedared erie etna SNE cher Ade alk AS W. M. Denison, B.A., M.A. FReAIEH PAU CRETOR ooh wis a ois) chute alae bd ste hie tere erate bre Charles H. Keene, B.A., M.D. Pre-Professional and Professional Credentials, C. D. Koch, M.A., Pd.D., Litt.D. RHPA ROIOAETON, Me sash pune a Matet do ate es ws ol ubaiiila ee eLV aM waiod, Candee, Lee L. Driver, M.A., Li.D. School Buildings, FAT RRR cs ne Er) Am MA II HuBert ©. Hicher, Se.B.. A.B., M.S. Schoo] Employes’ Rativeiienh Ue SATORU it Pee LRA ay a tok mr 1 o Bged S Oe 'Baish, M.A. MDECIAL: MOUCATIOR o's) Gare cient oie aw SNAPS A plaee eeS Ok Dieta line ys Francis N. Maxfield, Ph.D. PORCH OT Wicca! wiawue 4 gee ele Wea an alk oie! oe ute: Wlpeig Albert L. Rowland, M.A., Ph.D. MOCATIOM AL ee EO a igtie dere, area a WNataibieKiata tT ACL IPRS. a aang L. H. Dennis, B.S. DIRECTORS OF SUBJECTS Apt FUCUCRTLOM, 0 OS ais ou Cai 2 MOEA LARK VAIN CARS ALPE PEAS Be ©. Valentine Kirby, M.A. BOTA TIBD | ch GIR day ioc iZat be ure eat ak AEG Ctl ha tea al Co une amr Orton Lowe, BS. Foreign Gn guaeee es see weg welne eke Pa adit G. C. L. Riemer, M.A., Ph.D. RFGOCHADINY Ace acre ua UME ae as Nid Wa OK hcg ieee Erna Grassmuck, B.S. in Ed. High (School Inspections iy Caieideiels + been asec: weeeis James G. Pentz, B.A,, M.A. unior:' Ach: Schoolay ic Bievelscati omvelbrs cy aete Win tial ates wes pains M. ST aK: B.A., M.A. Mathems ies, ics oi sie ere wi ate loc nntare Sah Oh a eLe eke th ba anak care tpi edly aeenie HBG Krant J. A, Foberg, B.S. EBT Ue SR crate POS AR ras ta a cae a Se . -Hollis Dann, Mus.D. School Libraries, oi5c:6.iesjes Heda eTy ibe Sis bie Mh tos ara a evel Sera Wut oon Adeline B. Zachert Social Studies, ........ RS oR EE Marea Se. Edge Na J. Lynn Barnard, B.S., Ph.D. Speech Improvement, ........ Re REE GA ESS RUC EVAL EA Si aR AE Helen M. Peppard, B.A. i) “> EQw' “ MATH -COMMONWEALTH- OF PENNSYLVANIA - - e DEPARTMENT OF PUBLIC INSTRUCTION COURSE oF STUDY IN © MATHEMATICS YEARS VIL-XII HARRISBURG, PENNSYLVANIA. J. L. L. KUHN, PRINTER TO THE COMMONWEALTH 1983 (2) i SD 5} P38ea 3 % e =o 2 ei SYLLABUS IN MATHEMATICS FOR HIGH a on SCHOOLS Oo a wert Seal GENERAL STATEMENT \) X Algebra and geometry originally college subjects. A study of the } mathematical courses and entrance requirements of early American , colleges shows that algebra and geometry were originally taught in < the colleges, and that later both subjects were dropped down bodily into the high schools and academies, becoming college preparatory subjects. The high schools and academies then taught identically the same courses that the colleges formerly had given, using precisely the same textbooks. The mathematics curriculum for the secondary school thus came to be organized from the standpoint of the adult mind, resulting in a formal, logical classification of subject matter. ‘ Present organization is traditional. Our high school mathematics r> . courses have traditionally been organized so as to keep each subject > separate from the others: algebra first, then plane geometry, and so ; on. This organization delays the pupil’s introduction to very signi- ‘ ficant fields of elementary mathematics, such as numerical trigo- .= nometry, the use of logarithms, and other tables, graphical methods, and elementary notions of statistics, and denies him the stimulus and satisfaction that comes with real applications of elementary mathe- e matics to the solution of problems. The conventional organization, furthermore, carries the study of the technique and content of these isolated subjects to a point of completeness beyond that which can be justified for immature learn- ers, Recommendations growing out of recent studies. The world-wide movement for the reorganization of mathematics courses and teaching that has been in progress for a number of years has resulted (in our ewn country) in reports and recommendations by such important bodies as the National Committee on Mathematical Requirements, acting under the auspices of the Mathematical Association of America, and the Commission on the Reorganization of Secondary Ednuzation, appointed by the National Education Association. The recommendations of these two bodies agree that it is necessary to a give up “logical” arrangement of subject matter, especially for intro- ductory work, and to find instead an organization based upon the snecessful attack of projects and problems in connection with which the pupils already have hoth knowledge and potential interest. They 4 likewise agree in advocating the inclusion of arithmetic, algebra, and geometry in the course of study extending through the ninth school year. General mathematics. Large numbers of pupils leave the high school by the close of the ninth school year. For these pupils train- ing and instruction in “general” or “composite” mathematics, com- prising the fundamental ideas of the various branches of elementary mathematics, are more widely applicable to the experiences of every- day life than are the training and instruction limited to algebra alone. For the pupils who will remain in school and continue the study of mathematics, “general” or “composite” courses provide the best foundation for the study of mathematics in the second and later years. Fundamental principles. It is agreed that the emphasis should be placed throughout on the immediate values of the mathematics subjects studied—not preparation for the study of more mathematics at a remote date in the future, but the acquisition of knowledge and training that will enable the pupil to understand the relations of quantity and space entering into his daily life now. The course should be so planned that if the pupil is compelled to drop out of school at any time before the end of the term. the subjects studied up to that time shall have given him the most valuable mathematical training that he could receive. So planned, the course will likewise be the most natural and effective introduction to mathematics for those pupils who will continue the study of mathematics. RESPONSIBILITY OF THE TEACHER Knowledge of subject. It has been said that mathematics teachers “teach the textbook” to a greater extent than any other group of teachers. A reorganization of materials and methods of instruction can become effective only as the teachers assume responsibility for the choice and organization of the subject matter and presentation. The teacher must know his subject thoroughlv, and be able to adapt materials and methods to the varying needs of pupils. No method of presentation, and no particular choice of subject matter, can be permanently valid. Choice and organization of subject matter. In the absence of text- books embodying general agreement as to the choice and organization of the materials of instruction recommended for courses in composite mathematics, the teacher must assume responsibility for .choosing the needed arithmetic material from any good arithmetic text, the algebra material from any good algebra text, and the geometry mater. ial from any good geometry text. To teach high school mathematics so as to give the pupil the best mathematical training he is fitted to 5 receive—resulting in the broad acquaintance with mathematics as a whole that will give him an understanding of the relations of quantity and space growing out of his every-day environment, and that will enable him to decide wisely whether or not to continue the study of mathematics—calls for real skill in the teacher. To meet this de- mand is the inspiring task of the mathematics teacher of today. Professional contacts. No teacher of mathematics can do his work adequately unless he keeps in touch with the movement now going on everywhere in the interest of reform and improvement in the teaching of mathematics. This demands interest and participation in the work of such organizations as the Mathematical Association of America, the Association of Mathematics Teachers of the Middle States and Maryland, and the National Council of Teachers of Mathematics. The thoughtful reading of professional literature, including books and periodicals, is absolutely essential to successful teaching of high school mathematics. In particular, it is assumed that teachers will become familiar with standard practice material and with standard tests in high school mathematics subjects, and that they will employ both types of material in classroom practice. MATHEMATICS OF THE JUNIOR HIGH SCHOOL Fundamental principles. In the junior high school, comprising years seven, eight, and nine, the mathematics work should give the pupil as broad an outlook on the whole field of mathematics as he is able to comprehend, to the end that he may try out his capacities and aptitudes and learn whether or not he wishes to go more deeply into the study of mathematical topics and to take up the study of scienti- fic and technical subjects that demand extensive mathematical equip- ment. In addition, the mathematics work of these years should pro- vide the acquaintance with fundamental notions of elementary mathe- matics that has come to be regarded as essential to intelligent citi- zenship in the world of today. Content. The most authoritative opinions available, as expressed in the report and recommendations of the National Committee on Mathematieal Requirements, agree that the mathematics course in the junior high school should comprise arithmetic, the elementary notions of intuitive geometry and of algebra, and numerical trigo- nometry. The following paragraphs present, in outline, the topics that may appropriately be included under each of these headings. CONTENT OF JUNIOR HIGH SCHOOL MATHEMATICS Arithmetic A Practice in the fundamental operations applied to integers and fractions, common and decimal, should be continued until standard 6 proficiency is attained. When pupils attain standards, they should be excused from regular drill work. 1 In working with fractions, the emphasis is to be placed on simple fractions: such as 4, %, %, 14, 34, %, 1%, et cetera. 2 The decimal equivalents of fractions most commonly used should be fixed in mind, and the process of reducing any fraction to a decimal should be automatized. Rapid drill in one-step operations should be used for speed and accuracy. Material for this purpose may be of the following type: Ie SOLU eee 207 OL OU a 12 = What decimal part of 60? 12 = What fractional part of 60? > The accurate placing of the decimal point in the process of division should be made automatic by fixing the habit of writing the quotient over the dividend, and of making the divisor an integer by multiplying, if necessary, both divid- end and divisor by the appropriate power of ten. 4 Simple short cuts in multiplication and division: such as re- placing multiplication by 25 by multiplication by 100, and division of this result by 4. In the solution of problems, care is to be taken to pass froin the solution of the particular concrete problem to tue form- ulation of a general rule. The sequence, starting with the problem stated in numbers, through the problems stated in letters, the rule, the formula, and the problem without numbers, may be illustrated by the following series. The differences in degree of difficulty vary from step to step, and care must be taken that the pupils are not unduly hurried in the generalizations. a The length of a sidewalk is 50 ft., and its width is 3 ft. What is its area? b The length of a sidewalk is / ft. and its width w; What is A (the area) in terms of 1 and w? e State the rule for finding the area when length and and width are known. d. Write the formula for A, when ] and w are given. Write the formula for 1 when A and w are given. e The length and width of a sidéwalk are given—How may I figure the cost of laying it, if the cost of one square yard is given? 6 In general, problems involving long computations should not be done in class time. Cr t B Tables of weights and measures in common use. United States money, avoirdupois weight, dry measure, liquid measure linear measure, square measure, cubic measure, time measure. C Percentage 1 Fixing the equivalence of meanings of the symbols for frac- tion, decimal, and per cent. Graphs and diagrams to be used as helps in visualizing these ‘equivalences. 2 Interchanging common fractions and percentages, finding any per cent of a number, finding what per cent one number is of another, finding a number when a certain per cent of it is known; such applications of percentage as come within the pupil’s experience. 3 Much material must be provided for quick mental work, of the following type: 50% of 200=— ? terol ou == + 40 is what per cent of 200? Application of percentage a Interest The general method of figuring interest should be taught, and also the use of interest tables. Thrift and interest, studied in connection with savings accounts. Com- pound interest. b Profit and loss. Stress the need of care in choosing the base on which to compute the per cent of prifit or loss. Usually this base is the cost: sometimes the sel ing price is used. ¢ Commission. To be computed on the amount of purchases or sales handled by the agent. d Discount. Reasons for “2% off for cash” Successive discounts. 5 The following sequence illustrates desirable work to be done under the applications of percentage. Care must be taken to recognize the increase in difficulty as soon as general- ization is undertaken, and to anticipate these difficulties. a es eae pen alt Pda should first be simplified to —x-+8& = —4x-++9 and then, by adding 4x to both sides, and subtracting 8 from both sides, we have ox == 1 (b) Simple cases of quadratic equations that arise in solving prob- lems and in handling formulas. (c) Sets of equations in two unknowns, limited to pairs of linear equations. (d) Applications of ratio and proportion to simple cases of simi- larity and other problems of ordinary life. The proportion should al- ways be written as an equation between two fractions, and in solv- ing the proportion, it should be treated as an equation. The terms “means,” “extremes,” “antecedent,” “consequent,” should be discarded in favor of “numerator” and denominator.” 19 Algebraic Technique. The Four Fundamental Operations. Their connection with the processes of arithmetic should be made clear. Multiplication and division should rarely involve multipliers or divisors of more than two terms. “Nests” of parentheses should not be treated, because of their rare occurrence in practical applications. Literal equations should be treated only to the extent necessary for manipulating formulas. The habit of verifying solutions should be established. Factoring. The only cases that need to be treated are: 1. Monomial factors. 2. The difference of two squares. 3. The square of a binomial. Skill acquired in factoring other cases—as the sum of two cubes, for instance—is wasted because no opportunity is ever presented for its use in applications of mathematics to real situations. Fractions. The four fundamental operations should be applied only to simple cases, throughout the course, avoiding complicated forms that are never met with in practice. The use of “cancel” as a technical term should be postponed until late in the year, using instead the phrase “divide numerator and de- nominator by.” ax x Errors of the type— = — should be forestalled by in- ayt2 yz sistence upon division of both terms of the fraction in this simplifi- cation. Separate treatment of “least common denominator” and “highest common divisor’ should not be given. “Clearing an equation of fractions” should be rationalized as indicated on page 13. Somplex fractions should be restricted to such as are not more difficult than a C psa m Pp nm og Haponents and Radicals. The work done on exponents and radi- cals should be confined to the simplest material required for the treatment of formulas. Proofs of the laws for positive integral exponents should be in- cluded. Care should be taken in this connection to ensure under- standing of the facts dealt with, by illustrations from the arithmetic 20 \: side. Thus appreciation of the truth of 2? x 2? = 2°, by translat- ing into 4 x 8=382, should precede learning x* x°==x*T?, The meaning and use of fractional and negative exponents should | ‘be considered in connection with handling formulas. With some classes, it will be possible to include an elementary discussion of logarithms and the slide rule. The consideration of radicals should be confined to transforma- tions of the types: - fab = a/b and \/a/b =1/b\/ab. A process for finding the square root of a number should be taught (see page 14), but time should never be given to extracting roots of -algebraic polynomials. Problems. Most of the emphasis now frequently placed on formal exercises should be transferred to the solution of problems. Prob- lems should be “practical” so far as the maturity of the pupil per- mits. Problems should always be “real” to the pupil, should connect with his ordinary thought, and be within the world of his interests and experience. A conscious effect should be made, in the selection of problems, to correlate the work in mathematics with the other courses in the curriculum, particularly with the courses in science. Numerical Trigonometry. In view of the fact that this sub- ject has been recommended for inclusion in the College Entrance Examination Board Examination in Elementary Algebra (Part I), Algebra to Quadratics, it probably will be considered desirable to include numerical trigonometry with algebra in all schools devot- ing the whole ninth year to algebra. An outline will be found on page 14.’ TENTH YEAR MATHEMATICS Election and Guidance. It is assumed that the ninth year’s work in mathematics is the final year’s work in required mathematics. Later work in this subject will be elective, and elections will be made under the guidance of competent teacher-advisers. This assump- tion of adjustment to the individual needs and aptitudes of pupils is increasingly justified as proper recognition is given to the im- portance of guidance activities in the high school. Demonstrative geometry. On the basis of the work done in pre- ceding years, whether.in junior high school or in the elementary school and ninth year in the four-year high school, the pupil who elects to continue the study of mathematics will be ready to take up the study of demonstrative geometry. In some schools the training in intuitive geometry already secured will enable the pupil to cover 21 the work in plane and solid geometry in one year. In this ‘case the greater part of this year will be devoted to plane geometry, and the iesser part (say, about one-third) to solid geometry. The work in demonstrative geometry should always be preceded by work in intutive geometry, such as is outlined on page 9 above If the pupils beginning demonstrative geometry have not already had this type of work, a short time (say, three weeks) should be devoted to such introductory work before proceeding to the strictly demon- strative type of work. , Assumptions and Theorems for Informal Treatment. This list contains propositions which may be assumed without proof (postulates), and theorems which it is permissible to treat informally. Some of these propositions will appear as definitions in certain methods of treatment. Moreover, teachers should feel free to re- quire formal proofs of some, if they desire to do so. The precise wording given is not essential, nor is the order in which the pro- positions are here listed: 1 Through two distinct points it is possible to draw one | straight line, and only one. 2 A line segment may be produced to any desired length. The shortest path between two points is the line segment joining them. 4 One and only one perpendicular can be drawn through a given point to a given straight line. The shortest distance from a point to a line is the per- pendicular distance from the point to the line. 6 From a given center and with a given radius one and only one circle can be described in a plane. 7 A straight line intersects a circle in at most two points. S