First year algebra

This First Year Algebra, based on similar textbooks written by the senior author in the past forty-five years, has been produced by the joint efforts of its three authors, all widely experienced teachers. It furnishes: (a) A selection of objectives and teaching units that em­phasize the usefulness of algebra in solving significant problems rather than abstract algebraic computation. (b) An organization of these units into a basic course, supplemented by related optional topics, which teachers can follow with confidence. (c) Effective instruction and practice that will enable the pupils to master the objectives of the basic course. (d) A unique interesting format that invites and facilitates study of the book. A. The dominant objectives of the textbook are: 1) Mastery of the meaning and use of formulas, equations, and numerical trigonometry. This is accomplished by solving problems about perimeters (pp. 9,17,25, etc.), areas (pp. 19-25, etc.), volumes (pp. 34, 36, etc.), familiar relations in buying and selling property (pp. 31-33, 74, etc.), uniform motion (pp. 27, 151, 249, etc.), mixtures (p. 80), and heights and distances (pp. 261 ff.). Problems about number relations (pp. 55, 64, 98, 148, etc.) are employed to develop skill in formulating the equations by which to solve "word problems" (pp. 46, 55, 56, 64, etc.). Several traditional types of problems (pp. 155, 158, 182, etc.) are marked by asterisks to indicate that they need not be included in a basic course. 2) Acquisition of the necessary knowledge of and skill in using: literal numbers (pp. 9, 12, 14, etc.), formulas (pp. 9, 10, 18, 21, 48, etc.), signed numbers (pp. 77 ff.), equations (pp. 41- 47, etc.), monomials (pp. 16-18, 28, 54, etc.), polynomials (pp. 115 ff.), functional relations (pp. 10, etc.), factoring and fractions (pp. 193 ff.), and numerical trigonometry (pp. 261 ff.). These objectives are introduced as means of attaining the objectives listed in paragraph 1). B. The organization, designed to secure understanding, appre­ciation, and skill, is guided by pedagogical rather than by strictly logical considerations. 1) The basic course is suitable for all pupils. The optional topics, marked by stars, are included for superior pupils and classes (pp. 53, 73, 153, 155, etc.). 2) Mastery of formulas is secured by solving problems. Special attention is directed to the consistent use of the procedure described on page 21. A by-product of this instruction is the acquaintance with literal numbers (p. 9), certain notation and vocabu­lary (pp. 9, 12, 14, 16, 28), and combining terms (pp.: 16, 54) — all needed when using formulas. 3) Equations are introduced as the means of finding the value of an unknown which appears in the predicate of a formula (p. 41). The various laws for solving simple equations are introduced as these are needed (pp. 41, 44, etc.). Skill in solving equations is secured by spiral introduc­tion of. the several difficulties (pp. 9, 96, 130, 145, 212, 216, etc.). 4) Preparation for solving certain types of problems (pp. 55, 64, 70, 98) precedes solution of such problems (pp. 56, 57, 65, 71, 73, etc.). C. Numerous effective teaching procedures, perfected after long use in earlier textbooks, help the pupils to attain the objec­tives. 1) Simple non-technical vocabulary, when possible, and short paragraphs characterize a style that pupils can read with understanding. 2) The course as a whole, the several chapters, and many of the sub-units are motivated (pp. 1-5, 38, 76, etc.). 3) Each chapter is introduced by a preview that gives pupils a general idea of the purpose of the chapter. 4) Each chapter is broken down into sub-units that are taught with care (pp. 58, 61, 83, 91, etc.). 5) Arithmetical difficulties are anticipated and overcome by appropriate diagnostic tests and remedial instruction (pp. 11, 13). 6) Concepts, vocabulary, and definitions receive meticulous treatment (pp. 8, 9, 12, 31, 41, etc.). 7) Each sub-unit is taught inductively to promote the pupil's discovery and understanding of the desired conclusion (pp. 20, 27, 44, 82, 92, 128, 202, etc.). These lessons have been perfected after long use in earlier textbooks by the senior author. They suggest (not prove) conclusions that are printed conspicuously within black-line frames. Designation of - these conclusions as "rules" has been avoided. 8) Basic assumptions of algebra and the logical derivation of the basic laws (pp. 358-360), while optional, are a desirable innovation in an elementary algebra. 9) Each kind of computation has been illustrated by a solution printed within a colored frame. These solutions pro­vide patterns that promote neatness, order, and accuracy (pp. 46, 54, 68, 97, etc.). The symbols A, S, M, and D, introduced In early textbooks by the senior author, help to avoid the me­chanical solution of equations. "Transposing'' has been abandoned. 10) Instruction and practice for each sub unit appear on one page or on two facing pages, an arrangement that obvi­ously assists when the sub unit is studieded (pp. 64-65, 96-97, 108, etc.). 11) Abundant initial practice accompanied the introduction of each sub-unit, all of it graded and relatively easy.