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A sequential but non-axiomatic high school geometry course which includes Euclidean, transformation, and analytic geometry and vectors and matrices, and emphasizes the invariance property of transformations, is outlined. Sample problems, solutions, and comments are included. (MN)

Graphical solutions are illustrated for several algebra problems including finding roots of a quadratic equation, solving mixture and motion word problems, factoring the difference of two squares, and constructing the square root of a positive number. (MN)

Evidence on the decline in preparation of entering calculus students and the relationship to high school preparation is presented, focusing on the trend toward the de-emphasis of trigonometry and analytic geometry in favor of calculus. Data on students' perception of the adequacy of their preparation are also presented. (Author/MN)

Relationships between elementary analytic geometry and several types of number sequences are discussed. (JT)

Three constructions using thread are given for the hyperbolic paraboloid in addition to a method for constructing an approximation for the third-order saddle surface. (MN)

A method for finding the area of a circle is described which involves finding points on the circle with the aid of a calculator and then applying the formula for the area of a polygon. (MN)

The free-energy change, or binding energy, of an idealized bubble cluster is calculated on the basis of one mole of gas, and on the basis of a single bubble going from sphere to polyhedron. Some new relations of bubble geometry are developed in the course of the calculation. (BB)

Mathematical theorems about the properties of ellipsoids are developed. Included are Poisson's theorem concerning the magnetization of a homogeneous body of ellipsoidal shape, the polarization of a dielectric, the transport of heat or electricity through an ellipsoid, and other problems. (BB)

An approach to the proof that not all angles can be trisected which is elementary enough to present to senior level high school students as well as undergraduates is given. (MN)

Outlines mathematical topics of use to college geography students identifies teaching methods for mathematical techniques in geography at the University of Leeds; and discusses problem of providing students with a framework for synthesizing all content of geography education. For journal availability, see SO 506 593. (Author/AV)

The last 13 of 32 exercises of a course designed to give a wide range of skills in deductive, inductive, and indirect methods of proof in geometry are presented. (MP)

A mathematical derivation is given, developing the cardioid as an epicycloid locus. Curve-stitched designs are given for a family of epicycloids. (MP)

The solution of cubic equations is obtained by constructing intersecting conic sections. Details for a particular cubic are given. (MP)

Elucidated is the relationship among three threads of mathematical investigations: Kirkman's schoolgirl problems, finite geometries, and Euler's n-square officer problems. (JP)

The "Test Development Notebook" is a resource designed for the preparation of tests of literal comprehension for students in grades 1 through 12. This volume contains 200 multiple-choice cloze exercises taken from textbooks in mathematics, and the accompanying answer key. Each exercise carries the code letter of the section to which it belongs. The code letter is followed by the difficulty level of the passage as determined by application of a readability formula. Supplemental, small-print codes on the exercises give additional passage information as explained in SPPED Form 86, the "Handbook." The "Handbook" is the guide to the "Test Development Notebook," and users should review it before starting to work with the multiple-choice cloze exercises in this volume. (Author/BW)

Equations for the conic sections are given, and properties of the conics are discussed. The contributions of Greek mathematicians are reviewed. (DT)

Given two interesecting lines, the "pencil" of this interesection is the family of all lines passing through the point of intersection. Presented is a characterization of all lines both in the plane and lying in opposite sectors of the plane as determined by one of the original intersecting lines. (JP)

The 1959 Royaumont seminar "New Thinking in School Mathematics," having agreed on the need for modernization, recommended that a second group of experts work out detailed synopses of the entire subject matter of secondary school mathematics. This book is the report of the second seminar and contains the Dubrovnik Program which stimulated much curriculum development in Europe at the beginning of the last decade. The seminar concentrated on the needs of the most gifted students (the upper half of the European Lycee or Gymnasium). Synopses for algebra, geometry, and for probability and statistics are presented in two cycles, the first for ages 11-15 and the second for ages 15-18. A statement of the goals and prerequisite mathematics precedes the list of topics recommended for each subject in each cycle. This is followed by commentaries clarifying the spirit and scope of the newer topics. The major commentaries cover: the role of the theory of sets in secondary school mathematics; teaching first-cycle geometry to provide experience of a wide variety of concepts and properties, leaving their logical arrangement to the second cycle; groups, rings and fields; vector and linear algebra; axioms for affine space and for the affine plane; and the development of probability and statistics. Some aspects of the teaching of calculus are reviewed in an appendix. (MM)

The teacher's guide to Unit 9 covers the second chapter on perpendiculars and parallels in this SMSG series and the chapter on coordinate geometry. The overall purpose for each of the chapters is described, the prerequisite knowledge needed by students is specified, the mathematical development of each chapter is detailed, behavioral objectives are stated, and a time schedule is suggested. Background information for key concepts, answers for all exercises in each chapter, and suggested test items are provided. For other teacher's guides and student texts in this series, see SE 016 852 through SE 016 879. (DT)

The first chapter, Perpendiculars and Parallels (II), of the ninth unit in this SMSG series includes a discussion of the properties of triangles, circles and perpendiculars, parallels in space, perpendicular lines and planes, and parallel planes. The next chapter, on coordinate geometry, covers distance; midpoints; algebraic descriptions of subsets of the line, the plane, and space; separation; graphs of non-linear functions; parallel and perpendicular lines; coordinate proofs; and coordinate systems in three dimensions. For the teacher's guides and for other student texts in this series, see SE 016 852 through SE 016 879. (DT)

A generative computer-assisted instruction system is being developed to tutor students in analytical geometry. The basis of this development is the thesis that a generative teaching system can be developed by establishing and then stimulating a simplified, explicit model of the human tutor. The goal attempted is that of a computer environment adaptively responsive to the varieties of human personality. Housekeeping routines to handle input and output and format translations within the computer have been developed. Algorithms are used in the generation of problems; and answers are calculated, sometimes by simple expedients. Diagnostic dialogues to identify the specific causes of student errors are also produced by generative algorithms. Remedial routines to correct the deficiency in the student's repertoire of skills have not been prepared. Utility routines to assist the student upon his command are available. In summary, the notion of generative computer tutoring is thought to be a valid step forward for subject matters which have the degree of logical organization necessary for algorithmic evaluation. (MF)

Reported are the results of a study to determine if there is a difference in learning as measured by an achievement test between high school students who study plane analytic geometric topics via a vector approach and those who study the same topics via a traditional approach. Secondary objectives concerned the transfer to further topics in solid analytic geometry, and the effect of ability and achievement levels on learning. All instruction utilized programed units. Results on an achievement test following a unit on vector algebra determined high and low groups. Half of the high and low groups were assigned to each of the two treatment groups. The vector approach students performed significantly better (.05) than the traditional approach students on the transfer test and the criterion test (designed to measure achievement and understanding). The high and low grouped vector approach students did significantly better than the corresponding traditional groups in understanding and transfer. The high vector students did significantly better than the high traditional students on the achievement measure. (Author/RS)

PRESENTED IS A FUSED COURSE IN PLANE, SOLID, AND COORDINATE GEOMETRY. ELEMENTARY SET THEORY, LOGIC, AND THE PRINCIPLE OF SEPARATION PROVIDE UNIFYING THREADS THROUGHOUT THE TEXT. THE TWO CURRICULUM GUIDES HAVE BEEN PREPARED FOR USE WITH TWO DIFFERENT TEXTS. EITHER CURRICULUM GUIDE MAY BE USED DEPENDING UPON THE CHOICE OF THE TEACHER AND THE NEEDS OF THE STUDENTS. GUIDE A IS PRESENTED IN 11 UNITS WITH ONE INTRODUCTORY UNIT ON SETS. UNITS ARE AS FOLLOWS--(1) POINTS, LINES, PLANES, AND SPACE, (2) TRIANGLES, (3) ESSENTIALS IN PROOF, (4) INTRODUCTION TO FORMAL PROOF, (5) PARALLEL AND PERPENDICULAR LINES AND PLANES, (6) SIMILARITIES, (7) COORDINATE GEOMETRY, (8) POLYGONS AND POLYHEDRONS, (9) CIRCLES AND SPHERES, (10) AREAS OF PLANE FIGURES, AND (11) AREAS AND VOLUMES OF SOLID FIGURES. GUIDE B IS PRESENTED IN 13 UNITS AS FOLLOWS--(1) ELEMENTS OF GEOMETRY. (2) INTRODUCTION - A METHOD OF DISCOVERY,(3) DEDUCTION AND PROOF, (4) ANGLE RELATIONSHIPS AND PERPENDICULAR LINES, (5) PARALLEL LINES AND PLANES, (6) CONGRUENT TRIANGLES, (7) SIMILAR TRIANGLES, (8) CIRCLES, ARCS, AND ANGLES, (9) METHODS OF COORDINATE GEOMETRY, (10) PROOFS USING COORDINATE GEOMETRY, (11) AREAS OF POLYGONS AND CIRCLES (12) AREAS AND VOLUMES OF SOLIDS, AND (13) CONSTRUCTIONS AND LOCI. CROSS REFERENCES ARE GIVEN IN EACH OF TWO TEXTBOOKS USED. THE SYLLABUS IS BASED ON MUCH OF THE THINKING OF THE COMMISSION ON MATHEMATICS OF THE COLLEGE ENTRANCE EXAMINATION BOARD AND THE SCHOOL MATHEMATICS STUDY GROUP.

A GUIDE FOR A 1-YEAR COURSE IN DIFFERENTIAL AND INTEGRAL CALCULUS PREREQUISITED KNOWLEDGE IN ALGEBRA, ANALYTIC TRIGONOMETRY, AND ELEMENTARY ANALYSIS. EACH ASSIGNMENT CONTAINED BOTH NEW AND REVIEW WORK TO REINFORCE THE NEW WORK. THERE WERE ELEVEN UNITS OF STUDY USING THE FOLLOWING FOUR BOOKS--"CALCULUS AND ANALYTIC GEOMETRY, THIRD EDITIONS," BY G.B. THOMAS, "LIMITS" BY G.B. THOMAS, "CALCULUS" BY TEMAC. UNIT ONE TAUGHT THE RATE OF CHANGE OF A FUNCTION, UNIT TWO, THE DERIVATIVES OF ALGEBRAIC FUNCTIONS, UNIT THREE, THE APPLICATION OF MEAN VALUE AND OTHER THEOREMS TO CARVE PLOTTING, MAXIMA AND MINIMA, ETC, UNIT FOUR, INTEGRATION, UNIT FIVE, TRANSCENDENTAL FUNCTIONS, UNIT SIX, TECHNIQUES OF INTEGRATION, POPULAR COORDINATES ARE TAUGHT IN UNIT SEVEN, UNIT EIGHT, FURTHER APPLICATION, PLANE ANALYTIC GEOMETRY, VECTORS AND PARAMETRIC EQUATIONS, AND SOLID GEOMETRY MADE UP UNITS NINE THROUGH ELEVEN.