INTRODUCTION

The junior high school is recognized by its most ardent advocates as an immature institution. It has grown up during the last decade in all parts of the United States with astonishing rapidity. It came into being because there was a general demand for more productive teaching in the later years of the elementary school and a better administration of the social and personal lives of children in their early adolescent years. It came into being before suitable materials of instruction were at hand and indeed before those who recognized the need for a new institution were altogether certain what the characteristics of this new institution should be.

There are a few instances on record where the junior high school has been tried and abandoned because those who inaugurated it did not know what to put into the courses of study or how to relate this unit to the established units of the school system. There are numerous other instances where the junior high school has been only moderately successful because it was not adequately organized and equipped to fulfill its special functions.

As experience has accumulated, it has become increasingly evident that the ultimate success of the junior high school depends in a very large measure on the preparation of a new kind of teaching material.

This material must be made up of the fundamentals of high school subjects so arranged that they will gather up and review all of the results of elementary education and at the same time open the way to adult life either in the practical world or through higher and more completely differentiated studies of the senior high school. It is not enough that high school courses be carried back into the earlier years of the school's work, nor is it a solution of the junior high school's problem to mix, without genuine intellectual coördination, some of the materials which have heretofore been taught in the seventh and eighth grades with a few of the exercises which used to be given in the ninth grade or higher in the school curriculum. There must be a new and completely integrated body of instructional material capable of bridging the gap which has up to this time separated the elementary grades from later intellectual life.

The field of mathematics was one of the first in which teachers began to experiment in the effort to work out a true combination of elementary and higher materials. Numerous books have appeared which more or less successfully accomplished the purpose of introducing pupils to the fundamentals and applications of all the mathematical sciences.

Professor Breslich has a number of advantages in entering this field of experimentation. His books on combination mathematics for the senior high schools are far and away the most successful books of that type in the English language. They have been extensively used in the ninth year and upper years of the high school. Furthermore, Professor Breslich has had the

opportunity for the last six years of working with a seventh year class in the laboratory school of the University of Chicago where, through actual experimentation with a succession of classes which he has himself taught, he has refined his methods and materials to the point where they can now be offered to a wider constituency.

Two years ago, through a subsidy from the Commonwealth Fund, Professor Breslich was enabled to visit the leading junior high schools of the country and to make a study of their courses in mathematics. He has also conducted courses in the principles of teaching mathematics in the School of Education of the University of Chicago, and has in this way come into contact with experienced teachers and supervisors from all parts of the country.

This book is accordingly one of the maturest courses for junior high schools that has been prepared. It follows lines which Professor Breslich has long advocated in articles in the School Review and elsewhere and lines which the National Committee on Mathematical Requirements accepted in its report of 1923. It is full of exercises suited to the interests and intellectual abilities of adolescent children. It is a true fusion of arithmetic, geometry, and algebra. It has a background of careful sifting through practical use and criticism from a large number of teachers and students of the junior high school.

This book is to be followed by another, designed for use in the later years of the junior high school. The second book is completed and is equally based on trial and criticism. This and the second volume of the

series are presented by the institution which Professor Breslich represents, with the full confidence of his associates who have watched the development of his work and shared, to some extent, in its criticism.

August 7, 1924

CHARLES H. JUDD.

JUNIOR MATHEMATICS

CHAPTER I

WHAT IS MEANT BY A LINE SEGMENT

HOW TO MEASURE SEGMENTS WITH A RULER

1. The importance of measurement. Mathematics is one of the earliest sciences. All people have found it necessary to measure. Even the savages had to learn how to count and to measure their food supplies and other necessities. This was the beginning of mathematics. At first people measured in a very crude way by means of fingers or pebbles, but as civilization progressed, better and more accurate methods of measurement were needed and therefore developed. Today, everybody uses some kind of measurement in his daily tasks. Without the use of mathematics, we could not construct our modern machinery, our great buildings, railroads, bridges, or ships. We could not even carry on our daily business. Since modern civilization owes so much to mathematics and depends on it so largely, every pupil should study this subject.

Measurement plays a large part in mathematics because it is something everybody needs to know about. The two boys in Fig. 1 are using measurement in laying out a tennis court.