CHILDREN'S ARITHMETICS BY GRADES GLOBE SERIES SECOND BOOK THIRD YEAR FUNDAMENTAL OPERATIONS BY WILLIAM E. CHANCELLOR, A.M. GLOBE SCHOOL BOOK COMPANY NEW YORK AND CHICAGO 6. The proper use of the printed page is the greatest of all W. T. HARRIS, LL.D., - From Address before the National Educational Association, 1896. MANHATTAN PRESS NEW YORK PREFACE This book is intended for boys and girls who know the numbers from one to thirty thoroughly, who can count to one thousand, who know something of the multiplication tables of two, three, four, five, six, ten, and twelve, and who understand the simplest facts about ratio and fractions. For such boys and girls there is here about a year's work. How should boys and girls study numbers? The interrelations of number-facts and of number-principles are such as to make prog. ress very slow and very difficult through their intricate maze. Is there any Ariadne's thread to follow through the labyrinth of numbers ? Is number ratio or counting? Is it comparison, or magnitude, or multitude? Is it a logic of thought, which can be analyzed after the topical style, -- addition, subtraction, multiplication, division, rule of three, and so forth, — of which we may complete one part before beginning the next ? Shall we learn every discoverable fact about twenty before taking up twenty-one, or every conceivable fact about of g of of 13 before taking up liquid measure ? This book is both “topical ” and “spiral” in plan. Its substance is both ratio and counting. Its purpose is to conform numbers in their facts and principles to the usual processes and powers and interests of children's minds. The graded reader has opened the way for the graded arithmetic. Grading all books is part and parcel of the new education, which means to discover and to obey the facts of the child-mind, its methods, nascent periods, and order of growth. The core of the concentric theory is recognition of the value of finding something that is known even in the mass of the unknown. Let us not hesitate in schoolbooks as we do not hesitate in life to branch out into the new and to return again to the old. Because ratio is the root and numbering is the top, let us not forget reasoning, which is the main trunk of arithmetic. The child's knowledge of arithmetic should grow as evenly in all directions as the most careful and the most open-minded education can secure. Progress in education is largely a matter of progress in power to understand books. Oral instruction may be continued too long as the sole medium for imparting knowledge: This book is rather for reading and study than for the setting of many exercises in writing figures. It calls for oral expression far more than for written work ; but it is meant to call most for the quiet, studious effort of the child to think through the number-processes for himself in the light of the instruction of the teacher and of these pages. Many minds, of adults as well as of children, cannot at once comprehend principles and facts explained orally. We often need to see the printed words, and slowly and patiently to think out their truth and meaning for ourselves. We remember with more than twofold certainty what we have verified for ourselves after hearing from others. We cannot advance far in mathematics without giving ourselves to symbols wholly. No one can add 50 and 40 with the picture of 90 real objects in his imagination. In this book we are at the stage where we can properly think of 15 as a symbol only; hence we begin to use the singular verb in the English sentences, — 15 is what part of 50 ? and, 15 is 50 less ?. Yet while this treating of numbers abstractly is essential to progress in arithmetic, considered either as the science of nuinbers or as the art of computation, we must remember that we study arithmetic not only for culture, but also for utility. In our teaching we must frequently correlate numbers with real facts. This is especially necessary in our dealings with arithmetic as ratio. Our boys and girls must know quarts, yards, pounds, coins, square feet, as actual measures. To encourage interest in arithmetic as truth about real things, pictures and illustrations, tending to stimulate the activity of the imagination which gives men seeing eyes, have been introduced. Arithmetic is the chief instrument of science and the essence of certainty. It is concrete logic. In a world of flux and change and doubt it is of elemental importance for our boys and girls to know a kind of truth that is as positive as the very reason and the very mind of our humanity. Author and publishers desire to acknowledge the valuable suggestions of Principal W. B. Gunnison, Ph.D., of Erasmus Hall High School, Brooklyn, N.Y., in reviewing these pages. W. E. c. BLOOMFIELD, N.J., June 14, 1901. CONTENTS SUBTRACTION . . PAGE Reviews . . . 7, 11, 29, 48–49, 58, 61-62, 74, 84, 91, 93, 95-96, 109, 119, 120, 126 READING PROBLEMS . . . . . . . . . .8, 43 ADDITION . . . . . . . . . . 9, 14, 94, 118 WRITING PRACTICE . . . . . . . . 10–11, 43 . . . . 12, 13-14, 100-102 TELLING TIME . . . . . . . . . . . . 15, 88-90 MULTIPLICATION . . 16, 18, 22, 44, 53, 59, 63, 66, 75–76, 80-81, 105-108, 121 FRACTIONS . . . 17, 19, 23, 26, 28, 32, 34, 77, 41, 47, 52, 69, 70, 77-79, 82, 92 SQUARE MEASURE . . .. . . . . . . 20, 112, 117 ANGLES . i . . . . 21 Ratios . . . . . . . 27, 33, 69, 77, 92, 115, 125 Cubic MEASURE . . . . . . . 30, 116–117, 122 NUMBERS . . . . . . . . 31, 38-40, 42–43, 50–51 Music FRACTIONS . . . . . . . . . . 45 MANY-SIDED FIGURES . . . . . 46 DOLLARS AND CENTS. . 54-55, 64–65, 85 DATES . . . . 57 MEASURES. . . . 60, 67, 123, 127 DIVISION . . 71-72, 103-104 House NUMBERS . . . 73 ROMAN NUMERALS .. . 83, 98-99 SIMPLE CANCELLATION . . 83 PER CENTS .. .. · 111, 114 CIRCUMFERENCES . . . . . . . 113 DECIMALS . . . . . . . . . . . . 124 BILLS . . . . . . . . . . 125 TEST OF SUCCESS . . . . . . . . . . 128 . . . . . |