GIFT OF Mill- TEACHER'S MANUAL * AND =======^^ COURSE OF STUDY [Based upon and to accompany the use of the"! California State Series Texts in ArithmeticJ BY ALVA WALKER STAMPER 251757 I UNIVERSITY **$*< PREFACE. This course in arithmetic, which is in use in the Training School of the State Normal School, Chico, California, has been prepared as a manual for the schools of Chico and of Butte County, in accordance with an understanding with the two boards of education concerned. The size of the edition has, in consequence, been limited to local needs. It is the plan of the author to have the material of the manual appear as a chapter in his "A Text-book in the Teaching of Arithmetic," which will appear early in 1912. In this book a system of cross references will be used between the chapter referred to and the chapters on method. It is also probable that this manual will appear later as a regular bulletin of the Chico Normal School. The course contained in this manual is built around the central idea that children should learn to "figure." The appli- cation side of arithmetic has been given consideration but with- out going to the extreme in trying to vocationalize the subject. Perhaps not the least constructive side of the course is the de- tailed work outlined for the first two years. Inexperienced teachers are often at a loss to know how to systematize their work in arithmetic before the regular drills in the processes are begun. The present course arranges and adapts the ma- terial in Chapter I. of the State Primary Text for class use. A special effort has been made to unify the entire course with the State Series Texts. Teachers will find it necessary to learn thoroughly the primary text's special plans of teaching the four fundamental operations, especially addition and multiplication. There is no half-way method of using the book's plan of be- ginning column addition. This plan and other features of the State Texts are emphasized in this course. ALVA WALKER STAMPER. State Normal School, Chico, California. September, 1911. INTRODUCTION It is the aim in the first year and a half to give the founda- tions for the later computation work and the applied work in denominate numbers, measurements, and problems. Hence, there is a place in this early work for (1) counting, recognizing number groups, reading and writing numbers; and (2) compar- ing, measuring, and related work, that is sometimes classed under sense-training. There should be familiarity with most of the addition, subtraction, multiplication, and division facts in connection with the numbers 1 12, or possibly 1 20, this knowledge to be gained through the use of objects. No at- tempt should be made here to have pupils memorize the addi- tion combinations and other number facts, the idea being mere- ly to acquaint them with simple applications. The most im- portant thing for the teacher to remember in this preliminary work is that great care must be used to give pupils the neces- sary preparation for column addition, not through the learning of the addition combinations, but by having the pupils per- fectly familiar with the natural number series, so that they may know, for example, that 34 is in the thirties, that it comes before 35 and follows 33, that it is the next number after 29 that ends in a 4, and the like. The addition and subtraction combinations, column addi- tion, and subtraction are begun in the High Second and con- cluded in the High Third. The multiplication and division tables, mulplication, and division are begun in the High Third and finished in the High Fourth. The special plan of the pri- mary text in teaching the combinations and tables and in ap- plying them requires a deferring of certain combinations and tables until they are needed in examples. The primary text should be in the hands of the pupils in the Low Third. This is advisable since many of the exercises are especially prepared for class use by the pupils. Object work with fractions and the elements of denominate numbers enter into the earliest work under the aspect of com- parative magnitudes, measuring, and the like, and continue 2 " ' ' ' ' ARITHMETIC throughout the grades. Many problems in fractions in the form of partition enter into the problem work of the Fourth Grade. Serious work in common fractions is undertaken in the Fifth Grade, all the processes being completed with the most familiar fractions. Decimals are taken up and finished in the High Fifth, orders beyond thousandths being deferred to the next grade. The advanced text is introduced in the Low Sixth. The work of the Sixth Grade is essentially a review of all previous work, the aim being especially to give the pupils added facility in the use of common and decimal fractions. The Seventh Grade is primarily concerned with percentage and its applications and with mensuration. In the Low*ljighth, percentage is reviewed, powers and roots are introduced, and mensuration is completed. This concludes the standard topics of arithmetic. The work of the High Eighth gives a review of the most essential parts of arithmetic and the common com- mercial aspects of the subject, including the keeping of simple accounts. The one thing to be emphasized most throughout the grades is addition. If the plan of the primary text is followed, the work in addition should be in excellent shape by the Low Fourth. Teachers should see to it that the facility gained by this time is preserved and improved upon. It is only by con- tinued reviews that expertness is attained and maintained in any of the operations of arithmetic. Teachers should provide for reviews throughout every term. Ample drill should be given in the reading and writing of numbers, especially before taking up topics requiring this facility. Mental arithmetic, which is often neglected, should have an important place from the time the first number facts are memorized to the comple- tion of the course. A five-minutes rapid drill in mental arith- metic at the beginning of the period in the upper grades will be found most profitable. Require pupils to record their answers for correction. The early use of language forms in the mechanical work is desirable. This practice clears up ideas for the pupils. After the teacher is sure that the principles in any operation are un- derstood, the explanation of the process should be discontinued. The first mechanical process that needs the use of a language form is column addition, where "carrying" is involved. The pupil is taught to say, for example, "Write down the 3 and add the 1 to the next lowest figure." The analysis of problems, which is begun in the Low Fourth, is made easier by the habits of correct descriptions of processes ARITHMETIC 3 gained through the use of language forms in the earlier grades. The chief purpose of the explanation is to make plinciples un- derstood. It should never become a thing in itself. Pupils should work the great majority of examples and problems without explanations of any sort. It is expected that teachers will follow this course. If, how- ever, it seems better to sacrifice anything in order to cover the main topics of the grade, sacrifice, for example, speed but not accuracy; the solution and analysis of the harder types of problems but not the fundamental processes in mechanical work; the longer written exercises but not mental arithmetic; and the short processes but not the indispensable longer meth- ods in common use. Teachers will find it necessary to adhere very closely to the sequence of subject-matter in the primary text on account of the unique plans of developing the addition and subtraction combinations and the multiplication and division tables. There will be found much need of supplementary work for the pri- mary text. Bulletin No. 11, issued by the San Francisco State Normal School and edited by D. R. Jones, one of the authors of the State Texts, will be found very helpful. There will also be found need of supplementary work in the advanced text, beginning with the topic of percentage. A number of texts furnish good material for supplementary work, among these being the Wentworth-Smith Series and the Walsh Series. Among texts that make particular efforts to give problems having significance to daily living, teachers should refer to the Smith Series and to the Young and Jackson Series. For supplemental work in mental arithmetic, Bailey's American Mental Arithmetic is good. The Speer Text (Book I., for Teachers) will be found helpful in the first two grades in connection with sense-training and related work. LOW FIRST. The chief aim is the preparation for column addition, which to be taught a year and a half later, not by teaching now the addition combinations either with or without objects, but by teaching the children to know the places of the numbers in the number series. This presupposes counting and a knowledge of the number symbols. Much attention should be paid to the making of good figures and writing them in straight columns. The teacher will find the following outline of work for the Low First arranged so as to give an easy and logical development of the facts to be learned with respect to the place of numbers 4 ARITHMETIC in the series. This arrangement appears below under the cap- tions: Counting, Number Groups, Reading and Writing Num- bers, and the Place of Numbers in the Series. It is along this line that pupils are to be systematically drilled throughout the term. The other work that enters in the Low First is that usually given under the headings of sense-training, busy work, meas- uring, and comparisons. The order in which this is introduced may be varied by the teacher as occasions demand. It is in such work as this that pupils learn to express their thoughts in simple language based upon experiences gained within or without the class. Lessons I. -XIII. in Chapter I. of the primary text furnish material for much of this work. These les- sons from the text should not be taught in the order there given. Neither should all of some of the lessons be taught in this grade. The assignments below explain this. Only the simplest phases of the matter treated should be introduced in the Low First. Teachers should not be misled into magnifying the extent of the work expected in this line on account of the variety of topics presented below. Of the two lines of work mentioned in this and the preceding paragraph, the first should receive chief attention. Much time is needed especially in writing numbers. 1. Relative Position, Direction, Magnitude: Call on individual pupils and, in certain exercises, the whole class. Which is your right hand? Which is your left hand? Your right eye? Place your left hand on your right arm. Who is on the right of Mary? On the left of John? Point to your right. Your left. Behind you. In front of you. Write the figure 2 on the board. Write another 2 above the first 2. Write another below the first 2. Write a 1 on the board. Write another 1 close to this and to the right of it. Write another 1 close to the first 1 and to the left of it. Point to the top of the blackboard. Point to the bottom of the blackboard. Mary will stand to the left of John. Julia will stand between John and Mary. Place your finger on the middle of this ruler. Which of these blocks is the largest? Which is the smallest? Which is longer, this ruler or that ruler? Who is the tallest pupil in the class? Who is the shortest?. Who is taller, Mary or John? Who is shorter, Ida or Sue? The class will stand in a row. Who is the tallest? He may stand first in the row. Who is next to the tallest in the class? He may next take position. The pupils take position according to their respective heights. Compare weights of objects at hand. See Lesson I. in text and Speer, Book I. for Teachers. 2. Counting: 1 to 12; 1 to 20. With and without objects. There is little need of counting with objects beyond 20. Have pupils co,unt things they are interested in. How many pupils ARITHMETIC 5 in the class this morning? How many boys? How many girls? How many feet has a horse? Name other animals that have four feet. John may make three marks on the board. Mary may make some marks on the board. Sue may tell how many marks Mary made. See that the place of number in the series is not confused with number itself. Thus, ask Mary to fetch the first three of a number of blocks placed in a row. Then ask her to replace them and bring the third block in the row. Who is the third pupil from the left in line? Who is the second from the other end? The abacus furnishes a convenient means for count- ing with objects. See first part of Lesson II. of the text. 3. Number Groups: Unless specially arranged, pupils cannot, recognize more than four or five objects in a group. Teach the recognition of number groups as on dominoes. Have flash cards with spots thus arranged. Place the pupils two in a group and have them count the number of groups. The abacus is handy for this kind of work, where the aim is to prevent the idea of a fixed unit. How many 3's do you see? How many 2's? How many 1's? This refers to objects in groups, not to the number symbols. Do not yet count serial- ly by 2's, 3's, etc. 4. Reading and Writing Numbers: 1 to 12; 1 to 20. Board work only. No pencil throughout the term. A large class, only, should excuse its use. Relate the number (found by counting), the name, and the symbol (the figure). Read numbers written by the teacher. Use flash cards to teach quick recognition of the figures. Pupils write numbers at board under models set by the teacher. The crayon should be held not tightly, yet firmly. Short pieces of crayon are desirable for small hands. Aim toward large rather than small figures. Where pupils have difficulty let them trace over the copy made by the teacher. Then try without tracing. The teacher makes a 3 on the board. She asks the class to look at it carefully. She erases the 3 and asks the class to make one like it. Write 13 on the board. Which figure is on the right side? Which is on the left side? Which figure do we make first in writing 13? In reading the "teens" the teacher should place the pointer first on the right figure as the word is pronounced. This will help prevent the writing of 13 with the 3 first. Much individual attention should be given the pupils while they are learning to make the different figures. Be sure that the 8 is made toward the left, beginning at the top. Write the numbers in columns, vertical and straight. 5. Counting: 1 to 50. 6. The Place of the Numbers in the Series: 1 to 10; 1 to 20. Extend the limits of the series as indicated. The teacher should have on the board the series to be studied written from the bottom up (to prepare for the special plan of the primary text in teaching column addition). The numbers 1 to 50 may be printed on strong linen or heavy paper. A strip 9 feet long and 6 inches wide will answer the require- ments for figures 1 5/8 inches high. A set of stamps for figures of this size costs about 50 cents. The chart is to be 6 ARITHMETIC used in the following exercises. Do not force the children to visualize the figures. With the chart on the wall for their daily study, the power of visualizing will follow easily and naturally. a. Name the number next after 3, 7, etc.; and, later, after 13, 19, etc. b. What number before 6, 10, etc.?; and, later, before 13, 16, 20, etc.? c. Name the number between 5 and 7, 12 and 14, etc. d. Name the numbers between 4 and 7, 15 and 18, etc. 7. Reading and Writing Numbers: 1 to 50. 8 Time Measures: What day is today? What day was yes- terday? What day will tomorrow be? Name the days of the week. How many days in one week? How many school days? How many working days? Y/hat month is this? In what month is your birthday, Sara? What day of the month (date) is your birthday? What day of the month is today? At what hour in the morning does school begin? When does school let out in the morning? What time does school open in the afternoon? Refer to the clock face. The teacher draws a figure of a clock, using Arabic numerals. Where is the hour hand at 9 o'clock? Where is the minute hand? Sara was 5 minutes late this morning. Where was the min- ute hand? Do not go beyond the "time" that 'is of interest to the pupils. Compare the ages of the pupils. 9. Sense-Training: Review relative size, as previously taken up, thus training the sight judgment of the pupils. Train judgment of relative size through the sense of touch, the eyes being closed. For both of these purposes use various kinds and shapes of objects. Compare weights of objects. Thus far the pupils are not supposed to make exact com- parisons, although the "one-half" and "two times" may be brought out. Teach Forms as in Lesson IX. of the text. Train the pupils in the power of visualizing objects, the aim being to have them recall the properties of the objects and the number facts involved. Write a number on the board. Erase it and ask for the number erased. Write two numbers on the board. Erase them and ask for the num- bers erased. Ask questions about the position of the figures in certain numbers and see if the pupils have the correct mental picture. Test with 13 and 31. See Speer, pp. 37-48, for work in sense-training. 10. Counting and Reading and Writing Numbers: 1 to 50. Re- view previous work. How many figures in 25? In 6? Have pupils point out numbers on the chart as they count. Point out on the chart numbers that have been named. In looking for 43, for what number do you look first? Point out num- bers on the chart and have them named. How many num- bers written on the chart? Children count and discover that the last number written in the series tells how many have been written. Ask the pupils to write the first 5, 10, or 15 numbers in a column, this to be done without reference to the chart. Ask them to begin with a certain number and write two, three, five, or ten more. Count by 10's and 5's. ARITHMETIC 7 11. Comparative Magnitudes: Review previous work. Com- pare both unequal and equal magnitudes, using a great va- riety of objects and drawings. Most of the drawing should be done by the teacher and on the board, the square, rec- tangle, and circle being employed. The pupils may be given a number of blocks of equal sizes, from which they may build larger solids that will furnish means of comparison. Colored squares made out of cardboard will interest the children and serve a similar purpose. These objects and drawings will give sufficient material for bringing out the relation "one-half" and "two times," and perhaps a few other simple fraction facts. Develop the need of measuring to determine equality. Re- alte here the foot and the yard (Numbers 1 to 3) and the inch and the foot (Numbers 1 to 12). See Lessons IX., VI. (first part), and VII., in the text. Refer to Speer, but keep in mind that the work in comparing magnitudes there given extends over more ground than is contemplated for the Low First. Most of the material on pp. 37 to 60 of that text can be used here. 12.. The Place of the Numbers in the Series: 1 to 50. Use the chart with numbers written from the bottom up. Encourage the pupils in visualizing the numbers on the chart. After the position of the numbers is fixed in his mind, the pupil naturally can answer the teacher's questions more readily when not looking at the chart than if required to examine it. a. Drill as in (a), (b), (c) under (6) above. Emphasize especially asking for the numbers next after 9, 19, 29, 39, 49, and the numbers immediately before 50, 40, 30, 20, 10. b. The decades. After the first ten numbers are the "teens." After the "teens" are the twenties. After the twenties are the thirties, etc. Before the twenties are the "teens." Before the thirties are the twenties. Etc. c. Step half-way across the room and then stop, Charlie. Locate the middle of this stick, Mary. Point out the half- way point of this vertical line on the board, Arthur. Find the number on the chart that is half-way between 10 and 20. That is half-way between 20 and 30. Etc. Is 23 nearer 20 or 30? Is 37 nearer 30 or 40? Etc. The number table may be extended to 100 for counting, reading and writing numbers, and locating numbers in the series in case the pupils are ready. The counting and read- ing and writing numbers to 100 can be done almost as soon as the pupils can do this to 50, but the locating the num- bers in the series 50 to 100 may better be left for the High First. HIGH FIRST. The work of the High First is a review and continuation of the work of the previous term. The main emphasis should be on the reading and the writing of numbers and locating num- bers in the series. The study of comparative magnitudes is continued. The essentially new work consists of finding out 8 ARITHMETIC the number facts in connection with objects. In this connec- tion the numbers 1 12 are studied, the pupils employing addi- tion, subtraction, multiplication, and division, the latter in its dual form, division by measuring and division by partitioning. No effort is to be made here to teach the abstract number facts, for, according to the plan of the course, no use is to be made of these facts for a year to come. But it is essential that pupils learn the meaning of these processes in connection with things and hence the value of the work planned for this grade. Fur- thermore, we may remind ourselves here that an objective un- derstanding of 9 and 4 are 13 bears no immediate relation to the memorizing of that fact. When the pupil is ready to use the fact that 9 and 4 are 13 he should not be bothered then with the so-called reason involved. He needs to use it then as a tool. Number stories should accompany this early study of the operations. The abstract drills indicated below in connection with rec- ognizing the place of the numbers in the series should be given in the order as arranged. The rest of the work may readily be rearranged according to the wish of the teacher. Before teach- ing any topic, review the corresponding work in the Low First. 1. Counting: 1 to 100; 1 to 120. Count by 1's, 10's, 5's, and, later, by 2's. Introduce even and odd numbers. Count by naming the even numbers from 2 to 20; 20 to 50. Similarly with the odd numbers. Count objects, naming the place in the series. Thus, 1st, 2nd, 3rd, etc. 2. Reading and Writing Numbers: 1 to 100; 1 to 120. Much time should be spent on this work. It is preferable to write numbers entirely on the board. The class may write 37. Yours is good, John. You may now make a straight column of three 37's and be careful not to write too near the vertical line. Correct individual errors of the pupils. Write the even numbers up to 20. The odd numbers up to 21. Yours is not good, Julia, because your second column is too near the first. Write the numbers 1 to 20, placing the odd numbers on the left of a vertical line and the even numbers on the right. 3. Time Measures: Extend the work of the Low First. Tell time to the nearest 5 minutes, first reviewing counting by 10's and 5's as far as 60. Teach the reading (not writing) of the Roman numerals, I. to XII. Count 5 seconds, 10 sec- onds. Remain silent 10 seconds. Point out on the clock 5 minutes after the hour. 10 minutes. Etc. How many minutes in one hour? In a half hour? In a quarter of an hour? How long does it take you to walk home? How much time do we take for recess? For noon? 4. The Place of the Numbers in the Series: 1 to 100; and, later, 1 to 120. Use the charts referred to in the work of the Low First and follow the plan of (6) and (12). 5. Comparative Magnitudes: Bring out objectively the ideas ARITHMETIC 9 of 1/2, 1/4, and 1/3. Draw diagrams of squares and rec- tangles and compare lengths of sides. Compare areas (sizes). Also use blocks. Speer has good suggestions. Give the children squares of paper. Give directions for folding so as to get two equal parts; to get four equal squares. Bring out the 1/2 and 1/4 relations. Also twice as large and four times as large. Use Lesson X. of the text, omitting for the present the use of fractions in connection with num- bered objects, such as in the phrase, 1/2 or 10 balls. 6. Counting: 1 to 120. Emphasize the decades. Count by 10's, beginning with 10, 1, 2, 3, etc. For this purpose have a num- ber table already written on the board, the numbers 1 to 10 in the first column, 11 to 20 in the second, 21 to 30 in the third, and so on up to 100. Read the horizontal rows. Then count without looking at the table. See Lesson II. and the first part of Lesson X. in the text. 7 The Place of the Numbers in the Series: 1 to 120. Use the charts as an aid toward visualization. a. First explain, for example, what is meant when we say that 24 ends in a 4 or that 16 ends in a 6. What does 34 end in? 47? 50? Find a number among those that I write that ends in a 3? In a 9? b. What number in the twenties ends in a 3? In the thir- ties that ends in an 8? In the "teens" that ends in a 4? Etc. c. What is the first number after 20 that ends in a 0? Etc. What is the first number after 5 that ends in a 5? Etc. Similarly with other endings ten numbers apart. d. Bring out the ideas of greater than and less than (after and before the series). Which comes first, 15 or 17, 26 or 29, 38 or 40, 31 or 29? Notice the gradation. The last set of numbers carries the pupil into different decades. Which is greater, 12 or 10, 17 or 14, 23 or 27? Etc. e. Numbers after 10, 20, etc., that end with certain fig- ures but not ten numbers apart. What is the next number after 10 that ends in a 3, 5, etc.? The same after 20, 30, etc.? f. Numbers after any numbers in the same decade. What is the next number after 13 that ends in a 5, 7, etc.? After 23, etc.? g. Numbers after other numbers, the latter ending in 0. What is the next number after 16 that ends in a 0? After 26? Etc. h. From one decade into the next. What is the next num- ber above 7 that ends in a 1? Above 17 that ends in a 1? Above 27, 37, etc.? The same with other endings. This drill is very important since the ideas involved are essential in later work in column addition. 8. Complementary and Measure Contents of Numbers: Num- bers 1 to 6; 1 to 12. This work, in which the pupils are to learn the early number facts, is to be entirely objective. Use all sorts of objects. Splints blocks, and the balls of abacus are good. It is perhaps better to teach the complementary contents for any number, that is, the addition and subtrac- tion facts, first; but in concrete work with objects where no effort is made to have the pupils memorize results, teachers will find, for example, that a pupil can find the number of 10 ARITHMETIC 2's in 6 as readily as he can show the teacher with his sticks that 2 and 4 are 6. The addition and subtraction facts are to be taught for all numbers in the limits assigned above. The measure-contents' facts, that is, multiplication, division, and partition, are to be taught in close connection with the complementary-contents' facts, but do not at this time study the prime numbers, 3, 5, 7, and 11. As a plan of procedure, the following is suggested: Teach the complementary contents of the number 6. Here the pupil finds from the use of objects that 5 and 1, 1 and 5, 4 and 2, 2 and 4, and 3 and 3 make 6; and that subtracting in turn 1, 2, 3, 4, 5, from 6 leaves 5, 4, 3, 2, and 1, respectively. Then teach the measure contents of 6. Thus two 3's are 6, three 2's are 6; in 6 there are two 3's and three 2's; 1/2 of 6 is 3, and 1/3 of 6 is 2. There is no gain teaching now that six 1's are 6, one 6 is 6, and 1/6 of 6 is 1. Next teach the meas- ure contents of 4. Next the complementary contents of the number 7. Then all the number facts about 8 in the way number 6 was studied. Take up the other numbers as the term advances in the way that seems best to the teacher. Teach number stories in connection with the above work. This gives the pupils a chance to cultivate their powers of expression. The number story should follow the objective presentation and be based upon it. Teach Lessons III., IV., and VI. of the text, which are les- sons in comparative magnitudes, in connection with the above work. Lesson X. gives exercises in partition, pre- viously omitted. Use Lesson XII. for suggestions in giving number stories. 9. Comparative Magnitudes: Teach Lesson VIII. of the text, on pints, quarts, and gallons, if not already used in (8). Have the necessary measures in class and let the pupils compare capacities by filling with water. Review measure- ments, using feet, inches, and yards. Let the children make drawings, dividing up squares, rectangles, and the like. 10. Counting, Reading, and Writing Numbers: 1 to 200; 1 to 1000. Omit teaching of place value with the exception of naming and having the pupils name hundreds' place. Count by 10's and 100's. Write by 10's and 100's. LOW SECOND. Review the work of the previous grades. By the end of this grade the pupils should acquire considerable command of oral expression in the giving of number stories and the like. The placing of the numbers in the series should be continued and emphasized even more than in the previous grade, since in the next grade column addition will be begun . 1. Counting, Reading, and Writing Numbers: 1 to 10..000. Con- tinue serial counting as in the previous grade. Count by 10's, 5's, 2's (both the series of the odd and the even num- bers), 100's, and 1000's. Begin at any multiple of 10 in the series to -count in this manner. Thus, begin with 360 and ARITHMETIC 11 count by 10's to 400. Count by 100's from 300 to 1000. Count by 1000's from 1000 to 10,000. Write the higher numbers in the natural serial order and as above. Spend much time on the reading and writing of the higher numbers. Have both board and seat work. Teach the names of the different or- ders, or places. By the end of the term the pupils should learn how to build numbers of two places and of three places by using splints and rubber bands. Teach all of Les- sons V. and II. of the text. 2. Comparative Magnitudes: Extend the drawing work for fractional and multiple relations begun in the previous grade. Let the pupils draw diagrams freehand at the board and with short rulers at the seat showing relations specified by the teacher. Thus, draw a square. Divide it so as to show 1/4 of it. Again, draw a square and then a second square near the first that shall be 4 times the size of the first square. The fraction symbols may now be introduced but are not used in operations. Teach the 1/2, 1/4, 1/3. The pupils learn the relations that exist between 1/4, 2/4, 1/2, 3/4, and 4/4. Also between 1/3, 2/3, and 3/3; and between 2/2, 3/3, and 4/4. Relate to partition (See (4) below) with counted objects. Thus, find 1/3 of 12 counters; 2/3 of 6 blocks. 3. Place of the Numbers in the Series: Review the work of the previous grade, extending the numbers beyond 120. The number charts (1 to 50; 50 to 100) should be on the wall for reference. Among other things to emphasize is the naming of the number following another number and having a stated ending. Thus, give the next number after 83 that ends with a 1. This is vital in column addition, for, when the pupil has learned that 3 and 8 are 11 and that 8 added to any number ending in a 3 gives a number ending in a 1, he is in a position to tell that 91 is the sum of 83 and 8. 4. Complementary and Measure Contents: Review and extend the work of the previous grade. Carry the study of the numbers, still entirely objectively, to 20. Make no attempt to have facts memorized. Work of this kind is not intended so much to prepare for the later learning of the number facts, because this would be idle, but it is given so as to af- ford the pupils a first-hand application of the four funda- mental operations of arithmetic to things. See the sugges- tion at the end of (2) above. In teaching the measure con- tents, consider first exact multiples and divisors. Thus, in 12 there are three 4's. Later, in 14 there are three 4's and 2 "over." Continue the use of number stories. While it is suggested that the work under this topic be carried as far as 20, it is not intended that the work should be made scien- tifically exhaustive. The teacher should keep in mind above all the interest of the children. The abacus will be found very helpful in such work. Teach Lessons XI. and XIII. in the text. * 5. Addition and Subtraction That Is Permissible with Count- ing: It is generally agreed that children should not add by counting. Thus, in the next grade they will learn that 4 and 5 are 9 merely a.s a fact, not being permitted to count 12 ARITHMETIC 5 more beyond 4 to get the result. The addition and sub- traction of 1 and 10 and 5 as shown below is permissible and desirable: a. Add and subtract 1. What are 9 and 1, 19 and 1, 29 and 1, etc.? What is 1 less than 10, 20, 30, etc.? What are 4 and 1, 14 and 1, 24 and 1, etc.? What is 1 less than 5, 15, 25, etc.? Then give miscellaneous eexrcises without refer- ence to the decades 7 and 1, 23 and 1, 87 and 1; 1 less than 7, 16, 93. b. Adding to numbers ending with with the correspond- ing subtractions. What are 10 and 3, 20 and 3, 30 and 3, etc.? What are 10 and 9, 20 and 9, etc.? What are 10 and 10, 20 and 10, 30 and 10, etc.? What is 13 less 3, 23 less 3, 33 less 3, etc.? c. Adding 10 to numbers with the corresponding subtrac- tions. What are 3 and 10, 13 and 10, 23 and 10, etc.? What is 10 less than 13, 23, 33, etc.? d. Adding 5 to numbers ending with a 5 with the corre- sponding subtractions. What are 5 and 5, 15 and 5, 25 and 5, etc.? What is 5 less than 10, 20, 30, etc.? All the above questions are based on the pupil's previous experience in serial counting. Also associate 1 and 9 with 9 and 1, 1 and 4 with 4 and 1, etc. Emphasize the making up idea in subtraction, which is used later when subtraction is systematically taught. Thus ask, 9 and what are 10? 4 and what are 5? 20 and what are 25? Etc. In the review work of this term that relates to the text, review especially Lessons VII., VIIL, X., and XI. The pupils should have covered the material of Lessons I. -XIII. HIGH SECOND. Text. California State Series, "First Book in Arithmetic," pp. 34-56. Supplementary exercises from Bulletin No. 11, Hand- book for Teachers, Part I., State Normal School, San Fran- cisco, Cal., edited by D. R. Jones. Text in the hands of the teacher. Beginning Work in Formal Processes. Addition and subtrac- tion combinations of Lesson A of the text. Examples in addition and subtraction to correspond to the combinations learned. Defer the hard cases in subtraction. Language Forms and Technical Terms. Oral language forms in concrete problems and in column addition where "carry- ing" is involved. Use of terms sum and difference. Approximately the first half of the High Second should bv given to a complete review of the previous one and one -half years. If the previous work in counting ,in reading numbers, and in locating numbers in the series has been well done, this part of the review should be easily and quickly done. Continue to perfect the making of good figures and writing them in straight columns. It is sufficient to have the pupils read and ARITHMETIC 13 write numbers up to 10,000, the limit assigned for the Low Sec- ond. Pages 43-46 of Chapter III. of the text fit in well with the review. The review in the study of comparative magnitudes and the complementary and measure contents of the early numbers, still in connection with objects, should receive con- siderable attention. The responses of the children in story problems should now be readily given. This review should in- clude the material of Chapter I. of the text. Place Value: The pupils should already know the names of the orders in numbers having four figures. Test them by questioning. Which figure in 3,579 is in hundreds' place? Which figure is in tens' place? In units' place? In thous- ands' place? In 5,642, which place does the 2 occupy? The 6? Etc. Enumerate from the right, saying units, tens, etc. Have much practice in reading numbers and in answering questions like the above. The proper placing of the figures when writing numbers should now be more readily done. As a preliminary to work like the above or parallel with it, have the pupils build numbers with splints, using rubber bands to hold the bundles of tens together. Build 32. Write the number on the board. Identify the 3 in the written 32 with the 3 bundles of tens. In this exercise the children, not knowing in advance that 32 is being built, write the fig- ures that represent the number of splints used. Next write a number on the board and have it built. Build numbers of three figures. United States Money: See that the pupils can recognize the following coins: 5c, lOc, 25c, 50c, and $1. What other name has 25c? 50c? If a whole cake is worth $1, what is 1/2 of it worth? What is 1/4 of it worth? If a whole cake is worth 50c, what is 1/2 of it worth? Identify 75c as 3/4 of a dollar. What may be purchased with any of the above amounts? Have the children assemble the coins to show 15c, 20c, 30c, etc. Make change as the clerk in the store does. This, of course, must involve only those number facts that the chil- dren have memorized, namely the adding and subtracting of 10 and 5. For this purpose, toy money may be used. Ex- ercises like the following are within the powers of the pupils: I buy some candy worth 5c. I give the clerk lOc. He gives me 5c in change. I buy a pound of coffee worth 50c. I give the clerk $1. He gives me 50c in change. I buy lOc worth of apples. I give the clerk 25c. He gives me 15c in change. The teacher may play the clerk and, as in the last example, will give back the change as follows: She says lOc. Then she lays out lOc, saying 20c. She then lays out 5c more, saying 25c. The pupils should also be taught to do this. Preparation for Column Addition: The drills on the placing of the numbers in the series, begun in the Low First and extending into the High Second, have been planned with special reference to needs in column addition. Some of the most important things emphasized have been: a. Counting by 1's, 10's, and 5's. Counting by 10's, begin- 14 ARITHMETIC ning with any number. Counting the series of even and odd numbers. b. Naming the number after and before a number. Nam- ing the numbers between two numbers. c. Learning the order of the decades. The "teens" come before the twenties, the fifties follow the forties, etc. d. Naming the number immediately following 9, 19, 29, etc., and preceding 10, 20, 30, etc. e. Naming the number in any special decade that ends with a 1, 2, 3, etc. f. Naming which of two numbers comes first in the series and which of two numbers is the greater, or less. g. Naming the first number ending with a 1, 2, 3, etc., that follows some assigned number, especially where the answer introduces the next decade. Thus, name the next number after 38 that ends with a 4. h. Adding 1 to and subtracting 1 from any number. Add- ing 10 to and subtracting 10 from any number. Adding any number less than 10 to 10, 20, etc. Subtracting 3 from 13, 23, etc.; 7 from 17, 27, etc. Adding 5 to and subtracting 5 from any number ending with or 5. The pupils have perhaps already learned a few of the ad- dition and subtraction combinations in connection with ob- ject work, but have not been drilled in them as such. Column Addition: The teacher should read Chapter II. of the text carefully for the explanation of the method used in teaching the combinations and column addition. The meth- od is to teach a group of combinations and immediately ap- ply them in columns that have been specially prepared for the purpose. The method of the text is explained in the following steps: Step A. Teach the combinations 34326 + 2 +5 +9 +2 +4 Read 2 and 3, not 3 and 2, in order to conform to the plan of adding columns, at first, from the bottom up. Place the pointer on the upper figure as the numbers are added. No- tice that the lower figure of the second combination is the sum in the first and that the lower figure of the third is the sum in the second. See Step C below and note that the column is built according to this arrangement. The answer may be under the combinations at first, but for drill pur- poses the answers should not be written. Teach the reverses of these combinations and drill on both sets, taking the com- binations in any order 94225 + 3 +6 +2 +3 +4 Step B (The most important step). Establish the rela- tions that 2 and 3 are 5, 12 and 3 are 15, 22 and 3 are 25, etc. That is, adding 3 to any number ending with a 2 gives the next number ending with a 5. These may be written at first: 34326 34326 + 12 +15 +19 +12 +14; +22 +25 +29 +22 +24; ARITHMETIC 15 etc. In drills, do not write the tens' figures nor the sums. 34326 ( a ) -|-2 +5 +9 +2 +4 Give (a) as it stands. In (b), say 13, 22; 16, 20; 12, 14; etc. Return to (a) 94225 and say 22, 25; (b) Reverses: +3 +6 +2 +3 +4 25, 29; 29, 32; etc. In (b), 33, 42; 36, 40; 32, 34; etc. Again, the 40's with (a), the 50's with (b), and so on up to 120. Next give (b) as it stands, use the "teens" with (a), the 20's with (b), and so on up to 120. Extend the work beyond 120. This is the fundamental drill for Step B, since it uses each combination in every decade. A variation of this drill with (a) may be used just before or while the columns are being built. Thus, in (a) say 5; 9; 12; 12, 14; 14, 20; 22, 25; 25, 29; 29, 32; 32, 34; 34, 40; 42, 45; 45, 49; 49, 52; 52, 54; 54, 60; and so on through the decades. Again, repeat this, omitting the repetition of the sums that give the next lowest number. Thus, say 5, 9, 12, 14, 20; 25, 29, 32, 34, 40; etc. Next erase the lower numbers of (a) with the ex- ception of the first combination and add as just given. 34326 When the column is built + 2 from this arrangement, by writing the 4, 3, 2, 6, in turn above the 3 2, we have the column described in Step C that follows. Step C. The columns are built from the bottom up, using the combinations in Step A. Columns are added from the bottom up until, in the High Third, all the combinations are learned and applied in columns. The teacher places the pointer beside the 3 and the 6 pupil says 5. She then places the pointer beside 2 2 the 4 and the pupil says 9. Do not name the first 333 number in the column, but give the sum of the 4444 first two numbers. The combinations in Step A 3333 should be on the board for reference. 2222 In adding a column, never let a pupil pause to think, for in that case he is probably counting to get the answer. If, in adding a column, a pupil has said 29, for ex- ample, and stops, the next number being a 3, ask him for the sum of 9 and 3. Then ask for 19 and 3, 29 and 3, etc. Whenever a pupil stops in a column and after he has been corrected, have him begin again at the bottom, as many times as necessary, to carry him smoothly past the weak place. It may be necessary to caution some pupil not to look at a figure in the column after it has been added in. This will prevent the substitution of a wrong sum with which to continue the adding. Thus, if the pupil has come to a 3 in the column and has said 25, the sum up to that 16 ARITHMETIC point, there is danger that he may add the next figure in the column to 23 instead of 25 on account of looking at the 3 too long. Another corrective for helping some pupil in adding in the next figure is to place beside the column the sum up to that point. This device should be used sparingly, since pupils should not be encouraged to employ a crutch that hinders speed and spoils the form of the work. A good drill is secured by the teacher's naming a number, say 32, and writing a number, 3, for example, on the board. The pupil answers 35. The drill should be rapid to be ef- fective. This drill gives the pupils practically the same ex- perience which they encounter in adding a column, the hear- ing or thinking of one number and the seeing of a second number to be added to the first. A good drill in Step B is for the teacher to write, for example, on the board. The answer is given. Erase the 2 + 2 and substitue in turn 12, 22, 32, etc. Next vary from the serial order, using for the lower number 92, 42, 62, etc. In some cases it may aid to have the children count serially, 2 and 3 are 5, 12 and 3 are 15, 22 and 3 are 25, 32 and 3 are 35, etc. Do not teach Step C, the addition of the column, until the pupils have thoroughly memorized the combinations in Step A and respond readily in the drills in Step B. The pupils should be taught to study Step B alone. Then let them write the proper decades in the lower numbers and write the sums for the inspection of the teacher. The pupils should understand perfectly the relation between the col- umn in Step C and the arrangement of the combinations in Step A. They may also be taught to make their own columns from Step A and hence secure a very effective means of studying alone. Do not have at first more than six numbers in the single or double columns. The higher numbers can be reached by adding some multiple of 10 to the lowest figure in the single column, as in Step B. Thus, if 8 is the lowest figure, think 58, 78, or 128 in its place and add. Supplement from the Handbook for exercises in addition and subtraction. After a good start is made in Lesson A in addition have the pupils memorize and apply the corresponding subtraction combinations. To the addition combination 4 99 + 5 correspond 4 and 5 In subtracting, follow the plan of the text in using the Austrian, or addition, method, where one asks, 4 and how many are 9? Or we may say, "4 to make 9, five." This form is good on account of its retaining the mak- ing up idea and at the same time giving some distinction be- tween verbal addition and subtraction. Either of these forms is to be used in the written work that is to follow. In oral work use these and also the following: 12 less 3 is what? What is the difference between 12 and 3? Make up story problems to correspond. .Objects may prove of service here. Do not neglect to subtract like numbers in the written work, using the ARITHMETIC 17 Austrian method. Provide for the addition of 1 and In col- umn addition. After the addition of single columns is well in hand and also the subtraction of numbers of two figures each, begin the addition of two and three columns, which almost immediately involves the idea of "carrying." Use the following language form in column addition: The pupil says, in adding the first column: 12, 14. Write 42 down the 4 and add the 1 to the next lowest figure. 23 3, 5, 9, Write down the 9. The sum of the numbers is 29 94. This explanation may be discontinued after all the class understand the process of "carrying." In 4 adding the first column, the sum must be visualized. How many figures in 14? If we had but one column we should write down both figures in the sum for that column, but since there is a second column the 1 of the 14 is added to the second column. The 1 ten of the 14 belongs in the ten's column. Defer the hard case in subtraction until the Low Third. Also defer written tabulations by the pupils, such as is given on p. 55 of the text. Care must be used in making up exam- ples in addition and subtraction not to introduce combinations not already memorized. The work of the High Second in ad- dition and subtraction is restricted to Lesson A of the text supplemented by examples from the Handbook and others made up by the teacher. LOW THIRD. Text. California State Series, "First Book in Arithmetic," pp. 55-80. Supplementary exercises from Handbook, Part I. Text in the hands of the pupils. New Phases of Formal Processes. The hard case in subtrac- tion, deferred from Lesson A in the text. The addition and subtraction combinations in Lessons B, C, and D of the text and applications in column addition and subtraction. Language Forms. In the hard case in subtraction. Written tabulations in addition and subtraction problems. While the formal number processes are being developed, keep alive, by reviews throughout the term, the work in com- parative magnitudes and the related objectice work in frac- tions. Review telling time, the reading of the Roman numerals up to XIL, and the relations already learned in liquid measure and linear measure. The text provides for most of this work. Continue reading and writing numbers, using numbers as high as two periods. Insist on straight columns and good figures. There should be occasional speed drills in writing numbers. Test the class in enumerating the orders. Introduce the writing of dollars and cents, using the decimal point. The main line of emphasis for this grade should be on addition and subtrac- 18 ARITHMETIC tion. Before taking up Lesson B of the text, review the addi- tion and subtraction in Lesson A. Do not leave this until col- umns are added with great accuracy and smoothness and with considerable speed. Follow the same plan of teaching Lessons B, C, etc., as was employed in Lesson A, using Steps A, B, and C. See High Second. Follow the suggestion given in the High Second to utilize the higher numbers in a column by calling the lowest number of a single column; for example, 47 where the figure actually written is 7. Add orally two numbers of two figures each, as on pp. 60 and 66 (not p. 76) of the text. The method is that of adding the tens first. The examples for this grade are of the type 40 and 20, 46 and 20. It may be necessary to write these at first in column form. If so, insist, as in the second example, that the pupil says 46, 66. It should be the aim to add such numbers mentally without forming a mental image of one number under another. Any image formed should relate to the numbers in series. More time may be given to seat work than in the previous grade. Columns involving combinations not yet thoroughly learned should be studied at the board under the direction of the teacher. The tabulation of written problems may now oc- casionally be required of the pupils. The following tabulation of a concrete problem is sufficient for this grade: 28 boys. + 26 boys 54 boys Require the following language form in the hard case in subtraction until the results show that the process will not b? readily forgotten: 92 Since 3 is greater than 2, no number added to 3 can 33 give 2. Hence we think 12 in the place of the 2. 3 and 9 are 12. Write down the 9 and add 1 to the next 9 lower figure, making it 4. 4 and 5 are 9. Write down the 5. The difference between 92 and 33 is 59. In- stead of saying "3 and 9 are 21," "3 to make 12 nine," may be used. Flash cards on which are written the addition and subtrac- tion combinations that have been taught up to this point will prove of great help in fixing the combinations. The number charts, previously mentioned, should be on the wall for ready reference in case any pupil has forgotten the position of any number in the series. Impress on the minds of the children some conception of the magnitude of the numbers they are ARITHMETIC 19 using. Are there 500 children in this room? Are there 100? Are there 20? In making up examples in subtraction where "carrying" is involved, the teacher must use care to bring in no new combinations. Thus in 52 the second column depends primarily upon the addition 23 combination 3 and 2 are 5. In using the "borrowing" method, not employed in this course, the second column would depend upon the fact that 4 less 2 is 2. At the conclusion of Lesson B of the text, which is the limit of the work for the Low Third, there have been taught 20 addi- tion combinations, their reverses, and the corresponding com- binations in subtraction. HIGH THIRD. Text. California State Series, "First Book in Arithmetic," pp. 81-114. Supplementary exercises from the Handbook, Part I. New Phases of Formal Processes. The addition and subtrac- tion combinations given in Lessons E, F, G, and H of the text. Column addition and subtraction to correspond. This completes the learning of the combinations. Multiplication and division tables and related examples given in Lessons A and B on multiplication and division in the text. Multi- plier of one figure, multiplicand as many as three figures. Short division, without remainder in steps or at the end. Language Forms and 1 echnical Terms. Language form in multiplication where "carrying" is involved. Tabulations in addition and subtraction problems. Use of terms product, quotient, area. Use of symbols +, , =, in the equation. Review column addition and subtraction, employing the combinations learned in previous grades. Since the addition and subtraction combinations are completed in this grade, the work in column addition and subtraction should be thoroughly mastered. Whatever the rate of speed of the individual pupil in adding columns, he should add with a certain rhythm that only comes with a thorough knowledge of the combinations and the place of the numbers in the series. Towards the end of the semester, after all the combinations have been used in columns, have columns added from the top down. Then prove results by adding from the bottom up. Flash cards on which are written the combinations will be found helpful in quick drills. Also the circle device, common in most texts, will serve the same purpose. An excellent drill is secured by having the pupils write a list of numbers in a column at the left of a ver- tical line. Over a line at the top write the number (single 20 ARITHMETIC figure) to be added to the numbers in the column or subtracted from them. Place answers in a vertical column at the right. There is an excellent opportunity here for speed contests. An- other good review of the combinations is secured by asking for all the addition combinations that give, say, 15. Give mental drills like: 4, add 8, add 4, subtract 9, add 3, subtract 2, add 30, etc. Play store, using toy money and making change acced- ing to business methods (See High Second). Review reading and writing numbers of two periods. Enumerate the orders units, tens, etc. Point out, in 305, 964, the figure in hundreds' place. In hundred-thousands' place. Continue the addition of two numbers of two figures each after the manner suggested in the Low Third. Precede those given on p. 76 of the text by numbers used in making purchases. Thus, add 45 and 15, 45 and 35. In the latter, the pupil says 45, 75, 80 or 40, 70, 80, or perhaps he may learn to say 50, 80. Pupils should be able to add quickly, mentally, numbers like 13 and 17, 14 and 16, 24 and 16. The class should occasionally be drilled in tabulating addi- tion and subtraction problems according to the form given in the Low Third and also on p. 55 of the text. Before beginning the work in multiplication and division, review the measure contents of the early numbers objectively. See work for the High First and the Low Second. Read Chap- ter II. of the text, which gives the authors' plan of developing multiplication and division. Follow the plan of the text with respect to learning and applying the multiplication and division tables. Notice that parts of the tables are learned and then applied. Thus in Lesson A on multiplication the pupil learns the products: two 2's, three 2's, four 2's, three 3's, and the re- verses. After these are applied in examples, the correspond- ing division tables are taught and applied. The same plan holds In Lesson B in multiplication, where the pupil learns the prod- ucts: five 2's, six 2's, four 3's, five 3's, and the reverses. The pupils should identify, for example, five 3's are 15, with the result of adding 3 five times or counting by 3's as far as 15 This relation between addition or serial counting and multi- plication should be brought out from time to time while the tables are being developed, not so much as an aid for memor- izing the tables, for this may not generally assist, but as a means of associating equivalent mathematical facts with re- spect to the number series. Notice that when saying the mul- tiplication table of 3's one says five 3's are 15 and when saying the table of 5's, three 5's are 15. Notice that it is not the plan of the text to have the tables "said," since ARITHMETIC 21 some of the reverses bring in isolated products from the higher tables. Thus in Lesson B on multiplication the pupil learns the 3's as far as five 3's. The reverse of five 3's is three 5's, the first of the 5's that has been learned. Use the following language form in multiplication, where 326 "carrying" is involved, until the process is established: X2 Two 6's are 12. Write down the 2 and carry the 1. Two 2 2's are 4 and 1 are 5. Etc. 652 is the product obtained by multiplying 326 by 2. Do not permit the writing of the numbers "carried." The work in fratcions in the High Third is a continuation of the work of the previous grades. Halves, fourths, eighths, thirds, and sixth are studied in connection with diagrams and other objects. Memorize the relations suggested in the text, require the readnig and writing of the fractions studied, but do not give examples in the addition and subtraction of frac- tions, except to fill in the proper values when relations are ex- pressed in the equation form, thus: 1/2 -f 1/4 = x; 3/4 + x = 1. What is the fraction that should be in the place of x? LOW FOURTH. Text. California State Series, "First Book in Arithmetic," pp. 115-149. Supplementary exercises from the Handbook, Part I. New Phases of Formal Processes. Lessons C, D, E, F, and G of the text in multiplication and short division, applying the parts of the tables learned. Multiplication by numbers of two figures. In short division, remainders in the steps and at the end. Use of fractional parts in problems. The indi- cated fraction as an aspect of division. Language Forms and Technical Terms. Language forms in the mechanics of multiplication where the multiplier has more than one figure. In short division, where there are re- mainders, both in the steps and at the end. Oral analysis of multiplication, division, and partition problems, includ ing problems of one and two steps. Written tabulations for the same. Use of terms ratio, multiplier, multiplicand, divi- dend, divisor. Review column addition throughout the grade with the idea of gaining greater rapidity, but not at the expense of accuracy. Add columns in both directions. Encourage pupils to group when adding, especially the numbers that give 10. Follow carefully the suggestions of the text in this respect. Drill in 22 ARITHMETIC adding two numbers of two figures each mentally as suggested in the text. See suggestions for the High Third. Also drill in subtracting by the addition method after the manner of mak- ing change. Ex.: How many years have elapsed from 1888 to 1911? Solution: 2, 12, 23. The last number is the answer. In subtracting 39 from 76 we say: 1, 37. The 1 is the number that must be added to 39 to give 40. The 37 is the result of adding the 1 to 36, which is the number that must be added to 40 to give 76 Continue the learning of the multiplication and division tables and apply in examples. Use short division only. Follow the plan of the text. By the end of the Low Fourth, 25 prod- ucts, their reverses, and the corresponding division facts will have been memorized and used in examples. Flash cards and the circle device are excellent aids in fixing the tables. Insist on neat and orderly work in all examples, especially in multi- plication where the multiplier has more than one figure. Use language forms in the mechanics of multiplying by numbers of two figures and in short division where there are remainders in the steps and at the end. The text gives models. It is advisable to conclude the language form in multiplication by saying, for example, after multiplying 457 by 20, that 9140 is the product obtained by multiplying 457 by 20. This em- phasizes in the mind of the pupil what he has accomplished. Follow a similar plan in interpreting the quotient in exam- ples in division. Have analyses given in problems of one step and then of two steps. The text gives models. The teacher must use her discretion in the continuance of language forms in mechanical work and of analyses of problems. A main ob- ject of the description or explanation is to create correct think- ing. Use objective illustrations to help make clear the solu- tion of problems, especially where the fractional idea is in- volved. In all concrete problems, be ready to bring up in the minds of the pupils images of the numbered materials men- tioned in the problems, whether quarts, yards, or pairs of shoes. Defer until the High Fifth inverse problems of the type "If 2/3 of a yard of cloth costs 18c, what is the cost of a whole yard?" Hence omit p. 150. The idea of the fraction should be associated with problems in partition. Thus, what is the cost of 1/3 of a yard of cloth if a whole yard costs 15c? Or, what is the cost of 1 chair if 4 chairs cost $12? Emphasize, as does the text, that 2)12, 12/2, and 12-i-2 all indicate the same operation. Each means either 12 divided by 2 or 1/2 of 12, according to the nature of the problem. Advance in the study of fractions is made by ex- ARITHMETIC 23 pressing division examples in the form of 8/2, 11/2, and re- ducing to whole or mixed numbers. These are to be worked first as examples in division. Pupils should be able to tell what part of 18ft. are 3ft. Use the term ratio. What is the ratio of 3ft. to 18ft.? 18ft. to 3ft.? There should be much oral work in the simple fractional relations. HIGH FOURTH. Text. California State Series, "First Book in Arithmetic," pp. 151-187. Supplementary exercises from the Handbook, Part I. New Phases of Formal Processes. Multiplication and division tables completed and applied. Long division introduced and completed. Language Forms. Analyes of remaining types of problems in- volving multiplication, partition, and division. Language form in long division. Continue the reviews in column addition and subtraction along the lines suggested in the Low Fourth. Review the parts of the multiplication and division tables learned up to this point and apply in examples. Complete the multiplication and division tables, as outlined in Lessons H-M in the text and apply in examples and con- crete problems. Conclude the application of the multiplication tables by using 11 and 12 as multipliers and based upon the tables of ll's and 12's. Use Case I. of the text's special method for determining the quotient figure in long division. Also use the first principle under Case II. It may not be advisable to use the other phases of Case II. The text gives a model for a language form to be used in long division. Great care should be exercised in neatness and form in long division. The quo tient figures should be exactly over the proper figures in the dividend. The figures in the dividend when "brought down" to the remainders should be directly under their positions in the dividend. This makes it easy to tell which figure should next be "brought down" and avoids the practice of checking off the dividend figures as fast as they are used in the re- mainders. Do not permit the writing in of little figures in the dividend in short division. Have constant reviews of the multiplication and division tables. Use flash card sand the circle device. Begin the reci- tation period with the drill 4 times 5, add 4, divide by 6, multi- ply by 9, subtract 7. Use the drill suggested in the High Third for memorizing the addition and subtraction combinations, 24 ARITHMETIC writing a series of numbers in a column and placing above the column the number which is to be used either as the multiplier or divisor. Review the combinations in the same way. Where a record of mental drill is desired, have the pupils write their answers on paper. Test the pupils on multiplication and di- vision by asking for two numbers whose product is, say, 24. Name all the divisors of 24. This drill extended through the series 1-100 will be found productive of good results. Continue the analysis of problems according to the plans of the text. Do not neglect to refer to objects and real experi- ences within the knowledge of the pupils. Children cannot de- scribe things that they do not see or image. See suggestions in the Low Fourth. Defer until the High Fifth inverse prob- lems of the type referred to in the Low Fourth. Continue the concrete work in fractions as in the Low Fourth. The pupils should understand the meaning of frac tions as brought out in the analysis of problems involving par- tition. By the end of the High Fourth the class should under- stand examples like: How many fourths of a circle in 1 3/4 circles? 1/2 of a pie is how many fourths of a pie? 4 2/5 is how many fifths? No rules are yet to be taught. The pupils find the number of fifths in 4 2/5 by first learning that in a whole thing there are five 5ths. In 4 wholes there are then twenty 5th. Also continue the reduction of improper fractions to whole or mixed numbers, working the exercises at first as examples in division. Sum up with the pupils that which they have learned about denominate numbers in the earlier grades. Make the work concrete, using the actual liquid measures and the measures of length in connection with simple examples. Teach and ap- ply the completed table of linear measure, given on p. 174 of the text. Also develop the table of square measure. Make drawings to scale, such as is given on p. 176 of the text. Meas- ure the tennis court and make a drawing to scale. Measure some portion or all of the school garden and draw to scale. By the end of the High Fourth, the pupils should be thor- oughly grounded in the mechanics of addition, subtraction, multiplication, and division. They should read and write num- bers readily. They should be able to work and to anaiyze prob- lems in one and in two steps involving processes. They should know the simple fractional relations involving halves, fourths, eighths, thirds, and sixths. They should be able to reduce im- proper fractions to whole or mixed numbers and simple mixed numbers to improper fractions. They have memorized some of the simple addition and subtraction facts in fractions. They ARITHMETIC 25 can read and write the fractions they have studied. They know the relations between the pint .quart, and gallon. They know the tables of linear and square measure and can draw simple draiwngs to scale. LOW FIFTH. Text. California State Series, "First Book in Arithmetic," pp. 188-211; 227-230. Supplementary exercises from Handbook, Part II., of Bulletin No. 11, State Normal School, San Fran- cisco, Cal., edited by D. R. Jones; and from the advanced text. See Low Sixth. New Phases of Formal Processes. Addition, subtraction, and multiplication of simple common fractions with necessary work in reduction to lower terms, to higher terms, im- proper fractions to whole or mixed numbers, and mixed numbers to improper fractions. Reading and writing of decimals. Addition and subtraction of decimals. Continue the review of column addition and give mental drills in all the fundamental processes. Add and subtract men- tally any two numbers of two figures each by methods sug- gested in grades three and four. Review the multiplication and division tables and apply in examples. Distribute through- out the term problems of the character of those given in the fourth grade. Keep alive and extend the facility gained in analysis. Review especially the analysis involving partition since this work relates to the study of fractions. With respect to new work, take up the addition, subtraction, and multiplication of common fractions in the order named employing familiar fractions, such as given in the text. Then teach the reading and writing of decimals as far as thous- andths, followed by the addition and subtraction of decimals. The division of common fractions will precede the multipli- cation and division of decimals in the High Fifth. This ar- rangement secures for the pupil an early understanding of decimals and enables him to take up division of common frac- tions with less confusion than if it followed immediately the multiplication of common fractions. Teach all the cases in the multiplication of common fractions, including pp. 216 and 217 of the text. Make early use of concellation In connection with the multiplication of common fractions. The work of the previous grades, which has been oral and based on the use of objects, should have secured for the pupils a thorough understanding of the meaning of fractions. The symbolism of common fractions is the main cause of difficulty in this study. Teachers should spare no pains to have pupils 26 ARITHMETIC understand thoroughly the fundamentals of the beginning work in fractions by referring to diagrams and objects. The device of writing, say, 3/5 in the form 3 fifths secures a close correla- tion with the spoken form and may assist in making clear the fundamental processes. Make sure that the pupils can read fractions on the ruler and that they can draw to scale. Such work has an application in manual training. Do not teach here a "method" of finding G. C. D., the examples in reduction to lower terms being sufficiently simple to reduce by successive steps or by finding the G. C. D. by inspection. Neither should a "method" of finding L. C. M. be necessary for the addition and subtraction of the common fractions used in this grade. See the Handbook with reference to this matter. Teach the reduction of fractions to lowest terms before adding fractions whose answers need to be reduced. An answer should not be accepted as correct unless it has been reduced to its lowest terms. Precede the study of decimals by a review of the notation and numeration of integers, emphasizing the naming of the orders and periods and the relations between the units of the various orders, that is, the 10 times, 1/10 of, 100 times, 1/100 of, etc. Extend the principles here brought out into decimals. The work may be based on United States money or measure- ments with a decimal rule, tape, or chain. After the work is well in hand, relate the writing of decimals to that of common fractions. The practice of beginning decimals by saying, for example, that 7/10 is also written .7 is open to objections, the main one being that a misconception of the use of the point arises. It will be sufficient for this grade to read and wHta decimals having but three places to the right of the point. Secure a review of column addition in the addition of decimals. The problems in ratio and proportion (not formally treated) on pp. 224 and 225 of the text, should receive special emphasis. Any possible over emphasis of the method of analysis may be avoided by the use of ratio and proportion. Thus, why find the cost of 1 apple in the problem If 4 apples are worth 5c, 12 apples are worth c. HIGH FIFTH. Text. California State Series, "First Book in Arithmetic," pp. 212-226; 231-256. Supplementary exercises from the Hand- book, Part II.; also from the advanced text. See Low Sixth. New Phases of Formal Processes. Division of simple common fractions. Changing simple common fractions to decimal hundredths, and conversely. Multiplication of decimals. ARITHMETIC 27 Changing simple common fractions to per cents, and con- versely. Division of decimals. Review especially the work covered in common fractions in the Low Fifth. Review the operations with whole numbers as the needs of the class require. Continue tfle quick mental drills, including the simpler operations with fractions. Review the types of problems studied in the Fourth Grade. After tak- ing up the division of common fractions, teach the type of in- verse problems involving fractional relations deferred from the Low Fourth. See p. 150 of the text. Interpret the principles involved in the division of common fractions by diagrams, because much of the application work in fractions has to do with some sort of objects. Follow the suggestion of the text in dividing a mixed number by a whole number as in ordinary division. This plan is usually better than that of reducing to improper fractions. See that the class has facility in this work, for in manual training they are re- quired to find, say, 1/4 of 8 1/2. It is much easier to divide 8 by 4 and then 1/2 by 4 than to reduce 8 1/2 to an improper fraction and then divide. Examples equivalent to this are re- quired later in the sewing classes, where it is necessary "to measure 1/2 the width of the back plus 1 1/2 in."; or "out from 1 measure 3 in. less than 1/2 (or 1/4) the waist measure." Drawing rectangular areas to scale secures a good correlation between practice and principles. Perfect the work in common fractions by requiring written work to be as brief as possible. Restrict the work to simple fractions, such as given in the text. While percentage is not taken up as a topic for study in this grade, follow the plan of the text in the use of the symbol for per cent, applying it in such simple examples in interest that can be readily worked by the class. Terms like "1/2 off" and 25% off" are seen daily by the pupils in the store windows. Simple problems involving these terms can be readily worked in class. Follow the method of the text in pointing off in the division of decimals. Require the pupils to give at first all the steps, using a language form after the manner of the work of the lower grades. In this grade restrict all work with decimals to thousandths, possible higher orders appearing only in products. Conclude the work of the High Fifth by selecting from Chapter VI., on denominate numbers, such tables and exercises as constitute more of a review than the beginning of new work. The table of paper measure should be omitted. Also omit barrels and hogsheads in liquid measure. The reading ARITHMETIC and writing of the Roman numerals should now be reviewed and extended into the higher numbers. The teacher will use her judgment concerning the other topics of Chapter VI. Much depends upon the progress of the class in fractions. LOW SIXTH. Text. California State Series, "Advanced Arithmetic," pp. 1-112. Supplementary exercises from the Handbook, Part II. The work of the Low Sixth is a review and extension of the operations with whole numbers and fractions, both common and decimal. Review first addition, subtraction, bultiplication, and division of whole numbers. If pupils are weak in addition, drill on the combinations. If weak in multiplication and divi- sion, drill on the tables. The work in common fractions in the Fifth Grade was restricted to very simple fractions. While the present work should still be restricted to simple rather than complex or very small fractions, the pupils are expected to work with denominators that require the use of a method of finding L. C. M. The inspection method will, however : handle the great majority of examples. In the Fifth Grade the work in decimals did not employ orders beyond thousandths. The pupils should now be drilled on reading and writing deci- mals with orders up to millionths. The higher orders should appear in the applications only in products and, possibly, in dividends. Take up the work topically as outlined in the text, but keep alive in the minds of the pupils, through examples, any prin- ciples learned in the Fifth Grade that will not be systematical- ly reviewed until the last part of the term or in the High Sixth, for example, the multiplication and division of common frac- tions. In this review give more attention to rationalizing the processes than in previous grades, where reasons were gener- ally not given or required. Lay special emphasis on the deci - mal system as illustrated in whole numbers and in decimal fractions, reviewing the operations with decimals with those for whole numbers, as in the text. Emphasize oral and mental arithmetic. Work for speed in column addition, following the suggestions for grouping given in the previous grades. Pupils should multiply readily by 11 and 12 and use these numbers in short division. Encourage the memorizing of products like 13 times 2, 3, 4, 5; 14 times 2, 3, 4, 5; 15 times 2, 3, 4, 5, 6; 16 times 2, 3, 4, 5, 6; 17 times 2 and 3; 18 times 2; 19 times 2. Also encourage memorizing the corresponding division facts. Apply ARITHMETIC 29 in examples. Teach usable short processes, such as given in the text. Estimating answers before solving, as on p. 41 of the text, is important and should be freely used. Explain how to write a decimal to a specified degree of accuracy. Thus, 2.38 is written 2.40 or 2.4, correct to tenths. Also 3.572 becomes 3.57, correct to hundredths. Make use of this practice in in- terpreting answers. The solution and analysis of problems should be distributed throughout the term. HIGH SIXTH. Text. California State Series, "Advanced Arithmetic," pp. 113-165. Supplementary exercises from the Handbook, Part II. Complete the general review for the Sixth Grade, special attention being given to the multiplication and division of common fractions, left over from the topical review of the Low Sixth. Emphasize cancellation. Continue the use of mental approximations and the writing of decimal answers to specified degrees of accuracy. See the Low Sixth. Drill on the frac- tional relations that will occur in percentage. Thus, find the part one number is of another as well as how many times one number is another. Use the aliquot parts for short cuts in multiplication and division. Give abundant drills in changing common fractions to decimals and decimals to common frac- tions, the latter always being reduced to lowest terms unless otherwise specified. Change both common and decimal frac- tions to hundredths so as to prepare for the later work in per- centage. Work in denominate numbers has claimed attention from the early grades. Conclude the work of the High Sixth by re- viewing all the previous work and that given in the text as far as p. 165. Emphasize lumber measure. See the last chapter in the primary text. The Handbook gives abundant exercises. The tables and examples in denominate numbers as they are gradually developed in the primary and advanced texts are sufficient for ordinary needs. Pupils should know where to find in their text (in the appendix) the list of tables, including those not of common value. Omit the metric system, surveyors' measures, troy weight, apothecaries' weight, and apothecaries' liquid measure. Parts of some questionable tables may be taught. Thus, if one knows that 16 fluid ounces equal 1 pint, he is in a position to know the size bottle to call for at the druggist's. The selection of problems in denominate numbers 30 ARITHMETIC by the authors of the primary and advanced texts is well made. While the texts have a considerable number of exercises dis- tributed through the reviews, it is advisable to have some con- secutive drill in denominate number problems. Examples like "Multiply (or divide) 4 hhd. 5 bbl. 9 gal. 3 qt. 1 pt. 1 gill by 5," have little or no practical value in life. They are wisely omitted by the text. The longer examples in the addition and subtraction of denominate numbers should not be worked. The finding of differences between dates is of practical value, es- pecially in interest examples, but the addition of dates should not be taught. Instead of having the class work the traditional exercises in denominate numbers, it would be more profitable to have examples of more immediate interest to the class. Thus, the girls might be questioned on the following table of dry measure used later in the cooking class: If a recipe calls for a certain amount 3 tsp = 1 tbsp of flour and other ingredients, how 16 tbsp = 1 cup many tablespoons of flour are used if 2 c = 1 pt the recipe is reduced 1/3? Both the 2 pt = 1 qt boys and girls will be interested in problems involving local data. A chain of problems relating to a local hay business may be proposed and solved. Get data on the cost of seeding an 80-acre tract in alfalfa. The cost of keeping the ground in condition. The cost of cutting the four or more crops. The cost of baling. Figure the profit after learning the market price of alfalfa hay. Teach bills and accounts, such as the children see at home. Have a few bills made out by the pupils. Which person whose name appears on the bill is owing money? To whom is the money to be paid? Who receipts the bill? How is this done? Teach how to keep a cash account, such as is illustrated in the last chapter of the primary text. Use data within the knowl- edge of the pupils. The keeping of accounts in connection with the school garden could well be illustrated here. LOW SEVENTH. Text. California State Series, "Advanced Arithmetic," pp. 166-206. Supplementary exercises from the Handbook, Part II. The work of the Seventh Grade consists almost entirely of percentage and its business applications. The review of the Sixth Grade that is probably necessary is the changing of common fractions to , decimal hundredths and the changing of common fractions to common fractions with denominators 100. ARITHMETIC 31 The Low Seventh takes up the fundamental mechanical pro- cesses in percentage (See Handbook for additional eercises) and problems in profit and loss, commission, insurance, taxes, and trade discount, reserving problems involving time for the High Seventh. Success in percentage depends upon recogniz- ing familiar problems in common and decimal fractions when expressed in the language of percentage. The use of the "x" or "c" or some other letter is recommended in the solution of problems, this being especially advisable in inverse problems. This does not preclude, however, the advisability of giving mental work and the simpler written problems in which the three types in percentage are explained without the "x." Thus, after the class has obtained considerable understanding of the classes of percentage problems, introduce the use of letters in a problem of this type: "A man sold a lot at a gain of 16% above the cost. What was the cost if the selling price was $1,890?" Write the equation, 1.16 of c = $1,890. Then find c. The equation may well be used in problems where either the cost or per cent is to be found. An understanding of the equa- tion solves once and for all the question of the different cases in percentage. It should be remembered that practically all percentage problems in every-day life utilize the simplest case of per- centage, that involving multiplication. The finding of the per cent one number is of another also has considerable applica- tion. Pupils should be able to work inverse problems like the one illustrated above, but the following in commission is not practical and should be omitted: "Mr. Jones sent his agent $2000 to invest in flour, first taking out his commission at 3% for buying. What were the proceeds and what was the com- mission?" Taxes and insurance have a practical value, but customs and duties do not come within the experience of the ordinary individual and hence should be omitted. HIGH SEVENTH. Text. California State Series, "Advanced Arithmetic," pp. 207-236. Review the mechanical work in percentage, such as is re- quired in the three cases in percentage. Also review through- out the term the business problems of the Low Seventh. Percentage is now applied in problems where the time ele- ment enters. In simple interest, first work problems in which no "method" is required. Emphasize any one of the proportion methods in finding simple interest. The text gives two, the ARITHMETIC "Method of Aliquot Parts" and the "Sixty Days' Method." The cancellation method should also be thoroughly understood. The antiquated "Six Per Cent Method" has little to recommend its use and need not be taught. Compound interest deserves at- tention, but no great emphasis. Teach both the Mercantile and the United States Rules in partial payments. Avoid prob- lems having more than three payments. In work where prom- issory notes are involved, have the class write the notes and try to have the business transacted in the class. Bank discount is rarely used by the ordinary citizen. Choose problems in which the note is discounted on the date of issue. Familiarize I the class with the different ways of sending money and with the business that the ordinary citizen has to transact with a bank. Teach the mensuration of the familiar plane figures and \ solids as outlined in the text under Forms and Measurements. Let the class get data for problems from their own measure- ments either in the school-room or in the play-ground or in the school garden. Use the compasses in the simplest geo- metric constructions, such as bisecting angles and lines and erecting perpendiculars. Some attractive designs may be made with compasses. Enlarge upon the topics assigned to the appendix of the text according to the needs and capacity of the class. LOW EIGHTH. Text. California State Series, "Advanced Arithmetic," pp. 236-255. The review of the Low Eighth includes the mechanics of percentage, percentage problems, and miscellaneous examples in the work covered in the High Seventh. In advance, teach I longitude and time (correlated with geography), powers and roots, further work in mensuration, including the application of square root, and the application of proportion to similar figures, including the finding of heights and distances. Before leaving the review of percentage, bring together under one grouping topics that relate to investments. Dis- cuss with the class the various ways of investing money. Promissory notes will be recalled, together with the topics of simple and compound interest. Discuss the features of endowment life insurance policies and building and I loan associations. The subject of school and city improvement ARITHMETIC 33 bonds should be taken up and explained. Let the class com- pute the interest on any particular bond that has been re- deemed or is now drawing interest. The subjects of bonds and stocks studied as they have been treated in our texts in the recent past have little value for the pupil. The operations of a local stock company can well be explained to the class. The amount of any probable dividend or assessment can be easily computed by the class after the teacher has shown a certificate of stock and assigned a rate of dividend or assess- ment. The future citizen needs to know more about local bonds than local stock companies. The work of the class may be strengthened by the addition of material from the appendix to the text. In this connection may be mentioned "Public Lands," which should be correlated with civics, and "The Equation." HIGH EIGHTH. The essential principles and topics of arithmetic have been completed in the Low Eighth. The work of the High Eighth consists, first, of a review of the most essential parts of arith- metic, and, second, of business practice and the keeping of simple accounts such as may be needed by the average person. Give first attention in the review to mental arithmetic and accuracy and speed in addition. Drill on the combinations and the multiplication and division tables, as needed, to secure effi- ciency in the operations. Give rapid drills in the reading and writing of numbers. The prime thing in this general review is to give the class mechanical expertness in the operations with whole numbers and fractions, both common and decimal. Review the applications of arithmetic through lists of miscel- laneous problems. Employ usable short cuts in all the review. This aspect of the term's work should embrace from eight to ten weeks of intensive drill. Devote the remainder of the term to business practice and related work. The work of the previous grades should be re- viewed and extended. Some of this review relates to the writ- ing of business paper, such as letters, bills, receipts, and notes, and to the common business done with a bank. Among other things, the business associated with the following should be understood by the pupils: Bills, including merchandise tags heading, items, receipting of; notes heading, rate, mortgage, and other securities, other essential items; receipts and or- ders; checks and drafts; telegrams and telephone messages 34 ARITHMETIC rates; money orders; paper money kinds, meaning of. Pupils should practice writing the business paper associated with any of the above and should explain their essential features. To this should be added the keeping of a cash account and ac- counts with persons as the minimum requirements in book- keeping for the grades. Have the children write up personal cash accounts, the items consisting both of actual transactions in their own recent ex- perience and data furnished by teacher and pupils. Also teach the class how to keep an account with some person. The state- ment contained in a bill illustrates the fundamental idea, but require, in addition, the ledger form, one-half page for debits and the other half for credits, such as is used in ordinary book- keeping. The writing of most of the business paper and the keeping of accounts may be worked out in connection with activities of the pupils. Let them buy and sell, loan and borrow, and the like, using school money and merchandise cards. One way to , unify the work is to group the activities about .a school bank, with which the pupils do business. In case the bank idea is not used, the teacher should illustrate the business of deposit- ing money with and receiving and sending money through a bank. In like manner the use of the money order should bo illustrated. The new regulation concerning Postal Savings Banks should be explained and the pupils encouraged to make deposits when the local postoffice establishes a postal bank. In the meantime, the teacher or the school bank can receive deposits from the children and keep the necessary accounts. A study of household expenses may be associated with the study of bills and accounts. Classify the expenses of a house- hold of, say, five persons for one month. Under Household Maintenance include rent, table, electricity, gas, water, laun- dry, and miscellaneous needs of the house. Under Personal Expenses of the members of the family, taken individually, include wearing apparel, luxuries not shared in by the rest of the family, etc. Under Luxuries include theaters, concerts confectioners'-, etc. Under Unclassified Expenses include any- thing not included in the other headings, such as fees, dues, church, car-fare. The teacher and the pupils supply the neces- sary data for these items. An opportunity is offered in such work for the pupils to learn the prices of common articles of necessity. The ques- tions of values and economy are immediately concerned here. Pupils learn to read, gas, water, and electric light meters. If the teacher desires, a broader course in book-keeping ARITHMETIC 35 may be profitably worked out so as to provide for the above- mentioned activities and give the pupils practice in double- entry, not a complete system with three or more books to con- fuse the learner, but a simple plan which requires only a blot- ter and a ledger. (See the manual furnished by the depart- ment of mathematics of the Normal School.) UNIVERSITY OF CAITFOB* THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO S1.OO ON THE SEVENTH DAY OVERDUE. LD 21-100m-7,'39(402 A 251757 26m-6,'12