Pass £- u/S¥- f Book ^ ^ Gopyi$itN° COPYRIGHT DEPOSIT. MATHEMATICAL CONSTRUCTION INFORMAL NUMBER WORK FOR BUSY HANDS GRADES ONE AND TWO BY N. LOUISE LAFFIN OF THE CHICAGO SCHOOLS A. FLANAGAN COMPANY CHICAGO Copyright 1911 BY N. LOUISE LAFFIN $ C ©CI.A303189 TABLE OF CONTENTS Introduction 5 Plans of Lessons 12 All the combinations to 12, and the lessons in which they are found 16 FOLDING. LESSON. PAGE. I Shawl 23 II Booklet 25 III Wall-Pocket 26 IV Soldier's Cap 28 V Fireman's Cap 29 VI Envelope 30 VII Picture Frame 32 VIII Salt and Pepper Holder 35 IX Boat 36 X Square Prism 37 XI Cube 40 XII Cradle 42 XIII Chair 43 XIV Buggy 45 XV to XXV Baskets 46 FOLDING AND WEAVING. I Making 2 cuts on mat (16 squares), weaving with one strip 53 CONTENTS LESSON. PAGE. II Making 2 cuts on mat (12 squares), weaving with one strip 53 III Making 3 cuts on mat, weaving with 2 equal strips 54 IV Making 3 cuts on mat, weaving with 4 equal strips 54 V Making 5 cuts on mat, weaving with 2 equal strips 55 VI Making 5 cuts on mat, weaving with 4 equal strips &> VII Making 7 cuts on mat, weaving with 6 equal strips 56 VIII Making 5 cuts (relation 1 to 2), weaving with 4 strips (relation 1 to 2) 56 IX Making 5 cuts (relation 1 to 2), weaving with 5 strips (relation 1 to 2) 57 X Making 4 cuts (relation 1 to 4), weaving with 3 strips (relation 1 to 4) 57 XI Making 7 cuts (relation 1 to 2), weaving with 7 strips (relations 1 to 2 and 1 to 4) over 2 and under 1 58 XII Relation 1 to 2 cutting mat, 1 to 4 cut- ting strips, weaving over 3 and under 1. 58 FREE CUTTING AND WEAVING. I Relation 1 to 2 60 II Relation 1 to 4 61 III Relation 1 to 3 (4 cuts) 61 IV Relation 1 to 3 (6 cuts) 62 CONTENTS LESSON. PAGE. V Eelation 1 to 3 (8 cuts) 62 VI Relation 1 to 3 (7 cuts) 63 VII Eelation 1 to 5, 1 to 2 (6 cuts) 63 WALL-PAPER MADE BY FREE-CUTTING 65 PAPER RINGS MADE BY EREE-CUTTING 67 MEASURING AND WEAVING. I Bank Decoration. (Rulers: I", 2", 3", 4") 68 II Calendar Back. (Rulers: 1" ', 2", 3". 4", 7") 69 III Blotter. (Rulers: 6", 5", 4", 3", 2", 1") 70 IV Napkin Ring. (Rulers: 1", 2", 3", 4", 5", 6") 72 V Telephone Pad. (Rulers : 1", 4", 5", 8") 73 VI Wall-pocket for Letters. (Rulers: 1", 2", 3", 7", 10") 74 VII Circular Woven Basket. (Rulers: 1", 2", 5", 6", 9", 10") 76 VIII Needle-book. (Rulers: 1", 3", 4", 7", 8", 10") 78 PROGRESSIVE MEASURING LESSONS. Introduction of Single Unit Rulers. I Scissors Holder. (Ruler : 8") 80 II Mayflower. (Rulers : 8", 4") 81 III Cup. (Rulers : 6", 3") 83 CONTENTS LESSON. PAGE. IV Book-mark. (Rulers : 4", 2") 85 V Stamp Pocket. (Rulers : 2", 3", 5") . . . 86 VI Washcloth Pocket. (Rulers: 1", 7", 3", 4") 88 VII Pilgrim's Bonnet. (Rulers: 2", 4" ', 6") 89 VIII Stove (Rulers : 3", 6", 9") 91 IX Bank. (Rulers : 4", 8", 12") 93 X Envelope (for decoration units) (Rulers: 5", 1", 7", 2") 95 XI Taboret. (Rulers: 5", 3", 8", 1") 96 XII Father Bear's Chair. (Rulers: 3', 6", 9", 12") 97 XIII Mother's Chair. (Rulers: 2", 4", 6", 8") 99 XIV Baby's Chair. (Rulers: 1", 2", 3", 4"). 100 XV Father's Bed. (Rulers: 3", 6", 9", 12"). 101 XVI Mother's Bed. (Rulers: 2", 4", 6", 8"). 103 XVII Baby's Bed. (Rulers: 1", 2", 3", 4") . . 103 XVIII Father Bear's Bowl. (Rulers: 12", 4", 3", 1") 103 XIX Mother's Bowl. (Rulers: 12", 3", 2", 1") 104 XX Baby's Bowl. (Rulers: 12", 2", 1")... 105 XXI Jack's Pail. (Rulers: 9", 5", 4", 1") . . 106 XXII Jill's Pail. (Rulers: 9", 3", 2", 1").. 107 XXIII Fox's Dish. (Rulers: 5", 2", 1") 108 XXIV Stork's Dish. (Rulers: 5", 7", 6", 1"). 109 XXV Handbag. (Rulers: 9", 6", 3", 2", 1"). 109 XXVI Pencil-Box With Lid. (Rulers: 10", 2", 3", 5") Ill CONTENTS LESSON. PAGE. XXVII Match-Safe. (Eulers: 5", 4", 3", 2", 1") 112 XXVIII Wood-Box. (Rulers: 3", 6", 9", 12" 2", 4", 6", 8") 113 XXIX Sled. (Rulers: 5", 4", 3", 2", 1") .... 114 XXX Pushcart. (Rulers: 7", 5", 2", 1") 115 XXXI Gocart. (Rulers: 2", 4", 6", 8") 117 XXXII Cradle. (Rulers : V, 9", 5", 4") 119 XXXIII Bureau. (Rulers: 3", 6", 9", 12" 8", 2", 5") 121 XXXIV Chiffonier. (Rulers: 2", 4", 6", 8", 10") 123 XXXV A Large Envelope. (Rulers: 11", 1", 8", 4") 125 XXXVI Book for Cuttings — unfolded sheets — (Rulers : 1", 3", 6", 9") 127 XXXVII Book for Words— unfolded sheets— (Rul- ers: 6", 1", 7") 128 MATHEMATICAL CONSTKUCTION INTRODUCTION All the construction work should be carefully selected by the teacher with, a two-fold purpose. 1st. The appropriateness of the thing itself to the child's need; a co-relation to other studies, or a gift, plaything or actual necessity. 2nd. The appropriateness of the number relations to the child's mathematical discoveries; a proper sequence from the simplest of mathematical relations to the more complex, and a proper order of inferences. The possible number relations found in the process of the construction of an article should be brought out carefully by the teacher. While the child is making something he is in a receptive mood for this number work. Each step gives new surfaces or lines to compare. When the teacher artfully questions, giving concrete problems regarding these ever changing lines and surfaces, she gets answers from alert minds ; minds that are comparing and coming to conclusions then and there; minds that are not trying to remember what the teacher said yesterday or a week ago; minds that are seeing and drawing inferences at the present moment. In applying number work to the making of things in the 5 q MATHEMATICAL CONSTEUCTION first and second grades (by folding, free cutting or measur- ing), a teacher should let the child judge the relations of things himself. By repeated acts of judgment he gains the power to see relations correctly. Indefinite relations are seen before the definite. There- fore, the earliest lessons (found in the folding of articles) should deal only with words which express contrast; such as larger and smaller, longer and shorter, more or less. A few weeks of this indefinite relation work usually suffice. Then follow the relations, 1, 2, 3, 4, equal -J, J, J. In these lessons the teacher should only show the fold, the child imitating. This gives mental training through sight. After the child has imitated the teacher's silent direction, the teacher should ask questions bringing out the relations (indefinite or definite as the state of the child's mind permits). These questions should introduce technical words inci- dentally and informally to the pupil. By giving little concrete problems based on the relation of surfaces, lines and solids, she can do this. For instance: when a square is folded into equal oblongs she may say, "If this oblong is enough cloth (pointing to one oblong) for one doll's dress, what is this oblong?" (pointing to other). Or, "Play this oblong is a dresser scarf, which has lace all around it. If the lace on the upper edge costs a dime, what does the lace on the side edge cost?" Ans. "More than a dime." Child gets words oblong, upper edge, lower edge incidentally. Giving concrete problems in this way is interesting to the imaginative mind of a child. He plays that sand is sugar, buttons are money, and stones are potatoes or apples. Why can't he play that oblongs or squares are parks, play- INTRODUCTION 7 grounds, ceilings, bed quilts, tablecloths, articles of clothing, etc.? And why can't he also play that two of those are two parks, or three of them are three parks? If one park con- tains five acres, two parks contain two five acres, etc. Why can't he play that a triangle is : a doll's shawl, a pond of ice to skate on, a tile, a piece of glass, the leather corner of a desk blotter, etc.? Why can't he play that edges require fringe for rugs, bedspreads, napkins, doilies, the top of a carriage, etc.; fences for parks, farms, yards, empty lots, etc. ; binding for slates, books, pictures, cloth ; curbstones for streets ; hedges or trees in a row for yards, parks, roads ? When the children are familiar with the technical words, the teacher should give verbal directions instead of silently showing them, thus giving child mental training through hearing. These verbal directions are more difficult than the sight directions, for they involve a memory of the technical words and number relations learned incidentally through the concrete problem work. After a child can fold, by following the verbal directions of the teacher, he is ready to use a ruler. An ordinary 12-inch ruler is beyond the comprehension of the beginner. The inch is within the 2-inch measure, and within the 3-inch unit there is the 2-inch and also the 1-inch unit. The child cannot see the relation of 1 inch to 2 inches on the ruler; therefore, he should use single unit rulers. A teacher should have a complete set for each child in the room. A set consists of 12 pieces of strong, tough, smooth cardboard (pressboard is the proper name). Each piece is 1 inch wide, and respectively, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 inches long. MATHEMATICAL CONSTRUCTION UD Of course the child is not given the whole set at once. He would be confused. If the teacher intends making something which requires a 6-inch square, for instance, she gives him a 6-inch ruler. If that must be bisected he is given a 3-inch ruler. If the original paper is 9"xl2" let him estimate the width of it. He will tell you that the width is the sum of 3" and 6" or the sum of 6" and i/ 2 of 6" or three 3" He will say that the length is : four 3" or the sum of 6" and 6" or the sum of 6" and 3" and 3" or the sum of three 3" and l/ 2 of 6" INTRODUCTION 9 With these rulers one obtains mathematical relations of the measures which one could never get from little children with the ordinary ruler, which has the units within many other units. The children see: the sum of 3" and 3", the difference between 6" and 3", the division of 6"x2, the multiplication of 3"x2. When in a later lesson the 9-inch ruler is introduced, the children see: 6" 3'3"=9" $ of 9"=3" 3 // 9" 9" 9' -6" -3 3" 6" In making an article based on a square consisting of 16 squares, we use the dimensions : 3, 6, 9, 12, or - 2, 4, 6, 8, or 1, 2, 3, 4. Why should a child be bothered with all the units within units of an ordinary ruler when he can be given something which is definite to him? How easy it is to see that 6 is J of 12; 3 is J of it, etc., with these distinct units of measure. The making of an article based on a square consisting of 9 squares vividly shows the relation of 1, 2, 3; 2, 4, 6; 3, 6, 9; 4, 8, 12. " 10 MATHEMATICAL CONSTEUCTION One based on a square consisting of 4 squares gives the relations 1, 2 ; 2, 4 ; 3, 6 ; 4, 8 ; 5, 10 ; 6, 12. The first step in the using of the ruler is like the first step in folding. The teacher should show the direction and the child imitate (sight and mind training). The only verbal expression of the teacher is the naming of the ruler, viz. : "This is a 6-inch ruler." The second point in the use of the ruler is that the teacher directs verbally, after the child has learned how to use it. Third point: Child must do mental addition, subtrac- tion, division, or multiplication, by following teacher's verbal direction. Teacher asks questions like these : (Suppose child to have rulers 3", 6", 9", 12" long, and that he is familiar with their relations through former work in measuring, and that he has paper 9"xl2".) Show me the edge that is three 3" long. How many inches long is it ? Show me the edge that is the sum of 9" and 3" long. How many inches long is it? On the upper long edge mark off J of 12", One-half of 12" is how many inches? If it costs $12.00 to build a fence on this edge (pointing to the 12-inch edge) what part of $12.00 will it cost to build a fence on this part (showing 6") of it? Ans. -J of $12.00. How much money will it cost? ($6.00) Etc. This last question relative to cost is an inference. From actually seeing that 6" are -| of 12" the child easily infers that $6.00 are -J of $12.00. If questions were asked concern- ing time required to build the fence it would be easy for the child to infer that 6 days are J of 12 days ; INTRODUCTION n 6 hours are -| of 12 hours; 6 men are -J of 12 men. So all possible combinations, separations, products and parts of the numbers to 12 can be reached through the use of these rulers. When a child has seen that the sum of 3"+3"=6" and has inferred that 3c and 3c = 6c ; or 3 yds. and 3 yds. = 6 yds. ; 3 hours and 3 hours = 6 hours ; the teacher should go to the board and write 3 3 At some other time, other than the construction period, have a drill on writing the combination. Write it as a whole. Erase and let them write it. Write it again omitting one of the numbers. Erase. Let children write it inserting the one the teacher omitted, etc. Drills of this sort help to fix number relations which have been previously seen. MATHEMATICAL CONSTKUCTION PLAN OF THE LESSONS Folding I have in the folding lessons shown what variety of con- crete problems can be given; how many of them can be correlated to the finished article; how they do incidentally show the number relations 1, 2, 3, 4 and more, equal, §, J, J. At the end of some of the simple folding lessons, I have drawn surfaces which the child sees in making the article. These can be put on the blackboard and concrete problems based on them after the folding lesson. Folding and Free Cutting (Weaving and Decorating) After a child has dealt with the relations 1, 2, 3, 4, equal, •J, J, J in folding lessons, he is ready to do some free cutting. Weaving and Decorating give many opportunities for cutting a piece of paper into 2 equal pieces, or 3, or 4, as the case may be; or, for cutting a piece of paper twice as large, long, or wide as another. There is a natural sequence of the three topics folding, free cutting and measuring in the single subject Weaving. The weaving lessons are presented in this order. No concrete 12 PLAN OF THE LESSONS 13 problems have been written in connection with these lessons, but they should be given wherever there are surfaces or lines to compare. These woven mats can be utilized in many ways. They can be used in the doll house for mats, table cloths, dresser scarfs, sofa-pillow tops; they make a good background for wall-pockets, calendars, match scratchers; woven a certain way they make telephone pads; they also make a pretty top for blotters, or outer side of napkin rings when they are under transparent celluloid. Wall paper can sometimes be made by weaving. Woven mats can be folded and pasted into cornucopias, lanterns, and all sorts of baskets. They can be used to decorate books, outside of banks, envelopes, etc. In the folding and free cutting weaving lessons, the development of the relations 1, 2, 3, 4, has been shown. In the measuring lessons, besides showing how naturally and definitely number relations are felt in using these single unit rulers, a utility has been given for every mat. For progressive number work, however, carefully read the lessons given under the heading "Measuring with Single Unit Rulers/' Measuring with Single Unit Kulers In measuring with the Single Unit Rulers, I have shown how, through a series of lessons, the child finds all combina- tions, separations, products, and parts of numbers up to 12; also the great variety of lines, surfaces and solids there are to compare; how I correlate my concrete problems with the finished article. There is such a great variety of these surfaces and lines that the teacher must discriminate in comparing them. She 14 MATHEMATICAL CONSTKUCTION must not ask questions on every possible relation she sees; if she stops too often the child will lose interest in what he is making. That is worse than losing the number work. In making an article based on a square consisting of 4 small squares, the child sees the relation of 1 to 2; 2 to 4; 3 to 6; 4 to 8; 5 to 10; or 6 to 12. "5 /2. <0 ■ xj : \ PLAN OF THE LESSONS 15 From these squares the following articles can be made: Booklet. Picture frame. Wall-pocket. Salt and peppers. Soldier's cap. Boat. Bookmark. Basket. Fireman's cap. Cup. Scissors holder. In making an article based on a square consisting of 9 small squares, the child sees the relations of 1, 2, 3; 2, 4, 6; 3, 6, 9; or 4, 8, 12. From 9 squares the following can be made: Taboret. Dutch bonnet. Stove. Wheel-barrow. Cube bank. Box. Baskets (cut on straight folds or cut on diagonals). In making an article based on a square consisting of 16 small squares, the child observes the relations of 1, 2, 3, 4; 2, 4, 6, 8 ; or 3, 6, 9, 12. From 16 squares the following can be made: Baskets (cut on straight fold, or cut on diagonal). Cradle. Hallowe'en lantern. Table. Pilgrim cradle. Chair. Square prism. Lounge. Square pyramid. Bed. Cube. House. Wagons and carts of all kinds. Wood box. Sled. Box with lid. Pocket-book. 16 MATHEMATICAL CONSTEUCTION Dresser, sectional book case, or any kind of cabinet with drawers or compartments. Then there are all sorts of odd lengths and widths of oblongs to be used as sides for circular baskets, pails, bowls; backgrounds for calendars, match scratchers, wall-pockets; strips and mats for weaving. These give such combinations as: 2 3 5 9 7 3 4 4 2 3 5 7 9 11 10 etc. The following chart shows every combination of numbers from 1 to 12 inclusive, and the lessons in which the child sees these combinations : All the Combinations to 12 and the Lessons in Which They are Found Note. — These are in the measuring section unless otherwise specified. 1 Lessons : XIV, XVII, XX, XXII, XXVII. 1 2 Lessons: XIV, XVII, XIX, XXII, XXV, XXVII, 1 XXIX. PLAN OF THE LESSONS 17 3 Lessons : VI, XIV, XVII, XVIII, XXVII, VIII, 1 (Weaving, p. 78), II (Weaving, p. 69), XXIX. 4 Lessons: XXX, XXI, XXVII, XXIX. 1 5 Lessons: XXX, X, XXIV, VII (Weaving, p. 76). 1 6 6 Lessons: XXIV, XXXVII. 1 7 Lessons: II (Weaving, p. 69), VIII (Weaving, p. 78) 1 8 8 Lesson: XXXV. 1 9 18 MATHEMATICAL CONSTKUCTION 9 Lessons: XXXII, VII (Weaving, p. 76). 1 10 10 Lesson: VII (Weaving, p. 76). 1 11 11 Lesson: XXXV. 1 12 2 Lessons: IV, XIII, XVI, XXVII, XXXI, VII, XIV, 2 XVII, XXVIII, XXXIV. 3 Lessons: V, XXVI, XXVII, XXIX. 2 4 Lessons: XXX, VII, XIII, XVI, XXVIII, XXXI, 2 XXXIV, VI (Weaving, p. 74). 5 Lessons: XXX, X, XXXIII. 2 PLAN OF THE LESSONS 19 6 Lessons: XIII, XVI, XXVIII, XXXI, XXXIV. 2 8 7 Lesson : XXX. 2 8 Lessons: XXXI, XXXIII, XXXIV, VI (Weaving, p. 2 74). 10 9 Lesson: VII (Weaving, p. 76). 2 11 10 Lessons : XXXIV, XXXV. 2 12 3 Lessons : III, VIII, XII, XXVIII, XXXIII, XXXVI. 3 6 4 Lessons: VI, VIII (Weaving, p. 78), II (Weaving p. 3 69). 20 MATHEMATICAL CONSTEUCTION 5 Lesson : XI. 3 8 6 Lessons: VIII, XII, XV, XXV, XXVIII, XXXIII, 3 XXXVI. 9 7 Lessons: VI (Weaving, p. 74), VIII (Weaving, p. 3 78). 10 8 Lesson: VIII (Weaving p. 78). 3 11 9 Lessons: XII, XV, XXVIII, XXXIII. 3 12 4 Lessons: IX, XIII, XVI, XXVIII, XXXI, XXXIV, 4 XXXV. 8 5 Lessons : XXX, XXI, XXXII, XXXV. 4 9 PLAN OF THE LESSONS 21 6 Lessons : XXXI, XXXIV, VI (Weaving, p. 74). 4 10 7 Lessons: VIII (Weaving, p. 78). 4 11 8 Lessons : IX, XXVIII, XXXIV. 4 12 5 Lessons: XXVI, VII (Weaving, p. 76), 5 10 6 Lesson: VII (Weaving, p. 76). 5 11 7 Lesson : X. 5 12 6 Lessons: XII, XV, XXVIII, XXXIII, XXXIV. 12 22 MATHEMATICAL CONSTKUCTION 2, 1"— XIV, XVII, XXIX. 2f 2"— IV, VII, XIII, XIV, XVI, XVII, XXVIII, XXIX, XXXI, XXXIV. 2, 3"— III, VIII, XII, XV, XXV, XXVI, XXVIII. XXXIII. 2, 4"— II, IX, XIII, XVI, XXVIII, XXXI, XXXIV. 2, 5"— XXVI, XXXV. 2, 6"— XII, XV, XXVIII, XXXIII, XXXIV. 3 ? i"_xiV, XXXVI. 3, 2"— VII, XIII, XVI, XXV, XXVI, XXVIII, XXXI, XXXIV. 3 ? 3"_ VIII, XII, XV, XXII, XXV, XXVIII, XXXIII. 3 ? 4"— IX, XVIII, XXVIII, XXXII, XXXIV. 4, 1"— XIV, XVII. 4j 2"_ XIII, XVI, XXVIII, XXXI, XXXIV. 4, 3"— XII, XV, XVIII, XIX, XXVIII, XXXIII. 5, 1"— XXIX, XXIV, XXVII. 5, 2"— XXVI, XXXIV, XXXV. 6, 1"— XXXVI. 6, 2"— X, XIX, XXVIII, XXXIV. FOLDING LESSON I Shawl Note. — Early lessons should be given without verbal command. Teacher should only show what to do. Then let children imitate. Material : An oblong piece of paper. Give word oblong in this way: Show me a long edge of this oblong. Show me a short edge of this oblong. Show me an edge longer than a. Show me an edge shorter than b. Show me an edge shorter than c, etc. Play the oblong is a shawl. Which edge needs more fringe, a or b ? (Teacher points to edges.) If the fringe on a costs a nickel, what does the fringe on b cost? (Ans. More than a nickel.) If the fringe on b costs a nickel, what does the fringe on a cost? (Ans. Less than a nickel.) Teacher folds so that the edge a exactly touches b. Children imitate. Result: Figure 2. Children lay theirs on desks. Teacher shows large undi- vided surface to them. Play that it is a piece of cloth and 23 Fig. X. 24 MATHEMATICAL CONSTEUCTION repeat questions on relations of surfaces as before. Viz. : If it takes an hour to hem this side (pointing to d) will it take more or less to hem this (pointing to c) ? Ans. More than an hour, etc. Fold back oblong and tear off. Result: A doll's shawl. Surface Seen While Making Shawl Draw them on the blackboard to scale of 6" to 1". Suggestive comparisons on which to base problems A=3 of C. E=2 of C. =3 of B. =2 of B. =sum of B+E. =D. =sum of D+B. =sum of B+C. =sum of E+C. =sum of D+C. FOLDING 25 B=i of E. =4 of D. =i of A. D=2 of B. =2 of C. =E. =A-B. =sum of B+C. Note. — If child does not see the relations of the surfaces, cut pieces of paper the exact size of the blackboard figures. Dissect and put the pieces together in any necessary form to give the child a clear image of the relation of the surfaces. LESSON II Booklet Repeat lesson I but speak of the oblong being a cover for the library table, the edges of which are stenciled, hemmed, embroidered or fringed. When A is obtained the children have 2 surfaces each a triangle. Play one triangle is a library floor, what is opposite surface? (Ans. Another floor just as big.) If it takes 10 yds. of carpet for the first floor, how many yds. for opposite floor? The teacher just points to the sur- faces saying: If it takes 10 yds. for this floor, how many for this? (Ans. Just as many, 10 yds.) If this costs $25.00, what does this one cost? etc. Unfold the paper. Besult: A square. 26 MATHEMATICAL CONSTRUCTION Play this is a rug for the library. If the fringe on A costs a dollar, what does the fringe on C cost? on Df on B? How many dollars for fringe on A and C together? D and B together? How many dollars for all sides? Fold the square into 2 equal oblongs. Here are two oblong surfaces to compare as the triangles were compared. Result: A booklet. LESSON III Wall-pocket (New relation 1 to 2) From an oblong proceed to make a square as in previous lessons, but do not ask too many questions on the surfaces and lines already talked about. The same relations with different lengths of lines and size of surfaces can be found in the new folds. After the square has been folded into 2 equal oblongs the new folds begin. Fold so that the short edges touch. Result, 4 small squares. Children see only the outside ones. Teacher points to 1 square and says : If this window glass costs a dime, what does this one cost? (Pointing to opposite one.) (Ans. One dime.) If it costs a quarter? (Ans. One quarter.) FOLDING 27 If putty on one edge costs a penny, what does putty on another cost? What does putty on all cost? Hold square so that 4 free points point upward. Teacher moves one point from A S^K to B, thus making a wall-pocket. / N. Now child sees a whole triangular sur- <(C X face (0) and the opposite side which is a \^ q / square. \. / Now play C is a piece of glass for a B window. Teacher points to opposite surface (the square) and says: How many pieces just as big can be cut from this? (Ans. 2 pieces of glass.) If this (pointing to C) costs a nickel, what does this (pointing to opposite square) cost? (Ans. 2 nickels.) If it costs a dime? (Ans. 2 dimes.) If it costs a quarter? (Ans. 2 quarters), etc. Suggestive Comparisons of Surfaces Seen While Making Wall-pocket Draw surfaces on blackboard to any scale large enough to be easily seen by pupils. Base problems on these surfaces. A=sum of B+C. =E =sum of D+E. =3 E's. =3 Fs =3 B's. =2 G's G=2 H's. =4 of F. 4 of B. =6 G's. =J of D. B=i of A. =i of C. =4 of c. =J of A. =4 H's. F=2 G's. =± of D =4 H's. 28 MATHEMATICAL CONSTRUCTION LESSON IV Soldier's Cap Made just like wall-pocket with an additional fold at the end; the folding back of the three points which were left as the back of the wall-pocket. Eesnlt : Soldiers cap. Number relations not different than Lesson III; but this should be made of a differently sized paper, to give a new length of lines and size of surfaces not relative but actual. FOLDING 29 Play that the large square is a drill ground for soldiers. If the ground is 1 block long, how wide is it? (Ans. 1 block wide.) If one company of soldiers can stand next to one another on this side (pointing to one edge), how many can stand next to one another on this edge (pointing to another edge) ? When folded into oblongs these questions: If there is room for one company to drill here (showing 1 oblong), there is room for how many companies to drill here (show other oblong) ? After triangle is folded back repeat question and get relation 1 to 2. LESSON V Fireman's Cap No new number relation. A review of relations found in Lessons III and IV. Use a different size of paper. Make soldier's cap; then press the front and back points of the cap together. Hold so that 2 free points point up- ward. Fold one of them back until it touches bottom corner. Eesult : Fireman's cap. Play that the oblongs are windows in the engine house. Question on time it takes to wash them. Play that squares are floors. Question on scrubbing. Or play that squares are sheets on fireman's bed. Question on hemming of edges; or, in comparing two surfaces, question on length of time to iron them. 30 MATHEMATICAL CONSTRUCTION Play that triangle is a piece of rubber for making the hat. Problems on cost of it. If this piece of rubber (pointing to A) costs a nickel, what does this cost (pointing to opposite square) ? Ans. 2 nickels. If it costs 8 cents? Ans. 2, 8 cents. LESSON VI Envelope (New relation 1 to 4 and 1 to 3.) Play the square is the top of a desk. Compare cost of varnishing to another child's square just the same size. When folded on a diagonal, play that one triangle thus formed is a tile of the school vestibule floor. Compare cost of this with opposite triangle. Teacher holds her paper so that folded edge is down. She folds one of the upper points (A) to center of folded edge. Children imitate. Re- sult: Fig. 1. B is a whole surface itself. Compare it with opposite surface. Relation 1 to 4. FOLDING 31 If B is one tile opposite surface is 4 tiles. If B costs a dime opposite sur- face costs 4 dimes, etc. Fold point C to E. Result: Fig. 2. F is a whole surface, so is G; compare F to G. Relation 1 to 2. Fi 5 .2 Fold point D (Fig. 1) to E. Result: Fig. 3. Compare F and H. Relation equal. Compare F and G. Relation 1 to 2. Compare H and G. Relation 1 to 2. Compare G with opposite surface. Re- lation 1 to 3. Of course, give concrete problems to get Fig 3 these relations. Surfaces Seen While Making Envelope Draw them enlarged on blackboard. 32 MATHEMATICAL CONSTEUCTION Base problems on some of these comparisons A=2 of B. G =8 of E. =2 of G. =4 of D. =8 of D. =i of A. =8 of H. K=2 H's. C=3 of D. =F-D. =6 of E. =6 of J. =3 of H. =sum of H, D, E, J. =sum of K, E, J. E=J of D. =i of C. =i of F. =i of K. B=sum of C+D. D=i of F. =H. G=B. =2 of J. =4 of H. =1 of C. =8 of J. =4 of B. LESSON VII Picture Frame Relation 1 to 4; 1 to 5; 1 to 6. Fold square into 2 equal oblongs. Fold so that short edges of oblongs touch. Open. Large square is now divided into 4 small squares. (Fig. 1.) E Ftgt. FOLDING Fold so that point A touches center Fold so that point B also touches center (E.) (Result, Fig. 2.) Compare surface F with surface G, Fig. 2. Relation equal. Compare surface F with opposite sur- face. Relation 1 to 6. Compare surface G with opposite sur- face. Relation 1 to 6. Fold so that point D touches center E, Fig. 3. Compare H to F or G. Relation equal. Compare H to opposite surface. Rela- tion 1 to 5. Fold point C to center E, Result, Fig. 4 Compare surface J to opposite surface. Relation 1 to 4. Fold back point 1 to middle of edge 5. Fold back point 2 to middle of edge 6. Compare K and M (Fig. 5). Compare K and H, etc. Fold back point 4 to 8 and 3 to 7. Result : Picture frame for a doll's house. Base the concrete problems on things pertaining to a doll's house. E D E Viq.2. P'9 3 FsgS 34 MATHEMATICAL CONSTRUCTION The squares and oblongs can be lace curtains for the doll house. Question about length or cost of lace for the curtains. Some curtains having lace on 2 edges, others on 3 edges. Let triangle F (Fig. 4) be a doll's shawl. Let opposite surface be a piece of cloth. Ask how many shawls can be cut from it. Let small square (Fig. 1 before it is opened) be a hand- kerchief. Question on cost of lace on edges, or time to hem them. Surfaces Seen While Making Picture Frame Draw them enlarged on blackboard. Suggestive compari- sons: A=2 of B. =8 of C. =2 of I. A=sum of D+C. =sum of H+3 C's. =sum of 1+4 C's. FOLDING 35 A=sum of G+2 C's. =sum of D+E+F. =sum of G+C+E+F. C=l of B. =i of I. =sum of E+F. B=sum of C+E+F. =i of A. =1. D=B+3 C's. =7 C's. G=6 C's. =1+2 C's. H=I+C. =5 C's. 1=4 C's. F=3 E's. =C-E. LESSON VIII Fig. 1. Salt and Pepper Holder Fold square on its diagonals. Fold points a, h, c, d, to center E as in previous lesson, getting small tri- angles to compare with opposite surface. Eelations 7, 6, 5, 4, respectively. Turn Fig. 1 upside down. Fold the four points of the new square to center E again and turn up- side down. Insert fingers under points 1, 2, 3, 4, Fig. 2, and squeeze into the shape of salt and pepper holder. Fig. 3. I 2. 3 V Fig. 36 MATHEMATICAL CONSTEUCTION After the holder is made, inquire: If one little pocket holds lc worth of salt, how many cents' worth Will all hold? Fig ; . If it holds 2c worth, how many 2c worth will all hold? (Ans. 4, 2c worth.) LESSON IX Boat See Lesson IV, Soldier's Cap, for beginning of boat. Take the two side points of the soldier hat and make them touch one another. Now there are 2 free points of a square. Fold one of them back to opposite corner. Fold the other one back the other way. Result : A little soldier's hat. Press the two free points of this together. Fold back both in opposite directions to meet opposite corner. Eesult : A very small soldier's cap, with 3 points at the top. Put fingers on sides of middle point and pull the paper. Eesult: Boat. When boat is made ask questions about the prow and stern of it. Bind them with iron (imaginatively, of course). ^ ~~~~Z If prow is 5 yds. long, stern is \ & 5 yds. long. — is If it took 2 days to bind prow with iron, it took 2 days to bind stern with iron. If iron on prow costs $10.00, iron on stern costs $10.00. -rr FOLDING 37 If it takes a week to paint 1 side of ship, it takes a week to paint other side of it. If it takes $100.00 worth of wood to build one side it takes $100.00 worth to build other side. Compare cost of painting 1 side of cabin to 1 side of ship on the outside. (Eelation of 1 side of cabin to side of ship is*). Compare cost of painting 1 side of cabin outside and inside to 1 side of ship outside and inside, etc. LESSON X Square Prism (To be used for that game in which the teacher says: "I am thinking of something which looks like a square prism. Guess what it is.") Fold a square into two equal oblongs. Open. (Fig. 1.) Make edge A touch folded line. Compare oblong thus obtained to opposite surface. Eela- tion 1 to 3. Make edge B touch folded line. Compare oblong thus formed to opposite surface. Relation 1 to 2. Open whole paper and fold the other way into 2 equal oblongs. Fold long free edges back to long folded edge. Eesult: Large square is 9 ' divided into 16 small squares. 38 MATHEMATICAL CONSTRUCTION A B C D E F G H Cut line 1 and fold back whole upper row of squares. (Fig. 2). Compare surface A with surface bed. Eelation 1 to 3. Cut line 2. Compare surface a with surface cd. Compare surface cd with opposite surface. Eelation 1 to 6. Cut line 3. Cut line 4, fold back whole lower row. Compare surface ef with sur- s <+ <> face gh. ' 5 a Compare surface ef with opposite surface. Eelation 1 to 4. Compare surface gh with opposite surface. Eelation 1 to 4. Cut line 5. Compare surface e with opposite surface. Eelation 1 to 8. Cut line 6. Fold and paste into square prism shape. Play game mentioned. Concrete problems: In Fig. 1, after a is folded to middle fold ask: "If this wall (small oblong) needs two rolls of paper to paper it, how many rolls will this wall (opposite surface) need?" (Ans. 3, 2 rolls.) After B is folded to middle fold ask : "If one man paints this wall in 4 hrs. ? how long will it take him to paint this one (opposite surface) ?" (Ans. 2, 4 hrs.) Children should keep square prism in school for future comparison with the cube which is to be made next out of the same sized paper, as square prism was made. FOLDING 39 Surfaces Seen While Making Square Prism Enlarge and draw them on the blackboard. Base concrete problems on some of these comparisons: B A=2 of B. =4 of C. = 8 of G. =sum of D+C. =sum of B+2 Cs. B=2 of C. =D-C. =A-2 Cs. =A-B. B=sum of C-t-E+F. =sum of C+G+H. =sum of E+F+G+H. =4 of G. =§ of A. C=i of D. =F+E. B H =G+H. =A-D. =D-B. =2 times G. =4 times E. F=3 times E. =snm of G+E. 0=4 of A. =i of B. F=C-E. =i of D. G=2 Es. =H. =F-E. =C-H. a]'b[g 3 D ! » ! t 1 t E F | 15 if \(o • » ; » i i 40 MATHEMATICAL CONSTRUCTION LESSON XI Cube Fold 16 squares as in previous lesson. Cut off lower row. Compare this to surface which is left. If large surface will hold a loaf of cake, what will smaller surface hold? (Ans. ^ of a loaf.) If it takes 3 cups of flour for the former what will it take for the latter ? (Ans. -J of 3 cups). Fold back row of 3 squares. Com- pare this to opposite surface in same way getting result ^. Then put ques- tions the opposite way, getting result 3. Fold back other outside row of 3 squares and repeat ques- tions, getting answers J and 2. Open folds, making large flat surface with 12 small squares. Make cut 1. Fold back surface led. Compare it with opposite surface. Eelation ^ and 3. Make cut 2. Compare cd with opposite surface. Through questions get relation J and 4. Make cut 3. Make cut 4. Fold back ef. Compare with opposite sur- face. Eelation ^ and 3. Make cut 5. Make cut 6. Fold and paste into a cube. Compare Cube and Square Prism How many cubes like this can be cut from the square prism ? FOLDING 41 How many times larger is the square prism than the cube ? The cube is what part of the square prism? If the cube is a child's building block, how many can be made from the square prism? If the little block costs a nickel, what does the big one cost? (Ans. 2 nickels). If the prism costs 6c what does the cube cost? (Ans. J of 6c), etc. Surfaces Seen While Making Cube Enlarge and draw them on the blackboard. Base concrete problems on some of these comparisons: H J K C=i of A. =i of B. C=J of D. =i of I. 42 MATHEMATICAL CONSTRUCTION C=2 of K. D=sum of H+E E=J of F. I=B. =4 of G. =2 of C. =i of H. =4 of A. =4 of J. =4 of K. =i of D. =sum of J+K. =F-G. J=3 of K. =D-F. =G. H=sum of G+E. =2 of E. =sum of J+E. =C+K. LESSON XII Cradle Cut off lower row of squares. Compare this with large surface. If small surface is enough cloth for a pillow case for baby's bed, what is large surface? (Ans. Enough for 3 pillow cases). Make cut 1. Fold back b-c-d. Compare it with large surface. Eelation 1 to 3. Fold- ing back a-e-f and comparing it to what is left on opposite side gives the relation 1 to 2. Make cuts 2, 3 and 4. Fold and paste into box shape. Eock- ers to be made free-hand from row of squares cut off originally, then pasted on. 1/ 1 1 A B ; » D E ' • ' i ' ' 2 1 i F j . 4 1 : ; 1 » 1 1 • ^ FOLDING 43 New Surfaces Seen While Making Cradle (For other surfaces see lesson on Cube). Enlarge and draw them on the blackboard. Base concrete problems on the comparisons: A B A=4 of D. B = 3 of A. B=2 of A. =i of B. =3 of C. =2 of C. =C. C=i of D. =sum of A + C B=sum of D+C. =i of B. =B-C. =sum of D+A. =B-D. LESSON VIII =B-A. Chair Fold 16 squares. Cut off one row. Make cut 1. Fold back ah. Compare this with opposite surface. Play it is a blackboard. Pointing to opposite surface, this blackboard is how many times as large? (5 times.) If it takes one boy 1 minute to clean it, how long will it take him to clean the other? (5 minutes.) How many boys could clean it in 1 minute? (5 boys.) Make cut 2. Fold back c-d and repeat comparisons — getting result 4 in- ! „ 1* A • B | H E — I C I D 1 > 1 1 g] f 44 MATHEMATICAL CONSTEUCTION stead of 5. Make cuts 3 and 4. Fold back e and /. Com- pare c-d to what is left on opposite side. Result : Relation 3. By reversing questions get relations 1/5, 1/4, 1/3, respectively. Paste e under h. Paste / under g. Paste a under g and /. Paste c-d on i-j to strengthen back. Then chair is made. Legs may be cut out free-hand. B o D Chair Enlarge and draw on blackboard new surfaces not seen in previous les- sons. Give concrete problems based on these comparisons : A=l/5 of B. =1 of C. =i of D. B=sum of C+A. =sum of D+2 A's. C=4 of A. =B-A. =D+A. D=3 of A. =C-A. =B-2 A's. FOLDING 45 LESSON XIV Buggy From two equal oblong papers fold to get squares. From the little extra oblongs cut wheels and shafts free-hand. Fold one square into 16 small squares. Cut off 1 row. Compare this to 12 squares which are left. Base problems on things seen when driving. If a requires 10c worth of grass seed, b requires 3, 10c worth. If it takes 2 days to put a cement walk on a, it takes 3, 2 days to put one on b. If a yields 5 bu. corn b yields three 5 bu. If b yields 3 pk. potatoes a yields J of 3 pk., etc. Make cuts 1 and 2. Fold back left row and compare to opposite surface. Relation 1 to 3. Make cuts 3 and 4. Fold back right row and compare to left row (equal). Also compare to opposite surface. (Rela- tion 1 to 2.) Fold and paste into box shape. Do same with other square. it a • ! 46 MATHEMATICAL CONSTRUCTION Compare volumes of two box shapes. If one holds -J pk. oats for horse the other holds J pk., etc. Cut one of these as indicated in Fig. 2. Insert and paste this into other box shape. Use toothpicks for axles, adjust wheels and paste on the shafts. BASKETS This is a series of basket patterns more than anything else, just to show what a variety can be made from 16 squares cutting either on a straight fold or a diagonal. I have par- tially shown the variety of surfaces to be compared, but have not inserted any concrete problems, though of course they should be given whenever any comparing is done. • ,14... i — i- - i i i T i * _• ' i ! • i if. i : i 5 i ■r-i ... i i i i . !__•_ -J LESSON XV Ordinary box shape, based on 16 squares, straight cuts. LESSON XVI Oblong box shape. 1 row of squares cut off, straight cuts. Cutting for flaps different to give differently shaped sur- faces on which to base problems. FOLDING 47 i ; t , ' i . i ; i ; i i I •_ . • I ■ * i ' ■ ' ! • i LESSON XVII 16 squares; 2 rows cut off; straight cuts. Fig. 3 is the result when the free corners are cut off. LESSON XVIII. 5 , I I I I I I : ; •j-.l. .;...;- , i , . Like a cube with two sides cut out. 48 MATHEMATICAL CONSTRUCTION LESSON XIX Box shape; 9 squares; straight cuts. When bottom row is cut off number relation is 1 to 3. When right row is cut off number relation is 1 to 3. Make cuts a b. Fold back 2, 3, and 1. The sum of 2 and 3 can be cut how many times out of opposite surface? Sum of 2 and 3 is what part of opposite surface? Make cut C and fold back 4. Compare to opposite surface. Relation 1 to 5. Make cut d. Fold and paste into box shape. LESSON XX • \ \ • • * £ - - X \ \ i • 1 k y 1 __ J Make cuts indicated and fold on the d iagonals seen. Paste 1 on 2. Make handle of 3 strips cut free - hand. Eesult : Fig. 2. FOLDING 49 LESSON XXI Eight squares. Di- agonal cuts. Cut off free cor- ners if you so desire. LESSON XXII Six squares. Diagonal cuts. After a and b are cut out, the sum of a and &=J of whole surface left. a is £ of whole surface. b is i of whole surface. iff Open. Eold back on line e-f. Whole surface=4 of a or 4 of & or 2 times the sum of a and b. Fold on horizontal dotted line. Now a is J of surface seen. Now b is | of surface seen. Sum of a and b equals surface seen. Cut a on horizontal dotted line. Triangle 1 is J of b. B=2 of triangle 1. 50 MATHEMATICAL CONSTRUCTION Compare b to surface seen (^). Fold on horizontal dotted line again. Compare triangle 1 to surface seen (^). Open. Fold on lines g-h and e-f. Compare b to surface seen (^). Compare sum of A 1 and A 2 to surface seen (\). Compare sum of A 1 and A 2 and b to surface seen equal. Fold and paste. LESSON XXIII Fold back whole upper row after cuts 1, 2 are made. Compare Fig. a-b-c with opposite surface. Relation 1 to 3. Make cut 3. Fold back whole left row. Compare surface a-b-c to surface d-e-f. Compare d-e-f to opposite surface. Relation 1 to 2. Make cut 4. Compare g-h-i to opposite surface. They are equal. Make long edge of triangle / touch right edge of triangle g and paste. Treat other corners similarly. Cut two long strips free-hand for handles. E ; ; i 1 .-i ( 1 i • • FOLDING 51 LESSOR XXIV / 2 J I [ ' M * 1 1 3 « ! ! L— ! 1 J — . Make cuts 1 and 2. Fold back surface a-b-c-d. Compare this with opposite sur- face. Relation 1 to 3. Make cuts 3, 4. Fold back surface e-f-g-h. Compare this surface to surface a-b-c-d (equal). Compare it to opposite surface (relation 1 to 2). Fold back i-j-k. Compare i-j-k to opposite surface (relation 1 to 2). Fold back l-m-n. Compare it to i-j-k. Compare it to opposite surface (equal). Make long edge of A a touch lower edge of A i. Paste. Treat other corners similarly. Cut off points which pro- trude. Cut handle free-hand. This makes a nice basket for Eed Riding Hood, or a shopping basket. d, stopping half an inch from the top. Open. Cut an inch strip from the other 5- inch square. Teacher cuts first. Child imitates. This 1-inch strip is discarded after some concrete problems have been given causing child to compare it to sur- face which is left. (Relation 1 to 4.) 62 MATHEMATICAL CONSTRUCTION Cut this large surface into 3 equal oblongs, making cuts parallel with long edges of surface. Weave. LESSON IV into 3 equal parts. ones, then other wide one Material: Two 8-inch squares, con- trasting colors. Make foundation same as in Lesson III. When 3 strips have been cut, divide one of them into 3 equal oblongs cutting lengthwise. Divide middle strip of mat Weave, using wide strip first, then 3 small LESSON V Material: Two 9-inch squares, con- trasting colors. Foundation same as Lesson III. Divide two of the three strips for weaving into 3 equal oblongs each. Divide the left and right strips of the mat into 3 equal parts, leaving middle one as it is. Weave, using 3 narrow strips first, then the wide one, then 3 more narrow ones. FKEE CUTTING AND WEAVING 63 LESSOR VI Material : Two 10-inch squares, con- trasting colors. Fold one square into 2 equal oblongs, cut the half-inch strips for border (a, b) from folded edge. Divide folded edge into 2 equal parts at c. Divide a-c into 3 equal parts. Divide c-b into 3 equal parts. Cut an inch strip from other square. Divide large surface into 2 equal ob- longs, cutting parallel to long edges. Cut each of these into 3 equal parts. Weave in any desired way. Over one, under one ; over two, under one ; over two, under two; over three, under one; over three, under two. LESSON VII Material: Two oblongs 8"x5", contrasting colors. Fold the one which is for the mat so that the long edges meet. Cut as in- dicated at a, b, making strips for border. (Fig. 1.) Divide the fold between a and b into 5 equal parts, cut ting to half an inch from top. B 64 MATHEMATICAL CONSTEUCTION Cut an inch strip from one of the long edges of the other oblong from which strips are to be cut. Now divide this into 5 equal oblongs, cutting parallel to long edges. Cut 2 of these ob- longs into two smaller equal oblongs each. Weave thus: one wide strip, two narrow; one wide, two narrow, one wide. Eesult: Fig. 2. f FEEE CUTTING AND WEAVING 65 WALL PAPER MADE BY FREE-CUTTING Fig. 1. Strips any width, pasted on background at a distance from one another equal to their own width. Border, little oblong strips, same width as wall paper strips, laid as shown. Fig. 2. Strips any width, pasted on background so that distance between strips is twice the width of the strips. For border, cut strips same width as other strips, but have relation of length of small oblongs to be cut from these strips 1 to 2. Paste border as suggested. 1- I 1— I — I — I — I- T t i - a >>o: V >~ Cut ti' ie oblong 4"x2" into 4 equal oblongs lengthwise. Weave. Make six of these and paste them on the outside surfaces of the bank made in Ruler Lesson IX, page 93. 68 MEASURING AND WEAVING 69 LESSON II Calendar Back Material : One sheet colored paper 8"x5" ; one sheet paper 7"x5", of contrast- ing color, from which to cut strips. Rulers: 1"2"3"4"7". Tell children length and width of paper. Let them find out with their rulers what sums equal the width, 5"; or the length, 8" or 7". They will see. 3 2 2 2 - 1 5 — -H 8 — — 5' 1"=5" 4' 2"=8" Now on long edges of paper 8"x5" with 7-inch ruler make marks and draw line a-b. Make marks c, d with 1-inch ruler and draw line. Fold lower edge to line a-b. Open. Fold upper edge to line c-d. Open. Use 4-inch ruler to make marks g, h, measuring from left on upper and lower edges. Draw line g-h. Now with 3-inch and 2-inch rulers respectively make the two inner long-lines. With 1-inch ruler make marks e, f, and draw line. Fold lower to upper edge, having lines on the outside. Cut from folded edge up to the fold near the top on every line which was drawn. Now divide each one of these free-hand 70 MATHEMATICAL CONSTRUCTION into two equal parts, stopping at the fold near the top. mat is now divided into 10 half-inch strips and ready woven. To make the strips divide the 7"x5" paper into 7 one-inch strips, using single unit rulers to make points on the long edges, and draw lines connecting them. Divide each of these by free-hand cutting into 2 equal parts length- wise. Weave. Fig. 2. On this paste a little art picture and a calendar. Braid 3 8-inch strands of raffia together, slip it through two holes punched at the top, tie a knot, and calendar is ready to be hung. The to be LESSON III Blotter Material for mat, 2 sheets of paper 7"x3", contrasting colors. Rulers: 6"5"4"3"2"1". Make the vertical dotted lines in Fig. 1 with these rulers, using the 6-inch ruler first to make the line nearest the right; then the 5-inch ruler to make the next line to the left of that one, etc. f, s .i MEASURING AND WEAVING 71 Using the long 6-inch ruler first and then laying the 5-inch ruler in exactly the same way, let the child see the exact difference between the two. If the short ruler were used first and then the longer one there would be an impres- sion that the shorter one was covered up, and relationship between the two could not be seen. In building, we do not pile large blocks on top of small ones or the structure would be unstable and fall. Take these little rulers from 1-inch to 6-inches; pile them one on the other in regular order from 1 to 6, with the 1-inch underneath. What do you see? Well, nothing worth while. Now pile them the other way : 6-inches at the bottom and 1-inch on top. See the regular steps from 1 to 6. There is just as much difference between 6 and 5 as there is between 5 and 4 ; 4 and 3 ; 3 and 2 ; 2 and 1. The child feels these relation- ships if the rulers are handled in this way — always making the large measurement first and the small one on top of it. Fold so that the lower touches the upper edge and lines can be seen. Now begin to cut on each line on the folded edge. Now divide each of these parts into 4 equal parts and finish all cuts, cutting until the distance from the top is exactly the width of the little parts. Now, on the other paper draw the two vertical lines in Fig. 2, using the 2-inch and 1-inch ruler to make marks and 6-inch ruler to draw lines. Cut on lines. Subdivide each of these strips, cutting free-hand, into 4 equal long strips. Weave. Ttf - 3 f-Z Trrn-i[|!:::,:i|-ij.;ht.!.:.LL.: Pa-a-n-Dl ■:;^B"DI 1 B.;B. ,: 72 MATHEMATICAL CONSTRUCTION 3>3>S Formula for weaving Fig. 3. Over 3, under 3 1 Repeat Over 3, under 3 13J Over 1, under 1 I times. Formula for weaving Fig. 4. Over 4, under 4 "1 Repeat Over 4, under 4 L3J Over 2, under 2 | times. On top of this mat place a piece of transparent celluloid, underneath two or three sheets of blotting paper. Punch a hole in each corner and insert brass-headed fasteners. This makes a pretty blotter. LESSON IV Napkin Ring Two pieces of paper, contrasting color, 7"x2". Rulers: 1"2"3"4"5"6". Divide one piece of paper into 2 equal oblongs lengthwise, using the 1-inch ruler to make marks and the 6-inch ruler to draw line. Fold so that short edges touch. Cut a little dis- tance from folded edge on line. Now divide each half of this folded edge into 3 equal parts by making cuts. Continue all these cuts until the distance from the top equals the width of one of the strips. Open. On other paper draw lines indicated in Fig. 1, using 6"5"4"3"2"1" mlers respectively to do it. Cut on these lines. ti*-i. MEASUKING AND WEAVING 73 Divide each one of these strips into 3 smaller equal ones, cutting lengthwise free-hand. Weave in any desired way. 1 TT, 78 MATHEMATICAL CONSTRUCTION LESSON VIII Needle-book Eulers: l"3"4"7" Q Material: Two sheets stiff paper ll"x4", contrasting colors; 1 sheet thin lining paper ll"x4"; one or 2 oblongs of cloth 10"x3"; 10 inches of ribbon for fastening at hinge. 10". On background paper draw D lines a-b, c-d (Fig. 1), measur- ing with 1-inch ruler from B upper and lower edges, or with 3-inch and 1-inch ruler re- Fold so that short edges meet and lines can be seen. Cut from folded edge on the lines until the distance from the opposite end is \ the width of the narrow strips. Divide each of these strips into 3 equal parts, cutting parallel to other cuts. Give the other paper to the children. Tell children length of the paper (11")- Let them find out with their rulers what sums equal 11 inches. They will see : 7 8 4 3 4 " spectively from upper edge. ?'V 11 11 In comparing rulers they will see : 3 3 7 3 4 7 11 T,; r -a 10 MEASURING AND WEAVING. 79 Draw lines indicated in Fig. 2, measuring with rulers 10"8"7"4"3"1", respectively, from left edge. Cut on the lines. Divide each of these pieces into 3 equal parts, except the large middle one. Divide this into 2 equal parts. Weave. Paste down all the edges of the strips, and paste lining paper on back. Fold so that short edges meet. Insert one or two pieces flannel (edges pinked). Punch 2 holes on fold and lace with ribbon. T ' '> 3 MEASURING Introduction of SINGLE UNIT RULERS LESSON I Scissors Holder (8-inch ruler) Materia] : Use grey school paper, which is 9"xl2 Teacher goes to wall with her paper, lays her 8-inch ruler on it as indicated in Fig. 1, then draws a line the whole width of the ruler at point marked a. Children imitate at their seats. (Children must keep their papers in the same position on the desk until all ruler work is done.) Second step: Teacher lays 8-inch ruler on lower edge of the paper and draws a 1-inch line at b. Children imitate. Teacher draws line with her ruler connecting a and b. Children imitate. Teacher cuts on line. Children imi- tate. Now let children compare the two oblongs X and Y. How many sheets of cardboard as large as y can be cut from xf (Ans. Two f'* a sheets.) If this (showing y) costs 5c, 80 MEASURING: SINGLE UNIT EULEES 81 what does this cost (showing x) ? (Ans. Two 5c.) (Ans. If x Half costs a dime, what part of a dime does y cost? a dime.) Now teacher lays 8-inch ruler along left edge of large oblong from upper left corner as indicated in Fig. 2. Draw a line the width of the ruler at c. Children imitate. Now lay ruler on right edge from upper right corner in same way, drawing line at d. Children imitate. Draw line from c to d. Children imitate. Cut on this line. Eesult : 8-inch square. This is enough ruler work for a first lesson. Now fold into two equal oblongs. Open. Make point a touch middle of top line at b. (Fig. 3.) Now make c touch d. Eesult: Fig. 4. Cut on dotted line. Scissors holder now needs only to be pasted. LESSON II Mayflower (8-inch and 4-inch rulers) Make an 8-inch square as in previous lesson, but use white paper. At the end of the lesson let the children use crayola to color the hull of the boat black. The white sails will make a pretty contrast. Call the 8-inch square the tablecloth to be used for the Thanksgiving dinner. 82 MATHEMATICAL CONSTRUCTION If 4 people can sit on one side, how many can sit at 2 sides? (Ans. Two 4 people.) How many 4 sides? (Ans. Four 4 people.) Bisect the square, using a 4-inch ruler. Lay it on the top edge from upper left corner, drawing line at the end. Eepeat Beach 1 1 1 m 1 1 r Ni/ 1 1 r ~7 i ' i . p T"3 ' on lower edge, and draw bisecting line. Fold on this line. Now if the oblong is the table and there are 4 plates on the short end, how many on the long one ? (Ans. 2, 4 or 8 plates.) (Some will say 8 plates on account of ruler measurement.) That is enough ruler work for this early ruler lesson. Fold one long free edge to meet fold. If small oblong thus formed is cloth enough for a Pil- grim's dress, how many dresses can she get from opposite large oblong? (Ans. 3 dresses.) Fold other long edge to center fold. Open. Fold the other way, making 8 small oblongs out of the 8 inch square. During process in giving problems call surfaces cloth for aprons instead. Open. Fold so that point b touches center a. Fold so that point c touches center a. Compare triangles thus formed (e and /), calling them Pilgrims' shawls. If it took $1.00 worth of yarn to make e, how much did it take to make f? etc. Some Indians had a wigwam here (pointing to d). Some of them ran \ mile along here (pointing to edge d-g-h) to the beach to see the Mayflower. The others ran along here MEASURING: SINGLE UNIT RULERS 83 (pointing to edge d-k-m) to the beach. How far did the others run? (Ans. Just as far, ■J mile.) Fold so that point d touches a. Cut out triangles 1, 2, Compare size of triangles, which form sails. If large triangle shows number of people who came over in Mayflower, and Ti small one shows how many got sick, what ! part of them got sick? Many more questions can be asked if there is time ; questions about clearing the ground, planting corn, chopping down trees, building log-cabins, etc. 3,4. T.«3 2. LESSON III Cup Material : A sheet of manila paper 6"x9". Eulers: 6"3". Pass the 6-inch rulers saying, "This is a 6-inch ruler." Pass the 3-inch rulers saying, "This is a 3-inch ruler." Let the children put the rulers next to each other (Kg- 1). How many 3-inch rulers can be cut from the p.ji 6-inch ruler f 84 MATHEMATICAL CONSTEUCTION Show me the part of the 6-inch ruler which equals 3" ; or, show me the part of the 6-inch ruler which is just as much as the 3-inch ruler. (Children show a.) Now show me the difference between the 6-inch ruler and the 3-inch ruler. (Children show b.) Show me the sum of 6" and 3". Children show Fig. 2. When in a later lesson they use rulers 3"6"9", they will be able to say that the sum of 3" and 6" equals 9". In this lesson it is enough to have them get an idea of what the word "sum" means. Let them estimate the length of the paper which they have. They will tell you that it is the sum of 6" and 3" long, or 3'3" long, or 6" and J of 6" long. In estimating width, they will say: "The paper is 6" wide," or "The paper is 2'3" wide." Now teacher lays her 6-inch ruler on upper edge Ma- and makes a mark at a (Fig. 3). Children imitate. Eepeat on lower edge, marking at b. Draw line a-b. Cut on line. Compare sur- faces c and d. Play c is a towel. How many can be made from df If c costs a nickel, d costs 2 nickels. If d costs a $1.00, c costs J of a f/?.3. dollar. c Now using 3-inch ruler, bisect all the edges. Do not draw the bisecting lines. They are not necessary. Fold this 6-inch square on one diagonal with bisecting marks on the outside (Fig. 4). Make point a touch mark b. MEASUEING 35 Make point c touch point d. Turn back in opposite directions the two free points at e. Result : A drinking cup for use on a train, or in the park. LESSON IV Bookmark Material : A pretty thin cardboard, a piece of cord which harmonizes in color with the cardboard. Rulers: 2"4". Since cardboard comes in large sheets and must be cut up by the teacher, she may as well cut it the proper dimensions, 4"x2", to save material. But let the children have the two rulers and compare them as in previous lesson, showing sum, difference, etc. Let them estimate length and width of paper. Let them see that sometimes the paper is just wide enough and does not need marking. Lay 2-inch ruler on upper and lower long edges respectively. Make marks, but do not draw bisecting line. It is not necessary. D Compare triangle thus formed to oppo- site surface. If it is enough leather for the B corner of a book, the opposite surface is enough for how many corners? If it costs a dime, how much does opposite surface cost? (3 dimes.) Fold so that point d touches a. Compare triangle thus formed to opposite large triangle. Relation J. Punch holes 8C> MATHEMATICAL CONSTEUCTION on the free edges of the small triangles along line a-b. middle holes first equidistant from points a and b. Then the others equi- distant from center and points a and b, respectively. Lace these and you have a book mark which fits on the corner of the page. Punch LESSON V Stamp Pocket Get Material : A heavy, pretty paper. Eulers: 2"3"5". Make a 5-inch square, using a 5-inch ruler. Then give the 2-inch and 3-inch rulers to children, sum 2+3=5. Lay 3-inch ruler on upper edge from upper left-hand corner and make mark a. (Fig. 1). Eepeat on lower edge and make mark b. Draw the line. Lay 2-inch ruler on upper edge from upper left-hand corner and make mark c. Eepeat on lower edge, making mark d. Draw line. Lay 3-inch ruler along left edge from upper left corner and make mark e. Eepeat on right edge, making mark /. Draw the line e-f. C A G- \ I -... € -! \ MEASURING: SINGLE UNIT RULERS 87 Lay 2 -inch ruler along left edge from upper left corner and make mark g. Repeat on right edge and make mark h. Draw the line. Fold so that lower edge touches upper edge. Open. (Fig. 2). Fold so that left edge touches line c-d. Open. Fold so that right edge touches line a-b. Open. Cut from e to fold. Cut from / to fold. Fold so that point g touches h. Fold so that point i touches h. Fold so that d-b line touches h-h line. Fold so that upper left corner touches point I. Fold so that upper right corner touches point m. Fold so that c-a line touches l-m line. i c *l • 1 H K ; FI9.2. FI3.3 Fold so that line a (Fig. 3) touches line b. Fold so that line d (Fig. 3) touches line c. Eesult : Fig. 4. F<3- f Fold under the little triangles 1, 2, 3, 4 and paste. Result: A stamp holder with two pockets. Close it by making lower edge touch upper one. 88 MATHEMATICAL CONSTRUCTION LESSON VI Traveler's Pocket for Wash Cloth Material: Oilcloth (a 9-inch square) and a piece of tape 14" long. Rulers 1" 7" 3" 4". Handling these rulers the children find that: 3 4 3 4 13 3 1 The children have handled a 2-inch ruler in Lessons IV and V. They see a 2-inch length in the making of this wash- cloth pocket. So, even though they haven't the 2-inch length in their hands some will see : A c 2 t i; «r ;l 4 2 1 3 2 2 9 s: 2 .»• • .. . . . . ...«.., *! I 7 i i ! 12 Lay 7-inch ruler on left and right edges, respectively, from upper edge; make marks and draw line a-b. Use 2-inch ruler in the same way. Draw line c-d. With 1-inch ruler make marks, e-f-g-h. Fold line e-f. Fold line g-h. (The children have no ruler at their desks long enough to draw this line.) Cut out oblongs 1, 2, 3, 4. 96 MATHEMATICAL CONSTEUCTION Fold back oblongs 6, 7, and 5. Compare 5 to opposite surface. Eelation 1 to 5. Play oblong 5 is blotter. How many blotters just as large can be cut from opposite surface ? If large blotting paper costs a nickel, what does small one cost? (1/5 of a nickel or la), eta Put paste on flaps 6 and 7. Paste 8 on them. Cutting from c to e and from d to g makes the fold over flap look better. LESSON XI Taboret Enters: 8"5"3"1". Material : An 8-inch square of rather heavy paper. New numerical relations seen : 8 -3 8 -5 8 5 3 Lay 5-inch ruler on upper and lower edges, respectively, from left edge, making marks a, b. c Draw line a-b. Lay 3-inch ruler in same way and draw line c-d. Lay 5-inch ruler on left and right edges, respectively, from upper edge ; make marks e-f and draw line. Lay 3-inch ruler in same man- ner and draw line g-li. Make the 3i MEASUEING 97 marks which are necessary to draw the dotted edges of the ^^^~>. small oblongs 1, 2, 3, 4, with a 1-inch ruler. ^^V^J^J Draw the lines with the 3-inch ruler. i Cut out oblongs 1, 2, 3, 4. Make cuts indicated at marks e, f, g, h. Paste into box shape. (Fig. 2.) Cut out sides in any desired way to form legs. LESSON XII Chair for Father Bear (Story of Three Bears). Rulers: 3"6"9"12". Material: School paper 9"xl2". Numerical relations seen in comparisons of rulers and estimation of length and width of paper. 4 2 2'3"= 6" 3" are J of 6" 3)12" 6)12 3'3"= 9" 3" are ^ of 9" 3 2 4'3"=12" 3" are J of 12" 3)"~9~ 3)~6 2'6"=12" 3 3 6 6 3 6 9 9 12 12 12 6 9 6 3 3 -3 -3 -6 -9 -6 -3 9 12 12 3 12 3 3 6 3 9 3 6 9 98 MATHEMATICAL CONSTRUCTION 7 I ; »|* * j * i : -• -i i . / ' 2 3 5 G -- Lay 9-inch ruler on upper and lower long edges respectively, from left edge. Make marks a, b. Draw line a-b. Lay 6-inch ruler in same way and draw line c-d. Lay 3- inch ruler same way and draw line e-f. F u e Lay 6-inch ruler on left and right short edges, respec- tively, make marks g-h and draw line g-h. Lay 3-inch ruler in same way and draw line i-j. Make cut at b. Fold back surface 1-2-3. Compare to opposite surface. (Relation 1 to 3.) If it takes a roll of wall paper for this wall (indicating small surface), how many for this? (3 rolls.) If it takes 3 rolls for small wall, how many for large wall (3'3 rolls or 9 rolls-inference from rulers). Make cuts indicated at d and a. Fold back square 4. Now compare surface 1-2 to opposite surface. Eelation 1 to 4. If wall paper for small surface costs $3.00, paper for large pji "00 wall costs how much? (4 r $3.00 or $12.00- inference from ruler measurement.) If it took 12 pails of plaster for large wall, how many for small wall? (J of 12 pails or 3 pails), etc. Make cut indicated at i. Fold and paste so that square 5 is under 3 ; 4 under 6 ; and 7 under 3. MEASUEING: SINGLE UNIT RULERS 99 Fold and paste surface 1-2 under 8-9. Cut out sides to form legs in any desired design. LESSON XIII Mother Bear's Chair Eulers: 2"4"6"8". Material : Stiff paper, 6"x8". In comparing rulers and estimating length and width of paper, these numerical relations are seen : 4 2 3 2 , 2 , '=4 ,/ 2" are J of 4" 2)T 4)T 2)6 3'2"=6" 2" are J of 6" 2 4'2"=8" 2" are \ of 8" 2)4 2'4"=8" 4" are J of 8" 2 2 2 4 8 8 8 6 6 4 2 4 6 4-6-4 -2 -4 -2 -2 4688246242 Use rules 2"4"6" to divide surface into 12 squares, as the 3"6"9" rulers were used in previous lessons. Make cuts indicated at a and b. Fold back surface 1-2. Compare 3 it to opposite surface. Relation 1 to 5. Make cut indicated at c. Fold back surface 3-4. Compare this to opposite surface. Relation 1 to 4. 7 — 1 — r~ 10 M 100 MATHEMATICAL CONSTKUCTION If this floor (showing surface 3-4) requires 2 square yards of carpet to cover it; what does this floor require? (showing large surface.) (Ans. 4'2 square yards or 8 square yards- inference from rulers). Make cut indicated at d. Fold and paste so that square I is under 5 ; 6 is under 2 ; 3 is under 5. Fold back and paste 7-8 on 9-10 to strengthen back of chair. Make legs and back same design as Father Bear's Chair. LESSON XIV Baby Bear's Chair Rulers : 1"2"3"4". 8 ^ i Material: A 4-inch square of stiff paper. Divide this into 16 1-inch squares by using rulers 1"2"3". Cut off one row of squares. Play both surfaces are rugs with fringe on short edges. (Fig. 1.) If 1 yd. of fringe is on one short edge of rag B, how many yds. are on one short edge of rug A ? How many yds. on both short ends of a? (2'3 yds., or 6 yds.-inference from former ruler measurement.) How many yds. on both short ends of rug B? (2 yds.) How many times as much fringe must we have for the big rug? (3 times as much.) 3'2 yds. are how many yards? (Inference from ruler measurement in previous lessons.) MEASURING: SINGLE UNIT RULERS 101 D A Make cuts indicated at a, b, c, d. Fold and paste chair as in previous lessons. Make legs and back same design as father and mother bears' chairs. Number relations seen: 2'1"=2" 1" is i of 2" 3'1"=3" 1" is i of 3" 1112 12 3 2 2 2)T 4'1"=4" 1" is J of 4" 2'2"=4" 2" are J of 4" 4 4 4 -3 -2 -1 LESSON XV Father Bear's Bed — Bulers :3" 6" 9" 12". Material : 2 sheets stiff paper, 9"xl2". Divide paper into 12 small squares, using rulers 3"6"9", as in Lesson XII. Make cuts indicated at a, &. squares 1-2. Compare this surface to opposite large surface. (Belation 1 to 5). If it is material for 1 sheet, how many sheets can be made from large surface (5 sheets). Fold back 3. Make cut indicated at c. Fold back 4, 5, 6. Com- 102 MATHEMATICAL CONSTEUCTION pare 4, 5, 6 to opposite surface. Eelation 1 to 2. If this is material for a pillow-case, how many can be made from large surface? (Ans. 2 pillow-cases.) If it takes 3 minutes to hem one pillow-case, how long will it take to hem 2? (Ans. 2'3 minutes, or 6 minutes.) Make cut d. Fold and paste into box shape. This is enough for one lesson. Next day take the other 9"xl2" sheet of paper and make another box shape exactly the same. Compare volumes of these two. If one holds 3 lbs. of feathers, the other holds 3 lbs. of feathers. Now cut one box shape so that it will hold half the amount of feathers. Eye measurement only. Now if small box holds 3 lbs. of feathers, large one holds 6 lbs. If large one holds 12 lbs., small one holds J of 12 lbs., or 6 lbs. Ftf.fc. From small box shape cut so that the legs of the bed are formed. From large box shape cut half-way down on the edges a and b. Cut across on line c. This is the foot of the bed. Leave opposite end as it is. This is the head of the bed. Now to get the sides, cut down half-way again on what is left of edges a and b. Then cut across to head of bed. Paste Fig. 3 on Fig. 2 and bed is constructed. Numerical rela- tions found in Lesson XII are reviewed. Hr MEASURING: SINGLE UNIT RULERS 103 LESSON XVI Mother Bear's Bei>— Eulers : 2"4"6"8". Materials : 2 sheets stiff paper 6"x8". Make just as father's bed was made, but use rulers g"4"6"g" instead. Numerical relations found in Lesson XIII, reviewed. LESSON XVII Baby Bear's Bed— Rulers: 1"2"3"4". Material : 2 sheets stiff paper 3"x4". Make the same as Father's and Mother's Bed, but use rulers 1"2"3"4". Numerical relations found in Lesson XIV, reviewed. LESSON XVIII Bowl for Father Bear — Eulers : 12"4"3"1". Material : Piece of paper 12"x4" and a 4-inch square. Numerical relations seen: 4 3 3)12" 4)12" 3'4"=12" 4'3"=12" 3 4 4 1 -1 -3 104 MATHEMATICAL CONSTRUCTION On left and right edges, measuring from upper edge a [ | ,ujjU(J1U1.1./ _ .c. - .UI'utMimJ a with 3_inch rulei \ make marks a, b. Draw line a-b. Fold so that lower edge of paper touches line a-b. Cut on the fold. Compare this long narrow oblong to large surface. Relation 1 to 7. Now fold on line a-b. Then on the long narrow oblong c, make vertical cuts about J of an inch apart up to the line a-b. Paste right edge on left edge to* form cylinder, seeing that the little cuts on the bottom turn inward. These are the flaps. Put paste on the bottom of the flaps. Lay this cylinder on the four-inch square with flaps down. When the flaps are pasted to the square cut off the parts of the square which protrude. Bowl 3 inches high is made. LESSON XIX Mother Bear's Bowl— Rulers: 12"3"2"1". Material : Piece of paper 12"x3" ; a 4-inch square. Number relations seen in this lesson : 4 — 2 3 3 1 -2 -1 6'2"=12" 3 ) 12 4' 3"=12" _^_ 12'1"=12" 2)12 3 1 MEASURING: SINGLE UNIT EULEES 105 Lay 2 -inch ruler on left and right edges respectively. Make marks a, b. Draw line with 12-inch ruler. Fold so that lower edge touches line UJIdHJIULUi C.JLJUlLliJL'l'i a-b. Cut on fold. Compare this to large surface. Eelation 1 to 5. Fold on line a-b. Compare this long narrow oblong to opposite surface. Eelation 1 to 4. If it is a board in the floor what is the other? (4 boards). If it costs 3c, what does other cost? 4'3c or 12c. Make the little cuts for flaps up to line a-b. Paste right on left edge, seeing that flaps turn inward. Put paste on bottom of flaps. Set cylinder so that flaps have their paste side down on the 4-inch square. Cut away pro- truding surface of square. Mother Bear's Bowl 2" high is made. LESSON XX Baby Bear's Bowl— Rulers: 12"2"1". Material : Strip of paper 12"x2" ; a 4-inch square. On left and right edges measuring from upper edge make marks a, b with 1-inch ruler. Draw line a-b with ^ 12-inch ruler. Fold so that bottom edge touches line a-b. Cut on fold. Compare the oblong cut off to large surface. Relation 1 to 3. Fold on line a-b. Make vertical cuts up to line a-b on oblong c for flaps. Paste right edge on left edge, turning flaps inward. Put paste on bottom of mwimr. .g. . imimuim' e Q So 106 MATHEMATICAL CONSTRUCTION them, set on 4-inch square and cut off protruding part of square. Baby Bear's Bowl is 1" high. If baby's bowl contains a pint of porridge, what does papa's contain (3 pints) ? What does mamma's contain (2 pints) ? The child cannot say 1 quart, unless you have had the actual quart and pint measurements in your room and he has measured and found out that a quart equals two pints. The child should deal with the actual measures first. Then after he has seen the relations, let him draw infer- ences, by applying those relations in other ways. For instance: When we made a bowl for the baby bear, and another with twice its capacity for the mother bear, we did not make them with the actual capacity of a pint and quart. But we may assume that the baby's holds a pint. Then the child infers that the mother's (having twice the capacity) holds a quart, because he has previously discovered through actual measurement that a quart is twice as much as a pint. LESSON XXI Pail for Jack— Eulebs : 9"5"4"1". Material: Pretty colored paper 9"x5"; a 3-inch square. Number relations seen: MEASURING: SINGLE UNIT BULEES 107 5 4 4 4 14 5 5 -4 -1 9 5 1 Mld'JUL.c --UwlUWil On left and right edges, meas- uring from upper edge with 4-inch ruler make marks a, b. Draw line a-b. Fold so that lower edge touches line a-b. Cut on fold. Now fold on line a-b. Make little vertical cuts on oblong c free-hand about J of an inch apart up to line a-b. Paste right edge on left one, flaps turning in. Put paste on bottom of flaps, lay on 3-inch square and cut away part of square which is left on the outside. Use first strip x which was cut off as a handle for Jack's pail. LESSON XXII Jill's Pail— Rulers : 9"3"2"1". Material : Pretty colored paper 9"x3" ; a 3-inch square. Lay 2-inch ruler on left and right edges respectively from up- per edge. Make marks a, b and draw line a-b. Fold so that lower edge touches line a-b. Cut on fold. Use this strip x for handle. Fold on line a-b. Cut c into flaps. Paste right edge on- left one, turning flaps in. Put paste on bottom of flaps and paste the cylinder on to the a ]iLU11JM1 3 JUUiDUlf 108 MATHEMATICAL CONSTRUCTION 3-inch square. Cut off protruding parts of square. Paste on the handle. Jack's pail is 4" high. •s JilPs pail is 2" high. Compare vol- umes. If Jill's holds a quart of water, F-L—^J Jack's holds 2 quarts. If Jack's holds a gallon of water, Jill's holds J of a gallon. If Jill's holds a pint of water, Jack's holds 2 pints or 1 quart (if children have had pint and quart measurements). If a pint costs 3c, what does a quart cost? (2' 3c or 6c) etc. LESSON XXIII Fox's Dish— Eulers: 5"2"1". Material : Paper 5"x2" ; a 2-inch square. Using 1-inch ruler make marks a, b. Draw line a-b with 5-inch ruler. Fold so that lower edge touches line a-b. Cut on fold. Use pieces of this strip to make side handles for dish when it is finished. Fold on line a-b. Cut c into flaps. Paste right on left edges, flaps turning in. Put paste on bottom of flaps. Lay WMUJ. Mil MM MEASURING: SINGLE UNIT RULERS 109 cylinder flaps downward on 2-inch square. Cut off parts of square which project beyond cylinder. LESSON XXIV Stork's Dish— Eulers : 5"7"6"1". Material: Paper 5"x7"; a 2-inch square. Number relations seen : 5 6 6 7 6 7 11-1-1 -5 -6 < ) 6 7 5 6 11 Lay 6-inch ruler on left and right edges respectively. Make marks a, b. Draw line a-b. Fold so that lower edge touches line a-b. Proceed as in previous lesson to make the Stork's dish. Use strip cut off for side handles. Relation of Fox's dish to Stork's dish is 1 to 6. If Fox's dish contains lc worth of meat, what does Stork's contain? (6c worth.) If Fox's dish contains 2c worth, what does Stork's contain? (6' 2c worth or 12c worth.) LESSON XXV Handbag— Eulers : 9"6"3"2"1". Material : Paper 9"6" ; also a strip 9"xl". Number relations reviewed — a great many seen, in com- paring rulers and estimating length and width of paper. TJ 110 MATHEMATICAL CONSTRUCTION 9+3=6+6 6 2 3 3 12 9 3 1 6 2'3"=6" _* c 3'2"=6" 3'3"=9" £ 1 H D F J Lay 3-inch ruler on left and right edges respectively. Make marks a, b; ff]\ draw line a-b with 9-inch ruler. ' f -^» Lay 2-inch ruler on upper and lower edges, respectively, measuring from left edge make marks c, d, and draw the line. Lay 2-inch ruler on upper and lower edges respectively, measuring from right edge; make marks e, f and draw line. Lay 1-inch ruler on upper and lower edges respectively, measuring from left side; make marks g, li, and then draw line g-h. Doing same on right side make marks i, j, and then draw line i-j. Make cuts indicated at a and b. Fold on lines c-d and e-f. Now fold back on lines g-h and *-/. Fold on line a-b, so that all these small folds are inside. Paste oblongs 1 and 2 together. Paste oblongs 3 and 4 together. Cut the 9"xl" strips into two equal narrow strips for handles — eye judgment only. MEASUEING: SINGLE UNIT KULEKS 111 LESSON XXVI Pencil-box with Lid — Eulers: 10"2"3"5". Material : Thin, smooth cardboard 10"x6". Number relations seen : 2 5 3 2'3"= 6" 3 5 3 3'2"= 6" - _ 5'2"=10" 5 10 2 2 10 Lay 5-inch ruler on left and right edges, measuring from upper edges ; make marks a, b and draw the line. Lay 3-inch ruler in same manner; make marks c, d, and draw the line. Lay 2-inch ruler in same manner; make marks e, f, and draw the line. Make marks g, h, with 1-inch ruler; draw line with long ruler. Make marks i, ; with 1-inch ruler; draw line with long ruler. Cut out oblongs x and y. Make cuts indicated at a, b, c, d. Fold up and paste. Surfaces can be compared by giving concrete problems as in former lessons. 112 MATHEMATICAL CONSTRUCTION LESSON XXVII Match-safe Colored smooth, thin cardboard 10"x4" for back. Colored smooth, thin cardboard 6"x5" for box. Sandpaper, 3"x2". Kulers: 5"4"3"2"1". Let children estimate 10" length of cardboard with these rulers. They will say : 5 5 4 3 4 3 2 4 3 3 5 4 3 2 12 2 12 3 2 10 10 1 1 10 1 2 10 10 10 10 C 6 C A T On the cardboard which is 6"x5", lay 5-inch ruler on upper and lower edges, L respectively, measuring from left edge; j make marks a, h, and draw line. Lay 4-inch ruler in same way; make marks hp 6© c, d, and draw line. Lay 2-inch rule in same way; make marks e, f and draw line. Lay 1-inch ruler in same way ; make marks g, h and draw line. MEASUEING: SINGLE UNIT RULEES 113 Lay 3-inch ruler on left and right edges, respectively, measuring from upper edge ; make marks i, j and draw line. Lay 2-inch ruler in same way; make marks Jc, I and draw line. Cut out oblongs 1, 2, 3, 4. Make cuts indi- cated at i, j, k, I. Fold and paste into box shape. Paste box and sandpaper on large card- board — spacing agreeably. LESSON XXVIII Wood-box — Rulers 2" 9-" (o" r 3 used as supports for the axles, which are tooth- ! . picks. »j2 Now cut on dotted line, which is drawn be- tween marks made on right and left edges with 5-inch ruler. Cut rectangle b (Fig. 1) into two equal narrow strips to be used as shafts. From c (7"x5") the box shape is made. (Fig. 3.) With 1-inch ruler make marks a, b on upper and lower edges, measuring from left side. Draw line a-b. With 1-inch ruler make marks c, d on upper and lower edges, measuring from right side. Draw line c-d. With 4-inch ruler, measuring from top on the left and right edges, make marks and draw line e-f. With 1-inch ruler, measuring from top, make marks on left and right edges and draw line g-h. Make cuts indicated at e, f, g, h. Fold and paste into box shape. Paste the little supports for the axles on the under side of the box after the axles have been laid in place. Put on the wheels and paste on the shafts. (Fig. 4.) MEASURING: SINGLE UNIT RULERS 117 In comparing lines and surfaces in this lesson, base concrete problems on roads, trees, houses, stores or parks; things which the banana-man sees when he pushes his cart through the streets. Ask about time or number of men required or material needed to pave or sprinkle roads; to cut or sprinkle grass in the park; to plant trees in a row. Ask about amount of houses of equal size which can be built on compared surfaces. LESSON XXXI Gocart— Eulers : 2"4"6"8". Material : 2 sheets of stiff paper, one 10"x8" ; the other 8"x6". In estimating length of paper new number combinations seen are : 8 6 2 4 10 10 Some children ought to know that the paper is 10" long, for they talked about 10" in Lessons XXVI and XXVII. Measuring on upper and lower long edges from left side with the 8-inch ruler on the large paper, make, marks and draw line a-b. Cut on dotted line a-b. Com- pare c to dg, with questions about material and cost of material for doll quilts and mattresses. 118 MATHEMATICAL CONSTKUCTION From c measure and cut 4'2" squares, using the 6"4"2" rulers. From these squares cut free-hand the four wheels, keeping them 2" in diameter. Measuring on left and right edges A of 8" square from the upper edge with ' ' * the 6-inch ruler make marks, and draw A line e-f. Cut on line e-f. From g, U four equal long narrow strips are to be cut. Two of these, to be used for the handle, need a 2-inch mark at one -p, 3 ^_ end of each. Fold so that line a (Fig. 2) touches this mark. Place short-folded parts of these strips together and paste, forming handle. (Fig. 2.) Mark other two strips as indicated in Fig. 3. Fold on marks. Paste a on b. These are the springs to be pasted under ■^S^— the box-shape when it is made, putting toothpicks through the inch square sides of the springs for axles. To make box-shape use d. (Fig. 1.) It is 8"x6". c A Measuring with 6-inch ruler on | I T upper and lower long edges from left £...{ J. edge make marks and draw line a-b. In same way using 2-inch ruler draw line c-d. Measuring with 4-inch ruler, on left and right edges from r '3 h- upper edge, make marks and draw line e-f. In same way using 2-inch ruler draw line g-h. MEASTJBING: SINGLE UNIT EULEES 119 Make cuts indicated at a, h, c, d. Fold and paste into box-shape. This is enough for one lesson. Next day give other piece of paper 8"x6" to children. Make another box shape from it like Fig. 4. Compare volumes of these two which are equal. When mamma goes marketing she takes baby with her in the go-cart. If one volume holds a peck of potatoes, the other holds a peck of potatoes. If one holds half a peck of apples the other holds half a peck of apples. If one holds 5 pounds of sugar, the other holds 5 pounds, etc. Now stand last box up on end and inside of the first box which has the springs and axles under it. Put on the wheels and paste on the handle. LESSON XXXII Cradle— Rulers : 1"9"5"4". Materials : 2 sheets of stiff paper 10"xl6" stiff paper 12" square. Use one sheet of the paper, 10"x6", first. In estimating length, get new relations: 9 5 1 4 10 1 1 sheet of 10 120 MATHEMATICAL CONSTKUCTION Ti?( Measure on upper and lower edges, from left edge with 9-inch ruler; make marks and draw line a-b. Eepeat with 1-inch ruler, and draw line c-d. Measure on left and right edges from upper edge with 5-inch ruler ; make marks and draw line e-f. Eepeat with 1-inch ruler and draw line g-h. Make cuts indicated at e, f, g, h. Fold and paste into box shape. Take other sheet, 10"x6". (Fig. 2.) Measure on upper and lower edges from left edge with 4-inch ruler; make marks and draw line a-b. Measure on upper and lower edges again with 4-inch ruler, but measure from points a and b instead of left edge ; make marks and draw line c-d. Cut on dotted lines a-b and c-d. Compare rectangles, calling them rugs. Compare sides of rectangles by asking questions about length and price of fringe. Fold so that each rectangle is folded into 4 equal parts ; folding long edges so that they meet in each instance. The two larger rectangles e and / (they look like Fig. 3 now) are to be fashion- ed into rockers, after they are pasted in place under the box-shape. E ; F C TV z Ti'j'b MEASURING: SINGLE UNIT EULEES 121 '?¥ Cut off the protruding flaps indicated at 1, 2, 3 and 4. Slope bottom edge of vertical surface to look like rockers. Cut £ into strips and use them as braces to keep rockers in place. Next day make another box shape out of the 12-inch square thus: On upper and lower edges, measuring from left edge with 4-inch ruler, make marks and draw line a-b (Fig. 4), using same ruler on same edges, but measuring from right edges draw line c-d. Same ruler on left and right edges measuring from upper edge, draw line e-f. Same ruler, same edges, measuring from lower edge, draw line g-h. Make cuts indicated at e, f, g, h. Fold and paste into box shape. Compare this volume to the one made yesterday. Rela- tion 2 to 1. If one holds a quart of beans the other holds ^ of a quart. If one quart costs 10c, ^ quart costs J of 10c, or 5c. If one quart costs 12c, -| quart costs -J of 12c, or 6c, etc. Set large volume on one end inside the other one. (Fig. 5.) LESSON XXXIII Bureau— Rulers : 3"6"9"12"— 1st Day Material: 1st day, 12-inch square of stiff paper. Divide 12-inch square into 16 small squares, using the 3" 6" 9" rulers. Cut, fold and paste into box shape, having dimensions 6"x6"x3". ■F, f f 122 EATHEMATICAL CONSTRUCTION Material : 2nd day, stiff paper 10"x7". Kulers : 8"2"5". Number relations seen : 8 5 2 2 •S / 10 7 Lay 8-inch ruler on upper and lower edges, measuring from left edge; make marks and draw line a-b. (Fig. 1.) ' Lay 2-inch ruler in same way ; draw line c-d. c Lay 2-inch ruler on left and right edges, measuring from upper edge; make marks and draw line e-f. Lay 2-inch ruler on same edges, but meas- ure from lower edge, and draw line g-li. Make cuts indicated at a, b, c, d. Fold into box shape; dimensions 6"x3"x2". Make two more boxes with dimensions 6"x3"x2". Com- pare their volumes to each other and to the large one made the first day. Eelation 1 to 3. The small box shapes are drawers to be slid into large frame 6"x6"x3". If one drawer holds 3 tablecloths, how many will the whole bureau hold? (Ans. 3'3 table- cloths, or 9 tablecloths.) If the tablecloths in one drawer cost $4.00, how much does the bureau full cost? (Ans. 3'$4.00, or $12.00.) If they cost $2.00? (Ans. 3'$2.00, or $6.00.) Slide the drawers into the frame. Give the children an- r£ MEASURING: SINGLE UNIT EULEES 123 other small piece of the paper. From this let them make a back for the top free-hand; also the handles for the drawers. Silver paper is a very good play substitute for glass, if the children care to have a mirror at the top. Put on extra legs at bottom if desired, making them like tiny square prisms. LESSON XXXIV A Chiffonier Euleks: 10"2"8"; 2"4"6". 1 sheet stiff paper 12"xl0". 4 sheets stiff paper 8"x 6". 1 sheet stiff paper 6"x 6". 1 sheet stiff paper 10"x 8". Make line a-b with 10-inch ruler, measuring from top of paper, which is 12"xl0". (Fig. 1.) Make line c-d with 2-inch ruler, measuring from top. Make line e-f with 8-inch ruler, measuring from left. Make line g-h with 2-inch ruler, 3' measuring from left. Make cuts indicated at e, f, g, h. Fold and paste into box shape. Large frame is made. (Fig. 1.) Make line a-b (Fig. 2, paper 8"x6") with 6-inch ruler, measuring from left. Use 2-inch ruler same way to make line c-d. Use 4-inch ruler to make line e-f, measuring from top. Use 2-inch ruler same way to /r/ ^- 2 ' make line g-h. Make cuts indicated at e, f, g, h. Gr ! e- A: to • — • F 1 1 B[ 124 MATHEMATICAL CONSTRUCTION Fold and paste into box shape. Make four of these. They are the four drawers to the right. From 6-inch square make a 2-inch cube with one side open, using 2- and 4-inch rulers. This is the little drawer for the upper left-hand corner. a „ Use 8-inch ruler to get line a-b, measuring from top. (Fig. 3.) Use 2-inch ruler same way to get line c-d. Use 6-inch ruler, measuring from left, to get line e-f. Use 4-inch ruler same way to get line g-h. Use 2-inch ruler same way to get line i-j. Cut out upper and lower left-hand corners. Make cuts indicated at e, f, g, h. Fold and paste into square prism shape. The side which has three free edges is the door of the cupboard. (Fig. 4.) Make legs and mirror back as in pre- vious lesson if desired. Compare volume of a and b (equal). Compare volume of e and a (1 to 2). Compare volume of e and f (1 to 3). Compare volume of e to sum of a and b (1 to 4). Compare volume of e to sum of f and d (1 to 5). Compare volume of / to whole frame before any drawers were put into it (1 to 4). Compare volume of a to volume of large frame (1 to 6). If e holds two of baby's dresses, a holds 2 times 2 dresses, or 4 dresses ; / holds 3 times 2 dresses, or 6 dresses ; the sum of a and b holds 4 times 2 dresses, or 8 dresses; the sum of MEASURING: SINGLE UNIT EULEES 125 a, &, and c, holds 6 times 2 dresses, or 12 dresses; the sum of a and / holds 5 times 2 dresses, or 10 dresses, etc. In this lesson the children handled the rulers 2",4",6",8", 10", and one of the papers was 12" long. They have dealt with this length so often that they know it, even though the 12-inch ruler is not in their hands. Thev saw that: 6'2"=12" 2'4"= 8" 2'6"=12" 2'2"= 4" 3'2"= 6" 4'2"= 8" 5'2"=10" They also saw the combination, separation and division of these number facts. LESSON XXXV A Large Envelope (To hold weaving strips, or braids of cord or raffia until they are needed.) Killers: 11"1"8"4". Material: 2 sheets manila paper 12"x9". New numerical values seen: 11 8 1 1 126 MATHEMATICAL CONSTEUCTION Make line a-b, measuring from the left with the 11-inch ruler. Make line c-d in same manner, using 1-inch ruler to make marks c-d and long ruler to draw line. Using 8-inch ruler, measuring from top, make line e-f. Using 4-inch ruler, measuring from top, make line g-h. Cut out corners. Compare them to each other, giving problems. Fold on dotted lines. Next day use the other sheet of manila paper and rulers 2"4"5"10". Numerical values seen: 2x5"=10" 10" 5x2"=10" 2" 12" (paper is 12" long) 5" 4" 9" (paper is 9' wide). Measuring from left with 10-inch ruler, make line a-b. (Fig. 2.) Cut on line. Com- pare the two surfaces, giving problems on varnishing table tops. (Relation 1 to 5.) Measuring with 4-inch ruler from top, make marks and draw line c-d. Paste surface e on folded over flaps of Fig. 1, leaving x for cover flap. j* E c __ *n 1 • 1 1 1 1 !B MEASURING: SINGLE UNIT EULEES 127 LESSON XXXVI Book for Cuttings : Unfolded Sheets Rulers : 1"3"6"9". Materials: 10 sheets grey paper, 6"x9" ; 1 sheet grey or colored paper, 9"xl2"; 1 sheet manila or colored paper, 6"x9". Fifteen inches of cord, raffia or ribbon for lacing. Numerical relations seen : 3'1"=3" 2'3"=6" 6'i"=6" 3'3"=9" 9'1"=9" On each of the 10 sheets 6"x9" to be used as leaves, make dots A and B measuring with 1-inch ruler from the upper and lower left-hand corners (Fig. 1). Make dot C, using 3-inch ruler. Punch holes where the dots are. To make the cover, use the grey or colored paper, 12"x9" (Fig. 2). With the 6-inch ruler make x l °' ± marks A, B. Draw line and cut. Compare the two surfaces, giving problems on amount and cost of paper, linen or leather for book covers. (Relations 1 to 2). Measuring with 1-inch ruler make line C-D. Measuring with 3-inch ruler make line E-F. Fold left edge to meet line C-D. Cut on fold. Xow fold on line E-F. .3 a 128 MATHEMATICAL CONSTEUCTION J?iG»3- Using 1-inch and 3-inch rulers, as for pages, make clots A, B, C, and punch holes. To make decorations for cover, use manila paper, 6"x9". (Fig. 3.) Measuring with 3-inch ruler from left ^dge, make marks and draw line D-E. Cut on line. Compare the two sur- faces. Using 1-inch and 3-inch rulers, make dots and punch holes as before. This piece of paper slips under the flap of the cover. Use the 6-inch square which is left for any additional decoration desired. It can be cut into four equal oblongs. Bisect one of them and cut the pieces diagonally; use them to strengthen corners. Or, cut the pieces so that by making a cut J way down on the long edge and J way across on short edge, a corner is removed. Or, the pieces can again be bisected, making little squares, which can be cut in many ways. Another pretty decoration is the word "Cuttings" cut free-hand by the child, then pasted on the cover. Lace the book in any desired way. LESSON XXXVII Book With Folded Leaves for Words . Eulers: 6"1"7". Material: Six 7-inch squares of white, smooth paper; one 8-inch square of cover paper. MEASUKING 129 Sew together with cord 12" long. New combination seen 6 1 On each of the six squares of white paper draw the line a-b, rising the 1-inch ruler. Cut on the line. These narrow A c strips can be utilized later in a weaving lesson. Compare them with the large sur- face. Now fold each of these large surfaces so that their long edges touch, or use a 3-inch ruler to get the dividing line c-d. Measuring on this line from top and bottom, with 2-inch ruler make dots e, f. Half-way between these dots (eye judgment only) make another one, dot g. These are necessary only on the page which will be the center of the book, so the child can see where to place the needle. Fold the 8-inch cover paper into two equal oblongs. With the 2-inch ruler make dots as at e and /. Place center dot also. Sew and decorate cover in any desired way DEC 1 1911 One copy del. to Cat. Div. UtC I «»«' i M ill liii HHHni II up Pi I ill I if i WpHHi Mil if li fill I