1^3 37 UC-NRLF TUDY,ft >:^;^>;t>M-;>s^W^;-\^^i«'^^k;^\-s S5JSS5sS\ J ^S«11^S a\ «« ^cS>iS^.«;.^~;\S;,<.>;^ ii^v i* '**: it-' LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Received |ai\j f^ ^^93 . /^^ ^Accessions No.HQon^, class No. ^F#*; ,V-K •-:e^i^¥-^*-:. » '/r.-^. ^^= * IT-/:;- « » ^^***v^ ft .«•» ♦ ? ■.- 4 t ;v • ,'•.:/ « i X ■ ;.• a * Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/formlessonstopreOOspeerich FORM LESSONS. TO PREPARE FOR AXI) TO ACCOMPANY The Study of Number. F BY AY, ^y. SPEER TEACHEIi OK iMATHEMATICS, COOK COUXTV NOKMALi SCHOOL,. THIRD EDITION. C. C. N. S. SERIES. \.-<< \ s r-1 COPYRIGHT, 18«8. BY W. W. SPEER. DONOHUE & HENNEBERRV PRINTEilS AND BINDERS CHICAGO. PREFACE. In the first grade, this "book is designed to aid teachers in sys- tematizing their oral lessons. In the other grades, the book should be in the hands of each pupil. The constant test of the pupil's obser- vation must be the accuracy of his expression. The questions asked arc for the pupil to answer, and not the teacher. When his descrip- tions are not correct, he, through renewed observation, must discover and correct his error. It is not expected that all that can be dis- covered in tlie various forms will be seen by first or second-year pupils, but many tests have shown that the work is well adapted for these grades, while it furnishes sufficient material to profitably engage the attention of pupils of any grade. As an aid to mathematical investigation and for securing clear and concise expression, it will, I think, be found helpful even in the high school before beginning the study of scientific geometry, if pupils have not had previous training of this character. Where the programme is so full that there is not time to give special attention to teaching form, the lessons can be used as a means of teaching language. As the forms to be compared are definite, and demand accuracy of expression in description, and as all inaccuracy of statement can be readily detected, no study furnishes a better basis for language lessons. In the written exercises, penmanship, spell- ing, and punctuation can be taught. The attention of the reader is respectfully invited to a consid- eration of the remarks on page 73, on the value of form lessons as a preparation for number study, and to the opinions of some eminent educators and thinkers on this subject. Englewood, III. W. W. SPEER. OBSERYATION LESSON'S. " Lessons especially designed to cultivate the power and habit of oo^ervation appear to be less widely used than is desirable. It is probable that the erroneous methods of teaching, too often employed in such lessons, have led to meager results and consequent distrust of this branch of primary school work. The skill required to teach such lessons properly is apparently less common than skill in teach- ing other branches. The mistake often made is that of supposing a pupil is learning how to observe when he is merely listening to what his teacher tells him to remember about an object he may be looking at. So-called object lessons taught by such false methods have no tendency to cultivate the power and habit of observation, but rather to confuse and stultify the child's mind. On the other hand, obser- vation lessons, in which the children really do the observing, not only develop the observing powers, but also furnish the children's, minds with a stock of clear ideas which constitute the best possible material for language work." — Report of Massachusetts Teachers' Association, 1887. w LESSONS IN FORM. TO PREPARE FOR AND TO ACCOMPANY THE STUDY OF NUMBER. PART FIRST. The first exercises are to cultivate the habit of observation, to teach direction and position, and to train pupils to associate the terms to be used in the comparisons which follow with the corresponding ideas. SURFACES, LINES AND POINTS. Give each pupil a cubic inch, or have a large cube placed so that the class can observe it. Directions and Questions for Pupils : Place your finger on the surface of tlie cube. Place voui' hand on the surface of vour desk. 6 LESSONS IN FORM. Of what object do you see the surface ? Kecall objects that you have seen at home that have surfaces. Find other surfaces in the room. Of what object at home are you thinking that has a surface ? Phice your linger on one of the edges, or lines, of the cube. On one of the edges, or lines, of your desk. Find other lines, or edges, in the room. Of what have 3^ou found an edge, or a line? Name five objects that you have seen outside of the school-room that have edges or lines. I^ame five objects that 3^ou have seen outside tiie school-room whose surface is not bounded by lines. Example : The surface of a croquet ball is not bounded by lines. Drawing. — Look at the cu])e and draw four lines each as long as an edge of the cube. Use an edge of the cube and measure the lines you have drawn. Draw again and measure. Observe a face of the cube and draw it. Continue drawing and measuring until you can represent a face quite accurately. Place a finger on one of the corners, or points, of the cube. On a corner, or point, of your desk. Find other points in the room. Of what have you found a point ? Place a finger on one of the lines of the cube. What is at the end of this line ? Can you place a finger on a point which is at the end of one of the lines of the blackboard ? OPPOSITE, ADJACENT AND PARALLEL LINES. 7 OPPOSITE, ADJACENT AXD PAPtALLEL LINES.* The upper and lower lines, or edges, of the black- board are opposite lines of the blackboard. Can you find other edges, or lines, of the blackboard that are opposite ? Point to the other lines of the blackboard that are opposite. Place your finger on two lines of the cube that are opposite. Find opposite lines of your book. Find other opposite lines in the room. Of what have you found opposite lines? Drawing. — Look at two of the opposite lines of the cube and draw two lines Jis long and as far apart. Measure and see if they are the same length and as far apart as the opposite lines in tlie face of the cube. Draw again and measure. In Avhat direction do the opposite lines of the cube extend ? Find other lines in the room that extend in the same direction. Of what have you found lines extending in the same direction? Do the lines you drew extend in the same direc- tion? Find two lines of the blackboard that meet. Touch two edges, or lines, of the cube tliat meet. *If words are taught in connection with the ideas, primary inipils will have no difTiciilty, because of their lensrth, in learning'- to use iniraliel, parallelo^rram, perimeter, rectangle, rectangular, blackboard, grandlafher, or parallelopipcd. It is not wise to teach one set of terms in theiirimary and another in the higher grades. Teach the right terms from the begin- ning, so that i)upils may learn to think in the proper language. 8 LESSOITS IN FORM. Lines that meet are adjacent lines. Find other adjacent lines in the cube. Find other adjacent lines in the room. Of what have you found adjacent lines, or edges ? Deaaving. — Draw a pair of adjacent lines, making each line an inch long. Draw another pair, making each hne two inches long. Draw another pair, making each hne three inches long. Measure each pair and draw again. Find opposite lines and adjacent lines in the room and tell whether thej are opposite or adjacent. Ex- ample : This line and that line of the door are oppo- site. Lines that extend in the same direction are par- allel. Find parallel lines of the cube. Of a book. Of a chalk box. Drawing. — Draw three pairs of parallel lines each an inch long. Measure and draw again. Do lines need to be of the same length that they may extend in the same direction ? Do they need to be of the same length that they may be parallel ? "When are lines parallel ? Can you find opposite lines in the room that are not parallel ? Suggestion: If pupils can not find opposite lines in the room that are not parallel, draw several quadrilaterals on the blackboard of which one or both pairs of opposite lines do not extend in the same direction. Have pupils point to lines of the figures that are opposite but not parallel. ANGLES AND PERPENDICULAR LINES. 9 ANGLES AND PERPENDICULAR LINES. TlILSE ARE PeKFENDTCCLAII LiNES. Find lines in tlie r<,.oni tliiit aro perpendicular to each other. How many pairs of perpendicular lines are there in the blackboard ? Ans. The edges of tiie blackboard form four pairs of perpendicular lines. How many pairs of perpendicular lines are there in one face of the cube ? How many pairs of opposite lines are there in one face of the cube ? How many pairs of opposite lines are there in one of the Avails of the room I Remaek : "When the attention of pupils is first called to the fact that lines may extend in different directions, or form angles, you will do them a service by using for the first illustration lines that do not intersect. A majority of pupils, who think they know what an angle is, are very sure that unless the lines do intersect there is no angle. This prejudice would not exist had their first im- pression of an angle hecn gained when considering the difference in direction of lines that did not intersect. Point to lines extending in the same direction. Point to lines that extend in different directions. When two lines extend in different directions, the difference in direction is an angle. 10 LESSON'S IN FORM. Find five angles in the room. When you point to an angle, say that the difference in direction of this line and that line (indicating the direction each line extends, by moving pointer or hand) is an angle. Hold two sticks or slats so that they are perpen- dicular to each otlier. The difference in direction of two perpendicular lines is a right angle. What is a rio;ht ang-le ? Find perpendicular lines, and say that the differ- ence in direction of these perpendicular lines is a right angle. Hold two splints so that the difference in direction which they indicate is not so great as a right angle. When the difference in direction of two lines is not so great as that of a right angle, the difference in direction is called an acute angle. Find m the room differences in direction less than a right angle. What is a difference in direction less than a rio^ht anoxic called ? Hold the slats or splints so that the difference in direction is greater than a right angle. When the difference in direction of two lines is greater than the difference in direction of two perpen- dicular lines it is called an obtuse angle. What is an obtuse angle? Ans. An obtuse angle is a difference in direction greater than a right angle. Find obtuse angles in the room and tell Avhy they are obtuse. Example: The difference indirection of this line and that line is an obtuse angle because it is greater than a right angle. ANGLES AND PERPENDICULAR LINES. 11 How many differences in direction can you repre- sent with two lines? With three hnes? With four hnes? Represent three right angles by drawing lines. Can \^ou draw two right angles so that the differ- ence in direction of one is greater than the difference in direction of the other ? If you make long lines in representing a right an- gle, is the difference in direction greater than when represented by short lines ? Why not^? Can you make two right angles with two lines? Can you make three right angles with two lines? Can you make four with two lines ? How many right angles can you make with three lines? Draw tliree acute angles. Can you draw two acute angles so that the differ- ence in direction in one shall be greater than the differ- ence in direction of the other ? Draw three acute angles of different sizes. Draw three obtuse angles. Can you draw obtuse angles of different sizes, or magnitudes ? If you increase the length of the lines of an obtuse angle does it increase the difference in direction? Does it increase the size oP the angle ? Why not? There are what three kinds of angles ? What is a rio"ht ano-le? An acute anHe? An obtuse angle ? "^" * IlKMAUKS : Two straight lines may have cither of two relative posi- tions . 1st. Tlicy may extend in the same flireetion. 2(1. They mai' extcinl in . V r* 12 LESSONS IN FORM. DIEECTION AND POSITION. Place your linger on. the front face of the cube, that is, the face next to you. On the back face. On In the first case, instead of saying that the lines extend in the same direction, it is customary to say the lines are parallel. One form of expressing- this relation means no more nor less than the other. Parts of the same line thought of as distinct lines are parallel, for they can be thought of as extending in the same direction. In defining parallel lines, do not add either of the following state- ments to the definition, namely : If produced, they would never meet ; or, they are everywhere equally distant. It is not true that, when the parallels are parts of the same straight line, they would never meet if produced, and, it is an inference, from the definition proper, that they would never meet if they are not paints of the same line. An inference should not be a part of a definition. That parallels are everywhere equally distant, if they are rot parts of the same straight line, is a proposition to be demonstrated in geom- etry, and it ought not to be made a part of a definition. The second relation of lines, that of extending in different directions, is called an angle. An angle is simply a difference in direction. It may be a difference in direction of two lines, two or more surfaces, of two persons walking, etc. It is not necessary that the lines intersect in order that they indicate a difference in direction. Nothing is added to the difference in direction by the intersection. If the lines forming the angle do not inter- sect, and are in the same plane, it is an inference that they will intersect if produced in the direction of their convergence, and another inference that, if produced in the direction of their divergence, they will not intersect. In orderto measui-e an angle, or to pi-ove it equal to another angle by superposition, the lines indicating the difference in direction must inter- sect, and, therefore, lie in the same plane. The importan e of fixing right ideas of the two relations of lines will be recognized when it isimderstood that all of the reasoning in both plane and solid geometry is based on a comparison of the length of lines, the sameness of direction or difference in direction of lines. Hazy ^'mpressions of these elementary ideas will make the reasoning based on such impressions lack in clearness. SURFACES, LIKES AND POINTS OF A CUBE. 13 the left-hand face. On the upper face, or base. On the right-hand face. On the upper right-hand corner in front. On the upper left-hand corner at the back. On the lower left hand corner in front. On the upper edge of the front face. On the upper edge of the left- hand face. Suggestions: Have one pupil give a direction for pointing or placing a finger^ and other pupils follow directions. Have other pupils give similar directions. Have pupils point to different corners of (.bjects ana tell "where each is. Example: That is the upper right hand corner in the front part of the room. Place a finger on different points and faces of the cube or other objects, and have pupils tell where it is. Have pupils point to different corners of the room and tell to what corner they are pointing. To different windows in the room and tell where each is. ExamjJle: That is the east window in the north wall. To different pupils and tell where each is. Examjle: Mary Warner sits on the third seat in the second row from the right. Review exercises until pupils can follow directions in j^lacing finger on face, line, or point designated, and can give directions clearly and without hesitation. THE SUEFACES, LIXES AND POINTS OF A CUBE. Count the surfaces of a cubic inch. How many surfaces has a cubic inch? 14 LESSONS IN FORM. How many surfaces has a chalk box ? Find other objects in the room that have six surfaces. What object have you found that has six surfaces ? What objects have^^ou seen at home or in going to and from the school that have six surfaces? How many surfaces has the school-room ? (Count the Avails, ceiling and floor.) Place a finger on the upper face, or base, of the cubic inch. Count the edges of the upper base. How many edges has the upper base ? How many edges has the lower base ? Point toward the zenith, or the point in the heavens directly over 3^ our head. Point toward the center of the earth.' An upright line, or a line which extends from the zenith toward the center of the earth, is a vertical line. From what and toward what does a v^ertical line extend ? Find vertical lines in the school-room. Example : One of the right-hand edges of the door is an up-and- down line, or a vertical line. Kecall objects that are in a vertical position. Ex- amjple : Some telegraph poles are in a vertical posi- tion. Can you find edges of the cube that are vertical ? How many lines of the cube are vertical ? Point toward the horizon. Find lines in the room that extend toward the SURFACES, LINES AXD POIXTS OF A CUBE. 15 horizon. Exami)le : The upper edge of the blackboard extends toward the horizon. Lines that extend towara the horizon are horizon- tal lines. Find horizontal lines in the room. AVhat lines of a window are horizontal? In what direction do horizontal lines extend? If you place a cube so that four of its edges ai"e vertical, how many horizontal edges will it have? Can3^ou hold a cube so that none of its edges will be parallel? Why not? Can you hold a cube so that none of its edges will be perpendicular to each other? Does it change the relation of the lines of a cube to each other to hold the cube indifferent positions? Can 3^ou hold a cube so that its edges will be neither vertical nor horizontal? How many lines has the cubic inch ? How many lines has a cubic foot? Kame other objects that have twelve lines, or edges. How many lines bound the ceiling of the room ? How many lines bound the floor of the room? How many vertical edges have the walls of the room ? How many corners, or points, has the cubic inch ? How many points has the upper base? How numy p(jints has the lower base? How many points has a cubic foot ? How many points has a brick ? How many lines meet in one of the points of the cubic inch? In how many directions do the three edges extend from one point i 16 LESSORS i:n' porm. How many pairs of parallel lines are there in one face of the cube ? How many surfaces has the cube? How many two pairs of parallel Lnes has the cube? How many square inches are there in the surface of the cubic inch ? Can you tell eight things that are true of the cubic inch ? What is the shape of one of the surfaces of the cubic inch ? Find otlier squares in the room. What object have you found that has a square sur- face? Can you recall any squares that 3^ou have seen at home ? Examples : The tops of some collar boxes are square. The bottoms of some salt cellars are square. Can you recall any squares that 3^ou have seen in going to and from school ? Review, beginning with surfaces, lines and points of a cube. Dkawing. — Draw a two-inch square. Use a ruler and measure. Erase. Draw and measure again. Erase. Use the ruler and draw a two-inch square.^ * Remark : Manj^ of the pupils, in their efforts to make two-inch squares or to represent anything-, will not do well at first if the drawing- and not the effort is considered. Tlie effort made to represent the thing observed or recalled is worth a thousand times more than the drawing, ■which is the result of the effort. The question is not, Are they making ac- curate and beautiful drawings? but, ai'e they forming habits of observa- tion ? Drawing furnishes a means of expressing ideas, and man first resorted to it for that purpose; but when it is perverted and fails to accomplish this purpose, it does not produce the best results. Any method that teaches QUESTIONS OlST THE TWO-IXCH SQUARE. 17 THE SQUAEE. c d Tell as many things as you can about iliu two-inch square. QUESTIONS ON THE TWO-INCH SQUARE. 1. How many lines bound the square? 2. How many points has the square? 3. How many pairs of parallel lines has the square? How do parallel lines extend ? words before ideas is radically wrong-, ami any method that teaches dra wing- without usinjr it as a means of expressinj^or representing- ideas, is radically wrong-, because it leaves out that whch stimulates and dovch ps the powers of the mind. Iicpruducin^- aline Avithouta,considcring- its leu'rth or direc- tion does very little to increase one's power. That traininj? which leads puynls to be imitators only does little to develop thought and action. Drawing- ought to teach seeing, doing, and knowing. Drawing ought to cul- tivate the hand and the eye, and increase the knowledge of the object represented. '• As the first step in drawing is to learn to see correctly, it is"evident that all the exercises, both in gifts and occupations, prepare for the use of the pencil and chalk. As the mediation of word and object drawing is of vast importance in its reaction on tl-.e iiiind, and as th(> soul of all technical processes, it is the indispensable basis of industrial education." Susan E. Blow, 18 LESSONS 12^ FORM. 4. How many pairs of adjacent lines has the square ? (Each line is counted twice.) 5. How many pairs of perpendicular lines are there in the square ? G. The difference in direction of two perpendicu- lar lines is what kind of an angle? 7. How many right angles are there in the square ? 8. How many inches in the boundary, or perime- ter, of the two-inch square ? " That which the pupil knows thoroughly contains an explanation of what he does not know." DIRECTION AND POSITION. To Teacher: Give eacli pupil a 4 inch square, and have him place the letter a rear the upper left-hand point at the back, b near the right-hand point at the back, c near the left-hand point in front, and d near the right-hand point in front. QUESTIONS. 1. In what part of the square is the point a ? Ans. The point a is in the left-hand point at the back. 2. Where is the point near the upper right- hand point, ('. near the lower left-hand point, and d near the lower right-hand point. 1. Which is the point a \ Ans. It is the upper left-hand point of the sq. ft. 2. Which is the point dl The point M Tlie point G% 3. Which is the line (3^ h of the sq. ft. % 4. Which is the line a c oi the sq. ft. ? 5. Close your eyes. Near what point is each let- ter in the sq. ft. ? The letter d is near the lower right- hand point of the sq. ft. Suggestion: Have pupils tell as many different things as they can about the positions of lines and points of the blackboard, door, etc., without questioning them. 6. Call the edge of your desk next to you the front edge. 7. Place your hand on the front edge of your desk. On th ^Ag'^ at the back. On the middle of the right edge. 8. Toucn tlie left corner in front. The right cor- ner at the back. Place your hand on the middle of the edge at the back. Suggestion to Teacher: Have pupils give directions for other pupils to follow. Giving directions will force pupils to express themselves with precision. The necessity of saying exactly what they mean will make the exercise a valuable language lesson. REVIEW KROM THE BEGINNING OF THE BOOK. 20 LESSONS IN FORM. ExEEOiSES IN Compaeiso:n.* COMPARISON OF THE SQUARE RECTANGLE WITH THE OBLONG RECTANGLE. Remaek: The lieavy lines, iu the different cuts, indicate the forms to be compared made by folding the square. Get kindergarten 4-inch squares, or cut 4-inch squares out of paper. Give each pupil two of these squares. *" It is iDy comparisons that we ascertain the difference which exists between things, and it is by comparisons, also, that we ascertain the general features of thing-s, and it is hy comparisons that we reach general proposi- tions. In fact, comparisons are at the bottom of all philosophy. Without comparisons we never could go beyond the knowledge of isolated, discon- nected facts. Now, do you not see what importance there must be in such training,— how it Avill awaken the faculties, how it will develop them, how it will be suggestive of further inquiries and further comparisons; and as soon as one has begun that sort of study there is no longer any dullness in it. Once imbued with the delight of studying the objects of nature, the student only feels that his time is too limited in proportion to his desire for more knowledge. And I say that we can in this way become better aquainted with ourselves. . . . " The difficult art of thinking can be acquired by this method in a more rapid way than any other. When we study logic or mental philosophy in text- books, which we commit to memory, it is not the mind which we culti- vate ; it is the memory alone. The mind may come in, but if it does in that method it is only in an accessory way. But if we learn to think, by unfold- ing thoughts ourselves from the examination of objects brought before us, then we acquire them for ourselves, and we acquire the ability of applying our thoughts in life." Agassiz. COMPARISON OF RECTANGLES. 21 Direction for folding the square into the oblong rectangle: Place one of the squares so that its front edge will extend in the same direction as the front edge of the desk. Fold the paper so that the front edge will lie along or coincide with the edge at the back. Crease the paper. Place the oblong rectangle formed near the square for comparison. Remark: When the pupils are folding papers they should be ri'quired to keep themiu the same relative position. If they do not they can not follow directions. They should not lift the paj)ers oU the desks when folding. 1. Find the likenesses. 2. Find the differences. 3. Find square rectangles in room. 4. Recall square rectangles that you have seen at home or in going to and from school. 5. Find oblong rectangles in the room. Example: One of the windows is the shape of an oblong rect- angle. 6. Recall oblong rectangles that you have seen. Example: One of the surfaces oL' a brick is an oblong rectangle. 7. In going to and from school or at home, you may find five square rectangles, and five oblong rect- anoles, and in to-morrow's recitation vou mav tell me the names of the objects that have square surfaces and those that have oblong rectangular surfaces. The following questions are to be answered by pu])ils aft^r they have found all the likenesses and dif- ferences they can in comparing tlie forms above: Remark: Pupils should bo cncouraired to give the names of the colors of the dilfereut papers used in the comparisons and to 22 LESSOKS IK FORM. find like colors in the room, and to recall objects that are similar in color. QUESTIONS— LIKENESSES. 1. Each of the forms is what ? Ans. Each of the forms is a surface, or they are each surfaces. 2. Each surface has how many lines? 3. Each has how many points? 4. Each has how many pairs of parallel lines? 5. Each has how many pairs of adjacent lines ? 6. Each has how many pairs of perpendicular lines ? 7. Each has how many right angles ? 8. Each bas how many pairs of opposite points? 9. One of the long lines of the oblong rectangle is equal to what in the square? 10. The sum of the tw^o short lines of the oblong- rectangle is equal to what ? 11. The opposite lines in each are what ? 12. Give the likenesses without being questioned. Remarks : In elementary work, it is not well to-spend time ar- guing with pupils, or trying to force your views upon them, even if you are in the right. You can not correct false notions or nar- row views unless there arc ideas enough in the pupil's mind to en- able him to comprehend what you s^y. If you develop the dis- criminating power of the pupil by training him to observe, he will, in time, see for himself what you wish him to see. In observing the square, one pupil may see on]y two pair of perpendicular lines, while another sees the four. The former thinks that, as there are only four lines in the square, there can not be more than two pair of perpendicular lines. Trying 1o convince the first pupil that he is wrong may be worse than a waste of time. To-day, he sees two pair; leave his mind free, and to-morrow he may discover the four. COMPARISOIT OF RECTANGLES. 23 DIFFEEEIS^CES. 1. The square rectanole is bounded l)y how many lines? The oblong rectangle is bounded by two equal long lines, and by what? 2. One of the short lines of the oblong rectangle is equal to what part of one of the lines of the square rectangle ? 3. The oblong rectangle is equal to what part of the square rectangle? 4. The sum of the lines of tlie oblong rectangle is equal to how many of the lines of the square? 5. How many inches are there in tlie boundary of each form ? G. Can you find rectangles in the room that are not one-half of a square rectangle ? 7. Are the short sides of oblong rectangles always equal to one-half their long sides ? 8. What is the direction for folding the square into the oblono: rectano:le ? Cutting. — Direction for pupils : Cut out of paper an inch, a two-inch, a three-inch, and a four-inch square. Bring them to the class to- morrow. You may cut and measure as many as you wish before cutting those you bring to the class. Pin the squares you bring to the class together and write your name on one of them.^ * " Thrc/iitili their own productions children are slowlj' awakening to facts of form and relations of number and led to clear and concise use of language." Miss Susan Blow. "Almost invariablj' cliildren show a strong tendency to cut out things in j aper, to make, to build— a propensity which, if (Uily encouraged and directed, will not only i)icpare the wiiy for scientific conceptions, but will develop those powers of manipulation in which most people arc so de- ficient." Herbert Spencer. 24 LESSON^s i:n^ form. Drawing. — You may bring to the class to-morrow the different squares mentioned above, drawn on paper. You ma}^ draw and measure for a while, but do not measure those you bring to the class. " What we try to represent we begiu to understand." Froebel. Remark : The pupils should be encouraged to cut and meas- ure and to draw and measure many squares. Practice in this trains the hand and eye. Very little skill will be required by pupils who cut and draw only one square of each dimension given. These exercises may seem very simple, but neither a boy of five nor a man of fifty can cut or draw a four-inch square who has not been trained to observe. Upon our perceptions of form and of extent depends the cor- rectness of our ideas of objects and consequently our power of giv- ing true descriptions of things, of their location, their size, the rela- tions of one part to another, etc., etc. Give each pupil two four-inch squares. Give direc- tion for folding the squares into an oblong rectangle. Place the rectangle on one side of the desk. Take the other square and place it so that one of its edges Avill extend in the same direction as tiie front edge of the desk. Fold the square so that the right-hand point in front will fall upon or coincide with the left-hand point at the back. Crease the paper. Place the two folded figures near each other for comparison. COMPARISON OF THE OBLONG RECTANGLE AVITH THE TRIANGLE. RECTANGLES AKD TRIANGLES. 25 ^' The mistake often made is that of supposing a pupil is learning how to ohserve, when he is merely list- ening to what his teacher tells him to rememher about an object he may be looking at." 1. Find likenesses. 2. Find differences. 3. Find triangles in room.'^ 4. Recall forms that are triangular in shape. Examjyh: The gables of some houses are triangular. The lateral surfaces of pyramids are triangles. 5. In going to and from scliool or at home \o\\ may find five objects that have triangular surfaces, and you may tell me the names of the objects that have these surfaces. (). What are the names of the objects that you found having square surfaces ? 7. What are the names of those that you found having oblong rectangular surfaces? QUESTIONS— LIKENESSES. Suggestion: Have pupils write the answers to the following questions in complete sentences. In the recitation have answers read, and let pupils criticise one another's statements. 1. Each form is what? ♦Remark : Fin 'iii;^ formsof t'nos 'mo/rener dshape as thosetakon jjs .types, is of the hij^liestiniixii-tanuc. Uult-sstiiisi.sdc.ncpupilsaie nut learn- ing to pass from the partieuhiv to the general. They are not taught to see many things through the one, a .d the impression they gain is that the par- ticular forms observed i re tlie only forms of this kind. Unless that which the pupil observes ; ids him in interprctin.r something else, it is ('f no val.ie to him. Certain things arc taught that tlirouuh them (ther things may be seen. Pupilsshould not be trained tosee forthesake of the seeing, but that they may liave tlie power t) see. Il(;w diU'crent t!ie world up(.ears to a child who sees form in everything from what it d.>fcs to one who sees no definite form in anything, and to whom all is in a stiite of confusion. Teach- ing is leading pupils to discover the unity of things. \v >% 26 LESSONS IN FORM. Ans. Each form is a surface. 2. Each surface is bounded hv what? 3. The two short lines of the triangle are equal to what ? 4. The sum of the two short lines of the oblong- rectangle is equal to what ? 5. The sum of the two acute angles of the triangle is equal to what ? 6. Each form has, at least, one pair of what kind of lines ? 7. Each form has, at least, one angle of what kind ? 8. The area of the rectangle is equal to what part of the four-inch square ? 9. The area of the triangle is equal to what part of the four-inch square ? 10. The area of the triangle is equal to what? 11. Without observing the forms think of all the likenesses you can. DIFFERENCES. Write answers in complete sentences. 1. How many lines are there in the boundary of each surface ? 2. Each surface has how many pairs of adjacent lines ? 3. Eacn surface has how many angles or diifer- ences in direction? 4. Each has how many right angles ? 5. Each has how man}^ acute angles? Find the right angle of the triangle. Find the line opposite the right angle of the tri- angle. The line opposite the right angle of a right tri- angle is the hypotenuse of the right triangle. RECTANGLE AXD TKTAXGLE. 27 6. The hypotenuse of a right angle is opposite what ? 7. What is opposite the hypotenuse of a right triangle ? 8. The hypotenuse of the right triangle is longer than what in tlie rectangle? 9. Is there any line opposite another line in the triangle ?■ 10. How many pairs of opposite points are there in each form ? 11. Are there any points in the triangle opposite any other points in it ? 12. The sum of the angles of the rectangle is equal to how many right angles ? 13. The sum of the angles of the triangle is equal to how many right angles ? 14: The sum of the angles of the triangle is equal to what part of the sum of the angles of the rectangle? 15. What is there in the triangle that is not fomid in the rectano-le? IG. What is there in the rectangle that is not found in the triangle I 17 Without observing the forms think of all the differences you can ? 18. Write the directions for folding the square into the triangle. Cutting. — Cut a rectangle 2 inches long and 1 inch wide; another 3 inches by 2 inches; and another -t inches by 2 inches. Pin them together. Do not measure those you bring to the class. 28 LESSONS IN FORM. Cut a triangle having all its sides equal, another having only two sides equal, another of which no two sides are equal, and a fourth having a right angle. Write in the first, equilateral triangle ; in the second, isosceles triangle ; in the third, scalene triangle ; and in the fourth, right triangle. Can you cut a triangle having two right angles? Can you cut a triangle so that the sum of two sides of it shall be equal to the third side ? Drawing. — Practice trying to draw rectangles of the same dimensions as those you cut. Measure all that you draw except those that you draw to bring to the class. Cutting. — Observe a window at your home and cut a piece of paper in the same shape, and so that the edges sliall have the same relation to each other as the edges of the window. Example : If the window is 6 feet high and 3 feet wide, cut, as nearly as you can, a piece of paper 6 inches long and 3 inches wide, or 6 half inches long and 3 half inches wide, or so that the length of the paper shall be twice its width. Drawing. — Observe the window and draw it in proper proportion. Measure the window and write the measure on the paper on which you made the drawing, stating : The window is feet and inches high^ and feet and inches wide. Give each pupil two four-inch squares. Have one of the pupils give directions for placing a square and folding it into a triangle. RECTANGLE AND TRIANGLE. 29 Unfold the paper. 1. Do you see the line made by creasing the paper? 2. What does this line connect ? The hne joining the opposite points of the square is a diagonal line. 3. What does a diagonal line join ? 4. Show me opposite points of the blackboard. 5. What is a line called which joins the opposite corners, or points, of a blackboard ? 6. Show me what would be a diagonal line of the top of your desk. 7. What is a diagonal line of one of the surfaces of a pane of glass? Of one of the walls of the room? Fold the square again into a triangle. Place it at one side and take the other square. Place the square so that one edge shall be parallel to the right edge of your desk. Fold the square so that the right-hand point in front will coincide with the left-hand point at the back. Crease the paper. Open the paper. Do you see the diagonal line if Fold the paper so that the front edo^e will coincide with the diao'onal line. Crease the paper. Turn the paper over. The form you have folded is a trapezoid. COMPARISON OF THE TRIANGLE WITH THE TRAPEZOID. Remakk: The important part of the work suggested by this book consists in the simple exercises of finding and expressing the 30 LESSORS i:t^ FORM. likenesses and differences of the forms compared, and in finding sim- ilar forms in and outside of the school-room. These are the exer- cises that will foster habits of observation, and be of the greatest edu- cational value. At least nine-tenths of the time given to the study of form should consist, not in answering the questions of the book, but in discovering relations not suggested by the questions. Inces- sant questioning, on the part of teacher or text, fixes on the pupil the habit of waiting to be questioned, and, when this condition is induced, the pupil's thinking generally ends with the questioning. The teach- ers who will succeed in this work, are those that have the courage to wait, and who can make their practice harmonize with the theory that it is what the pupil docs for himself that educates him. 1. Find likenesses. 2. Find differences. 3. Find trapezoids in the room.* 4. Find, at least, five trapezoids in going to and- from school or at home. Remakk: Trapezoids can be found where building is being done. Parts of the roofs of many houses are in this shape. 6. What are the names of the five objects which you found having triangular surfaces? Write the answers to the following questions: 1. Each form has, at least, one pair of what kind of lines? 2. The difference in direction of two perpendicu- lar lines is what kind of an angle? 3. Each form has at least one — angle and one — angle. 4. What is the name of the line opposite the right angle in the triangle ? 5. The hypotenuse of the triangle is equal to what line of the trapezoid ? ♦Suggestion : If at any time forms are being compared which can not be found in the room, the teacher should have several of these forms drawn on blackboard for the children to discover when looking for forms. compauiso:n" of thiaxgle and trapezoid. 31 0. The sum of the two equal hues of the trapezoid and of the diagonal line is equal to what in the triangle ? 7. To wiiat is the sum of the two acute angles of the triangle equal? 8. The sum of tlie angles of the triangle is equal to what in the trapezoid ? 9. Can you find two equal lines and two equal angles in the triangle ? Which are they ? 10. Can you find two equal lines and two equal angles in the trapezoid? Which are they ? DIFFERENCES. 1. Which figure has the greater surface, or area? 2. How many lines bound each form? 3. How many angles has each ? 4. How many right, acute, and obtuse angles has each? 5. One of the acute angles of the triangle is equal to what part of one of the right angles of the trape- zoid ? 6. The acute angle of the trapezoid is greater than what in the triangle? 7. The acute angle of the trapezoid is greater than what part of the right angle of the triangle ? 8. The longer of the two lines forming the obtuse angle of the trapezoid is shorter than what line in the triangle, and longer than what line? 9. An acute angle of the triangle is less than one- half of what angle of the trapezoid? How do you know ? 10. The right angle in the triangle is greater than one-half of what angle in the trapezoid ? Why ? 32 LESSONS IN FORM. 11. If a right angle were equal to one-half of the obtuse angle of the trapezoid, to what, at least, would the obtuse angle be equal ? 12. Do any of the lines of the triangle extend in the same direction ? Any of the lines of the trapezoid ? What, then, is true of the trapezoid that is not true of the triangle ? 13. The sum of tne lines of the triangle is less than w^hat? 14. The shortest line of the trapezoid is greater than one-half of what, and less than one-half of what, in the triangle ? 15. The sum of the two acute angles of the triangle is greater than what in the trapezoid, and less than what in it ? 16. "Which figure has opposite points and lines? How many pairs of each ? 17. There are no pairs of opposite points or lines in which figure ? IS. If both pairs of opposite points of the trape- zoid were joined by lines, how many diagonal lines would it have? 19. Are there any diagonal lines in the triangle? Why not? 20. Write the direction for folding the square into a trapezoid. FINDING FORMS MADE BY FOLDING. Give each pupil a four-inch square. Place it for folding. Fold the square so that the right-hand point in front will coincide with the left-hand point at the back. Crease. Open the paper. Fold the square so that FIXDIXG FORMS MADE BY FOLDIXG. 33 the left-hand i)oint in front will fall upon the right- hand point at the back. Crease Open the paper. Fold the paper so that the front edge will fall upon one of the diagonals. Crease. Fold the left-hand edge so that it will coincide with the same diagonal. Crease. Fold the paper so that the right-hand point at the back will fall upon a diagonal, and so that an isosceles triangle will be formed. Crease paper carefully. Unfold the paper so that you will have the square again. Observe the forms made by the creased lines. 1. Observe the figure. How many triangles, each having a right angle, can you find? 2. How many triangles having two sides equal can yuu find { 3. How many triangles canyon lind having no two lines equal. 4. How many trapezoids can you InuH 34 LESSON'S IN" FORM. 5. How many different figures have you found in the square ? Give each pupil two four-inch squares. Have some pupil give direction for placing the square and folding it into a trapezoid. Take the other square. Fold it so that the right- hand point in front will fall upon or coincide with the left-hand point at the back. Crease the paper. Open it. Fold the paper so that the edge in front will coin- cide with the diagonal. Crease. Fold the paper so that the edge at the back will coincide with the diag- onal. Crease. Turn the paper over. This form is called a rhomboid. OBLIQUE ANGLES AND LINES. Obtuse and acute angles are called oblique angles. 1. How many oblique angles are there in the rhomboid? Find oblique angles in the room. 2. What angles are called oblique angles? 3. Are there any oblique angles in the square ? 4. Are there any oblique angles in the trapezoid? How many ? 5. The lines forming either acute angles or obtuse angles are called oblique lines. How many pairs of oblique lines are there in the rhomboid ? 6. Find oblique lines in the room. Are any of the lines of a rectangle oblique lines ? 7. "What kind of angles are formed by oblique lines ? COMPARISON OF TUAPEZOID WITH RHOMBOID. :^5 COMPARISON OF THE TRAPEZOID WITH THE RHOM- BOID. 1. Find likenesses. 2. Find differences. 3. Find rhomboids in room. 4. Find, at least, five rhomboids, and report the names of the objects that have surfaces of this shape. 5. What objects did you find having a surface or surfaces in the shape of a trapezoid ? QUESTIONS— LIKENESSES. Write answers. 1. Each surface is bounded by how many lines and has how many angles ? 2. How many pairs of opposite points and lines are there in each ? 3. How many pairs of adjacent lines has each ? 4. How many diagonal lines can each have ? 6. The longer diagonal of the rhomboid is equal to what in the trapezoid ? 6. The obtuse angle of the trapezoid is equal to wliat in the rhomboid ? Y. One of tlie acute angles in the rhoml)oid is c(jual to wliat in tlie trapezoid ? 8. Tlie sliortest lines of the trapezoi:! is equal to what in the rhomboid 'i 36 • LESSORS i:n" form. 9. The longest boundary line of the trapezoid is equal to what in the rhomboid ? 10. The sum, of the longer diagonal line, the shortest line and the longest line of the trapezoid is equal to what in the rhomboid ? 11. There is, at least, one pair of opposite lines extending how in each form ? 12. That part of the trapezoid bounded by its longest line, shortest line and the longer diagonal line is equal to what in the rhomboid ? 13. Can you find two pqual angles and two equal lines in each ? What or which are they ? DIFFERENCES. 1. How man}^ different angles are there in each form ? 2. What kind of lines are found in the trapezoid that are not found in the rhomboid ? 3. How many pairs of parallel lines are there in each ? 4. One of the two equal lines of the trapezoid is longer than what and shorter than what in the rhom- boid ? 5. The shorter diagonal of the trapezoid is longer than what in the rhomboid and shorter than what in the same figure ? 6. Which form has the greater area? 7. Write the directions for folding the rhom- boid. Cutting. — Observe and cut two straight line fig- ures in proper pro])ortion, such as doors, windows, floors, walls, blackboards, tops of tables, sides and ends COMPAiaSOX Ob' i:iI().M!5Ull) Wnu TRAPEZIUM. 37 (.A boxes, covers of books, etc., etc. Measure the two dimensions of the objects observed and write the meas- ure on the paper cut. - Dkawixg. — Observe and draw two straight line figures. Write the measures of the two dimem-ions on the drawing. Can you (h'aw a trapezoid, having only one right anoie ? Give eacli pupil two squares. Have some pupil give directions for placing and folding a square into a rhomboid. Directions for folding the square into a trapezium : Place a square so that its right edge will extend in the same direction as the right edge of the desk. Fold the square so that the right point in front will fall upon the left point at the back. Crease. Open the paper. Fold the paper again so that the front edge will lie along the diagonal line. Crease. Fold the pa])er so that the left edge will lie along the same diagonal. Crease. Turn the paper over and place it for comparison with the rhomboid. COMPARISON OF THE RHOMBOID WITH THE TRA- PEZIUM. 1. Find likenesses. 38 LESSONS IK rORM. 2. Find differences. 3. Find trapeziums in room. 4. Find five objects that have surfaces or a surface in the shape of a trapezium. Any four-sided figuro having none of its edges parallel is a trapeziura. 5. What objects did you find having a surface or surfaces in the shape of a rhomboid? QUESTIONS— LIKENESSES. Write answers to the following questions: 1. The two long lines of the trapezium are equal to what in the rhomboid ? 2. The two short lines of the trapezium are equal to what? 3. What is true of one of the diagonals of each 9 4. Into what does the longer diagonal of each figure divide it? 5. The area of half of the trapezium is equal to the area of what ? 6. The area of the rhomboid equals what ? 7. Each form has how ma;ny obtuse angles ? 8. The sum of the right angle and the acute angle of the trapezium equals what in the rhomboid? 9. If the two obtuse angles of each form are equal, and if the sum of the right angle and acute angle of the trapezium equal the sum of the two acute angles of the rhomboid, to what is the sum of the angles of the trapezium equal ? DIFFERENCES. 1. What is true of two adjacent lines of the tra- CLASSIFICATION OF FORMS. 89 peziiim that is not true of any of the adjacent lines of the rhomboid? 2. AVhat is true of the two short Hnes of the tra- pezium that is not true of any of the lines of the rhom- boid ? 3. What kind of an angle is found in the trapezium that is not found in the rhomboid? 4. What is true of the opposite lines of the rhom- boid that is not true of the opposite lines of the trape- zium ? 5. Are all trapeziums kite-shaped ? 6. Must a trapezium have a right angle ? 7. Can a trapezium have two right angles ? 8. Can a trapezium have two adjacent right angles ? 9. Write the direction for folding a square into a trapezium. CLASSIFICATION OF FORMS. Quadri-lateral , Parallelogram , Kectangle , Quadrilateral , rarallelogram , Rectangle, Quadrilateral , Parallelogram , a Quadrilateral, Trapezoid . In wliat respects are all these iigurcs alike? 40 LESSONS IN I'OUM. 2. What general name can be used in speaking of any of the above forms ? 3. In what are a, h and e ahke ? 4. By what common name can you speak of a, h and c? 5. What common name hav^e a and h ? 6. Are all quadrilaterals parallelograms ? Are all parallelograms quadrilaterals ? 7. Are all rectangles parallelograms? Are all parallelograms rectangles ? 8. Are all squares rectangles? Are all rectangles squares ? Why not ? 9. What is true of an oblono: rectano^le that is not true of the square? What is true of a square that is not true of an oblong rectangle ? 10. In what respects are the square, oblong rect- angle and trapezoid alike ? 11. How many pairs of parallel lines has a square, an oblong rectangle, and a trapezoid, respectiveljT' ? 12. A parallelogram is a quadrilateral whose opposite sides extend in the same direction, or are par- allel. 13. Why is a square a parallelogram ? 14. Is a rhomboid a parallelogram ? Why ? 15. Canyon draw a quadrilateral that is not a parallelogram ? 16. Can you draw a quadrilateral that is neither a parallelogram nor a trapezium ? W^hat is its name? 17. Is a trapezoid a quadrilateral ? Is a trapezoid a parallelogram ? Are both pairs of opposite sides in the trapezoid ]mrallel ? 18. Can you find five angles, or differences in coMPArasox of scalene triangle and tiial'ezh m. 41 direction, in the trapezoid? Can you iiiul six in a trapezium? 19. Draw a figure having at least one right angle and four equal sides. What is its name? 20. Draw a figure liavin^: at least one pair of perpendicular lines and four equal sides. What is it? TRAPEZIUM AND SCALENE TPtlANGLE. Give each pupil two squares. Have some pupil give the directions for folding the trapezium. Place the trapezium at one side, and have another pupil give the directions for folding the other square into a trapezium. Observe the trapezium. Do you see an obtuse angle? Do you see the point a,t which the two lines forming the obtuse angle meet? This point is the vertex of the angle. How many angles has the trape- zium ? How many vertices ? Directions for folding tlie trapezium into a scalene triangle : Fold the trapezium so that the vertex of one of the obtuse angles will fall upon the vertex of the other obtuse angle. Crease the paper. Write a comparison of the scalene triangle with the trapezium. 1. Give likenesses. 42 LESSOKS 11^ FORM. 2. Give differences. 3. Find scalene triangles in the room. 4. Write the names of five objects that have sur- faces in the shape of trapeziums. 5. Find five scalene triangles at home or in going to and from school. 6. Write the directions for folding the square into the scalene triangle. Have pupils cut and bring to the class to-morrow one of each of the forms compared. Have them cut trapezoids and trapeziums differing in shape from the ones that have been compared. When the forms are brought in, mix them and have pupils give name and description of the form selected. Example : This quad- rilateral is a trapezoid as it has only two sides parallel. This quadrilateral is a rectangle as its angles are right angles. It is an oblong rectangle, because it is longer than it is wide. Have pupils find forms in the collection, from descriptions given by other pupils. Have just enough of a description given to enable the one who selects to find the form. Let the pupil w^ho gives the descrip- tion name the one who is to find the form. Ex- am>ples : 1. Find a quadrilateral having two pairs of par- allel lines. What is its name ? Find another liaving a different name but having two pairs of parallel lines. What is its name ? 2. Find a quadrilateral having only tAvo lines parallel. What is its name ? COMPATHSON" OF SCALENE TRIAN-QLE AND RHOMBUS. 43 3. Find a parallelogram of which any two of its adjacent lines are equal Wliat is its name? 4. Find a rectangle that is longer than it is wide. What is its name? Remaiik: These exercises will lea(\ to close observation, which is the basis of correct expression and exact reasoning, THE SCALENE TRIANGLE AND RHOMBUS. Give pupils two squares. Have a pu[)il give the direction for folding a square into a scalene triangle. Have another pupil give the direction for folding a square into a trapezium. Direction for folding the trapezium into a rhom- bus : Fold the trapezium so that one of its short lines will coincide with the diagonal. Crease. Fold the paper so that the other short line will fall upon the diagonal. Crease. Write a comparison of the scalene triangle with the rhombus : 1. Give likenesses. 2. Give differences. 3. Find rhombuses in room. 4. What objects did you find having a scalene triangle for a surface ? 44 LESSONS IN FORM. 5, Find live objects that have a surfacG or sur- faces that are rhombuses, and give the names of the objects to-morrow. 6. Write the directions for fokling the rhombus. FINDING FORMS MADE BY FOLDING. Give each pupil a square. Fold the square so that the right-hand point in front will fall upon the left-hand point at the back. Crease. Open the paper. Fold the paper so that the left-hand point in front will fall upon the right-hand point at the back. Crease and open the paper. Fold each point to the center and crease. Open the paper. How many squares, oblong rectangles, trapezoids and triangles can you find in the figure ? Draw each fio:ure. Trv to draw the fificures the same size, and in the same proportion, as those you see in the square. RHOMBUS AND ISOSCELES TRIANGLE. 45 THE RHOMBUS AND THE ISOSCELES TRIANGLE. Give each pupil tAvo scpiares. Have pupils give directions for folding each square into a rhombus. The following are the directions for folding the rhombus into an isosceles triangle : Fold the rhombus so that the vertex of one of the acute angles will fall upon the vertex of the otlier acute angle. Crease the paper. Write a comparison of the rhombus with the isosceles triangle. 1. Find likenesses. 2. Find differences. 3. Find isosceles triangles in room. 4. Write the names of the objects which you found having a surface or surfaces like a rhombus. 5. Find five objects which have a surface in the shape of an isosceles triangle. C). Write the direction for folding the isosceles triangle. 46 LESSONS IN" FORM. CLASSIFICATION OF FORMS. Suggestion: Omit work on classification of forms uniil third year. Quadrilateral , Trapezoid , Quadrilateral , Parallelosram, Rhomboid. uadrilatcral, Parallelogram, Rhomboid, Rhombus. Quadrilateral , Paral 1 e 1 ogram , Rectangle. Quadrilateral, Parallelogram, Rectangle, Square . Write such a description of each of the above forms that the person reading it can select the form described, l^.Iake the descriptions as short as you can. Example : It is a quadrilateral with no sides parallel. What is the name of the figure described ? 1. What is wrong in the following description of a square ? A square is a figure bounded by four equal lines. Of how many of the forms is the description true? 2. What is wrong? A square is a figure bounded by four lines and having four right angles. CLASSIFICATIONS' OF FORMS. 47 Of how many figures is the description true? 3. What is wrong? A rhombus is an oljlique-angled parallelogram. How many figures does this describe ? 4. What is wrong ? A rhombus is a figure having four equal lines. How many figures are included in this description? 5. A quadrilateral is a plane figure bounded by four straight lines. How many of the above figures are quadrilaterals? 6. A quadrilateral whose opposite lines are paral- lel is a parallelogram. How many of the figures are parallelograms? 7. A rhomboid is a parallelogram whose angles are oblique angles. How many of the figures are rhomboids? Is a rhombus a rhomboid ? Are all rhomboids rhombuses ? 8. A rectangle is a parallelogram whose angles are right angles. How many of the figures are rectangles? Is a square a rectangle ? Are all squares rectangles ? Are all rectangles squares ? Call attention to the different figures cut out of paper, drawn on the blackboard, or found in the room, and have pupils tell what they are and define them. Ex- amjjle: The blackboard is a rectangular parallelogi-am because its opposite lines are parallel and its angles are ri.. # w >!V' « I ^fJ^»: '■■,; % »;^^:Y»±\»xh^'i:<**\v^ f . .♦r^-;?*;;^' •, ♦ ,«>.-.'n-,^vv.». Mt * ♦ >\\^ ^.^^|^^^ >\^\\«\\a: