The Elementary Rules of Arithmetic. 'T is a familiar experience of business men that when they take into their offices a boy fresh from school he can rarely be depended on to perform with accuracy the simpler arithmetical operations - those which involve nothing but the four fundamental rules. And as these are the only operations to which a boy is at all likely to be set, one cannot wonder that there should be a feeling in many quarters that there is too great a difference between proficiency in arithmetic when judged from the school examiners' standpoint, and when tested by the requirements of practical business. The fact is that boys are not sufficiently drilled in the elementary rules; their later years at school are spent in the study of questions which, although valuable enough as mental exercises, do not call for heavy additions, subtractions, multiplications, or divisions. Nor is it always easy to provide examples in this class of work; a sub_ traction sum takes at least twice as many figures to state as it does to work out, an addition sum much more than twice as many; and should a teacher undertake to set examples of his own, he has all the trouble of checking the results. It is with a view of getting over this difficulty that the following table is submitted: it will be found to furnish the means of setting an endless number of examples in the elementary rules, which examples possess the not inconsiderable advantage of being self-checking; and it is suggested that even after a pupil has passed the more advanced work, frequent opportunities should be taken of exercising him in the way now to be described. It is only by constant practice in adding, subtracting, multiplying, and dividing, that proficiency in these processes can be attained. of the two products is to be placed below the greater, no line being drawn, and subtracted therefrom, the remainder to be deducted from the minuend, and so on successively until, after twenty-three subtractions, two lines the same are reached, each line being of course 12345, the original common factor; that the work is then to be added up; and that finally the sum is to be divided by 12345. The resulting quotient should be 196417, viz. the twenty-seventh term of the series diminished by unity, and its correctness will be a sufficient test of the accuracy of all the intermediate steps, the chance of so highly improbable an occurrence as a balance of errors being disregarded. The possibility of obtaining this check depends upon the fact that in a series constructed in the way described, the sum of n terms is always equal to 1 less than the (n + 2)th term. The scheme of the work would be as follows: (25th term) 75025 × 12345 = 926183625 (24th term) 46368 × 12345 = 572412960 In addition, if numbers contain two figures, add the tens first and then the units, as:-32+43-70+5=75; 44X57=90+11=101. If the numbers contain three figures, add the hundreds first, then the tens, and last the uuits, as:-432+357=700+80+9=789; 644+296= 800+130+10=940. In subtraction, 46-13-3 tens+3=33. 946-334-6 hundred+ ten+2=612. Problems of any length, when the figures in the minuend are larger than the corresponding ones in the subtrahend, can be performed easily and read with ease. Problems in which one or more figures in the minuend are smaller than the corresponding ones in the subtrahend will tax the child a little more. For instance, 64-39 solve thus: 39 and 20=59 and 20+5=25. 7629 solve thus: 29 and 40=69, and 40 and 7=47. Take three figures in numbers, as: In the series set forth above it will be observed that the first and second terms are unity, and that each succeeding term is obtained by adding together the two which precede it. Thus, the third term, 2, is the sum of the first and second terms; the thirteenth term, 233, is the sum of the eleventh and twelfth, and so on. The teacher (who will retain the table in his own hands) will select two consecutive lines - say the twenty-fourth and twenty-fifth and announce them to the pupil, with the instruction that they are each to be multiplied by some one number-say 12345; that the lesser DRAWING In Grammar Schools. By NATHANIEL L. BERRY, Supt. of Drawing, Newton, Mass, EVELOPMENT or Pattern-making, to be of the greatest educational value, should be studied, not directly from models* or objects, but from sketches or working drawings. For, in this way, the pupil must see in the drawing, the image of the thing itself. He must think the arrangement, and relation of its faces, and calculate the extent of its surface. Success in this work, then, will depend, to a great extent, on what has been attained in Working Drawing. The cube, and simple modifications, are adapted to this study in the fourth grade; and in the fifth grade, work on this line may be continued, using as models the square prism, and square plinth, and as modifications of these, various millinery and bon-bon boxes, etc. All the above will prove simple in drawing and construction having plane faces and straight edges, and these either parallel of at right angles. Place on the blackboard, two views of the cube. Ask the children for their opinions as to the number of pieces which it would be necessary to cut out, were we to make the cube of paper or of card-board. Accept the suggestions received, and if Johnny says six pieces, cut out six, and let the little people see the difficulties involved in fastening so many pieces into position. When some one has discovered that several faces may be joined continuously, tell pupils to think of a strip of paper of just the right shape to make the four vertical faces of the cube. Have this represented on the blackboard, and its division into faces shown by light lines. "What is necessary to complete the surface? Who can add the top and bottom, so that all faces may be cut in one piece? Make the best arrangement possible." Now state the size of one face, and call for the dimensions of the whole; and when the measurements have been correctly given, have the development drawn on paper, using the rule. Let that boy who is always anxious to do something, cut out the result of his work, and bend it into the form of the cube. Show to the class, and have them observe which edges meet. Consider the different methods of joining these and discuss the value of having "laps" and their most convenient arrangement. Have these added to the drawing, and in completing the pattern, finish in full all lines which are to be cut, and leave light all lines to be folded. * Preliminary work on this line should have been done by pupils in primary grades; tracing, cutting, and arranging in their proper rela tions the faces of the different solids. Sheet VIII. Topics for Nature Study. Pupil's Name. "Draw two views of this card case from my sketch. "How large shall we make it? When the views are completed we will draw its pattern, and if this is well done, we will construct it." Date Trees. Shrubs and vines. Points to notice. 1. Bark smooth or rough, thick or thin, whole or broken. 2. Branches, how arranged, opposite or alternate. 3. Twigs, old and new growth compared. 4. Buds, how protected for winter. Select large scaly buds, as of balm-of-gilead, horse-chestnut or plane trees. Collect and examine. See the varnish on the outside, and the woolly substance enveloping the leaves like a blanket. Low Plants. Three kinds. 1. Annual, dies each year, reproduced from seed. 2. Biennial, dies down first year to ground, comes up next, blossoms and dies after ripening seed. 3. Perennial, comes up year after year from roots. Find examples of each. Animals. Fur-bearing,; squirrels, foxes, raccoons, field mice, woodchucks, beaver, etc. Show adaptability to season, and food. Resident Birds. What do you find? These vary according to locality. We find them all the year. Domestic fowl, turkeys, chickens, geese, etc. School. Pupils Name. A Suggestion. Write on the blackboard the following: 12 X $0.65; 23 X $1.34; 6 × $2.07; 14 X $0.98. Tell the class to make out a bill containing a group of these numbers. I have found this exercise a great help in many ways. It saves time for the teacher, gives the pupils busy work, requires several arithmetical operations, teaches spelling, and cultivates the imagination. A. M. B. into a very small space it would change it into a liquid. It is also odorless and is much lighter than air for if we should lift the cover off the bottle the oxygen would rise up, and the particles separate. We all know that (O.) is transparent or the air would not be clear, as it is; and the amount is or 20%. Oxygen is made from heating the mixture of Chlorate Potash and Black Oxide Mawganese; it is found in the air. Science Teaching in Milton. By C. H. MORSS, Supt. of Schools, Milton, Mass. N all grades the first step in science work should be observation. These observations should be the result of the child's unaided effort. When the teacher performs an experiment before the class, and calls on pupils to tell what they see, the keen and quick ones get all the benefit. The lazy and mentally slow pupils simply absorb knowledge without exerting their own powers. Many a science lesson fails of attaining its object, by the teacher asking questions to draw the children out. One of the most difficult things for a teacher to learn, and it must be the first if he would become a successful science teacher, is to keep from talking. After all have had a chance to find out all they can, by their own unaided efforts, from the specimen or experiment under consideration, a discussion may take place, during which the teacher may introduce the information considered necessary, to accompany the lesson. But it cannot be emphasized too strongly that the first step in observation on the part of the child, is silence on the part of the teacher. The simple experiments under the head of Elementary Chemistry can scarcely be dignified by the name of science. They aim to introduce the child to a few phenomena that he is familiar with superficially from infancy, and include a study of fire, air and water. Under the head of fire various experiments to prove the nature of flame are made. The study of air introduces us to the elements of oxygen and nitrogen; water furnishes an additional one, hydrogen. The experiments include such as will illustrate the properties of oxygen, nitrogen, carbon and hydrogen. These open the way to much information. The subject of ventilation necessarily enters, charcoal and its manufacture, the foundation of our coal deposits, the chemical fire extinguisher, etc. This work is designed for the fifth grade with a repitition and enlargement in the seventh. As all grades entered upon this work at the same time, the seventh grades used the same experiments as the fifth. The following extracts are from the note-book of a seventh grade pupil : Oxygen. In the mouth of a test-tube we put a cork with a hole in it. We put in the hole in the cork a bent tubing. Then we put the water in a dish and put the bent part of the tubing in the water and put the mouth of the (test)-tubing in the bottle. In the test-tube we had some Black Oxide Mawganese and Chlorate Potash. We lighted an alcohol lamp and kept moving the flame around at the end of the test-tube. The flame was mixing the stuff in the bottle and causing a gas to go up the tubes into the bottle. When the bottles were full of smoke, we put a piece of glass over the top. Experiment I. In the bottle with the gas in it we dropped a piece of burning charcoal. We burnt this by holding it one side of the flame, while on the other side, we held a blow-pipe, the end of which was in the blue part of the flame. When we dropped it into the gas (the gas burnt all up) and charcoal glowed very brightly. Experiment II. In a combustion spoon we put some sulphur and took the blow-pipe and put it in the blue part of the flame. The sulphur burned with a very blue flame, but when we put it in the bottle of gas it burned with a very bright flame. This proved that the gas in the bottle was oxygen. Oxygen is colorless and the state of it is gas; if we should press it Coal and the Coal Mines. Greene. Houghton, Mifflin & Co. Natural History Object Lessons. Ricks. D. C. Heath & Co. Our Common Birds. Grant. Scribner's Sons. Oral Training Lessons in Natural Science. Barnes & Co. Barnard. A. S. Realm of Nature. Mill. Published by Chas. Scribner's Sons. School and Field Botany. Gray. American Book Company. IYSIC V. Experiment 20.- Plunge your finger into water; press your finger on a piece of gold-leaf; press your hand on the top of your desk for a few seconds; write on paper with a lead pencil. Obs.-Water adhered to the finger and wet it; the gold-leaf adhered to the finger; the hand adhered slightly to the desk; the lead from the pencil adhered to the paper. Inf.- There must bave been an attractive force which caused the substances to adhere. In our last two lessons we considered the attractive force between molecules of the same kind of matter. In this lesson we are to study an attractive force that exists between molecules of different kinds of matter - a sticking together force. Definition 16.- The attraction between unlike molecules is called Adhesion. Experiment 23.- Put two bricks in water. Mix half a tumbler of plaster of Paris with water till it has the consistency of thick cream. Remove the bricks from the water, spread the plaster over the face of one brick, and press the other firmly over it, and let it remain till it "sets." Inf.- The adhesive force holds the plaster to the bricks, and cohesion holds together the molecules of plaster. Try the same experiment with cement such as is used by masons. The cement should be mixed with twice its volume of fine, sharp sand. Try the same experiment with mortar,- some boy will tell or find out how mortar is made. Call upon the pupils to illustrate the uses of cement, mortar, putty, glue, etc. Experiment 24.- Press two shingles together-sawed ones - and take them apart; plunge the shingles into water, press them together, and then pull them apart. Obs.-? Inf.- The liquid filled up the spaces where the shingles did not touch, and made adhesion possible. What may be done to make two rough surfaces adhere? What do we put between paper to make it adhere? Between bricks? Between the pieces of a broken dish? Experiment 25.- Make a filter of strong white blotting paper. Prepare an 8-inch square of blotting-paper; fold this square through the middle; and again, at right angles to the first fold; and then diagonally from the folded corner. Trim the edges to convert the square into a circle; open the paper so as to make a hollow cone having three thicknesses on one side and one on the others. Place this cone in a wide-mouthed bottle, moisten it with water, and bend the edges down so as to keep it in place. Color a half tumbler of water with any vegetable color; stir into the water a third of a teaspoonful of bone-black; then pour the mixture into the filter. Obs.-? Inf.- The coloring matter adhered to the bone black more strongly than to the water. The color may be taken from cider vinegar in this way. Explain "solution" to the pupils. Experiment 26.-Fill a saucer one quarter full with water, and color it with a little ink cochineal is better. Hold a strip of glass in a vertical position in the saucer, resting on the bottom. Observe the liquid next the glass. Inf.-The adhesion of the liquid to the glass is stronger than the cohesion of the liquid, hence the liquid next the glass rises above the surrounding level. Experiment 27.- With a piece of quartz cut two strips of glass about 2 by 5 inches. Press the two strips between the thumb and fingers of one hand and hold them vertically in the saucer of colored water. Obs. The liquid passes up between the plates of glass in the small meandering spaces, where the two surfaces do not touch, in a beautiful manner. Experiment 28.- Dry the strips of glass and press them as before, with a bit of business card between two of their edges, so that the strips will come together at two edges, but the other two edges will be kept apart by the bit of card, and hold them vertically in the colored liquid. Obs.-The liquid instantly rises in the tubes, the finer the bore the higher the liquid rises. Experiment 30.- Hold the end of a card of matches in the colored water, brimstone up. Definition. The adhesive force which causes liquids to move in tubes and hairlike spaces, and penetrate porous solids, is called Capillary Attraction. PRACTICAL QUESTIONS.- Why is it so difficult to lift a board from the surface of water? If you should spill some ink on the edge of your book ought you to press the leaves together? Why must the wood work of a kitchen be washed with soap before it be painted? Why must an alchohol lamp have a cap? Why will the whole towel be wet if one corner be left in a basin of water? Why is a new pen more likely to blot than an old one? Why do figures drawn with the finger upon the window pane, become visible if we breathe upon them? What are some of the disadvantages of adhesion ? |