3. Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, and begin to toll at the same instant. When will they next toll together?
4. A, B, C, and D start together, and travel the same way around an island which is 600 mi. in circuit. A goes 20 mi. per day, B 30, C 25, and D 40. How long must their journeying continue, in order that they may all come together again?
5. The periods of three planets which move uniformly in circular orbits round the sun, are respectively 200, 250, and 300 da. Supposing their positions relatively to each other and the sun to be given at any moment, determine how many da. must elapse before they again have exactly the same relative positions.
(6) Rare and unimportant words.—The bonds between rare or unimportant words and their meanings should not be formed for the mere sake of verbal variety in the problems of the textbook. A pupil should not be expected to solve a problem that he cannot read. He should not be expected in grades 2 and 3, or even in grade 4, to read words that he has rarely or never seen before. He should not be given elaborate drill in reading during the time devoted to the treatment of quantitative facts and relations.
All this is so obvious that it may seem needless to relate. It is not. With many textbooks it is now necessary to give definite drill in reading the words in the printed problems intended for grades 2, 3, and 4, or to replace them by oral statements, or to leave the pupils in confusion concerning what the problems are that they are to solve. Many good teachers make a regular reading-lesson out of every page of problems before having them solved. There should be no such necessity.
To define rare and unimportant concretely, I will say that for pupils up to the middle of grade 3, such words as the following are rare and unimportant (though each of them occurs in the very first fifty pages of some well-known beginner's book in arithmetic).
| absentees account Adele admitted Agnes agreed Albany Allen allowed alternate Andrew Arkansas arrived assembly automobile baking powder balance barley beggar Bertie Bessie bin Boston bouquet bronze buckwheat Byron camphor Carl Carrie Cecil Charlotte charity Chicago cinnamon Clara clothespins collect comma committee concert confectioner cranberries crane currants dairyman Daniel David dealer debt delivered Denver | department deposited dictation discharged discover discovery dish-water drug due Edgar Eddie Edwin election electric Ella Emily enrolled entertainment envelope Esther Ethel exceeds explanation expression generally gentlemen Gilbert Grace grading Graham grammar Harold hatchet Heralds hesitation Horace Mann impossible income indicated inmost inserts installments instantly insurance Iowa Jack Jennie Johnny Joseph journey Julia Katherine | lettuce-plant library Lottie Lula margin Martha Matthew Maud meadow mentally mercury mineral Missouri molasses Morton movements muslin Nellie nieces Oakland observing obtained offered office onions opposite original package packet palm Patrick Paul payments peep Peter perch phaeton photograph piano pigeons Pilgrims preserving proprietor purchased Rachel Ralph rapidity rather readily receipts register remanded | respectively Robert Roger Ruth rye Samuel San Francisco seldom sheared shingles skyrockets sloop solve speckled sponges sprout stack Stephen strap successfully suggested sunny supply Susan Susie's syllable talcum term test thermometer Thomas torpedoes trader transaction treasury tricycle tube two-seated united usually vacant various vase velocipede votes walnuts Walter Washington watched whistle woodland worsted |
(7) Misleading facts and procedures.—Bonds should not be formed between articles of commerce and grossly inaccurate prices therefor, between events and grossly improbable consequences, or causes or accompaniments thereof, nor between things, qualities, and events which have no important connections one with another in the real world. In general, things should not be put together in the pupil's mind that do not belong together.
If the reader doubts the need of this warning let him examine problems 1 to 5, all from reputable books that are in common use, or have been within a few years, and consider how addition, subtraction, and the habits belonging with each are confused by exercise 6.