twos? How many threes in nine? Three threes, then? How mauy threes in twelve? Four threes, then? Three fours? Threes in fifteen? Five threes, then? Three fives? Threes in eighteen? Six threes, then? Three sixes? Threes in twenty-one? Seven threes, then? Three sevens? twenty-four? Eight threes, then? Three eights? twenty-seven? Nine threes, then? Three nines? thirty? Ten threes, then? Three three? Eleven threes, then? Three elevens? Threes in thirty-six? Twelve threes, then? Three twelves? Threes in Threes in Threes in tens? Threes in thirty Four 3. How many fours in eight? Two fours, then? twos? Fours in twelve? Three fours, then? Four threes? Fours in sixteen? Four fours, then? Fours in twenty? Five fours, then? Four fives? Fours in twenty-four? Six fours, then? Four sixes? Fours in twenty-eight? Seven fours, then? Four sevens? Fours in thirty-two? Eight fours, then? Four eights? Fours in thirty-six? Nine fours, then? Four nines? Fours in forty? Ten fours, then? Four tens? Fours in forty-four? Eleven fours, then? Four elevens? Fours in forty-eight? Twelve fours, then? Four twelves? 4. How many fives in ten? Two fives, then? Five twos? Fives in fifteen? Three fives, then? Five threes? Fives in twenty? Four fives, then? Five fours? Fives in twentyfive? Five fives, then? Fives in thirty? Six fives, then? Five sixes? Fives in thirty-five? Seven fives, then? Five sevens? Fives in forty? Eight fives, then? Eight fives, then? Five eights? Fives in forty-five? Nine fives, then? Five nines? Fives in fifty? Ten fives, then? Five tens? Fives in fifty-five? Eleven fives, then? Five elevens? Fives in sixty? Twelve fives, then? Five twelves? 5. Two sixes are how many? Three sixes; another six? Four sixes; another six? Five sixes; another six? Six sixes; another six? Seven sixes; another six? Eight sixes; another six? Nine sixes; another six? Ten sixes; another six? Eleven sixes; another six? Twelve sixes; another six? 6. Two sevens, how many? Three sevens; another seven? Four sevens; another seven? Five sevens; another seven? Six sevens; another seven? Eight sevens; another seven? Nine sevens; another seven? Ten sevens; another seven ? Eleven sevens; another seven? Twelve sevens; another seven? 7. Two eights, how many? Four eights; another eight? Three eights; another eight? Six eights; another eight? Seven eights; another eight? 8. Two nines, how many? Three nines; another nine? Four nines; another nine? Five nines; another nine? Six nines; another nine? Seven nines; another nine? Eight nines; another nine? Nine nines; another nine? Ten nines; another nine? Eleven nines; another nine? Twelve nines; another nine? 9. Two tens? tens? Seven tens? Twelve tens? Three tens? Four tens? Five tens? Six 10. Two elevens? Three elevens? Four elevens? Five elevens? Six elevens? Seven elevens? Eight elevens? Nine elevens? Ten elevens? Eleven elevens? Twelve elevens ? 11. Two twelves? Three twelves; another twelve? Four twelves; another twelve? Five twelves; another twelve? Six twelves; another twelve? Seven twelves; another twelve? Eight twelves; another twelve? Nine twelves; another twelve? Ten twelves; another twelve? Eleven twelves; another twelve ? Twelve twelves; another twelve? [This section will require more frequent repetition than the others. Omit, in reviewing, the words" another six," "another seven," &c.] SECTION XV.- Explanatory. NUMBERS are not always expressed in words. What are called figures, are frequently used for that purpose. These figures are only nine in number, as may be seen below. They should be well studied, so as to be readily known, wherever they may appear. 1, stands for one. 6, stands for six. 5, (C "four. "five. These are all the figures that stand for numbers. But how, then, do we manage, when we wish to use a number larger than nine? The same figures are used, only they are put in a dif ferent place. Every figure becomes ten-fold greater by being removed one place to the left. Thus, the figure 1 stands for one, when alone, or at the right hand of other figures; for ten, when placed the second from the right; and for a hundred, or ten times ten, when it stands the third from the right. Thus, the three figures below, 111 stand for one hundred and eleven (or one-teen): the first figure on the left standing for one hundred, the second for one ten, the third for a single one. It is the same with all the other figures. Thus, 444 stands for four hundred and forty- (or four tens) four; and 666 stands for six hundred and sixty-six. These places for the figures are called ranks, or orders, and are reckoned from the right. Every figure placed in the first order, stands for as many ones, or units, as it represents; when placed in the second order, for as many tens, or teen; and for as many hundreds when it stands in the third order. A figure placed in the next order to the left (the fourth order) would stand for so many thousands, each of which is equal to ten hundred. Thus, in the following number, 4536 the 4 stands for so many thousands, the 5 for hundreds, the 3 for ty, or tens, the 6 for units, or ones. The whole number should be read thus: four thousand, five hundred, and thirtysix. This is very much like the arrangement of the Frame. [Exemplify on the frame.] A single bead on any of the upper ten wires stands for one. Each row of beads stands for ten, any one of which is called teen, if units be added to it. Each bead on the lower row stands for 100, and the whole row, of course, for ten hundred, which is a thousand. Each of these numbers increases tenfold, just as the figures do from the place in which they stand. [Let the following figures now be written vertically on the slate or blackboard, and named repeatedly by the class till they are familiar. 7 9 6 3 8 5 1 4 2.] But it is frequently necessary to write a number in which one or more of the orders is wanting: for example, two thousand and fifty-four. Here we must have four places, or orders, to represent thousand, and yet we have only three figures, viz., 2 for two thousand, 5 for fifty, and 4 for four. In all such cases, we use this character, 0, which is called cipher or nothing, because it stands for nothing. Our number, two thousand and fifty-four, becomes 2054. There are no hundreds, you perceive, and the O fills that place. Had it not been put there, the 2 would have stood in the third order, and thus represented 2 hundred instead of two thousand. The cipher, accordingly, is sometimes called figure of place, because it is only used to show the place of the other figures. Take notice, however, that a cipher is useless unless it occupies the place of units, or stands between a significant figure. and the place of units. Thus, if we wish to write three hundred and seventy-four, the cipher is not wanted, although there are only three figures, because each figure can stand in its proper order, 374, without any cipher. But a cipher must be used in expressing two hundred and five, since we have only two figures, while the hundred is in the third order. Accordingly the number is written 205. For a similar reason, the number three thousand and forty-five must be written with a cipher, 3045. Write the following numbers in figures on the slate or blackboard, and then read them over without the book: 1. Four hundred and thirty-five. 2. Two thousand, six hundred, and four. 3. Three thousand, and forty-two. 4. Six thousand, three hundred, and seventy-six. 5. Four thousand, four hundred, and forty-four. 6. Two hundred and three. 7. One thousand and twelve. Sometimes one thousand is considered as ten hundred, as in the following: 8. Fifteen hundred and sixty. 9. Eighteen hundred and two. [Specimen of questions to the class on the above numbers, when they have changed them from words to figures on the blackboard or slate.] For No. 1. What does the 4 stand for? [Point to the figures as they are spoken.] Why hundreds? The 5? Why units? The 3? Why ty, or tens? No. 2. What is the value of the 2? The 6? The 4? What rank does the cipher occupy? Why? Ans. Because there are no No. 3. What is the value of the 3? The 4? The 2? What rank does the cipher occupy here? Why? No. 4. Why is there no cipher in this number? No. 5. What is the value of the first figure on the right? Why? The fourth from the right? Why thousands? The third? Why? How many times is the third greater than the second? The third than the first? The fourth than. the second? The fourth than the third? The fourth than the first? How many times is the first contained in the second? In the fourth? In the third? How many times is the second contained in the fourth? In the third? How many times is the third contained in the fourth? No. 6. What is the use of the cipher here? Why is there none in the place of thousands? Ans. Because the cipher is useless, unless it stands, &c. [Show this principle by an example on the blackboard.] Does the cipher stand for any number? What would this number be, if the cipher were omitted? If another cipher were placed beside the first, thus: [place one] what effect would it produce on the 2? Ans. Its value would be fold. What effect would be produced on the 3? If a cipher were placed after the 3 [place one], what effect would be produced on the number? Would both the 2 and 3 be increased tenfold? No. 7.-- If another cipher were introduced between the two Is, what effect would be produced, that is, what figures would change their value? Add a cipher after the 2, and then say What change is thus produced, on each figure severally, and on the whole number? No. 8.- What effect would a cipher produce on this number, if placed to the left of the 1? To the right of the 13 Between the 5 and 6? After the 6? No. 9. What effect would a cipher produce on this num ber, if placed to the left of the 1? On its right? Beside the other cipher? To the right of the 2? How many are 10 times 26? How many tens in 2050? How many hundreds in 2500? How many tens in 3700? How many hundreds in 2000? Tens in 540? Tens in 270 ? [While proceeding with the following sections, the class should still be exercised in notation and numeration, as above, varied till the subject is perfectly familiar.] |