A. 36 = .... 12s B. 44 = .... 11s C. 50 = .... 25s 60 = .... 12s 88 = .... 11s 125 = .... 25s 24 = .... 12s 77 = .... 11s 75 = .... 25s 48 = .... 12s 55 = .... 11s 200 = .... 25s 144 = .... 12s 99 = .... 11s 250 = .... 25s 108 = .... 12s 110 = .... 11s 175 = .... 25s 72 = .... 12s 33 = .... 11s 225 = .... 25s 96 = .... 12s 66 = .... 11s 150 = .... 25s 84 = .... 12s 22 = .... 11s 100 = .... 25s Find the quotients and remainders. If you need to use paper and pencil to find them, you may. But find as many as you can without pencil and paper. Do Row A first. Then do Row B. Then Row C, etc. __ __ __ __ __ __ Row A. 11|45 12|45 25|45 15|45 21|45 22|45 __ __ __ __ __ __ Row B. 25|55 11|55 12|55 15|55 22|55 30|55 __ __ __ __ __ __ Row C. 12|60 25|60 15|60 11|60 30|60 21|60 __ __ __ __ __ __ Row D. 12|75 11|75 15|75 25|75 30|75 35|75 ___ ___ ___ ___ ___ ___ Row E. 11|100 12|100 25|100 15|100 30|100 22|100 __ __ __ __ __ __ Row F. 11|96 12|96 25|96 15|96 30|96 22|96 ___ ___ ___ ___ ___ ___ Row G. 25|105 11|105 15|105 12|105 22|105 35|105 __ __ __ __ __ __ Row H. 12|64 15|64 25|64 11|64 22|64 21|64 __ __ __ __ __ __ Row I. 11|80 12|80 15|80 25|80 35|80 21|80 ___ ___ ___ ___ ___ ___ Row J. 25|200 30|200 75|200 63|200 65|200 66|200 Do this section again. Do all the first column first. Then do the second column, then the third, and so on. Consider, from the same point of view, exercises like (3 × 4) + 2, (7 × 6) + 5, (9 × 4) + 6, given as a preparation for written multiplication. The work of 48 68 47 3 7 9 -- -- -- and the like is facilitated if the pupil has easy control of the process of getting a product, and keeping it in mind while he adds a one-place number to it. The practice with (3 × 4) + 2 and the like is also good practice intrinsically. So some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication. In some cases the bonds are purely propædeutic or are formed _only_ for later reconstruction. They then differ little from 'crutches.' The typical crutch forms a habit which has actually to be broken, whereas the purely propædeutic bond forms a habit which is left to rust out from disuse.