not perform any exercise with accuracy and firmness, she should not proceed. It is only the full conviction of the child's being perfect master of the preceding step, that should determine the teacher to lead him on to the next. The same exercises which have been given with material objects, may afterwards be given without them. Thus, the mother seeing that the child, with the aid of real objects, has so far advanced as to know: that 8 times 1 are 4 times 2, and 4 times 2, 8 times 1, may propose, without the aid of them, questions similar to those which follow: Moth. S times 2 and the half of 2, how many times 1? Child. 3 times 2 and the half of 2 are 7 times 1. Moth. Why? • Child. S times 2 are 6 times 1—the half of 2 is 1, 6 times 1 and once 1 are 7 times 1. Moth. 5 times 1, how many times 2? Child. 5 times 1 are twice 2 and the half of 2. Moth. Why? F. Child. 4 times 1 are twice 2-; once 1 is the half of 2; 4 times 1 and once 1 are 5 times 1. Similar questions are applicable to all ordinary objects of life; for instance: Moth. Two sixpences make 1 shilling, how many shillings do 7 sixpences make? Child. 7 sixpences make 3 shillings aud the half of a shilling. Moth. Why? Child. 2 sixpences make 1 shilling; 4 sixpences make 2 shillings; 6 sixpences make 3 shillings; 1 sixpence is the half of a shilling; 7 sixpences are 3 times 2, and the half of 2 sixpences; 7 sixpences, therefore, are 3 shillings, and the half of a shilling. Moth. 2 pair of shoes and half a pair, how many single shoes? Child. 2 pair of shoes and half a pair are 5 single shoes. Moth. Why? Child. 1 pair of shoes consists of 2 single shoes; twice 2 single shoes are 4 shoes; the half of a pair is 1 single shoe; 4 shoes and 1 shoe make 5 shoes. After several questions of this nature, the mother may proceed to the combined unity of 3 and of 4, continuing the use of cubes, or of other objects. Moth. (Placing 3 cubes in a straight line at equal distances.) How many times 1 are here? Child. 3 times 1. Moth. (Lifts up 1 of the 3 cubes, shews it to them, and places it at some distance from the two others.) □ □□ which lie close together, and placing it next to the single one.) □ □ □ Twice 1 are twice the third part of 3. Child. Twice 1 are twice the third part of 3. Moth. (Moving the third cube nearer to the two first, so that all 3 are lying in the same line and at equal distance.) □ □□ 3 times 1 are (moving all 3 close together, so as to form a rectangle I I I I) once 3. Child. 3 times 1 are once 3. Moth. (Placing a fourth cube below the first of the 3 former, so that with the fourth a new row begins, as represented here.) LMJ □ 4 times 1 are once 3 and the third part of 3. Child. 4 times 1 are once 3 and the third part of 3. Moth. 5 times 1 are once 3, and twice the third part of 3. TTTj m Child. Repeat. Moth. 6 times 1 are twice 3. Child. 6 times 1 are twice 3. Moth. 7 times 1 are twice 3, and the third of 3. TtTl □ Child. 7 times 1 are twice 3 and the third of 3. Moth. 8 limes 1 are twice 3, and twice the third part of 3. 1 I I I IJJ Child. 8 times 1, &c. Moth. 9 times 1 are 3 times 3. fTTl FTP Child. 9 times 1 are 3 times 3. The various exercises which the Mother has givtin to the children relative to the combined unity of 2, may be repeated with the combined unity of 3, the Mother continuing to form rows of 3 cubes one after another, till she has placed 10 times 3 cubes before them. As soon as the children have advanced so far as to answer, and to prove without the aid of cubes or other objects, to the mother's question: 26 times 1, how many times 3? 26 times 1 are 8 times 3, and twice the third part of 3; and inversely, to the question: 7 times 3 and the third part of 3, how many times 1? 7 times 3 and the third part of three are 22 times 1; she then proceeds to the combined unity of 4, 5, &c. in the same manner. |