SUBJECT MATTER AND METHODS 153 



many times are there in 3 ? Thus there is developed the 

 rule for changing an integer to a fractional form. 



Fractions are next reduced to a common denominator, as 

 J and J. This is a repetition of the process of finding 



higher terms for a fraction, and is devel- 

 Common . .. . . 



Denominator oped with lines and circles m a similar man- 



ner. This includes fractions whose de- 

 nominators are prime to each other, as | and ^. The drills 

 in factoring immediately suggest 6. By means of circles 

 divided into sixths the common denominator is illustrated. 

 Fractions having different denominators are compared 

 as a preparation for the operations for which the pupils are 



about ready. A pupil is given ten circles, 

 Comparison 



of Fractions cut mto halves - From these i l> f are 



taken and fitted into wholes, f are taken 

 up and f- are removed from them. After many exercises 

 of this sort, fourths are made from a portion of the halves, 

 and twelfths from some of the fourths. Fourths and 

 twelfths are added, subtracted; combinations are also made 

 with the halves. Five circles are cut into thirds; some of 

 these are cut into sixths; some of these into twelfths. 

 Thirds are added and subtracted; sixths; twelfths; combi- 

 nations are made, and the pupils .work out the results. 

 Mental and oral drills are given. Addition and subtraction 

 of fractions now follow with comparatively little hesitation 

 or confusion. 



Multiplication of a fraction by an integer is first pictured 

 with lines. One line is divided into eighths, and one part 



is taken. A second line, of equal length, 

 Multiplication {s drawn t ^ ^ fi it . g 



of Fractions 



divided into eighths, three of which are 



taken. The pupils readily see that the second line that is 



