The purpose of this application is to give a model of fraction multiplication and to at least hint at the meaing of multiplication in general. The setup is the same as the simple fraction wheel model except for the addition of a multiplication button which is initially disabled. After selecting an initial fraction pulley to connect to the paddle wheel, the multiplication button changes color indicating that it is enabled. After clicking it the student is offered the chance to select a second fraction pulley to connect to the first. The result is a double compound pulley. In the example pictured above, a pulley representing 1/4 is connected to a pulley representing 1/6. The initial pulley reduces each turn of the paddle wheel by 4 and the second reduces each of the quarter-turns by six for a net reduction of 24. Thus, the compounded system represents the fraction 1/4 × 1/6 = 1/24. As for the simple fraction model, this model ties in nicely with operation of mechanical devices like the pair of gearng systems in a typical 10-speed bicycle. As shown in the diagrams at the bottom, it also ties in nicely with mathematical concept of “composition”.  Fraction Wheel Multiplication

 

 

 

 

 

 

 

 

 

 

 

 

 

Multiplication as “composition” We can view a pulley as a kind of “input-output” machine. It takes “inputs” in the form of “turns” and produces “outputs” also in the form of “turns”. The first diagram below shows the behavior of a pulley represent ½. It takes a full turn as input and produces a ½ turn as output. The next diagram shows the result of connecting two such machines. The first takes a whole and produces a ½ turn which is input to the next which takes the ½ turn and produces a quarter turn. The result is a machine representing ¼ = ½ x ½. More generally if we have any two “input/output” machines where the output of the first is acceptable as input to the second we can connect them to produce a new machine. The operation of connecting input/output machines in this way is called “composition.” Multiplication is a special case of composition! As indicated in the final diagram, addition is also a special case of composition.