The Fraction Wheel Multiplication Machine
Note: This pure HTML file is completely self-contained. You do not need access to the Internet in order to use it. To run it all you need is a reasonably up to date browser like Chrome or Firefox.
This is the fourth in a series of interactive applications that use pulleys to create animated models of fractions. The idea is analogous to the way bicycle gears work. A pulley of a given size is attached to a fraction wheel in the same way that the gears on the back of bicycle are attached to the rear wheel of the bike. A crank, analogous to the pedals of the bike, is connected by a belt, analogous to the chain on a bike, to the pulley so that turning the crank turns the pulley which turns the fraction wheel. How much the fraction wheel turns depends on the ratio of the size of the crank pulley to the size of fraction wheel pulley. This ratio is analogous to the gear ratio on a bike. For example if the fraction wheel pulley is 4 times as large as the crank pulley, corresponding to a ratio of 1:4, 4 full turns of the crank will be required to produce 1 full turn of the fraction wheel. Thus one turn of the crank produces a quarter turn of the fraction wheel providing both a mechanical and a visual model of 1/4. By changing pulleys students can observe the behavior of different fractions.
In the first application, 'The Fraction Wheel Pulley Machine', students observed fraction families for different base fractions by turn the crank multiple times for a given pulley until the fraction wheel had turned completely. In the second application, 'The Fraction Wheel Addition Machine', students were able to change pulleys without restarting in order to model fraction addition. The third application, 'The Fraction Wheel Comparison Machine', modeled subtraction and division by allowing the students to compare two fractions on the fraction wheel. In this fourth application use the fraction wheel machine to model multiplication of fractions.
The model for multiplication is based on a model of a given fraction as an input/output machine that given a quantity as input produces that fraction of the quantity as output. For example, we represent 1/2 as a machine that outputs a half of whatever quantity is input. Then we model multiplication as "compounding". For example, we model 1/3 x 1/2 as the result of connecting a 1/3 machine to a 1/2 machine. When we input an amount to this machine the 1/3 machine reduces it to a third and feeds this third of the quantity into the 1/2 machine which reduces the input by another 1/2 for a total reduction to 1/6 of the quantity. This model is illustrated in the first of the thumbnails for this product.
This model translates directly to our pulley model. A pulley can be viewed as an input/output machine that operates on rotation. For example, given a certain rotation as input, the '1:3' pulley outputs 1/3 of the rotation. Thus we can represent 1/2 x 1/3 by connecting a '1:3' pulley to a '1:2' pulley. The model allows students to observe directly the meaning of fraction multiplication on 2 or more fractions.