The Fraction Wheel Comparison Machine
Note: This pure HTML application is completely self-contained. Running it does not require internet access. All that is required is a reasonably up to date browser like chrome or firefox.
This is the third 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 could mix pulleys of different ratios thus modeling addition. In this application we use the pulley machine to compare two fractions in three ways. The first is just to determine which fraction is larger. The second is to compute the difference, i.e. the result of subtracting the smaller from the larger, and the third is to compute the ratio of the two fractions, i.e. the result of dividing one by the other.
The thumbnails for this product illustrate how it works. We want to compare 4/9 with 2/5. The first image shows the result of laying out 4/9 on the fraction wheel. The second shows the result of clicking on the 'Compare' button. This repaints the sections representing 4/9 dark gray and resets the fraction wheel so that we can paint over the image. We then lay out 2/5 on the fraction wheel. In laying out 2/5 the fraction wheel is automatically marked off into units of 1/45, the least common denominator of 1/9 and 1/5. By comparing the two images we can see that 4/9 > 2/5, 4/9 - 2/5 = 2/45, and 4/9 divided by 2/5 is 20/18 = 10/9.