Fractions. Ugh! I could just hear the squeals coming from my students any time we entered the realm of these nasty little demons. Anytime we embarked on an area of mathematics that would require heavy fraction work, students would act as though we were entering Hades after an arduous crossing of the river Acheron, led by the fearless ferry-man Charon and his three-headed dog Cerberus. Ouch! It was that bad. Yet in all reality, these bugbears we call fractions are not nearly so demonic as they are made out to be.
And when we consider how important they are in the study of all areas of mathematics, we best give them their proper place--and respect. At the early ages, children stumble over these entities because they are inherently difficult to reckon with. Unlike whole numbers, which consist of one part, fractions (or rationals, as they are called) consist of two: the numerator, or top part, and the denominator, or bottom part. Pretty much everyone knows this. And these monsters are quite friendly when we perform the arithmetic operations of multiplication or division (which will not be discussed here; you'll just have to wait until I write that article).
However, add or subtract--now we're talking serious business. Students would cringe at the thought of adding two fractions with unusually different denominators, not to mention three fractions with different bottoms. I guess "bottoms up" would not apply here. At any rate, truth be told: adding fractions is not difficult.
We just need to get on a common playing field and by that I refer to the common denominator. Specifically, we want the lowest common denominator, or LCD, for short. Once we have the LCD, we do a quick conversion on the numerators and then add them together.
Case closed. Yet getting to this LCD is what gives students the most trouble. Now I could go into the method of getting the LCD by first decomposing each bottom into primes--a process known as decomposition into primes--and then obtaining the LCD by taking out the all the distinct primes as well as the common primes to the highest power--ugh, I'm already getting confused by all this mumbo jumbo. Hey wait, isn't there an easier way? Yes. Thankfully, there is.
Since most students learn to get a common denominator (not necessarily the LCD, though) by multiplying the two bottoms together, we will base our method on that procedure. The only problem with this method is that they might need to multiply two large numbers together. By large, I mean perhaps 12 x 18 or 24 x 16.
Most students have a calculator to resort to so this is really not an issue. (Although if they learn my techniques, they won't need the calculator.) Okay, let's get to the meat of this method. Let's take a specific example. Suppose we needed to add 5/18 and 5/12 together.
First, we need to get the LCD of 12 and 18. Before we multiply these numbers together, we need to observe that the greatest common factor of 12 and 18 is 6. The greatest common factor, or GCF of two numbers, is the largest number that divides evenly both given numbers. To get the LCD, all we need do is multiply the two given numbers together, 12 x 18 = 216, and then divide this result by the GCF of 6, to get 216/6 = 36. Presto! The LCD of 12 and 18 is 36.
No prime decompositions, no taking out distinct primes, no worry about highest powers. Finally, to add the two fractions, we need to multiply the numerators by an appropriate factor to get the adjusted fraction. For example, since 36/18 = 2, we need to multiply the 5 of 5/18 by 2 to get 5/18 = 10/36; similarly, since 36/12 = 3, we multiply 5 by 3 to get 15; thus 5/12 = 15/36. Finally, 5/18 + 5/12 = 10/36 + 15/36 = 25/36.
Try this method out for size, and I'm sure you won't be taking any boat rides with Charon or Cerberus any time soon. Till next time. .
By: Joe Pagano