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Teach Your Kids Arithmetic – Fractions, Those Devils!
Fractions. Ugh! I could just hear the screams coming from my students every time we entered the realm of those nasty little demons. Whenever we embarked on an area of mathematics that would require intensive work on fractions, the students would act as if we were entering Hades after an arduous crossing of the Acheron River, led by the intrepid ferryman Charon and his dog. three-headed Cerberus. Ouch! It was so bad.
Yet in all reality, these scarecrows we call fractions are not as demonic as claimed. And when you consider how important they are in the study of all areas of mathematics, they better be given their rightful place – and respect. At early ages, children stumble over these entities because they are inherently difficult to deal with. Unlike whole numbers, which consist of one part, fractions (or rationals, as they are called) consist of two: the numerator, or upper part, and the denominator, or lower part. Almost everyone knows this. And these monsters are quite nice when we perform the arithmetic operations of multiplication or division (which won’t be covered here; you’ll just have to wait for me to write this article). However, add or subtract – now we are talking about serious business. Students would recoil at the thought of adding two fractions with unusually different denominators, let alone adding three fractions with different bottoms. I guess “bottom up” wouldn’t apply here.
In any case, to tell the truth: adding fractions is not difficult. We just need to put ourselves on a common playing field and by that I mean 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. Case closed. However, accessing this LCD screen is what causes the most problems for students. Now I could get into the method of getting the LCD by first decomposing each background into primes – a process known as prime decomposition – then getting the LCD by removing all the distinct primes as well as common primes to the highest power – ugh, I’m already confused by all this gibberish. Wait, isn’t there an easier way?
Yes. Fortunately, there are. Since most students learn to get a common denominator (not necessarily the LCD, though) by multiplying the two funds together, we’ll base our method on this procedure. The only problem with this method is that they might need to multiply two large numbers together. Basically, I mean maybe 12 x 18 or 24 x 16. Most students have a calculator to use, so that’s really not a problem. (Although if they learn my techniques, they won’t need the calculator.)
Alright, let’s get to the meat of this method. Let’s take a specific example. Suppose we need to add 5/18 and 5/12 together. First, we need to get the LCD of 12 and 18. Before multiplying 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 equally divides the two given numbers. To get the LCD, just multiply the two given numbers together, 12 x 18 = 216, then divide that result by the GCF of 6, to get 216/6 = 36. Presto! LCD of 12 and 18 is 36. No prime number decompositions, no deletion of distinct primes, no worry about higher powers.
Finally, to add the two fractions together, 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 obtain 15; so 5/12 = 15/36. Finally, 5/18 + 5/12 = 10/36 + 15/36 = 25/36.
Try this method for size, and I’m sure you won’t be boating with Charon or Cerberus anytime soon. Till next time…
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