"I do not believe that the men who served in uniform in Vietnam have been given the credit they deserve. It was a difficult war against an unorthodox enemy."William Westmoreland
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A Turning Point
19 years ago, I was a young working mother at 28, independent and working for a prestigious company, until circumstances made me choose between family and career. After going through weighing my options, I chose to become a full-time Homemaker because I ...
How to Mastermind your Destiny through Self-Coaching
Reading how to books and self-coaching is an excellent way of getting to know yourself, quietly receiving answers you need, or resolving a secret issue you've been stuck on. Not only that, the only challenge will come from you and the only thing standing ...
Why Self-Love is Critical in Operating a Successful Home Business, Part 2
We ended Part 1 with two questions: If self-love is so crucial to building a healthy self-esteem and helping us to be happy, what prevents more of us from deepening our self-love? What keeps so many of us stuck in painful, hurtful, sad scenes where ...
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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...
About the author:
Joe is a prolific writer of self-help and educational material. Under the penname, JC Page, Joe authored the classic of mathematical ABC's Arithmetic Magic. Joe is also author of the charmingly pithy and popular ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the seminal collection of verse, Poems for the Mathematically Insecure. For more information, visit his website at www.mathbyjoe.com.
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Whizkids message: Self-help is the best helpIndian Express"It is because both my parents were there to support me at every step, along with my teachers, that I was able to achieve such a good score," she added. "Not once did we have to tell her to study and we are happy that her sincerity and dedication to ...and more » |
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Tokii Brings Self-Help to Tech Forefront with Mobile AppMarketWatch (press release)Canada-based online startup Tokii releases its mobile application for iOS devices today (the Android application will be available soon), revolutionizing the self-help space by bringing it from the popular Tokii.com web 2.0 platform into the rapidly ...and more » |
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How Self-Talk Can Help Accomplish GoalsHuffington PostSelf-talking aloud can have benefits, too. A recent study in the Quarterly Journal of Experimental Psychology showed that talking to yourself as you search for something could actually help you to find that item faster. In that study, researchers found ... |
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