Teach Your Kids Arithmetic - Fractions, Those Devils!
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