Fractions are a common source of problems for many students. There are difficulties that arise when we work with fractions. In particular, finding common denominators when we add or subtract fractions can be a messy process. Because of these difficulties, practical arithmetic is usually done with a limited set of fractions where finding common denominators is much easier than it is in the general case.
When we discuss length in inches, the fractions we use are halves, quarters, eighths, etc. When working with these fractions, the lowest common denominator will always be just the largest denominator among the group of fractions.
When we use the metric system, our fractions will be tenths, hundredths, thousandths etc. Again the lowest common denominator will just be the largest denominator among the group--although in this case our fractions will usually be written as decimal fractions rather than common fractions.
Common denominators are important because we can only add or subtract like things. If we want to add a half gallon and a quart we need to re-express the half gallon as 2 quarts--and then we can add 2 quarts and 1 quart to get 3 quarts.
Add 1 fifth and 2 fifths and we obviously get 3 fifths. Write the same problem as 1/5 + 2/5 and it is easy to see how some people might mke a mistake and think the answer is 3/10. When we multiple fractions we do multiply both the tops and the bottoms of the fractions--but applying a similar rule to additon will give us the wrong answer. Writing the denominator in words is probably a good way to get used to the fact that the denominator does not change when we add like fractions. Indeed, if we could simply add numerators and denominators there would be no need to find common denomionators when we add. The only reason we have to find common denominators is because the denominators have to be the same--and they stay the same when we add fractions. All of this is also true when we subtract like fractions.
The best way to start adding or subtracting unlike fractions is to keep it simple. Work with halves, quarters, and eighths. You don't even have to learn general rules for converting fractions. It is enough to know that 1 half is 2 fourths is 4 eighths, 1 fourth is 2 eighths, and 3 fourths is 6 eighths. Just convert everything to thelargest denominator you see, then add or subtract the numerators.
From this beginning we can proceed to consider more general problems where the largest denominator will work as a common denominator, or where we are given a common denominator that will work. We can convert fractions to other equivalent fractions by multiplying top and bottom by the same number, Start with 5 sixths of a pizza, divide each piece in half and we wind up with 10 pieces each half as large. Instead of six of these pieces to make up a whole pizza it now takes twelve--so we see that 5 sixths is equivalent to 10 twelfths.
So how do we find a common denominator if the largest denominator doesn't work and we aren't given a denominator we can use? Consider a dish of brownies. We cut the dish into individual brownies by making 5 rows and 6 columns for a total of 30 individual brownies. A row is 1/5 and the 6 columns divide the row into 6 brownies or 6/30. A column is 1/6 and the 5 rows divide the column into 5 brownies or 5/30. We see that 30, which is 5 x 6 will work as a commmon denominator for fifths and sixths--and this same idea will work whenever we want to find a common denominator for two fractions.
We often tell students they have to find the lowest common denomminator when they add or subtract fractions. We teach students complicated rules using factor trees for finding prime factorizations, and then teach them other complicated rules for so they can use these prime factorizations to find the lowest common denominator. But this is really overkill if all we want to do is is add or subtract unlike fractions. Multiplying the denominators will always give us a common denominator. Even if this isn't the lowest common denominator, it will work.
As a practical matter there is no need to worry if our common denominator is really the lowest. But it usually makes things easier if we can keep our common denominators as small as we can. There are practical methods that can help keep our common denominators small, and will often let us find the lowest common denominator.
One approach simply relies on experience. If you encounter problems with fourths and sixths often enough you will remember that 12 is the LCD or lowest common denominator. A more general method we can use is the following: Just multiply the two denominators and then divide by the largest number you can see that will divide into both denominators. In the case of fourths and sixths, we would multiply 4 x 6 and then divide the product by 2. With eighths and twelfths we would multiply 8 x 12 to get 96. Dividing by 4 we would get 24. These methods can help us keep down the size of our common denominator--but if we happen to use a denominator that is not the lowest possible, we can still solve the problem.
No comments:
Post a Comment