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Topic: To meet a challenge
Replies: 18   Last Post: Oct 6, 2012 10:30 AM

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Paul A. Tanner III

Posts: 5,920
Registered: 12/6/04
Re: To meet a challenge
Posted: Oct 5, 2012 12:53 AM
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We teach teachers to teach the adding and subtracting of fractions (and this includes rational functions in algebra) using the least common denominator with a method so clunky, long, and complicated that it cannot be written as a single equation, the left side written as the sum/difference of the fractions, and the right side being a single fraction with no fraction in the numerator or denominator. (We do not do this with respect to the teaching of the multiplying and dividing of fractions - they are taught with methods that are elegant and concise enough to be written as single equation. Imagine how much worse the outcome would be compared to the present situation if we taught students the multiplying or dividing of fractions with methods that are so clunky, long, and complicated that they cannot be written elegantly and concisely as a single equation.)

To address this problem:

Here is an algorithm for fraction addition/subtraction that is written as a single equation, the left side being the sum/difference of the fractions, and the right side being a single fraction with no fraction in the numerator or denominator. (Where I use the "/" symbol surrounded by a space to be the main dividing line):

a/b +- c/d = (m/b)a +- (m/d)c / m.

(Each of (m/b) and (m/d) is an integer when each of the variables is an integer - when we are in the rationals. And so the right side is a single fraction with no fraction in the numerator or denominator.

This generalizes to any number of fractions elegantly and concisely, letting n be any number of fractions:

a_1/b_1 +- ... +- a_n/b_n = (m/b_1)a_1 +- ... +- (m/b_n)a_n / m.

How to teach this equality as an algorithm to students? I like the (easy to memorize) verbalization "m over the bottom times the top" for each addend. And since this method generalizes so elegantly and concisely to any number of fractions, try introducing the method on three fractions to show how easy it is - if they see that it's easy on three or more fractions, it of course is easy on just two. Here's a favorite set of three fractions I like to use, one with an easy LCD:

3/4 + 5/6 - 7/8 = 18 + 20 - 21 / 24.

The rest is just any needed simplification of the single non-complex fraction on the right side, which is what we have to consider anyway with the usual way of fraction addition/subtraction and with fraction multiplication and fraction division when we get to a single noncomplex fraction.

If we wanted to write out the pattern that they would use in algebra in the adding/subtracting of rational functions, we would do everything exactly the same, keeping the same pattern, but doing only "m over the bottom" in our heads, resulting in

3/4 + 5/6 - 7/8 = (6)(3) + (4)(5) - (3)(7) / 24.

Note that where we replace each of these integers with polynomials, this last pattern on the right side is exactly what we typically have with the usual method on rational fractions, but with this equality-based method arrived at faster, in one written step.

A major reason we teach fraction addition/subtraction with the LCD is because it is an anticipation of rational function addition/subtraction. Once we have an LCD, we can in one written step go straight to a single noncomplex fraction in fraction addition/subtraction, just as we can with fraction multiplication/division. In algebra, being able to handle expressions of which fractional forms are a part is an important skill that many have a hard time with. So my ultimate motivation here is about making this aspect of algebra easier.

Message was edited by: Paul A. Tanner III

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