Why can't we reduce?
Why do we multiply 2x3 for the lowest common denominator of, but not multiply 2x6 for the l.c.d. of ?
Why can't we combine ?
Why do we rationalize immediately, but not ?
Why is the lowest common multiple of sin x and cos x their product?
Why can we "cross-multiply" to get equivalent terms for ?
The answer of course is a greatest common factor of 1. As in the true nature of a silent partner, the greatest common factor is always there when we need it, but never in the forefront. A g.c.f. of 1 is almost always taken for granted although without it, we couldn't justify our steps.
A g.c.f. of 1 is the reason why we can or cannot proceed with some of math's most fundamental steps. It is our back-up and solid answer to all of the above questions: Quite simply we cannot reduce because g.c.f. (3,7) = 1. We multiply 2x3 for a common denominator because g.c.f. (2,3) = 1, but g.c.f. (2,6) = 2. We cannot combine because g.c.f. (3,7) = 1 and we rationalize right away but not because g.c.f.
(1,6) = 1 while g.c.f. (2,6)1. The same holds true for common denominators in higher
stages of mathematics like trigonometry and calculus.
For example, g.c.f. (sin x, cos x) = 1 and g.c.f. (x + h,x) = 1 as well.
My argument is that the nature of and calculation of the greatest common factor should be much more dominant in the curriculum and hence in the classroom because it is so fundamental. It's time to reinforce basic skills so students can understand why they do things the way they do and then match this understanding with basic skills such as the division algorithm.
Now let's pursue our discussion of g.c.f. by examining its role when reducing fractions and rational expressions later in the curriculum. For example, reduce to lowest terms without a calculator. Many students would first wonder whether it was possible, and even if it were, how then to proceed. The quickest method is to calculate g.c.f. (57,95) using Euclid's Algorithm.
Step 1) Divide the smaller number into the larger keeping track 57 of the remainder. 38
Step 2) Divide the remainder (38) into the previous divisor (57) 38 again keeping track of 19 the remainder.
Step 3) Repeat step 2 until the remainder is zero. 38 0
Step 4) The divisor which yields a remainder of zero is our g.c.f. Here g.c.f. (57,95) = 19 and so to reduce , we simply divide top and bottom by 19. This gives .
Some may think this skill is no longer necessary until we run into rational expressions in Grades 10 and 11. For example, reduce to lowest terms. Here, we are instructed to factor both top and bottom and then reduce. Logically it would be better to know whether factoring is worthwhile, as the expression may not reduce after all. Finding the g.c.f. for polynomials reinforces the division algorithm and answers a key question. Will the expression reduce at all? If g.c.f. then no, it won't. However, if the g.c.f. is not 1, then we should proceed.
Applying Euclid's Algorithm again, we have
Here necessarily and so (x-3) has to be common to both and . If students are having difficulty with factoring, half the work of reducing is now already done. That is, we know the expression can be reduced and we know the term needed to start the process. The above is another way of approaching a common algebra problem, but it is not for every student. Some may have forgotten the division algorithm altogether while others would rather try their luck at factoring. On the other hand, some students will have added to their understanding of how things work and become more independent learners.
They may even realize the power of division and Euclid's Algorithm. A high-energy honors class may appreciate the power of reducing fractions without a calculator and the natural lead into calculating lowest common multiples using l.c.m. .
This is the method used for calculating l.c.m. in Asia. However, a teacher who is going back to the basics may appreciate another use of the division algorithm.
In short, the calculation of the greatest common factor is simple and direct and knowing that g.c.f. of any two numbers is 1 explains many steps. This partner need not be silent anymore.