A textbook I am currently using in my College Algebra course has a number of boxes with notes to students like this one:
“CAUTION! Clearing fractions is a valid procedure when solving rational equations but not when adding, subtracting, multiplying or dividing rational expressions.”
The remainder of this cautionary box goes on to clarify the meaning of this statement, but does nothing to explain why we use one procedure when solving and a different procedure when adding, etc.
Are we worried that the real mathematical story is too hard for College Algebra students? I don’t think that it is (and more on this shortly). Do we believe that this represents the true nature of mathematics? I know that it is not.
Mathematics is not an arbitrary set of unrelated procedures. It is a coherent body of knowledge that includes a number of really useful procedures. But when these procedures dominate the public face of the subject, there remains nothing left to connect them to each other and we have to issue all sorts of cautions and warnings to our students so that they will remember which procedures to use in which situations.
My College Algebra students (pity them) have been privy to my first attempts at building these sorts of connections. So here is the real mathematical story as I see it…
Rational expressions are things like this: (x+1)/(x-3). They have a polynomial in the numerator and one in the denominator. Rational equations are things like this: (x+1)/(x-3)=2. They have rational expressions in them and an equal sign, so one thing is equal to another. Often we want to solve rational equations, which means that we want to find the value(s) of x that make the equation true. One procedure for doing so is commonly referred to as “clearing fractions”. To clear fractions, we multiply both sides of the equation by the least common denominator of the fractions involved. In the example, there is only one denominator (x-3), so we multiply both sides by this. After simplifying, the result is (x+1)=2(x-3). This is now a linear equation in one variable and solving it is a well-practiced skill for College Algebra students.
But what is going on when we clear fractions? We maintain equality by multiplying by the same expression on both sides of the equal sign, but what’s really going on? Consider an alternate way of thinking about solving the equation (x+1)/(x-3)=2. An equivalent form of the equation is, (x+1)/(x-3)-2=0. If we think about the left-hand side as a function, f(x)=(x+1)/(x-3)-2, then we are trying to find the value of x that makes the function equal to zero. We are trying to find the roots of the f(x)-the places where the graph of f(x) crosses the x-axis.
When we multiply both sides of the equation (x+1)/(x-3)-2=0 by (x-3), we get this: (x+1)-2(x-3)=0. On the left-hand side we have “cleared” the fraction and on the right-hand side 0 times anything equals zero. Notice that if we think of the left-hand side as a function it is a different one, g(x)=(x+1)-2(x-3). We are looking for the roots of a different function. But g(x) has something important in common with f(x). They have the same roots. In the graph below, f(x) is in blue and g(x) is in red.
Indeed, we can multiply a function by any meaningful value or expression (the various forms of 1/0 do not count as meaningful!) and we will get a new function, but we can be sure that the roots of both functions will coincide because anything times zero is equal to zero.
So why can we clear fractions when we are solving equations but not when we are simplifying expressions? Because when we multiply by the least common denominator we are changing the function. The new function has the same roots so the new equation has the same solutions. But the new function has a different, and non-equivalent expression.
There are other ways to think about the relationship between these procedures, of course. I do not mean to advocate for this function-based approach as the solution to a complex problem of teaching and learning. But I do believe that operations on functions (which are a standard topic in College Algebra courses anyway) can provide reasons for these procedures, and that these reasons are within the grasp of College Algebra students. I believe my students can learn to think mathematically. CAUTION boxes imply that they cannot.